Active vibration control of ship mounted flexible rotor-shaft-bearing system during seakeeping

Journal Pre-proof Active vibration control of ship mounted flexible rotor-shaft-bearing system during seakeeping Tukesh Soni, A.S. Das, J.K. Dutt PII:

S0022-460X(19)30609-1

DOI:

https://doi.org/10.1016/j.jsv.2019.115046

Reference:

YJSVI 115046

To appear in:

Journal of Sound and Vibration

Received Date: 3 March 2019 Revised Date:

15 August 2019

Accepted Date: 25 October 2019

Please cite this article as: T. Soni, A.S. Das, J.K. Dutt, Active vibration control of ship mounted flexible rotor-shaft-bearing system during seakeeping, Journal of Sound and Vibration (2019), doi: https:// doi.org/10.1016/j.jsv.2019.115046. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. 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. © 2019 Published by Elsevier Ltd.

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Active Vibration Control of Ship Mounted Flexible Rotor-Shaft-Bearing System during Seakeeping

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Tukesh Soni1 UIET, Panjab University, Chandigarh, India 160014 A S Das Department of Mechanical Engineering, Jadavpur University, Kolkata, West Bengal, India J K Dutt Department of Mechanical Engineering, IIT Delhi, Hauz Khas, New Delhi, India Abstract:

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Rotor-shaft-bearing system is an integral part of the engine and propulsion system of a ship. Ships are subject to water-waves which cause large rigid body motion of the ship hull involving all six degrees of freedom. This large time-varying ship-motion causes parametric excitation to the flexible rotor mounted on the ship, and may generate high vibratory response of the rotor, although fairly balanced. This paper proposes active control of lateral vibration in such rotors with a suitably placed electromagnetic actuator and compares simulated performance (response amplitude and control current) of different control laws, namely, (i) PD, (ii) PID and (iii) two novel control laws inspired by the mechanical models of a viscoelastic semi-solid. Realistic ship motion during sea-keeping conditions is generated by numerically solving the governing differential equations of motion of a ship under the action of water waves, using indigenously developed code. The equations of motion of the discretized rotor continuum subject to forces from conventional bearings, base motion and the actuator are obtained with respect to a noninertial reference frame attached to the moving rotor base. Multi-objective optimization of control gains is carried out to obtain minimum rotor-disk response at the expense of the optimum control current. Numerical simulations reveal that the novel control law proposed in (iii) is the most efficient in terms of vibration response and control cost.

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Keywords: Rotor vibration during sea-keeping, Active vibration control, Electromagnetic actuator, Viscoelastic model based control law

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1.

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In rough sea conditions, ships are subject to powerful water waves, which impinge upon the ship hull and cause large generic time varying motion of the ship. Therefore, the flexible rotor shaft bearing system mounted on ships (in the form of turbo engines and propeller shaft) are subject to large time varying base motion. Such base motions are large as compared to the flexible vibratory motion of the rotor shaft. The base motion in the case of ship mounted rotor, generally involves all six degrees of freedom, namely, surge (forward motion), sway (lateral motion), heave (vertical motion), roll (rotary motion about surge axis), pitch (rotary motion about sway axis) and yaw (rotary motion about the heave axis). For the purpose of monitoring and control of the rotor shaft vibration, it is crucial to derive the equations of motion of the rotor with respect to a frame which is fixed to the ship hull. Such a reference frame is noninertial, and therefore the resulting equations of motion contain parameters varying with time. These time varying parameters cause parametric excitation to the system and may result in excessive vibrations, even in a fairly balanced rotor shaft system.

Introduction

1

Corresponding author, Email: [email protected] (Tukesh Soni) [email protected] (A S Das), [email protected] (J K Dutt).

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Although the use of rotors on-board a moving vehicle, namely, an aircraft, a ship etc., is very common, there seems to be few reported research activities investigating the vibration response of a rotor mounted on a moving base, in comparison with the number of existing literature on rotors with fixed bases. The following discussion highlights some of the available literature dealing with numerical modelling and experimentation of rotor-dynamic behavior on moving bases.

