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Mitigation of nonlinear oscillations of a Jeffcott rotor System with an optimized damper and nonlinear energy sink
Accepted Manuscript Mitigation of nonlinear oscillations of a Jeffcott rotor System with an optimized damper and nonlinear energy sink Ghasem Ghannad Tehrani, Morteza Dardel
PII: DOI: Reference:
S0020-7462(17)30254-8 https://doi.org/10.1016/j.ijnonlinmec.2017.10.011 NLM 2915
To appear in:
International Journal of Non-Linear Mechanics
Received date : 2 April 2017 Revised date : 16 October 2017 Accepted date : 16 October 2017 Please cite this article as: G.G. Tehrani, M. Dardel, Mitigation of nonlinear oscillations of a Jeffcott rotor System with an optimized damper and nonlinear energy sink, International Journal of Non-Linear Mechanics (2017), https://doi.org/10.1016/j.ijnonlinmec.2017.10.011 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Mitigation of nonlinear oscillations of a Jeffcott rotor System with an optimized damper and nonlinear energy sink Ghasem Ghannad Tehrani and Morteza Dardel1 Abstract Contact occurrence between disk and stator as result of undesirable vibrations produced by eccentricity of the disk is one of the most destructive and common phenomena in rotor dynamics systems. In this work, utilizing tuned mass damper (TMD) and nonlinear energy sink (NES) are suggested as a solution for preventing contact occurrence. The mass and angular position of absorbers determine their efficiency for resisting the eccentricity force produced by the disk, and their stiffness and damping coefficients determine the displacement scope of the absorber. In order to efficiently design absorbers, optimization is proposed. In this suggested optimization process, complex averaging method is used in order for deriving the equations of motion of the system in presence of dynamic absorbers at the steady state condition. Afterwards, for determining trustworthiness of each absorber’s performance, system’s behavior is studied for different values of its parameters such as rotational speed, stiffness, clearance and eccentricity in presence of each absorber. From the obtained results, it can be perceived that TMD and NES are as efficient as possible and they have exactly the same positive influence on the system’s vibrations. The reliability of the proposed optimization process can be determined by the results. Key word: Jeffcott rotor, Contact, TMD, NES, Complex averaging method, Optimization, rotor dynamics. 1Corresponding author: Department of Mechanical Engineering, Babol Noshirvani University of Technology, P.O. Box 484, Postal Code: 47148‐71167, Shariati Street, Babol, Mazandaran, Iran. [email protected].
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1‐ Introduction Disk’s eccentricity is one of the commonest reasons of undesirable behavior in turbomachines. In addition to eccentricity, small clearance between disk and stator is another reason which leads to destructive consequences like contact phenomena. All of the mentioned problems result in undesirable vibrations which makes different faults like disk–stator contact phenomena. Removing contact and reduction of whirling vibration amplitudes in rotor dynamics systems are very important. Choy and Padovan [1] and Beatty [2] have done the first complete nonlinear disk–stator contact models. Chu and Zhang [18] investigated the nonlinear vibration characteristics of a rub‐impact Jeffcott rotor. Also they found, with increasing rotating speed, the grazing bifurcation, the quasi periodic motion and chaotic motion occur after the rub‐impact. Goldman and Muszynska [4] found the chaotic behavior of rotor/stator systems with rubs. Yu et al. [5] presented the results of experimental and analytical studies on rotor/seal full annular rub. Wan et. al. [7] worked on a rotor dynamics system with rub‐ impact effect and applied hydrodynamic theory for oil film bearings as the supports. They derived orbit diagram of the disk’s motion in order to investigate chaos phenomena. Samantaray [8] analyzed a four‐degrees‐of‐freedom rotor driven by a DC motor considering Sommerfeld effect. The steady‐state spin frequency and whirl orbit amplitude are derived as functions of the drive and rotor system parameters. Z. Shang et al. [9] studied a rotor dynamics system by taking into account rotor/stator rub impact and dry friction effect (the previous works and is the main factor for the self‐excited dry friction backward whirl). The results confirmed the coexistence of different rubbing responses observed in tests. Richardet et al. [11] worked on numerical models and experimental setups in order to highlight the phenomenology involved in different rotor to stator contacts. Chavez and Wiercigroch [12] studied a non‐smooth model of Jeffcott rotor which bearing clearance and rub‐impact effect was taken into account. In this paper the bifurcation analysis is applied using software TC‐HAT and it complex dynamics occurring is investigated.
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Najafi et al. [13] investigated the stability of a bladed rotor with cylindrical and conical whirling using Krein’s theorem, they showed that if a stagger angle of the blades be equal to 0 and 90 , the blades vibrations are coupled only with cylindrical whirling for blades stagger angle of 0 and only with conical whirling for blades stagger angle of 90 . For other stager angles on the other hand, the blades vibrations are coupled with both cylindrical and conical whirling altogether. Vlajic et. al. [14] are analytically and numerically studied a nonlinear torsional vibration of Jeffcott rotor subjected to continuous stator contact for both forward and backward whirling. They gave special consideration to the torsional response during high‐ speed backward whirling, and found that although the rotor is subject to discontinuous friction, during high‐speed backward whirling, full system and reduced‐order models for the torsional oscillations are smooth in nature, and centrifugal stiffening effect is experienced by the torsional motions during both forward and backward whirling. Varney [15] investigated different nonlinear dynamic behavior of an asymmetrically supported Jeffcott rotor by considering contact between disk and stator. Saeed and Kamel [16] worked on controlling vibrations of a Jeffcott‐rotor using an active control technique. In this paper after observing high amplitudes and nonlinear behavior in the system, a Proportional‐Derivative (PD)‐Controller using two pairs of electromagnetic poles is applied to the system. The results provided by bifurcation analyses demonstrated that using this controller is highly efficient for vibration attenuation. Saeed and El‐Gohary [17], studied the vibration analysis of a horizontally supported Jeffcott rotor system. They used multiple scales perturbation technique to obtain a second‐order approximate solution at the resonance conditions, and obtained the stability of system by applying Lyapunov first method. The influences of all the parameters on the system behavior are explored. Also the effect of both the negative and positive values of the nonlinear stiffness coefficient is studied. Eissa and Saeed [18], applied a positive position feedback controller for a horizontally supported Jeffcott‐rotor system. The multiple scales method is applied to obtain the approximate solution of the system. The results are shown in the form of bifurcation diagrams and also the effects of different controller parameters on the system frequency – 3
response curves are studied. Saeed and Kamel [19], utilized active magnetic bearing for reducing the vibration of Jeffcott‐rotor system. In this work, the controller is attached to the primary system through four electromagnetic poles. Then, the influence of different parameters such as disk eccentricity, air‐gap and others are investigated on the vibration amplitudes of the system. The analytical and numerical results are in a good agreement with each other. Saeed and El‐Ganaini [20] showed how to utilize time‐delays to mitigate the oscillations of a two‐degree‐of‐freedom nonlinear model for a horizontally supported Jeffcott‐rotor system. The extracted the slow‐flow modulating equations of both the amplitudes and phases, and the conditions that make the time‐delayed controller works as a damper or exciter are clarified. Also Saeed and El‐Ganaini [21] used time‐delayed position‐velocity controller for vibration control of a horizontally suspended Jeffcott‐rotor system having cubic and quadratic nonlinearities. They showed that the time‐delay increases the vibration amplitudes and can destabilize the system solution in the case of negative position feedback control, while at positive position feedback control it improves the vibration suppression performance. In continuo of their previous works, Saeed and El‐ Gohary [22] used a modified version of the controllers based on the saturation phenomenon which is connected to the system via quadratic coupling nonlinearity, for nonlinear vibrations control of a vertically supported Jeffcott‐rotor system. Time‐delays in the control loop are included in the considered model to explore their influences on the controller performance and stability chart. They showed that the system has chaotic and quasi‐periodic motions in addition to the periodic ones. In recent years, researches on application of nonlinear energy sinks absorbers for vibration attenuation of discrete and continuous systems has been raised, instead of tuned mass dampers (TMD) or weakly nonlinear absorber. Bab et. al. [23] worked on attenuating vibrations of rotor dynamics system using smooth non‐linear energy sinks. In this paper the analytical solution is obtained using Multiple scales – Harmonic balance method (MSHBM). It is shown that for medium magnitudes of external force NESs collection demonstrates an impressive effect since larger area of SMR has been obtained in the domain of the system’s parameters. Bab et. al. [24] worked on vibration reduction of rotor dynamics system supported with journal bearings using NES. They attached NES on the
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bearings and optimized its parameters utilizing genetic algorithm. Taghipour and Dardel [25] investigated the effect of single and double DOFs NESs on the behavior of a two‐DOF dynamical system in steady state condition. It is concluded that dynamical system with single DOF of NES shows less robustness to change in parameters and amplitude of external forces in comparison to double. Guo et. al. [26] mounted NES & TMD separately on the cross‐section of a hollow shaft in a rotor system and individually studied the influence of them on the vibrations of the system. Taleshi, et. al. [27] studied the behavior of a simply supported plate using NES. By changing the parameters and location of the NES using complex averaging and arc‐length continuation method, an optimized passive targeted energy transfer of the plate was obtained. Furthermore, the performance of NES and TMD was compared with each other. Parseh et. al. [28] worked on the influence of NES on a non‐ linear beam, modeled in according to Euler‐Bernoulli and Timoshenko theories. Using complex averaging with continuation method, the steady state response of the system based on resonance condition is investigated. Vlajic et. al. [29] worked on a modified model of Jeffcott roto, in which the dynamic behavior of the system including torsional deformation and contact phenomena is studied with considering forward and backward whirling. For forward whirling two rotor – stator friction models are taken into account and analytical and numerical analysis are done. For forward synchronous whirling fold, Hopf, lift‐off, and period‐doubling bifurcations are studied. Furthermore, transition from pure sticking to stick – slip regions is demonstrated for backward whirling. In previous studies for designing passive absorbers, numerous simulations were done in order to provide the efficient parameters of the absorbers which these simulations are so time consuming and are not coherent. Therefore, proposing an efficient, accurate and inclusive yet simple method of optimization is one of the main purposes of this paper. In this work, the simple model of the rotor dynamics system which is known as Jeffcott rotor is adopted as the case study. In the first section, the equations of motion of the system are derived both with and without attached absorbers. Then, a method of optimization is applied to provide the optimum parameters of each absorber. Finally, the efficiency of each absorber for different values of the system’s parameters like rotational speed, stiffness etc. is investigated.
5
2‐ Modeling and equations of motion In this section, the equations of motion of the system with different vibration absorbers are derived.
2.1 Modeling of Jeffcott rotor Before preceding the simulation of the system, it is assumed that: a) stator is to be rigid, b) the shaft is massless and has a uniform stiffness, c) gyroscopic effect of the shaft is neglected, and d) longitudinal and torsional vibrations of the shaft is not considered. Figs. 1 and 2 show the primary model of rotary machine. This model is known as Jeffcott rotor. The geometrical center of the disk is and the mass center is which has an eccentricity of geometric center. The mass of the disk is denoted by
, is the rotational
speed of the disk and is the angular displacement of mass center of the disk with respect to horizontal axis which is equal to
.
is the shaft’s stiffness in and directions. It
should be noted that the two ends of the shaft are clamped. In Fig. 2 the contact between disk and stator with the contact forces are shown. The clearance between disk and stator is and the radial displacement of the disk is shown by
.
Figure 1: Jeffcott rotor fixed in both end. Figure 2: a) Geometric and mass center of the rotor, b) the contact force components in normal and tangential directions. Normal and tangential forces contact stiffness is assumed to be
and
are generated at the disk–stator interface. The
and the tangential friction force is proportional to the
normal force by the friction coefficient.Since the contact occurs when
6
, in order to
satisfy this condition a Heaviside function is applied in the expression of normal contact force[5]. (1)
(2)
With respect to Fig. 2, the components of impact force in and directions are: sin
cos
cos
sin
(4)
are the displacements of the geometric center of the disk in horizontal
,
where
(3)
and vertical directions. The equations of motion of the system are [5]: cos
(5)
sin t
(6)
In order to analyze this rotor dynamics system, it is assumed that the shaft is massless and rigid, the disk is rigid, and axial and torsional vibrations are negligible. In order to write equations in dimensionless form, following quantities are introduced.