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Brown and Shabana [1] applied the multi-body methodology to a rotating shaft problem and numerically studied the response of a rotating shaft with base excitation and a strong coupling between shaft dynamics and base excitation was established. However, the equations of motion were derived with respect to an inertial reference frame and therefore are not suitable from vibration-control point of view. The influence of aircraft maneuver on the dynamic response of mounted rotor rotating with constant angular velocity was studied by Lin and Meng [2]. The authors considered a simple Jeffcot rotor and the aircraft was assumed to maneuver only in the vertical plane. Duchemin et al. [3] analyzed the dynamics and stability of a simply supported rotor with base excitation. A general model of the rotor system mounted on a moving base was derived, but the model was simplified using the Rayleigh-Ritz method to apply the Method of Multiple Scales (MMS) for stability analysis. Driot et al. [4] applied the method of normal form to the simplified model based on Rayleigh-Ritz method as derived by Duchemin et al. [3]. It was concluded that the method of normal form is very much suitable for problems with parametric excitation and nonlinearities. Active control action on the rotor shaft by means of an actuator was not considered by Duchemin et al. [3] and Driot et al. [4].

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Lee et al. [5] studied the effect of base shock excitation of a rotor bearing system, through numerical simulation using Newmark-
method and also reported experiments. Bachelet et al. [6] investigated the stability of rotor under random rotational base motion by numerically computing the largest Lyapunov exponent with an iterative formula. Dynamics of a rigid rotor in the presence of mass unbalance and support excitation was simulated by El-Saeidy and Sticher [7]. Non-linear dynamics and vibrations of rotors mounted on hydrodynamic bearings and subject to base motion was reported by Dakel et al. [8]. Geometric asymmetry of the shaft was considered in the study and stability of the rotor system was discussed. The authors noted significant effect of the frequency and amplitude of the support motion on the shape and size of the rotor orbits. Later on, Dakel et al. [9] also analyzed the steady state dynamic behavior of an on-board rotor under combined base motion. Liu et al. [10] analyzed the effect of base motion on a hydrodynamic bearing supported rotor. Dynamic response of asymmetric rotors subject to periodic angular base motion were studied by Yi et al. [11]. Xinghui et al. [12] considered the dynamics of planetary gear drive subject to harmonic base pitch motion.

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All of the literature discussed above is concerned with the effect of base motion on the rotor response and its stability. However, active control action by means of an actuator has not been considered in any of the work discussed above. As a distinct deviation compared to the above authors, Das et al. [13] attempted to numerically simulate the active vibration control of a flexible rotor-shaft system on a base undergoing large generic motion. Das et al. [14] studied the influence of the active vibration controller proposed above to improve the stability of the rotor shaft system. These activities, however, considered only a PD control law to decide the control force from the actuator and did not attempt any optimization of the vibration control scheme.

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It has been shown by Schweitzer et al. [15] that the PD control law generates a 2-element suspension model, which is the same as Voigt model [16]. Dutt and Nakra [17], [18] reported the superiority of 2

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viscoelastic materials as supports to improve the stability and response of rotor-shaft systems due to disk unbalance for static base rotor-shaft systems. Inspired by this, Roy et al. [19] reported generation of a particular viscoelastic support characteristics by modifying the control law of an Active Magnetic Bearing (AMB), and validated the utility of such supports experimentally, but for a static base rigid rotor-shaft system.

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Having surveyed the recent literature on dynamics and vibration control of flexible rotor-shaft systems on moving bases, it is seen that

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•

• •

Vibration control of rotors on moving bases needs more attention in finding an efficient control law, as in the existing literature, only the PD control law has been tested for its application to control rotor vibration. The research should not only compare different control laws but also find the efficient control law with optimum control parameters for the purpose of design. More so, the authors could not find in the existing literature, any research activity, which considers realistic base motion in studying the effect of base motion on the dynamics of the rotor shaft system. Only simple sinusoidal base motion have been used; the references [2] to [12] may be seen as examples.