,
,
, 2
, (7)
̅
,
̅ ,
,
, ̅
,
,
̅
Hence:
̅
2 2
̅
̅
1 1
̅ ̅
̅ ̅
̅
̅
̅ ̅
̅
̅ ̅
(8)
(9)
̅ where the superscript of ′ denotes differentiation with respect to dimensionless time τ.
7
2.2 Dynamic absorber Dynamic absorber is a mechanical system which is mounted on a mechanical structure for the purpose of vibration mitigation of the structure. Two types of absorbers are well known: active and passive. The active dynamics absorbers operate by an external input, while passive absorbers do not need external input. In this study, two passive dynamic absorbers of TMD and NES are applied. In TMD is kind of passive dynamic absorber which consists of a mass, linear spring and a linear damper. These absorbers are tuned to a particular frequency of the main system, and reduce the system’s vibration amplitude by its own out of phase vibrations [6]. A schematic application of TMD to a system is shown in Fig. 3. NESis another kind of passive dynamic absorber with a mass, nonlinear spring and a linear damper. These absorbers are able to mitigate the vibration of the main system in a wider range of frequencies. The process in which the energy of the system is absorbed by such absorbers is call passive targeted energy transfer [10]. Due to essential nonlinearity of NES, it is not tuned to a particular frequency and can absorb vibration in wider frequency range. A schematic application of NES to a system is shown in Fig. 4. Figure 3: A schematic of the applied tuned mass damper. Figure 4: A schematic of the applied nonlinear energy sink. In order to place the absorbers in the primary system, a rod is used in which from one end is tied up to the disk and from another end is tied up to the shaft’s lateral surface. The rod placement makes two limitations for the absorber’s displacement: first of all, the absorber is bound to displace in a space between disk and shaft. Secondly, it is limited to just move along the rod.
2.2.1 Rotor dynamics system with TMD
8
In this section, the equations of motion of the system in presence of TMD are derived. For this system
,
and
are the mass, damping coefficient, and stiffness of the
absorbers, respectively. Parameter is the angle between rod and mass center of the disk. It can vary from 0 to . Here is an angular displacement of the rod.
represents shaft’s
is the displacement of the TMD’s mass along the rod. All the
radius. Furthermore,
and
assumptions were made in section 2.1 are valid here. Here displacements of the mass center of the disk, and
and
are the
are the displacement
components of the absorber. The displacements of the springs and dampers of TMD are and respectively. Figure 5: Rotor dynamics system with single TMD. In order to achieve the equations of motion Lagrange method is applied. The kinetic and potential energies, and Rayleigh dissipation functions are as follows. 1 1 1 (10) 2 2 2 1 1 1 (11) 2 2 2 1 1 (12) 2 2 where the displacements of disk and TMD in horizontal and vertical direction are: cos
,
sin
cos sin
,
sin
,
cos
sin
cos
cos
sin
(13)
Following additional dimensionless quantities are introduced. ,
̅
,
,
2
9
(14)
Substituting Eqs. (7, 14) into Eqs. (10 ‐ 12) and taking
,
as independent ,
parameters and applying Lagrange – Euler method dimensionless equations of motion in following form are derived. ̅
1
̅ cos ̅
1
cos ̅ ̅
̅ ̅ sin
1 ̅
1 ̅
̅ cos
sin
̅
cos 2
2
̅
(15) ̅
(16)
2 sin
̅
2
cos
sin ̅
̅
sin
cos
sin ̅
̅
2
(17)
2.2.2 Rotor dynamics system with NES In order to obtain the equations of motion, the only change in comparison to TMD is in the potential energy term, which is in following form. 1 2 where
,
1 2 ,
1 4 have the same definition previously mentioned for TMD.
,
(18)
Figure 6: Rotor dynamics system with single NES. With introduced following dimensionless quantities: e
, ̅
,
, 2
dimensionless equations of motion are obtained as following.
10
(19)
̅
1
̅ cos ̅
1
̅ cos ̅ ̅
̅ ̅ sin
1 1
̅
̅
̅
cos
̅
̅
2
cos
̅ sin
̅
̅ sin
2
̅ cos
2
(20) ̅
(21)
2
̅ ̅
2
̅
̅
(22)
2.3
Optimization the parameters of TMD and NES
In the following sections, it is observable that variations in structural or working parameters of the system cause contact between disk and stator. This undesirable phenomenon, results in emerging some residual stresses and deflections in disk and bearing and in some cases unwanted crack in the disk. So, eliminating such unwanted vibrations and improvement the working condition of the rotor dynamics systems is vital. Changing of the absorber’s parameter as a method to reach an optimum behavior of the system is not reliable since this parametric variation causes nonlinear system involve with bifurcations and many behavioral changes and optimization process less efficient. Hence, in this paper optimization is accomplished by a predefined purpose which is described in the following paragraph. It is assumed that there is no contact between disk and stator. This assumption is made for removing the nonlinearity due to contact force expressions. The considered rotor dynamics system is a self – excited system and dynamics of such systems is dependent on time which makes the optimization process so difficult. Hence, it is better to rewrite the equations of motion independent on time. To this end, complex averaging method is adopted.
2.3.1 Complexification of the equations of motion Complex averaging method is similar to harmonic‐balance method, in which the equation of motion is written in terms of amplitude and phase of the systems response. 11
This method is an analytical method and gives an approximate solution of the problem. The complex variables , and are defined as below. ̅ ̅ ̅
̅
̅
in which ̅
,
̅ , and ̅
,
(23)
are static part of displacements.
The displacements and their derivatives in terms of complex variables of , , , and are expressed as follows: ̅ ̅
̅
, ̅
2 2
2 ̅ ̅
̅
2
2 ,
2
̅
, ̅
2
(18)
2
2 ̅
(24)
(19)
2
̅ 2 2 where , , and ̅ are the complex conjugates of , , and respectively. Introduced ,
complex variable , , and are written as
, and
,
which from solving the obtained equations, the amplitudes and the phases of the system’s response are: | |
, tan
, | |
̅
tan
| |
,
(27)
, ̅
tan
Then, replacing latter complex variables in Eq. (24 ‐ 26) and Substituting resulted displacements and their derivatives into equations of motion of the system with TMD Eq. (15‐17) and averaging of the obtained equation with respect to the exciting frequency , , equations in terms of phase and amplitude are obtained(See Appendix section B). Then the equations are set into a following standard form:
12
(28)
(29)
where Eq. (28, 29) represents dynamics and static parts of the equations of motion, respectively. These matrices are given in appendix section B. The static response vector can be obtained from Eq. (29): (20)
Substituting Eq. (30) in Eq. (28), the equations of motion in complex averaging form has following form:
Where
(31)
′
(21)
The matrices
,
,
,
,
,
,
are all defined in
Appendix section A. Separating the real and imaginary parts of Eq. (28, 29), a system of nine unknowns and nine equations is obtained which consists of dynamic and static parts which are as following: Dynamic part: ̅ 2
2
2
2 2
2
2
2
2
2 0
2 2
̅
2
(33) ̅
2
2 2
2 2
̅ 2
2 2
13
0
2
Static part: 0
(22) 2
2
2
The same procedure for providing optimum parameters of TMD is adopted for NES and the standard form of equations of motion is obtained as follows (23)
The static and dynamic parts of the equations are as follows: Dynamic part: ̅ 2
2
2 3
2
3 3
3
3
8
2
2
3 8
2
2 3 3 3
3
̅
3
3
2
2 03
3
2
2 3 3 3
8
2
3
2
2
2
2 2
3 3
2 ̅
8
2
2
2 3
2
0
(36)
̅ 2 03
2
2 0
Static part: 0 3 2 2
2
3 2
2
2
2
(37)
It is clear that the nonlinearity due to nonlinear stiffness of the NES makes the dynamic and static parts equations dependent on each other. It means that unlike TMD it is not possible to distinguish static expressions vector
14
and provide them from static equation
and replace them in dynamic part. Therefore, the dynamic and static parts are needed to be solved altogether.
3 Optimization According to the outputs in the following sections, it is observable that variations in some parameters like rotational speed increase the amplitude of the system’s vibrations and causes instability in the system which is absolutely undesirable. There are variety of and
methods for optimum design of absorbers such as
that are the most famous and
applicable ones among the others. The integral of the amplitude in terms frequency is minimized in
method. In
method on the other hand, the maximum amplitude in
terms of frequency is minimized. In
and
optimizations, linear and nonlinear
constraints, linear and nonlinear in‐equalities can be considered, they are relevant to design on TMD and NES design, in the current study. Also codes for solving these optimization functions are available in commercial software such as Matlab. Here following procedure for optimization is proposed: At the beginning the structure without absorber is considered. A design point is determined from the simulations or based on the experience. For instance, this valuecan be chosen at a situation when resonance happens in the system or the system has large amplitudes of vibrations. The goal is to determine the system’s parameters under following conditions:
The amplitude of the primary system reaches a desirable value: Objective function
The equation of motion must be satisfied: Nonlinear constraint function
The stability of the system should be guaranteed: Nonlinear inequality constraint function, since the eigenvlaues of linearized equation of motion must be located in the left half of complex plane.
The absorber’s parameters should be within the allowed interval (upper and lower bound): Constraint function
15
The objective function is determined as follows: min
(24)
The refers to the amplitude of the disk in vertical and horizontal direction and the amplitude of the absorber. The purpose is to mitigate the amplitude of the disk and the absorber altogether and furthermore, the fmincon function of Matlab has just one scalar as the output, the objective function is determined as minimizing total amplitude of disk and absorber minus desired values
.
is selected as a feasible small value.It should be
mentioned that for the steady state response, and ′ must be set to zero, but for stability analysis it is necessary to obtain the eigenvalues of Eq. (31, 35). The amplitudes | | and | | which represent the amplitude of the disk’s center in the vertical and horizontal directions are considered as the purpose function of the optimization process. Then, using fmincon function in Matlab, the values of each absorber’ parameters can be computed. It should be noted that the resulted parameters values should satisfy the condition of stability which is negativity of the real part of the eigenvalues. A sample Matlab code to implement the mentioned algorithm is presented in appendix C.
3‐1
Jeffcott rotor without absorber
At the beginning, the influence of rotational speed on system’s behavior is taken into account.In Fig. 7, the orbit diagram and radial displacement of the disk is demonstrated for 0.5, 0.8 and 1. For deeper understanding of the rotational speed influence on the system, the bifurcation diagram of the radial displacement in terms of rotational speed is shown in Fig. 8.It should be noted that, the dash line (red color) in following diagrams determines the position of the stator. Whenever the trajectories intersect with this line, the contact is happened. From this figure, it can be understood that, before and after contact occurrence
0.5 ,
0.8 the system has a periodic behavior and the contact just
happens at the underside part of the disk. However, as the rotational speed goes up,
16
system’s vibrations get more intense and multi – periodic and more contacts take place. These claims can be fully understood from Fig. 6 where the radial displacement of the disk for different values of the rotational speed is shown. With regard to this diagram, contact starts near to of 1
0.8 and the operating system has its most crucial behavior in the interval
1.8. However after
1.8 the vibrations of the system is considerably
reduced. This sudden reduction of vibration amplitude can be understood from the linear vibration theory of rotary systems with eccentricity. As we know, form frequency response of linear vibratory single degree of freedom system, at
2
, the amplitude of
oscillation will be reduced due to dominance of inertial force to centrifugal force. In this range the inertial force increase with square of exciting force, and accordingly the amplitude of vibration reduces and reaches to constant small value. Figure 7: Orbit and radial displacement of the disk for
0.5, 0.8, 1 respectively.
Figure 8: Bifurcation diagram for radial displacement of the disk for 0.5
2.
A delicate point that can be extracted from the time‐response diagram is that the system has an asymmetrical displacement in horizontal and vertical directions. The reason is that when a vibratory system with eccentricity works in a rotational speed approximately two times more than natural frequency, a static amplitude emerges in the system’s displacements both in horizontal and vertical directions. This is because of the fact that the force due to eccentricity meets a static equilibrium with spring force which always exists in the system motion. Due to nonlinearity of the equations of motion, the static responses vary at different rotational speeds. As a result of centrifugal force of eccentricity, static response is produced which give static response and varies with rotation of disk. This static displacement remains with the system response for different rotational speed with respect to Fig. 7. Figure 9: Bifurcation diagram for static displacement of the disk for 0.5 17
2.
In the previous section the effect of different rotational speed on the system’s behavior was taken into account. In the following section the influence of another parameter would be studied. It should be noted that the rest of the system’s parameters have the values of Table 1. Moreover, the effect of each parameter variations on the system’s behavior is considered for two rotational speed values: contact beginning rotational speed and resonance
0.8
1 .
Table 1: Primary system parameters. The influence of shaft’s stiffness variations on system’s behavior is shown in Fig. 8 (a) and (b).According to the results, at
0.8 the amplitude of system decreases as
increases. However, the frequency content is not regular. So many bifurcations can be obserevd and the system containes peroidic windows. On the other hand, at lower values of
the system shows almost a two peridic behaviour. But, as
1 for
increases,the
system executes a multi‐periodic behaviour with fixed frequency content. Furthermore, the amplitude of the system increasesas
increases.
Figure 10: Bifurcation diagram for radial displacement of the disk for 5 2.5
10 , (a)
0.8, and (b)
10
1.
The amount of clearance between disk and stator is important in designing rotor dynamics systems. In fact, increasing the efficency of such devices requires the small values of clearance. On the other hand, little gap between disk and stator can raise the probality of contatc happening. Fig. 9a and 9b show the system’s behaviour at
0.8 and
1,
respectively,with respect to disk and stator clearance variations. It is evident that as expected for lower clearnces, radial displacment is higher. It is evident from this figure that at
1 the lower clearance results in higher vibrations
18
amplitude in comparison to
0.8. In addition, it can be understood that for lower values
of the system has a larger frequency contetnt; however, greater values of leads the system’s behaviour to a single periodic responses. Due to importance of clearance, it is of interest to provide a safe condition in which enables the rotary system to operate in small values of clearance. 10 , (a)
Figure 11: Bifurcation diagram for radial displacement of the disk for 10
0.8, and (b)
1.
Another important parameter that should be take care of, is the eccentricity of the disk. In Fig. 10 it can be percievd that the growth in the ecentricity results in the growth of radial displacement of the disk. For
0.8 Fig. 10 (a), if the eccentricity of the disc is lower that
0.5, there will not be any contact; however, for the values ̅
0.5 an undesirbale behvaiour
of the system will be emerged. But at reseonance frequency Fig. 10 (b), disk is more sensetive to the eccentricity, and in the entire cosidered interval of eccentricity 0.1
̅
1 the contatc is occurred between disk and stator. Furthermore, based on the
results augmentation of ̅ gives a multi‐periodic behaviour of the system. Figure 12: Bifurcation diagram for radial displacement of the disk for 0.1
0.8, and (b)
̅
1, (a)
1.
Now after investigation of the characteristics of selected rotor dynamics system with variations in its parameters, in the following section, at first an optimum TMD and NES are designed for this system, and the effectiveness of TMD and NES on the system’s vibrations at different situations are discussed.
3‐2
Jeffcott rotor with TMD and NES
19
As was mentioned before, dependency of rotor dynamics system on time and presence of nonlinearities in its dynamic and equations of motion make it hard suggest a suitable and well – organized procedure for designing dynamic absorbers such as TMD and NES. To this end, an optimization process is proposed in section 2.3. Applying the proposed optimization algorithm, the obtained optimum values of TMD and NES are presented in Table 2. Table 2: Optimum parameters of TMD and NES at different dimensionless rotational speeds of disk. According to Table 2, equal values are obtained for TMD and NES through the optimization process. The angular position of both absorbers is equal to 3.14 or which meansthat both of them should be placed in the opposite direction of the disk’s eccentricity, where In this position the absorber has its best performance.
3.2.1 Effectiveness of absorbers with variation of rotational speed Using the optimum values of TMD and NES from Table 2, the orbit and radial displacement diagram of the disk at
1 are illustrated in Fig. 11. It is evident that both
absorbers are able to perfectly resist the effect of the eccentricity force and entirely reduces the vibrations. Figure 13: Orbit and radial displacement of the disk with TMD and NES at
1.
Furthermore, the effect of TMD and NES on the system’s performance for different values of rotational speed is shown in Fig. 12. According to the results, using TMD not only reduces the vibration of the rotor dynamics system in resonance frequency
1, but also
it can be observed that absorbers are able to meliorate the behavior of the system in critical interval of 1
1.8 which is a region of undesirable vibrations, according to Fig.
20
6. It is evidentthat both TMD and NES have greatly decreased the system’s vibrations and static displacements just remain within the system. Figure 14: Radial displacement of the disk with TMD and NES at different rotational speeds. In previous section, the infulence of absorbers on system’s behavoiur were analysed based on different rotaional speed values. Now, other parameter’s variations of the the system is taken into account. Accoring to Figs. 8 ‐ 10 it can be realized that changing the values of such parameters had a considerable effects on the system without absorber. In the follwing sections, it will be shown that utilising TMD and NES in the rotor dynamics system equip the system with better resistance to parameters variations such as
, ̅ , .
3.2.2 Effectiveness of absorbers with variation of shaft’s stiffness In this section the influences of TMD and NES on the system’s behaviour at contatc beginning frequency
0.8 and resonance frequency
1for different values of
are
taken into account. The obtained results are shown in Fig. 13. This figure demonstartes that the presence of absorbers enables the system to work in a wide range of shaft’s stiffness. Besides, there is no period changing in the system’s response and that contact is omitted entirely. This results show that disk with absorbers have sufficient robustness with respect to the change in the stiffness of the shaft. Fig. 15: Bifurcation diagram for radial displacement of the disk with TMD and NES, (a) at 0.8, and (b) at 1.
3.2.3 Effectiveness of absorbers with variation of rotor – statorclearance The clearance between disk and stator is important in designing rotor dynamics systems. In fact, increasing the efficency of such devices requires the small values of clearance. On the other hand, little gap between disk and stator can raise the probability of
21
contatc happening. In this section the goal is to determine whether TMD and NES can provide a relaible condition for lower values of clearance or not. According to Fig. 14 it can be perceived that TMD and NES are able to provide the condition of using the smaller clearance values for designing a rotary system in addtion to mitigate undesirable vibrations of the system. Also this result shows that the performance of TMD and NES at
0.8and
1 are the same, and absorbers can effectively tolerate
the variation in the disk– statorclearance.
Figure 16: Bifurcation diagram for radial displacement of the disk with TMD and NES: (a) at 0.8, (b) at 1.
3.2.4 Effectiveness of absorbers with variation of disk’s eccentricity In the following part of study, TMD and NES performances are investigated with variation on eccentricity of disk at rotational speeds of
0.8 and
1. With using
optimum parameters of absorbers, the obtained results are shown in Fig. 15. According to this diagram, at
0.8 both absorbers operate efficient and are able to attenuates the
vibrations of the system. However, for very small and high values of eccentriciy, the amplitude of vibrations are increased and there are problem of contacts. What can be percieved this figure, is that TMD and NES have the best performances in the eccentriciy point ̅
0.5 that the optimized parameters of them are obtained and in a vivinity of this
point. The legnth of this vicinity can be approxiamted about
0.5 , 0.5 accoring to TMD
and NES diagrams. This figure show that both absorbers can compensate the variation in the eccentricity of disk. The degredesion in the performance of both absorbers in small and high values of eccentriciy can be described as follows. At higher disk's eccentriciy, since the mass of absorbers are small, in the confined space between the disk and stator, the centrifugal force of absorer cannot copensate the disk unbalance force, hence the contact of disk and stator will occure. At small values of disk's eccentricity, opposite conditions are presented. In this case the mass of absorber is higher than disk eccentricity and absorber acts as an uballance force, hence contact will occure again.
22
Also the performances of the absorbers at resonance frequency
1 are shown in this
figure. Both absorbers have the same performances, and are effective in the domain of 0.42
̅
0.58, and the best performance of both of them is at ̅
0.5. Unlike for
0.8,
in resonance both absorbers are able to just mitigate the vibration of the system in a limited vicinity of ̅
0.5.
Figure 17: Bifurcation diagram for radial displacement of the disk for with TMD and NES, (a) 0.8, (b) 1.
3‐3
Redesign of TMD and NES for maximizing eccentericity tolerance
More investigation is done in the following section in order to evaluate the efficiency of TMD and NES, with respect to the change in value of eccentricity. Here, the optimum values of TMD and NEs are provided based on ϵ
1 rather that ϵ
0.5 and new parameters as
presented in Table 3, are obtained. Table 3: Optimum parameters of absorbers for maximazing eccentericity tolerance. Fig. 16 (a) shows that the best performance of TMD for
0.8 is in ̅
1. This result
does approve the claim that absorbers work well in the ̅ which is cosidered for providing its optimum performance. However, in the interval of ̅
0.5 which no contatc happens
according to Fig. 18, contatc phenomena is observed in this case. What can be conclude from Fig. 18 (a) and Fig. 18 (b) is that at optimum parameter for TMD and NES is ̅
0.8 the best value of eccetricy to obtain the 0.5.
Figure 18: Bifurcation diagram for radial displacement of the disk for with TMD and NES, (a) 0.8, (b) 1.
3‐ Conclusion 23
The main objective of this paper is suggesting a method of optimization for obtaining optimum parameters of TMD and NES in order to attenuating the vibrations of a simple model of an operating rotary machine (Jeffcott rotor), and investigation of their performance with respect to the change in the parameters of system. For the sake of determining the influence of each absorber on the rotor dynamics system a study on the system’s behavior is done for different values of its parameters such as rotational speed , clearance
, stiffness
and eccentricity of disk
̅ at contact beginning and
resonance frequency. What can be concluded are: i.
TMD and NES perform an efficient and positive influence on reducing the amplitude of the studied rotor dynamics system. Their efficiency is quite observable through bifurcation analysis of the system for the control parameters such as: rotational speed
ii.
, shaft’s stiffness
and clearance
At resonance frequency
.
1 , TMD and NES are able to just mitigate the
vibrations of the system in a specific amount of eccentricity
̅ . It means that the
design of these absorbers should concentrate on a particular value eccentricity. On the other hand, at contact beginning frequency
0.8 , TMD and NEs are more
reliable to provide a safe working condition for variety of eccentricity values. iii.
The application of this kind of placement of TMD and NES (alongside a radial rod) in rotational low and high speed operating machines, can be trusted.
Appendices Appendix A The matrices in Eq. 26and Eq. 27for TMD are as follows. 1 0 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
24
0 0 0 0 0
1
A.1
1 2
0
1 2
0 0
0 1 2
0
0
2
2 0
1 2
0
0
2
2
0
0 0
,
0 0
2 ̅ 2 0
̅
2 0 0
0 0
0 0 0 0
0
,
A.3
2 ̅
2 0
0 0
0 0
̅ 2
2 0 0 0
2
0
0 0 0 0 0 0
A.2
2 0
0
0 0
2
0
2
2 0
0 0
2 0 0
0
2 1 0 0 1 0 0
,
0 0
A.4
0 0 0 2 2 For the NES due to the nonlinear stiffness all the stiffness matrices has nonlinear 0
expressions in terms of
,
, ̅
except
which is the same as for TMD.