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These research gaps have inspired the present research work. The present work considers the vibration control of a flexible rotor-shaft borne by a ship, subject to large generic base motion due to waves impinging on the ship hull. A work of this kind is missing in the literature on this topic. The control force is generated by an electromagnetic actuator situated at a suitable location, away from the disk, along the shaft. Data for base motion to the ship is generated for sea keeping using a indigenously developed MATLAB code, based on the theory obtained in literature [20], [21]. Performance and suitability of the four control laws to govern the control force of the actuator has been compared; the control laws compared are (i) the PD control law, (ii) the PID control law and (iii) two novel control law based on the mechanical model of a linear viscoelastic semisolid. The control laws in (iii) are based on the ThreeElement and Four-Element viscoelastic material model, to represent creep and stress relaxation [16] and are therefore referred to as, TE control law and FE control law. These TE and FE control laws may also be interpreted as modified forms of real PD and real PID control laws respectively. Therefore, the control law based upon Three-Element model (TE control law) can also be termed as, Modified Real Proportional Derivative (MRPD) control law and the control law based upon the Four-Element material model (the FE control law) can also be termed as, Modified Real Proportional Integral Derivative (MRPID) control law. Before comparison of the control laws, nonlinear multi-objective optimization of the controller gains is also carried out, for the sake of a fair comparison.

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Numerical simulations reveal that the FE/MRPID control law results in the minimum vibration amplitude at rotor-disk compared to PD, PID and TE/MRPD control laws. During the process, the FE/MRPID control law utilizes minimum amount of control current and thus the FE/MRPID control law is energy efficient, economical (lesser number of coil turns) and has light weight control hardware. The FE/MRPID control law is also found to be more robust or the least sensitive to small changes in optimum controller gains. A work similar to this is not reported in the literature surveyed so far and so this paper is written.

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2.

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The study of the motion of ships, under the influence of water waves can be broadly classified as seakeeping. Figure 1 shows a ship and its six degrees of freedom expressed in body fixed reference frame. In ship dynamics, the coordinates, ζ , ζ , ζ , ζ , ζ and ζ are referred to as Surge, Sway, Heave, Roll, Pitch and Yaw motion respectively. The details of derivation of equations of motion of ship under the influence of water waves can be found in Chapter 5 of [20]. A simplified form of the same is taken from [21] and is given below:

Ship Motion under Sea-keeping Condition

 +    +   =  −   −   +     −   

 +   +   +    =  −   −   −    +  −   +   

 +   +   =  −   −   −   −   −  −  −     +   +   +    =  −   −   −    −   + !! − "" #   !! +  # +   +   =  −  −   −   −   + "" −    

"" + 

 +    +    =   −    −    − 

  +  − !! # 

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(1) (2)

(3) (4)

(5)

(6)

& &$% ' are the components of the added mass matrix (or the hydrodynamic mass), $% ' are the components

of hydrodynamic damping matrix and $%& ' are the components of hydrodynamic stiffness matrix. These Eqs. (1) – (6) are solved numerically using MATLAB, to find the motion of the ship hull subject to water waves. This motion of the ship is then considered as a base motion to the rotor-shaft-bearing-actuator system, the dynamics of which is discussed in the following section.

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3.

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Generalized kinematics and dynamics of a flexible rotor bearing system mounted on a moving base has been dealt with, in detail, in [13]. Some features of the same analysis which is relevant to the present work is revisited in this section. Lagrange’s principle is used to find the governing equations of motion for the system. To this end,

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Equations of motion of a flexible rotor bearing system mounted on a moving base

• • •

First the reference frames and kinematic of a generic point along the rotor shaft is presented. Based on the linear and angular velocity found using the kinematic relations, the expressions for the kinetic and the potential energy of the rotor disk and the rotor shaft element is then written. Using these expressions and the Lagrange’s principle, the final equations of motion for the rotor mounted on moving base is found.

The dynamic modeling of the rotor mounted on a moving base (ship) is based on the assumption that the base motion directly reaches the rotor-bearing-shaft system, unabated.

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3.1.

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For the case of a rotor mounted on a moving base, it is important from monitoring and vibration control point of view, to find the relative motion of the rotor shaft with respect to a frame attached to the moving rotor base. The reason for this is that the actuator is assumed to be fixed to the rotor base. To this end, three coordinate frames are defined, which are:

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Reference frame and kinematics

• • •

Inertial frame of reference $ : )$ − *$ − +$ A non-inertial frame attached to rotor base (or somewhere in ship, assuming ship hull to be a rigid body), ,- : ),- − *,- − +,Frame attached to rotor shaft and rotating with it,  : ) − * − + (shown in Figure 2).

Figure 1: Ship with six degrees of motion Figure 2: A generic rotor bearing system on a moving base. expressed in ship body fixed frame.