25
0
1
0 0 1
1
0 1
2
2
2 1
2
2
1
2
2
0
0 0
0 0
0
0
0
0
0 0
2
1
8
2 3
9 3
0 0
0
0
1
0
0
1
0
0
0 3
0
3 3 3
4
2
0 1
0
2 1
0
0
3
0
0
0
0 0 0
8
2 3
̅
3 3
2
2
A.5
0 2 3
3 8
2 3
9 1
3
3
8
̅
3 2
1
2
2
0 0 3
3 3 3
2
4
2 3 ̅ 0 0
1 0
,
0 1
0
0 0 3 2 ̅
0 3
3 2
2 3 ̅
0 0
0
0 0 3
2
A.6
0 0 ̅ 2
0 0 0
0 0 3 ̅
Appendix B For deriving the complex averaging form of the equations of motion of the system with TMD the eq. 24 – 26 are replaced in eq. 15 – 17. The resulted equations are as follows:
26
1
2
2
2
̅
2
2
̅ ̅
2
2
2
2
̅
̅
1
2
̅
2
̅ 2
2
̅
2
sin
cos
2
2
2
2 ̅ 2
2
27
2
2 ̅
̅
It should be noted that , sin
2
̅
̅
2
2
2
2 ̅
2
2
2
̅
2
cos
̅
2
2
2
2
2
̅
̅
1
̅
2
2
2
2
̅
2
2
2
̅
2
2
̅
1
̅
2
2
Then with retaining the coefficients of
and ignoring all other terms,the equations are
transformed to the complex averaging form.
Appendix C A sample code in Matlab format to implement the proposed optimization algorithm is given here. Generally the required code must has the following form. function Complex_averaging_Amplitude_optimization clc, clear all, close all, format long, %% specifying the constant parameters of system such as Mass of disk, stiffness of spring and etc. %% Specifying the design rotational speed a specific value for optimization such as the rotational speed in which contact or resonance is occurred. %% specifying the desired eigenvalues to guarantee the stability of linear part of equation eig_o_d = specifying a stable eigenvalues : This eigenvalue can be determined from the state of original system, in which system is stable without any contact %% specifying the desired amplitude of vibration for rotor and absorber such as: rd_d = the desired dynamic amplitude for rotor; qs_d = the desired static amplitude for rotor; %% setting the parameters to be optimized such as Par = setting the parameters of TMD or NES and their location as the parameters to be optimized; %% setting the upper and lower values of parameters and the initial guess LB = lower bound of parameters; UB = upper bound of parameters; x0 = an initial guess; %% Use of fmincon command for optimization in the following form [x, fval, exitflag] = fmincon(@fun_TMD_eigen_value_optimization, x0, [], [], [], [], LB, UB, @mycon, options) %% specifying the optimization function or cost function function Out = fun_TMD_eigen_value_optimization(y) Out = sum(abs(rd.^2‐rd_d.^2)); 28
% Note that these stiffness and mass matrices are dependent to the dynamic and static amplitude of vibration and are determined by rd_d and qs_d. end %%imposing the linear and nonlinear constraint function [c_nonl , ceq] = mycon(y) % determination of mass and stiffness matrices and from then the eigenvalues of linearized system eig_o = eig(Stiffness matrix, ‐Mass matrix); c_nonl = real(eig_o); % This condition guarantee that eigenvalue of system has negative real part and system has the required stability. ceq = the equation of motion; % The equation of motion must be satisfied, and it can be considered as an equal nonlinear constraint. end end
References
[1] F. K. Choy, and J. Padovan, Nonlinear transient analysis of rotor‐casing rub events, journal of Sound and Vibration (1987) 113:529–545 [2] R. F. Beatty, Differentiating rotor response due to radial rubbing, Journal of Vibration, Acoustics, Stress and Reliability in Design (1985) 107:151–160. [3] F. Chu, Z. Zhang, Bifurcation and chaos in rub‐impact Jeffcott rotor system, Journal of Sound and Vibration (1998) 210 (1):1–18. [4] P. Goldman, A. Muszynska, Chaotic behavior of rotor/stator systems with rubs, Journal of Engineering for Gas Turbine and Power (1994) 116:692–701. [5] J. J. Yu, P. Goldman, D. E. Bently, A. Muzynska, Rotor/seal experimental and analytical study on full annular rub, Journal of Engineering for Gas Turbine and Power (2002) 124:340–350. [6] J. J. Connor, Introduction to structural motion, Prentice hall, England, First edition, (2002) [7] C. Wan, C. Jian, C. K. Chen, Chaos and bifurcation of a flexible rub‐impact rotor supported by oil film bearings with nonlinear suspension. Mechanism and Machine Theory (2007) 42:312–333
29
[8] A.K. Samantaray, Steady‐state dynamics of a non‐ideal rotor with internal damping and gyroscopic effects, journal of Nonlinear Dynamics (2009) 56: 443 [9] Z. Shang, J. Jiang, L. Hong, the global responses characteristics of a rotor/stator rubbing system with dry friction effects, Journal of Sound and Vibration (2011) 330: 2150–2160 [10] O. V. Gendelman, Analytic treatment of a system with a vibro – impact nonlinear energy sink, Journal of sound and vibration (2012) 331: 4599 – 4608. [11] G. Jacquet‐Richardet, M. Torkhani, P. Cartraud, F. Thouverez, T. Nouri, Baranger, M. Herran, C. Gibert, S. Baguet, P. Almeida, L. Peletan, Rotor to stator contacts in turbomachines. Review and application, Mechanical Systems and Signal Processing (2013) 40: 401–420 [12] J. P. Chavez, M. Wiercigroch, Bifurcation analysis of periodic orbits of a non‐smooth Jeffcott rotor model, Communications in Nonlinear Science and Numerical Simulation (2013) 18: 2571–2580 [13] A. Najafi, M.R. Ghazavi, A.A. Jafari, Application of Krein’s theorem and bifurcation theory for stability analysis of a bladed rotor, Meccanica (2014) 49:1507–1526. [14] Nicholas Vlajic, Xianbo Liu, Hamad Karki, and Balakumar Balachandran, Torsional oscillations of a rotor with continuous stator contact, International Journal of Mechanical Sciences, (2014) 92: 102–110. [15] P. Varney, I. Green, Nonlinear phenomena, bifurcations, and routes to chaos in an asymmetrically supported rotor–stator contact system, journal of Sound and Vibration, (2015) 336: 207–226. [16] N. A. Saeed, M. Kamel, Nonlinear PD‐controller to suppress the nonlinear oscillations of horizontally supported Jeffcott‐rotor system, International Journal of Nonlinear Mechanics, (2016) 87: 109‐124. [17] N. A. Saeed, H. A. El‐Gohary, On the nonlinear oscillations of a horizontally supported Jeffcott rotor with a nonlinear restoring force, Nonlinear Dynamics., (2017) 88: 293‐314. [18] M Eissa and N.A. Saeed, Nonlinear vibration control of a horizontally supported Jeffcott‐rotor system, Journal of Vibration and Control, (2017), DOI: 10.1177/1077546317693928. [19] N. A. Saeed and M. Kamel, Active magnetic bearing‐based tuned controller to suppress lateral vibrations of a nonlinear Jeffcott rotor system, Nonlinear Dynamics., (2017) pp:1‐22. [20] N. A. Saeed and W. A. El‐Ganaini, Utilizing time‐delays to quench the nonlinear vibrations of a two‐degree‐of‐freedom system, Meccanica, (2017), 52: 2969–2990. [21] N. A. Saeed and W. A. El‐Ganaini, Time‐delayed control to suppress the nonlinear vibrations of a horizontally suspended Jeffcott‐rotor system, Applied Mathematical Modelling, (2017), 44: 523‐539.
30
[22] N. A. Saeed and W. A. El‐Ganaini, Influences of time‐delays on the performance of a controller based on the saturation phenomenon, European Journal of Mechanics ‐ A/Solids, (2017), 66: 125‐142. [23] S. Bab, S.E. Khadem, M. Shahgholi, Lateral vibration attenuation of a rotor under mass eccentricity force using non‐linear energy sink, International Journal of Non‐Linear Mechanics, (2014) 67: 251‐266. [24] S. Bab, S. Khadem, M. Shahgholi, Vibration attenuation of a rotor supported by journal bearings with nonlinear suspension under mass eccentricity force using nonlinear energy sink, Meccanica(2015) 50:2441‐2460. [25] J. Taghipour, M. Dardel, Steady state dynamics and robustness of a harmonically excited essentially nonlinear oscillator coupled with a two‐DOF nonlinear energy sink, Mechanical Systems and Signal Processing (2015) 62‐63:164–182. [26] C. Guo, M. A. AL‐Shudeifat, A. F. Vakakis, L. A. Bergman, D. M. Mcfaral, J. Yan, Vibration reduction in unbalanced hollow rotor systems with nonlinear energy sinks, Nonlinear Dynamics, (2015) 79:527–538. [27] M. Taleshi, M. Dardel, M. H. Pashaie, Passive targeted energy transfer in the steady state dynamics of a nonlinear plate with nonlinear absorber, Chaos, Solitons and Fractals (2016) 92:56–72. [28] M. Parseh, M. Dardel, M. Hasan Ghasemi, Steady state dynamics of a non‐linear beam coupled to a non‐linear energy sink, International Journal of Non‐Linear Mechanics (2016) 79:48–65 [29] Nicholas Vlajic, Alan R. Champneys, Balakumar Balachandran, Nonlinear dynamics of a Jeffcott rotor with torsional deformations and rotor‐stator contact, International Journal of Non‐Linear Mechanics, (2017) 92: 102–110.
31
Figu ures:
Figure 1: Jeffccott rotor fiixed in both h end.
a)
b)
Figurre 2: a) Geo ometric and d mass cen nter of the d disk, b) the contact forrce compon nents in normal and tangentiaal direction ns.
32
Figure e 3:A schem matic of thee applied tu uned mass d damper.
Figure 4: A schem matic of the applied no onlinear eneergy sink.
33
Figu ure 5: Rotorr dynamics with singlee TMD.
Figu ure 6: Rotor dynamicss with single NES.
34
Fig gure 7: Orb bit and radiial displaceement of thee disk for
0.5, 0.8, 1 respectiively.
Fig gure 8: Bifu furcation diiagram for rradial displlacement of the disk fo or 0.5
35
2.
Fig gure 9: Biffurcation diiagram for static displlacement off the disk fo or 0.5
(a)
36
2.
(b) Figu ure 10: Bifu urcation diaagram for rradial displaacement off the disk fo or 5 2.5 5
10 , (a))
0.8, aand (b)
37
1.
10
(a)
(b) Figure 11: Bifurcaation diagraam for radiial displaceement of thee disk for 1 10
0 0.8, and (b) ) 38
1.
10 , (a)
(a)
(b) Figu ure 12: Bifu urcation diaagram for rradial displaacement of f the disk fo or 0.1
0 0.8, and (b) ) 39
1.
̅
1, (a)
Fig gure 13: O Orbit and raadial displaccement of tthe disk witth TMD and d NES at
1.
Figu ure 14: Rad dial displaccement of th he disk witth TMD and d NES at diffferent rotational speeds.
40
(a)
(b) Fig. 15 5: Bifurcatiion diagram m for radiall displacem ment of the d disk with TMD and NE ES, (a) at 0..8, and (b) aat 1.
41
(a)
(b) Figure e 16: Bifurccation diagrram for rad dial displacement of th he disk with h TMD and NES: (a) at 0.8, (b) att 1.
42
(a)
(c) Figure e 17: Bifurccation diagrram for rad dial displacement of th he disk for w with TMD aand NES, (a) 0.8, (b) 1.
43
(a)
(b) Figure e 18: Bifurccation diagrram for rad dial displaceement of th he disk for w with TMD aand NES, , (a) 0.8, (b) 1.
44
Tables: Table 1: Primary system parameters.
.
0.1
9.81
. 2100
10
2.5
10
7.5
2
10
5
10
10
10
Table 2: Optimum parameters of TMD and NES at different dimensionless rotational speeds of disk.
Optimum parameters of TMD
0.8 1
0.0025 0.0025
5 5
0.185 0.185
3.14 3.14
Optimum parameters of NES
0.8
0.0025
5
0.185
3.14
1
0.0025
5
0.185
3.14
Table 3: Optimum parameters of absorbers for maximazing eccentericity tolerance
Optimum parameters of TMD
0.8 1
0.005
5
1.85
3.14
0.005
6
2.15
3.14
Optimum parameters of NES
0.8
0.005
15
1.85
3.14
1
0.005
15
1.85
3.14
45
Highlights ‐
Tuned mass damper (TMD) and nonlinear energy sink (NES) are suggested for preventing contact occurrence of rotor system.
‐
In order to efficiently design absorbers, optimization is proposed.
‐
For optimization process, complex averaging method is used in order for deriving the equations of motion.
‐
The results show that TMD and NES are efficient in vibration reduction.
PII: DOI: Reference:
S0020-7462(17)30254-8 https://doi.org/10.1016/j.ijnonlinmec.2017.10.011 NLM 2915
To appear in:
International Journal of Non-Linear Mechanics
Received date : 2 April 2017 Revised date : 16 October 2017 Accepted date : 16 October 2017 Please cite this article as: G.G. Tehrani, M. Dardel, Mitigation of nonlinear oscillations of a Jeffcott rotor System with an optimized damper and nonlinear energy sink, International Journal of Non-Linear Mechanics (2017), https://doi.org/10.1016/j.ijnonlinmec.2017.10.011 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Mitigation of nonlinear oscillations of a Jeffcott rotor System with an optimized damper and nonlinear energy sink Ghasem Ghannad Tehrani and Morteza Dardel1 Abstract Contact occurrence between disk and stator as result of undesirable vibrations produced by eccentricity of the disk is one of the most destructive and common phenomena in rotor dynamics systems. In this work, utilizing tuned mass damper (TMD) and nonlinear energy sink (NES) are suggested as a solution for preventing contact occurrence. The mass and angular position of absorbers determine their efficiency for resisting the eccentricity force produced by the disk, and their stiffness and damping coefficients determine the displacement scope of the absorber. In order to efficiently design absorbers, optimization is proposed. In this suggested optimization process, complex averaging method is used in order for deriving the equations of motion of the system in presence of dynamic absorbers at the steady state condition. Afterwards, for determining trustworthiness of each absorber’s performance, system’s behavior is studied for different values of its parameters such as rotational speed, stiffness, clearance and eccentricity in presence of each absorber. From the obtained results, it can be perceived that TMD and NES are as efficient as possible and they have exactly the same positive influence on the system’s vibrations. The reliability of the proposed optimization process can be determined by the results. Key word: Jeffcott rotor, Contact, TMD, NES, Complex averaging method, Optimization, rotor dynamics. 1Corresponding author: Department of Mechanical Engineering, Babol Noshirvani University of Technology, P.O. Box 484, Postal Code: 47148‐71167, Shariati Street, Babol, Mazandaran, Iran. [email protected].
1
1‐ Introduction Disk’s eccentricity is one of the commonest reasons of undesirable behavior in turbomachines. In addition to eccentricity, small clearance between disk and stator is another reason which leads to destructive consequences like contact phenomena. All of the mentioned problems result in undesirable vibrations which makes different faults like disk–stator contact phenomena. Removing contact and reduction of whirling vibration amplitudes in rotor dynamics systems are very important. Choy and Padovan [1] and Beatty [2] have done the first complete nonlinear disk–stator contact models. Chu and Zhang [18] investigated the nonlinear vibration characteristics of a rub‐impact Jeffcott rotor. Also they found, with increasing rotating speed, the grazing bifurcation, the quasi periodic motion and chaotic motion occur after the rub‐impact. Goldman and Muszynska [4] found the chaotic behavior of rotor/stator systems with rubs. Yu et al. [5] presented the results of experimental and analytical studies on rotor/seal full annular rub. Wan et. al. [7] worked on a rotor dynamics system with rub‐ impact effect and applied hydrodynamic theory for oil film bearings as the supports. They derived orbit diagram of the disk’s motion in order to investigate chaos phenomena. Samantaray [8] analyzed a four‐degrees‐of‐freedom rotor driven by a DC motor considering Sommerfeld effect. The steady‐state spin frequency and whirl orbit amplitude are derived as functions of the drive and rotor system parameters. Z. Shang et al. [9] studied a rotor dynamics system by taking into account rotor/stator rub impact and dry friction effect (the previous works and is the main factor for the self‐excited dry friction backward whirl). The results confirmed the coexistence of different rubbing responses observed in tests. Richardet et al. [11] worked on numerical models and experimental setups in order to highlight the phenomenology involved in different rotor to stator contacts. Chavez and Wiercigroch [12] studied a non‐smooth model of Jeffcott rotor which bearing clearance and rub‐impact effect was taken into account. In this paper the bifurcation analysis is applied using software TC‐HAT and it complex dynamics occurring is investigated.
2
Najafi et al. [13] investigated the stability of a bladed rotor with cylindrical and conical whirling using Krein’s theorem, they showed that if a stagger angle of the blades be equal to 0 and 90 , the blades vibrations are coupled only with cylindrical whirling for blades stagger angle of 0 and only with conical whirling for blades stagger angle of 90 . For other stager angles on the other hand, the blades vibrations are coupled with both cylindrical and conical whirling altogether. Vlajic et. al. [14] are analytically and numerically studied a nonlinear torsional vibration of Jeffcott rotor subjected to continuous stator contact for both forward and backward whirling. They gave special consideration to the torsional response during high‐ speed backward whirling, and found that although the rotor is subject to discontinuous friction, during high‐speed backward whirling, full system and reduced‐order models for the torsional oscillations are smooth in nature, and centrifugal stiffening effect is experienced by the torsional motions during both forward and backward whirling. Varney [15] investigated different nonlinear dynamic behavior of an asymmetrically supported Jeffcott rotor by considering contact between disk and stator. Saeed and Kamel [16] worked on controlling vibrations of a Jeffcott‐rotor using an active control technique. In this paper after observing high amplitudes and nonlinear behavior in the system, a Proportional‐Derivative (PD)‐Controller using two pairs of electromagnetic poles is applied to the system. The results provided by bifurcation analyses demonstrated that using this controller is highly efficient for vibration attenuation. Saeed and El‐Gohary [17], studied the vibration analysis of a horizontally supported Jeffcott rotor system. They used multiple scales perturbation technique to obtain a second‐order approximate solution at the resonance conditions, and obtained the stability of system by applying Lyapunov first method. The influences of all the parameters on the system behavior are explored. Also the effect of both the negative and positive values of the nonlinear stiffness coefficient is studied. Eissa and Saeed [18], applied a positive position feedback controller for a horizontally supported Jeffcott‐rotor system. The multiple scales method is applied to obtain the approximate solution of the system. The results are shown in the form of bifurcation diagrams and also the effects of different controller parameters on the system frequency – 3
response curves are studied. Saeed and Kamel [19], utilized active magnetic bearing for reducing the vibration of Jeffcott‐rotor system. In this work, the controller is attached to the primary system through four electromagnetic poles. Then, the influence of different parameters such as disk eccentricity, air‐gap and others are investigated on the vibration amplitudes of the system. The analytical and numerical results are in a good agreement with each other. Saeed and El‐Ganaini [20] showed how to utilize time‐delays to mitigate the oscillations of a two‐degree‐of‐freedom nonlinear model for a horizontally supported Jeffcott‐rotor system. The extracted the slow‐flow modulating equations of both the amplitudes and phases, and the conditions that make the time‐delayed controller works as a damper or exciter are clarified. Also Saeed and El‐Ganaini [21] used time‐delayed position‐velocity controller for vibration control of a horizontally suspended Jeffcott‐rotor system having cubic and quadratic nonlinearities. They showed that the time‐delay increases the vibration amplitudes and can destabilize the system solution in the case of negative position feedback control, while at positive position feedback control it improves the vibration suppression performance. In continuo of their previous works, Saeed and El‐ Gohary [22] used a modified version of the controllers based on the saturation phenomenon which is connected to the system via quadratic coupling nonlinearity, for nonlinear vibrations control of a vertically supported Jeffcott‐rotor system. Time‐delays in the control loop are included in the considered model to explore their influences on the controller performance and stability chart. They showed that the system has chaotic and quasi‐periodic motions in addition to the periodic ones. In recent years, researches on application of nonlinear energy sinks absorbers for vibration attenuation of discrete and continuous systems has been raised, instead of tuned mass dampers (TMD) or weakly nonlinear absorber. Bab et. al. [23] worked on attenuating vibrations of rotor dynamics system using smooth non‐linear energy sinks. In this paper the analytical solution is obtained using Multiple scales – Harmonic balance method (MSHBM). It is shown that for medium magnitudes of external force NESs collection demonstrates an impressive effect since larger area of SMR has been obtained in the domain of the system’s parameters. Bab et. al. [24] worked on vibration reduction of rotor dynamics system supported with journal bearings using NES. They attached NES on the
4
bearings and optimized its parameters utilizing genetic algorithm. Taghipour and Dardel [25] investigated the effect of single and double DOFs NESs on the behavior of a two‐DOF dynamical system in steady state condition. It is concluded that dynamical system with single DOF of NES shows less robustness to change in parameters and amplitude of external forces in comparison to double. Guo et. al. [26] mounted NES & TMD separately on the cross‐section of a hollow shaft in a rotor system and individually studied the influence of them on the vibrations of the system. Taleshi, et. al. [27] studied the behavior of a simply supported plate using NES. By changing the parameters and location of the NES using complex averaging and arc‐length continuation method, an optimized passive targeted energy transfer of the plate was obtained. Furthermore, the performance of NES and TMD was compared with each other. Parseh et. al. [28] worked on the influence of NES on a non‐ linear beam, modeled in according to Euler‐Bernoulli and Timoshenko theories. Using complex averaging with continuation method, the steady state response of the system based on resonance condition is investigated. Vlajic et. al. [29] worked on a modified model of Jeffcott roto, in which the dynamic behavior of the system including torsional deformation and contact phenomena is studied with considering forward and backward whirling. For forward whirling two rotor – stator friction models are taken into account and analytical and numerical analysis are done. For forward synchronous whirling fold, Hopf, lift‐off, and period‐doubling bifurcations are studied. Furthermore, transition from pure sticking to stick – slip regions is demonstrated for backward whirling. In previous studies for designing passive absorbers, numerous simulations were done in order to provide the efficient parameters of the absorbers which these simulations are so time consuming and are not coherent. Therefore, proposing an efficient, accurate and inclusive yet simple method of optimization is one of the main purposes of this paper. In this work, the simple model of the rotor dynamics system which is known as Jeffcott rotor is adopted as the case study. In the first section, the equations of motion of the system are derived both with and without attached absorbers. Then, a method of optimization is applied to provide the optimum parameters of each absorber. Finally, the efficiency of each absorber for different values of the system’s parameters like rotational speed, stiffness etc. is investigated.
5
2‐ Modeling and equations of motion In this section, the equations of motion of the system with different vibration absorbers are derived.
2.1 Modeling of Jeffcott rotor Before preceding the simulation of the system, it is assumed that: a) stator is to be rigid, b) the shaft is massless and has a uniform stiffness, c) gyroscopic effect of the shaft is neglected, and d) longitudinal and torsional vibrations of the shaft is not considered. Figs. 1 and 2 show the primary model of rotary machine. This model is known as Jeffcott rotor. The geometrical center of the disk is and the mass center is which has an eccentricity of geometric center. The mass of the disk is denoted by
, is the rotational
speed of the disk and is the angular displacement of mass center of the disk with respect to horizontal axis which is equal to
.
is the shaft’s stiffness in and directions. It
should be noted that the two ends of the shaft are clamped. In Fig. 2 the contact between disk and stator with the contact forces are shown. The clearance between disk and stator is and the radial displacement of the disk is shown by
.
Figure 1: Jeffcott rotor fixed in both end. Figure 2: a) Geometric and mass center of the rotor, b) the contact force components in normal and tangential directions. Normal and tangential forces contact stiffness is assumed to be
and
are generated at the disk–stator interface. The
and the tangential friction force is proportional to the
normal force by the friction coefficient.Since the contact occurs when
6
, in order to
satisfy this condition a Heaviside function is applied in the expression of normal contact force[5]. (1)
(2)
With respect to Fig. 2, the components of impact force in and directions are: sin
cos
cos
sin
(4)
are the displacements of the geometric center of the disk in horizontal
,
where
(3)
and vertical directions. The equations of motion of the system are [5]: cos
(5)
sin t
(6)
In order to analyze this rotor dynamics system, it is assumed that the shaft is massless and rigid, the disk is rigid, and axial and torsional vibrations are negligible. In order to write equations in dimensionless form, following quantities are introduced.