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Since the  frame rotates and moves with the rotor shaft, for convenience, the ) axis is chosen to be along the axis of rotation. Three Euler angles which relate coordinate frames $ and ,- are graphically represented in Figure 3 and are defined as: • • •

Rotation of ./ about *$ axis leading to first intermediate orientation of ,- , ) − * − + . Rotation of
1 about + axis leading to second intermediate orientation of ,- , ) − * − + . Rotation of 23 about ) axis leading to the final orientation of ,- .

The position vector of any generic point ′5′ situated on the shaft axis at a distance, 6/ from the origin of the base-fixed co-ordinate system, ,- , is given by C<= 8 9 6 = 7 8 + 6/ :̂<= + >6/  ? @<= + A6/ B 7

(7)

C<= # + > 6/ ?̂<= + A 6/ B C <= 8 HI J 6/ :̂<= + >6/ ?̂<= + A6/ B 8 9 6/  = 7 8 + G F <=

(8)

The absolute velocity of the point ‘5’ can then be obtained by differentiating the Eq. (7), and is given below:

5

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The absolute angular velocity vector of the cross section of the rotor-shaft at point 'P' is given by:

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H 8 HI + K 8 H<= K9I = K 9 <=

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Having defined the linear and rotational velocities at a generic point on the rotor shaft, the kinetic energy and potential energy for the rotor disk element and the rotor shaft element can be written.

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3.2.

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Energy equation for a rotor disk element

Assuming that a generic rotor disk on the rotor shaft is situated at a distance of 6L along the M̂,- unit vector from the origin of ,- the expression for kinetic energy is given by, 1 1    NO,P = L 6 L + > L + AL  + / SLT + L SL1 + L SLW  U V U 2 2

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(9)

(10)

Potential energy due to disk gravity can be found by using the following expression: 5O,P = L XAL

(11)

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Figure 3: Graphical representation of Euler angles used.

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3.3.

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Since, in the present work, the effect of the base motion on the axial and torsional vibrations are not considered, therefore, a two-noded Rayleigh beam finite elements are used to discretize the rotor-shaft continuum [22]. There are four degrees of freedom (linear displacements along two perpendicular directions and slopes about each) associated with each node, given as,

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Energy equation for a finite shaft element

{Z}]\ = [_`

_&`

a&`

_b

ab

_&b

a&b ]

d

(12)

The transverse displacement at any point other than the node is approximated using a shape function matrix, [Ψ] d [_ a] = [f]{Z}]g

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a`

(13)

So, the expression for slopes of transverse displacement can then be easily found as, [_&

d i −` a& ] = h j [f& ]{Z}]g ` i

(14)

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A detailed expression for shape function matrix [f] is given by Nelson [23]. The expression of kinetic energy for a shaft element, is given by NO = k

m

n

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1 1  6 l + >l + Al  + op STU + oP S1U + oP SWU  q6 2 2

(15)

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Potential energy due to gravity of the shaft element can be found in a similar manner as in the case of a disk element and is given by,

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5O = k XAl q6

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m

n

Strain energy due to bending is given by [22],

m

rO = k O >l& + Al&  q6

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3.4.

Contribution from bearings

n

(16)

(17)

For rigid end supports the zero displacement boundary conditions, >-,s = 0 and A-,s = 0, are imposed on the assembled finite element equation of the rotor bearing system. For flexible bearings the expression for force applied on the rotor-shaft by the bearings is given as, H=
(18)

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where, Π is the assembled global degrees of freedom of the rotor shaft.

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Using the above expressions for kinetic and potential energy of the rotor-shaft-disk system and applying the Lagrange’s principle the assembled equations of motion for the uncontrolled system (i.e. without the electromagnetic actuator) can be found and is given as,

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3.5.

Assembled equations of motion for the uncontrolled rotor-shaft system on moving base

where, • •

•

•

 + [v]{Z} + [w]{Z} = {H} [y]{Z}

(19)

[y] is the assembled mass matrix (complete expression is given in Appendix A). [v] is the assembled matrix coefficient to the global velocity vector, which includes damping components, gyroscopic effects, Coriolis components of acceleration, damping from the bearings etc. (complete expression is given in Appendix A). [w] is the assembled stiffness matrix which apart from bending stiffness matrix, also includes the circulatory matrix, the parametric stiffness matrices due to base motion and bearing stiffness matrix (complete expression is given in Appendix A). {H} is the global load vector due to inertia forces due to base motion and force due to gravity (complete expression is given in Appendix A).