,
,
, 2
, (7)
̅
,
̅ ,
,
, ̅
,
,
̅
Hence:
̅
2 2
̅
̅
1 1
̅ ̅
̅ ̅
̅
̅
̅ ̅
̅
̅ ̅
(8)
(9)
̅ where the superscript of ′ denotes differentiation with respect to dimensionless time τ.
7
2.2 Dynamic absorber Dynamic absorber is a mechanical system which is mounted on a mechanical structure for the purpose of vibration mitigation of the structure. Two types of absorbers are well known: active and passive. The active dynamics absorbers operate by an external input, while passive absorbers do not need external input. In this study, two passive dynamic absorbers of TMD and NES are applied. In TMD is kind of passive dynamic absorber which consists of a mass, linear spring and a linear damper. These absorbers are tuned to a particular frequency of the main system, and reduce the system’s vibration amplitude by its own out of phase vibrations [6]. A schematic application of TMD to a system is shown in Fig. 3. NESis another kind of passive dynamic absorber with a mass, nonlinear spring and a linear damper. These absorbers are able to mitigate the vibration of the main system in a wider range of frequencies. The process in which the energy of the system is absorbed by such absorbers is call passive targeted energy transfer [10]. Due to essential nonlinearity of NES, it is not tuned to a particular frequency and can absorb vibration in wider frequency range. A schematic application of NES to a system is shown in Fig. 4. Figure 3: A schematic of the applied tuned mass damper. Figure 4: A schematic of the applied nonlinear energy sink. In order to place the absorbers in the primary system, a rod is used in which from one end is tied up to the disk and from another end is tied up to the shaft’s lateral surface. The rod placement makes two limitations for the absorber’s displacement: first of all, the absorber is bound to displace in a space between disk and shaft. Secondly, it is limited to just move along the rod.
2.2.1 Rotor dynamics system with TMD
8
In this section, the equations of motion of the system in presence of TMD are derived. For this system
,
and
are the mass, damping coefficient, and stiffness of the
absorbers, respectively. Parameter is the angle between rod and mass center of the disk. It can vary from 0 to . Here is an angular displacement of the rod.
represents shaft’s
is the displacement of the TMD’s mass along the rod. All the
radius. Furthermore,
and
assumptions were made in section 2.1 are valid here. Here displacements of the mass center of the disk, and
and
are the
are the displacement
components of the absorber. The displacements of the springs and dampers of TMD are and respectively. Figure 5: Rotor dynamics system with single TMD. In order to achieve the equations of motion Lagrange method is applied. The kinetic and potential energies, and Rayleigh dissipation functions are as follows. 1 1 1 (10) 2 2 2 1 1 1 (11) 2 2 2 1 1 (12) 2 2 where the displacements of disk and TMD in horizontal and vertical direction are: cos
,
sin
cos sin
,
sin
,
cos
sin
cos
cos
sin
(13)
Following additional dimensionless quantities are introduced. ,
̅
,
,
2
9
(14)
Substituting Eqs. (7, 14) into Eqs. (10 ‐ 12) and taking
,
as independent ,
parameters and applying Lagrange – Euler method dimensionless equations of motion in following form are derived. ̅
1
̅ cos ̅
1
cos ̅ ̅
̅ ̅ sin
1 ̅
1 ̅
̅ cos
sin
̅
cos 2
2
̅
(15) ̅
(16)
2 sin
̅
2
cos
sin ̅
̅
sin
cos
sin ̅
̅
2
(17)
2.2.2 Rotor dynamics system with NES In order to obtain the equations of motion, the only change in comparison to TMD is in the potential energy term, which is in following form. 1 2 where
,
1 2 ,
1 4 have the same definition previously mentioned for TMD.
,
(18)
Figure 6: Rotor dynamics system with single NES. With introduced following dimensionless quantities: e
, ̅
,
, 2
dimensionless equations of motion are obtained as following.
10
(19)
̅
1
̅ cos ̅
1
̅ cos ̅ ̅
̅ ̅ sin
1 1
̅
̅
̅
cos
̅
̅
2
cos
̅ sin
̅
̅ sin
2
̅ cos
2
(20) ̅
(21)
2
̅ ̅
2
̅
̅
(22)
2.3
Optimization the parameters of TMD and NES
In the following sections, it is observable that variations in structural or working parameters of the system cause contact between disk and stator. This undesirable phenomenon, results in emerging some residual stresses and deflections in disk and bearing and in some cases unwanted crack in the disk. So, eliminating such unwanted vibrations and improvement the working condition of the rotor dynamics systems is vital. Changing of the absorber’s parameter as a method to reach an optimum behavior of the system is not reliable since this parametric variation causes nonlinear system involve with bifurcations and many behavioral changes and optimization process less efficient. Hence, in this paper optimization is accomplished by a predefined purpose which is described in the following paragraph. It is assumed that there is no contact between disk and stator. This assumption is made for removing the nonlinearity due to contact force expressions. The considered rotor dynamics system is a self – excited system and dynamics of such systems is dependent on time which makes the optimization process so difficult. Hence, it is better to rewrite the equations of motion independent on time. To this end, complex averaging method is adopted.
2.3.1 Complexification of the equations of motion Complex averaging method is similar to harmonic‐balance method, in which the equation of motion is written in terms of amplitude and phase of the systems response. 11
This method is an analytical method and gives an approximate solution of the problem. The complex variables , and are defined as below. ̅ ̅ ̅
̅
̅
in which ̅
,
̅ , and ̅
,
(23)
are static part of displacements.
The displacements and their derivatives in terms of complex variables of , , , and are expressed as follows: ̅ ̅
̅
, ̅
2 2
2 ̅ ̅
̅
2
2 ,
2
̅
, ̅
2
(18)
2
2 ̅
(24)
(19)
2
̅ 2 2 where , , and ̅ are the complex conjugates of , , and respectively. Introduced ,
complex variable , , and are written as
, and
,
which from solving the obtained equations, the amplitudes and the phases of the system’s response are: | |
, tan
, | |
̅
tan
| |
,
(27)
, ̅
tan
Then, replacing latter complex variables in Eq. (24 ‐ 26) and Substituting resulted displacements and their derivatives into equations of motion of the system with TMD Eq. (15‐17) and averaging of the obtained equation with respect to the exciting frequency , , equations in terms of phase and amplitude are obtained(See Appendix section B). Then the equations are set into a following standard form:
12
(28)
(29)
where Eq. (28, 29) represents dynamics and static parts of the equations of motion, respectively. These matrices are given in appendix section B. The static response vector can be obtained from Eq. (29): (20)
Substituting Eq. (30) in Eq. (28), the equations of motion in complex averaging form has following form:
Where
(31)
′
(21)
The matrices
,
,
,
,
,
,
are all defined in
Appendix section A. Separating the real and imaginary parts of Eq. (28, 29), a system of nine unknowns and nine equations is obtained which consists of dynamic and static parts which are as following: Dynamic part: ̅ 2
2
2
2 2
2
2
2
2
2 0
2 2
̅
2
(33) ̅
2
2 2
2 2
̅ 2
2 2
13
0
2
Static part: 0
(22) 2
2
2
The same procedure for providing optimum parameters of TMD is adopted for NES and the standard form of equations of motion is obtained as follows (23)
The static and dynamic parts of the equations are as follows: Dynamic part: ̅ 2
2
2 3
2
3 3
3
3
8
2
2
3 8
2
2 3 3 3
3
̅
3
3
2
2 03
3
2
2 3 3 3
8
2
3
2
2
2
2 2
3 3
2 ̅
8
2
2
2 3
2
0
(36)
̅ 2 03
2
2 0
Static part: 0 3 2 2
2
3 2
2
2
2
(37)
It is clear that the nonlinearity due to nonlinear stiffness of the NES makes the dynamic and static parts equations dependent on each other. It means that unlike TMD it is not possible to distinguish static expressions vector
14
and provide them from static equation
and replace them in dynamic part. Therefore, the dynamic and static parts are needed to be solved altogether.
3 Optimization According to the outputs in the following sections, it is observable that variations in some parameters like rotational speed increase the amplitude of the system’s vibrations and causes instability in the system which is absolutely undesirable. There are variety of and
methods for optimum design of absorbers such as
that are the most famous and
applicable ones among the others. The integral of the amplitude in terms frequency is minimized in
method. In
method on the other hand, the maximum amplitude in
terms of frequency is minimized. In
and
optimizations, linear and nonlinear
constraints, linear and nonlinear in‐equalities can be considered, they are relevant to design on TMD and NES design, in the current study. Also codes for solving these optimization functions are available in commercial software such as Matlab. Here following procedure for optimization is proposed: At the beginning the structure without absorber is considered. A design point is determined from the simulations or based on the experience. For instance, this valuecan be chosen at a situation when resonance happens in the system or the system has large amplitudes of vibrations. The goal is to determine the system’s parameters under following conditions:
The amplitude of the primary system reaches a desirable value: Objective function
The equation of motion must be satisfied: Nonlinear constraint function
The stability of the system should be guaranteed: Nonlinear inequality constraint function, since the eigenvlaues of linearized equation of motion must be located in the left half of complex plane.
The absorber’s parameters should be within the allowed interval (upper and lower bound): Constraint function
15
The objective function is determined as follows: min
(24)
The refers to the amplitude of the disk in vertical and horizontal direction and the amplitude of the absorber. The purpose is to mitigate the amplitude of the disk and the absorber altogether and furthermore, the fmincon function of Matlab has just one scalar as the output, the objective function is determined as minimizing total amplitude of disk and absorber minus desired values
.
is selected as a feasible small value.It should be
mentioned that for the steady state response, and ′ must be set to zero, but for stability analysis it is necessary to obtain the eigenvalues of Eq. (31, 35). The amplitudes | | and | | which represent the amplitude of the disk’s center in the vertical and horizontal directions are considered as the purpose function of the optimization process. Then, using fmincon function in Matlab, the values of each absorber’ parameters can be computed. It should be noted that the resulted parameters values should satisfy the condition of stability which is negativity of the real part of the eigenvalues. A sample Matlab code to implement the mentioned algorithm is presented in appendix C.
3‐1
Jeffcott rotor without absorber
At the beginning, the influence of rotational speed on system’s behavior is taken into account.In Fig. 7, the orbit diagram and radial displacement of the disk is demonstrated for 0.5, 0.8 and 1. For deeper understanding of the rotational speed influence on the system, the bifurcation diagram of the radial displacement in terms of rotational speed is shown in Fig. 8.It should be noted that, the dash line (red color) in following diagrams determines the position of the stator. Whenever the trajectories intersect with this line, the contact is happened. From this figure, it can be understood that, before and after contact occurrence
0.5 ,
0.8 the system has a periodic behavior and the contact just
happens at the underside part of the disk. However, as the rotational speed goes up,
16
system’s vibrations get more intense and multi – periodic and more contacts take place. These claims can be fully understood from Fig. 6 where the radial displacement of the disk for different values of the rotational speed is shown. With regard to this diagram, contact starts near to of 1
0.8 and the operating system has its most crucial behavior in the interval
1.8. However after
1.8 the vibrations of the system is considerably
reduced. This sudden reduction of vibration amplitude can be understood from the linear vibration theory of rotary systems with eccentricity. As we know, form frequency response of linear vibratory single degree of freedom system, at
2
, the amplitude of
oscillation will be reduced due to dominance of inertial force to centrifugal force. In this range the inertial force increase with square of exciting force, and accordingly the amplitude of vibration reduces and reaches to constant small value. Figure 7: Orbit and radial displacement of the disk for
0.5, 0.8, 1 respectively.
Figure 8: Bifurcation diagram for radial displacement of the disk for 0.5
2.
A delicate point that can be extracted from the time‐response diagram is that the system has an asymmetrical displacement in horizontal and vertical directions. The reason is that when a vibratory system with eccentricity works in a rotational speed approximately two times more than natural frequency, a static amplitude emerges in the system’s displacements both in horizontal and vertical directions. This is because of the fact that the force due to eccentricity meets a static equilibrium with spring force which always exists in the system motion. Due to nonlinearity of the equations of motion, the static responses vary at different rotational speeds. As a result of centrifugal force of eccentricity, static response is produced which give static response and varies with rotation of disk. This static displacement remains with the system response for different rotational speed with respect to Fig. 7. Figure 9: Bifurcation diagram for static displacement of the disk for 0.5 17
2.
In the previous section the effect of different rotational speed on the system’s behavior was taken into account. In the following section the influence of another parameter would be studied. It should be noted that the rest of the system’s parameters have the values of Table 1. Moreover, the effect of each parameter variations on the system’s behavior is considered for two rotational speed values: contact beginning rotational speed and resonance
0.8
1 .
Table 1: Primary system parameters. The influence of shaft’s stiffness variations on system’s behavior is shown in Fig. 8 (a) and (b).According to the results, at
0.8 the amplitude of system decreases as
increases. However, the frequency content is not regular. So many bifurcations can be obserevd and the system containes peroidic windows. On the other hand, at lower values of
the system shows almost a two peridic behaviour. But, as
1 for
increases,the
system executes a multi‐periodic behaviour with fixed frequency content. Furthermore, the amplitude of the system increasesas
increases.
Figure 10: Bifurcation diagram for radial displacement of the disk for 5 2.5
10 , (a)
0.8, and (b)
10
1.
The amount of clearance between disk and stator is important in designing rotor dynamics systems. In fact, increasing the efficency of such devices requires the small values of clearance. On the other hand, little gap between disk and stator can raise the probality of contatc happening. Fig. 9a and 9b show the system’s behaviour at
0.8 and
1,
respectively,with respect to disk and stator clearance variations. It is evident that as expected for lower clearnces, radial displacment is higher. It is evident from this figure that at
1 the lower clearance results in higher vibrations
18
amplitude in comparison to
0.8. In addition, it can be understood that for lower values
of the system has a larger frequency contetnt; however, greater values of leads the system’s behaviour to a single periodic responses. Due to importance of clearance, it is of interest to provide a safe condition in which enables the rotary system to operate in small values of clearance. 10 , (a)
Figure 11: Bifurcation diagram for radial displacement of the disk for 10
0.8, and (b)
1.
Another important parameter that should be take care of, is the eccentricity of the disk. In Fig. 10 it can be percievd that the growth in the ecentricity results in the growth of radial displacement of the disk. For
0.8 Fig. 10 (a), if the eccentricity of the disc is lower that
0.5, there will not be any contact; however, for the values ̅
0.5 an undesirbale behvaiour
of the system will be emerged. But at reseonance frequency Fig. 10 (b), disk is more sensetive to the eccentricity, and in the entire cosidered interval of eccentricity 0.1
̅
1 the contatc is occurred between disk and stator. Furthermore, based on the
results augmentation of ̅ gives a multi‐periodic behaviour of the system. Figure 12: Bifurcation diagram for radial displacement of the disk for 0.1
0.8, and (b)
̅
1, (a)
1.
Now after investigation of the characteristics of selected rotor dynamics system with variations in its parameters, in the following section, at first an optimum TMD and NES are designed for this system, and the effectiveness of TMD and NES on the system’s vibrations at different situations are discussed.
3‐2
Jeffcott rotor with TMD and NES
19
As was mentioned before, dependency of rotor dynamics system on time and presence of nonlinearities in its dynamic and equations of motion make it hard suggest a suitable and well – organized procedure for designing dynamic absorbers such as TMD and NES. To this end, an optimization process is proposed in section 2.3. Applying the proposed optimization algorithm, the obtained optimum values of TMD and NES are presented in Table 2. Table 2: Optimum parameters of TMD and NES at different dimensionless rotational speeds of disk. According to Table 2, equal values are obtained for TMD and NES through the optimization process. The angular position of both absorbers is equal to 3.14 or which meansthat both of them should be placed in the opposite direction of the disk’s eccentricity, where In this position the absorber has its best performance.
3.2.1 Effectiveness of absorbers with variation of rotational speed Using the optimum values of TMD and NES from Table 2, the orbit and radial displacement diagram of the disk at
1 are illustrated in Fig. 11. It is evident that both
absorbers are able to perfectly resist the effect of the eccentricity force and entirely reduces the vibrations. Figure 13: Orbit and radial displacement of the disk with TMD and NES at
1.
Furthermore, the effect of TMD and NES on the system’s performance for different values of rotational speed is shown in Fig. 12. According to the results, using TMD not only reduces the vibration of the rotor dynamics system in resonance frequency
1, but also
it can be observed that absorbers are able to meliorate the behavior of the system in critical interval of 1
1.8 which is a region of undesirable vibrations, according to Fig.
20
6. It is evidentthat both TMD and NES have greatly decreased the system’s vibrations and static displacements just remain within the system. Figure 14: Radial displacement of the disk with TMD and NES at different rotational speeds. In previous section, the infulence of absorbers on system’s behavoiur were analysed based on different rotaional speed values. Now, other parameter’s variations of the the system is taken into account. Accoring to Figs. 8 ‐ 10 it can be realized that changing the values of such parameters had a considerable effects on the system without absorber. In the follwing sections, it will be shown that utilising TMD and NES in the rotor dynamics system equip the system with better resistance to parameters variations such as
, ̅ , .
3.2.2 Effectiveness of absorbers with variation of shaft’s stiffness In this section the influences of TMD and NES on the system’s behaviour at contatc beginning frequency
0.8 and resonance frequency
1for different values of
are
taken into account. The obtained results are shown in Fig. 13. This figure demonstartes that the presence of absorbers enables the system to work in a wide range of shaft’s stiffness. Besides, there is no period changing in the system’s response and that contact is omitted entirely. This results show that disk with absorbers have sufficient robustness with respect to the change in the stiffness of the shaft. Fig. 15: Bifurcation diagram for radial displacement of the disk with TMD and NES, (a) at 0.8, and (b) at 1.
3.2.3 Effectiveness of absorbers with variation of rotor – statorclearance The clearance between disk and stator is important in designing rotor dynamics systems. In fact, increasing the efficency of such devices requires the small values of clearance. On the other hand, little gap between disk and stator can raise the probability of
21
contatc happening. In this section the goal is to determine whether TMD and NES can provide a relaible condition for lower values of clearance or not. According to Fig. 14 it can be perceived that TMD and NES are able to provide the condition of using the smaller clearance values for designing a rotary system in addtion to mitigate undesirable vibrations of the system. Also this result shows that the performance of TMD and NES at
0.8and
1 are the same, and absorbers can effectively tolerate
the variation in the disk– statorclearance.
Figure 16: Bifurcation diagram for radial displacement of the disk with TMD and NES: (a) at 0.8, (b) at 1.
3.2.4 Effectiveness of absorbers with variation of disk’s eccentricity In the following part of study, TMD and NES performances are investigated with variation on eccentricity of disk at rotational speeds of
0.8 and
1. With using
optimum parameters of absorbers, the obtained results are shown in Fig. 15. According to this diagram, at
0.8 both absorbers operate efficient and are able to attenuates the
vibrations of the system. However, for very small and high values of eccentriciy, the amplitude of vibrations are increased and there are problem of contacts. What can be percieved this figure, is that TMD and NES have the best performances in the eccentriciy point ̅
0.5 that the optimized parameters of them are obtained and in a vivinity of this
point. The legnth of this vicinity can be approxiamted about
0.5 , 0.5 accoring to TMD
and NES diagrams. This figure show that both absorbers can compensate the variation in the eccentricity of disk. The degredesion in the performance of both absorbers in small and high values of eccentriciy can be described as follows. At higher disk's eccentriciy, since the mass of absorbers are small, in the confined space between the disk and stator, the centrifugal force of absorer cannot copensate the disk unbalance force, hence the contact of disk and stator will occure. At small values of disk's eccentricity, opposite conditions are presented. In this case the mass of absorber is higher than disk eccentricity and absorber acts as an uballance force, hence contact will occure again.
22
Also the performances of the absorbers at resonance frequency
1 are shown in this
figure. Both absorbers have the same performances, and are effective in the domain of 0.42
̅
0.58, and the best performance of both of them is at ̅
0.5. Unlike for
0.8,
in resonance both absorbers are able to just mitigate the vibration of the system in a limited vicinity of ̅
0.5.
Figure 17: Bifurcation diagram for radial displacement of the disk for with TMD and NES, (a) 0.8, (b) 1.
3‐3
Redesign of TMD and NES for maximizing eccentericity tolerance
More investigation is done in the following section in order to evaluate the efficiency of TMD and NES, with respect to the change in value of eccentricity. Here, the optimum values of TMD and NEs are provided based on ϵ
1 rather that ϵ
0.5 and new parameters as
presented in Table 3, are obtained. Table 3: Optimum parameters of absorbers for maximazing eccentericity tolerance. Fig. 16 (a) shows that the best performance of TMD for
0.8 is in ̅
1. This result
does approve the claim that absorbers work well in the ̅ which is cosidered for providing its optimum performance. However, in the interval of ̅
0.5 which no contatc happens
according to Fig. 18, contatc phenomena is observed in this case. What can be conclude from Fig. 18 (a) and Fig. 18 (b) is that at optimum parameter for TMD and NES is ̅
0.8 the best value of eccetricy to obtain the 0.5.
Figure 18: Bifurcation diagram for radial displacement of the disk for with TMD and NES, (a) 0.8, (b) 1.
3‐ Conclusion 23
The main objective of this paper is suggesting a method of optimization for obtaining optimum parameters of TMD and NES in order to attenuating the vibrations of a simple model of an operating rotary machine (Jeffcott rotor), and investigation of their performance with respect to the change in the parameters of system. For the sake of determining the influence of each absorber on the rotor dynamics system a study on the system’s behavior is done for different values of its parameters such as rotational speed , clearance
, stiffness
and eccentricity of disk
̅ at contact beginning and
resonance frequency. What can be concluded are: i.
TMD and NES perform an efficient and positive influence on reducing the amplitude of the studied rotor dynamics system. Their efficiency is quite observable through bifurcation analysis of the system for the control parameters such as: rotational speed
ii.
, shaft’s stiffness
and clearance
At resonance frequency
.
1 , TMD and NES are able to just mitigate the
vibrations of the system in a specific amount of eccentricity
̅ . It means that the
design of these absorbers should concentrate on a particular value eccentricity. On the other hand, at contact beginning frequency
0.8 , TMD and NEs are more
reliable to provide a safe working condition for variety of eccentricity values. iii.
The application of this kind of placement of TMD and NES (alongside a radial rod) in rotational low and high speed operating machines, can be trusted.
Appendices Appendix A The matrices in Eq. 26and Eq. 27for TMD are as follows. 1 0 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
24
0 0 0 0 0
1
A.1
1 2
0
1 2
0 0
0 1 2
0
0
2
2 0
1 2
0
0
2
2
0
0 0
,
0 0
2 ̅ 2 0
̅
2 0 0
0 0
0 0 0 0
0
,
A.3
2 ̅
2 0
0 0
0 0
̅ 2
2 0 0 0
2
0
0 0 0 0 0 0
A.2
2 0
0
0 0
2
0
2
2 0
0 0
2 0 0
0
2 1 0 0 1 0 0
,
0 0
A.4
0 0 0 2 2 For the NES due to the nonlinear stiffness all the stiffness matrices has nonlinear 0
expressions in terms of
,
, ̅
except
which is the same as for TMD.
25
0
1
0 0 1
1
0 1
2
2
2 1
2
2
1
2
2
0
0 0
0 0
0
0
0
0
0 0
2
1
8
2 3
9 3
0 0
0
0
1
0
0
1
0
0
0 3
0
3 3 3
4
2
0 1
0
2 1
0
0
3
0
0
0
0 0 0
8
2 3
̅
3 3
2
2
A.5
0 2 3
3 8
2 3
9 1
3
3
8
̅
3 2
1
2
2
0 0 3
3 3 3
2
4
2 3 ̅ 0 0
1 0
,
0 1
0
0 0 3 2 ̅
0 3
3 2
2 3 ̅
0 0
0
0 0 3
2
A.6
0 0 ̅ 2
0 0 0
0 0 3 ̅
Appendix B For deriving the complex averaging form of the equations of motion of the system with TMD the eq. 24 – 26 are replaced in eq. 15 – 17. The resulted equations are as follows:
26
1
2
2
2
̅
2
2
̅ ̅
2
2
2
2
̅
̅
1
2
̅
2
̅ 2
2
̅
2
sin
cos
2
2
2
2 ̅ 2
2
27
2
2 ̅
̅
It should be noted that , sin
2
̅
̅
2
2
2
2 ̅
2
2
2
̅
2
cos
̅
2
2
2
2
2
̅
̅
1
̅
2
2
2
2
̅
2
2
2
̅
2
2
̅
1
̅
2
2
Then with retaining the coefficients of
and ignoring all other terms,the equations are
transformed to the complex averaging form.