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4.

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Electromagnetic actuator (shown schematically in Figure 4) consists of four pairs of electromagnetic poles placed symmetrically on the stator around the circumference of the rotor at a suitable transverse plane along the axis of the rotor-shaft. Each pole pairs exerts radial control force on the rotor and is a function of the air gap and the current in the coil. The actuator force originates due to the reluctance force generated by the change in the air-gap length between the rotor surface and the pole surface of the electromagnet.

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The expression for the force exerted by an electromagnetic pole pair on the rotor-shaft is given in this section. A detailed derivation is given by Das et al. [13]. The electromagnetic force exerted on the rotor in the Y and Z direction after linearization is given by,

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Electromagnetic Actuator

1 = z$ o{1 − z1 > ; W = z$ o{W − zW A

(20)

where, z$ is the force-current factor or the current stiffness, z1 and zW are the displacement stiffness and o{1 and o{W are the control current values along }̂,- and z~,- unit vectors respectively. > and A are the rotor shaft displacement values along }̂,- and z~,- unit vectors respectively at the electromagnetic actuator node location. In the present work, the aim is to analyze the flexural vibrations of the rotor shaft, therefore, any effect of the electromagnetic actuator on the axial motion of the rotor shaft is not considered. Expressions for z$ , z1 and zW are given as [15], z$ = 4z€ls

$ ; z1 s‚

= zW = −4z€ls

$ƒ s‚

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Figure 4: Electromagnetic actuator with four pole pairs mounted on rotor-shaft.

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It should be noted that the linearization of the electromagnetic force (Eq. (20)) is only valid for the values of > and A in very close neighborhood of the nominal position of the rotor shaft and for the values of control current o{1 and o{W very small as compared to the bias current values. It is assumed that when the cross-section of the shaft is at the nominal position, i.e. equidistant from the poles of the electromagnet, then the value of the control current is zero. Following [13] a value of radial displacement from the nominal position not exceeding 10% of the air-gap and a control current not exceeding 10 % of the bias current in the coil have been considered to be permissible to admit linearization without appreciable error.

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Moreover, the electromagnetic actuator has been considered to be working well below its magnetic saturation limits.

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5.

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Control law is used to decide the relationship between the rotor displacement at the actuator location and the control current input to the electromagnetic actuator. The nominal position of the shaft with respect to the base fixed reference frame is taken as the reference position, with which the error signal is calculated.

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Multi-element support/suspension models are the models or networks with more than two elements (spring and dashpot being considered as an element each). They are used to represent the viscoelastic material models. The viscoelastic material is well known to dissipate vibrations effectively and provide better stability as compared to Two-Element models [17], [24]. The Three and Four-element viscoelastic model is shown in Figure 5(a) and (b) respectively. The functional form of the transfer function between the force and the displacement for these mechanical models is used to formulate the TE/MRPD and FE/MRPID control laws. To derive a control law based on the constitutive relationship of the four element viscoelastic material, first the force-displacement relationship for such a model is written below (for details see Appendix B):

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Control Architecture

„… =

 z z + [† + † z + † z ]‡ˆp + † † ‡ˆp ‰… z + † ‡ˆp

(21)

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Figure 5: (a) Three element and (b) Four element viscoelastic material model. A similar expression for the constitutive relationship for the three element model can be written as,

z z + z + z † ‡ˆp (22) ‰… z + † ‡ˆp where, „… is the force applied to the multi-element model and ‰… is the resulting displacement of the model. The above transfer function in Eq. (21) is used to generate the Four-element (FE) control law and is given in Eq. (23). Using Eq. (20), the expressions of the force exerted by the electromagnetic actuator along the Y and Z directions is found and is given in Eq. (24). „… =

o{1,${W = −zŠ$

 z z + [† + † z + † z ]‡ˆp + † † ‡ˆp >l{ , Al{ z + † ‡ˆp

(23)

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1,W = ‹z$ Œ−zŠ$

 z z + [† + † z + † z ]‡ˆp + † † ‡ˆp  − z1,W Ž >l{ , Al{ z + † ‡ˆp

(24)