Appendix C A sample code in Matlab format to implement the proposed optimization algorithm is given here. Generally the required code must has the following form. function Complex_averaging_Amplitude_optimization clc, clear all, close all, format long, %% specifying the constant parameters of system such as Mass of disk, stiffness of spring and etc. %% Specifying the design rotational speed a specific value for optimization such as the rotational speed in which contact or resonance is occurred. %% specifying the desired eigenvalues to guarantee the stability of linear part of equation eig_o_d = specifying a stable eigenvalues : This eigenvalue can be determined from the state of original system, in which system is stable without any contact %% specifying the desired amplitude of vibration for rotor and absorber such as: rd_d = the desired dynamic amplitude for rotor; qs_d = the desired static amplitude for rotor; %% setting the parameters to be optimized such as Par = setting the parameters of TMD or NES and their location as the parameters to be optimized; %% setting the upper and lower values of parameters and the initial guess LB = lower bound of parameters; UB = upper bound of parameters; x0 = an initial guess; %% Use of fmincon command for optimization in the following form [x, fval, exitflag] = fmincon(@fun_TMD_eigen_value_optimization, x0, [], [], [], [], LB, UB, @mycon, options) %% specifying the optimization function or cost function function Out = fun_TMD_eigen_value_optimization(y) Out = sum(abs(rd.^2‐rd_d.^2)); 28
% Note that these stiffness and mass matrices are dependent to the dynamic and static amplitude of vibration and are determined by rd_d and qs_d. end %%imposing the linear and nonlinear constraint function [c_nonl , ceq] = mycon(y) % determination of mass and stiffness matrices and from then the eigenvalues of linearized system eig_o = eig(Stiffness matrix, ‐Mass matrix); c_nonl = real(eig_o); % This condition guarantee that eigenvalue of system has negative real part and system has the required stability. ceq = the equation of motion; % The equation of motion must be satisfied, and it can be considered as an equal nonlinear constraint. end end
References
[1] F. K. Choy, and J. Padovan, Nonlinear transient analysis of rotor‐casing rub events, journal of Sound and Vibration (1987) 113:529–545 [2] R. F. Beatty, Differentiating rotor response due to radial rubbing, Journal of Vibration, Acoustics, Stress and Reliability in Design (1985) 107:151–160. [3] F. Chu, Z. Zhang, Bifurcation and chaos in rub‐impact Jeffcott rotor system, Journal of Sound and Vibration (1998) 210 (1):1–18. [4] P. Goldman, A. Muszynska, Chaotic behavior of rotor/stator systems with rubs, Journal of Engineering for Gas Turbine and Power (1994) 116:692–701. [5] J. J. Yu, P. Goldman, D. E. Bently, A. Muzynska, Rotor/seal experimental and analytical study on full annular rub, Journal of Engineering for Gas Turbine and Power (2002) 124:340–350. [6] J. J. Connor, Introduction to structural motion, Prentice hall, England, First edition, (2002) [7] C. Wan, C. Jian, C. K. Chen, Chaos and bifurcation of a flexible rub‐impact rotor supported by oil film bearings with nonlinear suspension. Mechanism and Machine Theory (2007) 42:312–333
29
[8] A.K. Samantaray, Steady‐state dynamics of a non‐ideal rotor with internal damping and gyroscopic effects, journal of Nonlinear Dynamics (2009) 56: 443 [9] Z. Shang, J. Jiang, L. Hong, the global responses characteristics of a rotor/stator rubbing system with dry friction effects, Journal of Sound and Vibration (2011) 330: 2150–2160 [10] O. V. Gendelman, Analytic treatment of a system with a vibro – impact nonlinear energy sink, Journal of sound and vibration (2012) 331: 4599 – 4608. [11] G. Jacquet‐Richardet, M. Torkhani, P. Cartraud, F. Thouverez, T. Nouri, Baranger, M. Herran, C. Gibert, S. Baguet, P. Almeida, L. Peletan, Rotor to stator contacts in turbomachines. Review and application, Mechanical Systems and Signal Processing (2013) 40: 401–420 [12] J. P. Chavez, M. Wiercigroch, Bifurcation analysis of periodic orbits of a non‐smooth Jeffcott rotor model, Communications in Nonlinear Science and Numerical Simulation (2013) 18: 2571–2580 [13] A. Najafi, M.R. Ghazavi, A.A. Jafari, Application of Krein’s theorem and bifurcation theory for stability analysis of a bladed rotor, Meccanica (2014) 49:1507–1526. [14] Nicholas Vlajic, Xianbo Liu, Hamad Karki, and Balakumar Balachandran, Torsional oscillations of a rotor with continuous stator contact, International Journal of Mechanical Sciences, (2014) 92: 102–110. [15] P. Varney, I. Green, Nonlinear phenomena, bifurcations, and routes to chaos in an asymmetrically supported rotor–stator contact system, journal of Sound and Vibration, (2015) 336: 207–226. [16] N. A. Saeed, M. Kamel, Nonlinear PD‐controller to suppress the nonlinear oscillations of horizontally supported Jeffcott‐rotor system, International Journal of Nonlinear Mechanics, (2016) 87: 109‐124. [17] N. A. Saeed, H. A. El‐Gohary, On the nonlinear oscillations of a horizontally supported Jeffcott rotor with a nonlinear restoring force, Nonlinear Dynamics., (2017) 88: 293‐314. [18] M Eissa and N.A. Saeed, Nonlinear vibration control of a horizontally supported Jeffcott‐rotor system, Journal of Vibration and Control, (2017), DOI: 10.1177/1077546317693928. [19] N. A. Saeed and M. Kamel, Active magnetic bearing‐based tuned controller to suppress lateral vibrations of a nonlinear Jeffcott rotor system, Nonlinear Dynamics., (2017) pp:1‐22. [20] N. A. Saeed and W. A. El‐Ganaini, Utilizing time‐delays to quench the nonlinear vibrations of a two‐degree‐of‐freedom system, Meccanica, (2017), 52: 2969–2990. [21] N. A. Saeed and W. A. El‐Ganaini, Time‐delayed control to suppress the nonlinear vibrations of a horizontally suspended Jeffcott‐rotor system, Applied Mathematical Modelling, (2017), 44: 523‐539.
30
[22] N. A. Saeed and W. A. El‐Ganaini, Influences of time‐delays on the performance of a controller based on the saturation phenomenon, European Journal of Mechanics ‐ A/Solids, (2017), 66: 125‐142. [23] S. Bab, S.E. Khadem, M. Shahgholi, Lateral vibration attenuation of a rotor under mass eccentricity force using non‐linear energy sink, International Journal of Non‐Linear Mechanics, (2014) 67: 251‐266. [24] S. Bab, S. Khadem, M. Shahgholi, Vibration attenuation of a rotor supported by journal bearings with nonlinear suspension under mass eccentricity force using nonlinear energy sink, Meccanica(2015) 50:2441‐2460. [25] J. Taghipour, M. Dardel, Steady state dynamics and robustness of a harmonically excited essentially nonlinear oscillator coupled with a two‐DOF nonlinear energy sink, Mechanical Systems and Signal Processing (2015) 62‐63:164–182. [26] C. Guo, M. A. AL‐Shudeifat, A. F. Vakakis, L. A. Bergman, D. M. Mcfaral, J. Yan, Vibration reduction in unbalanced hollow rotor systems with nonlinear energy sinks, Nonlinear Dynamics, (2015) 79:527–538. [27] M. Taleshi, M. Dardel, M. H. Pashaie, Passive targeted energy transfer in the steady state dynamics of a nonlinear plate with nonlinear absorber, Chaos, Solitons and Fractals (2016) 92:56–72. [28] M. Parseh, M. Dardel, M. Hasan Ghasemi, Steady state dynamics of a non‐linear beam coupled to a non‐linear energy sink, International Journal of Non‐Linear Mechanics (2016) 79:48–65 [29] Nicholas Vlajic, Alan R. Champneys, Balakumar Balachandran, Nonlinear dynamics of a Jeffcott rotor with torsional deformations and rotor‐stator contact, International Journal of Non‐Linear Mechanics, (2017) 92: 102–110.
31
Figu ures:
Figure 1: Jeffccott rotor fiixed in both h end.
a)
b)
Figurre 2: a) Geo ometric and d mass cen nter of the d disk, b) the contact forrce compon nents in normal and tangentiaal direction ns.
32
Figure e 3:A schem matic of thee applied tu uned mass d damper.
Figure 4: A schem matic of the applied no onlinear eneergy sink.
33
Figu ure 5: Rotorr dynamics with singlee TMD.
Figu ure 6: Rotor dynamicss with single NES.
34
Fig gure 7: Orb bit and radiial displaceement of thee disk for
0.5, 0.8, 1 respectiively.
Fig gure 8: Bifu furcation diiagram for rradial displlacement of the disk fo or 0.5
35
2.
Fig gure 9: Biffurcation diiagram for static displlacement off the disk fo or 0.5
(a)
36
2.
(b) Figu ure 10: Bifu urcation diaagram for rradial displaacement off the disk fo or 5 2.5 5
10 , (a))
0.8, aand (b)
37
1.
10
(a)
(b) Figure 11: Bifurcaation diagraam for radiial displaceement of thee disk for 1 10
0 0.8, and (b) ) 38
1.
10 , (a)
(a)
(b) Figu ure 12: Bifu urcation diaagram for rradial displaacement of f the disk fo or 0.1
0 0.8, and (b) ) 39
1.
̅
1, (a)
Fig gure 13: O Orbit and raadial displaccement of tthe disk witth TMD and d NES at
1.
Figu ure 14: Rad dial displaccement of th he disk witth TMD and d NES at diffferent rotational speeds.
40
(a)
(b) Fig. 15 5: Bifurcatiion diagram m for radiall displacem ment of the d disk with TMD and NE ES, (a) at 0..8, and (b) aat 1.
41
(a)
(b) Figure e 16: Bifurccation diagrram for rad dial displacement of th he disk with h TMD and NES: (a) at 0.8, (b) att 1.
42
(a)
(c) Figure e 17: Bifurccation diagrram for rad dial displacement of th he disk for w with TMD aand NES, (a) 0.8, (b) 1.
43
(a)
(b) Figure e 18: Bifurccation diagrram for rad dial displaceement of th he disk for w with TMD aand NES, , (a) 0.8, (b) 1.
44
Tables: Table 1: Primary system parameters.
.
0.1
9.81
. 2100
10
2.5
10
7.5
2
10
5
10
10
10
Table 2: Optimum parameters of TMD and NES at different dimensionless rotational speeds of disk.
Optimum parameters of TMD
0.8 1
0.0025 0.0025
5 5
0.185 0.185
3.14 3.14
Optimum parameters of NES
0.8
0.0025
5
0.185
3.14
1
0.0025
5
0.185
3.14
Table 3: Optimum parameters of absorbers for maximazing eccentericity tolerance
Optimum parameters of TMD
0.8 1
0.005
5
1.85
3.14
0.005
6
2.15
3.14
Optimum parameters of NES
0.8
0.005
15
1.85
3.14
1
0.005
15
1.85
3.14
45
Highlights ‐
Tuned mass damper (TMD) and nonlinear energy sink (NES) are suggested for preventing contact occurrence of rotor system.
‐
In order to efficiently design absorbers, optimization is proposed.
‐
For optimization process, complex averaging method is used in order for deriving the equations of motion.
‐
The results show that TMD and NES are efficient in vibration reduction.