Similar expressions for TE control law for the control current and the actuator force can be derived. The block diagram for the rotor bearing actuator system with feedback control action in the Y direction is shown in Figure 6. However, the TE and FE control laws can also be interpreted as modified versions of real PD and real PID control laws. Therefore, they can be also termed as, Modified Real PD (MRPD) and Modified Real PID (MRPID) control laws respectively. The controller block transfer function (shown in Figure 6) for the case of the four controllers is given below: PD control law: PID control law: TE/MRPD control law: FE/MRPID control law:

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zp + zP ‡ˆp N N/ + + NL ‡ˆp ‡ˆp z z + z + z † ‡ˆp z + † ‡ˆp  z z + [† + † z + † z ]‡ˆp + † † ‡ˆp z + † ‡ˆp

(25)

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It should however be noted that no viscoelastic material has been used for vibration control in the present work. Only inspiration is drawn from the constitutive relationship (force-displacement relationship) of a viscoelastic material to formulate the different control laws, and two forms of constitutive relationship used in this work are the Three Element model and the Four Element model.

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5.1

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Though, the overall transfer functions of FE and MRPID may look similar, yet, when seen term-wise, they look different. In generic form, the overall and the term-wise expressions of the transfer function of a FE/MRPID control law can be written as,

Difference between ‘FE/MRPID’ control law and Conventional or Real PID control law

‘’ =

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  “ + “ ‡ˆp + “ ‡ˆp “ ‡ˆp “ ‡ˆp “ = + + ‡ + ‡ ‡ˆp ‡ + ‡ ‡ˆp ‡ + ‡ ‡ˆp ‡ + ‡ ‡ˆp

From the above, it may be seen that the proposed FE/MRPID control law is different from the PID control law (both conventional or a real PID control law with low pass filter) in the sense that: • •

•

In FE/MRPID control law, the proportional term also has a low-pass filter, as compared to conventional or real PID control. There is no pure integral term of error signal in FE/MRPID control law, as compared to the  conventional or real PID control. Instead, a term containing second derivative (‡ˆp  of the error signal is present with a low pass filter (for reducing noise in the feedback signal). This term enables the controller to decide upon the control action based on second derivative of the error signal. The controller transfer function in the case of conventional or real PID control law has a pole at the origin of the s-plane. This is not the case with FE/MRPID control law. 10

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Although the controller transfer function for the FE/MRPID and TE/MRPD control laws can be obtained as modifications of the PD and PID control laws, the FE/MRPID and TE/MRPD control laws are novel in the sense that they have been inspired by the multi-element viscoelastic models. Again, different such control laws may be constructed by arranging different creep and relaxation fields; TE/MRPD and FE/MRPID control laws proposed here show only two possibilities out of many.

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Figure 6: Block diagram for rotor shaft bearing actuator system with feedback control.

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6.

Results and Discussion

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6.1.

Finite Element Model Validation

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Results from the paper by Dakel et al. [9] are used to validate the finite element model of the rotor bearing system with moving base developed for this work. The work by Dakel et al. [9] is used because it reports steady state dynamic behavior of simply supported rotors subject to base motion and the work is significant because the results reported in the work are validated with the experimental and analytical results present in the literature. However, none of the existing literature contains experimental study of rotor system subject to base motion under the action of active control. Therefore, active control aspect of this work could not be validated. The work by Dakel et al. [9] reports frequency response of the rotor system with unbalance and subject to constant base roll motion with three different speeds. The phenomena of shifting of the natural frequency of the rotor system due to the presence of base roll motion is reported. Three different cases of constant base roll speed namely, 0 Hz, 5 Hz and 10 Hz, were used. The same results are re-generated with the finite element model developed in this work and are shown in Figure 7. The results are in close agreement with Figure 7 of [9] and therefore the developed finite element model of rotor shaft bearing system mounted on a moving base is validated.

(a)

(b) 11

312

Figure 7:Results for validation. Frequency response of rotor with different constant roll speed of the base. Both (a) and (b) are in close agreement with Figure 7 of Dakel et al. [9] Table I: Details of VLCC ship Particular Length between perpendiculars Breadth Draft Block coefficient Displacement

Value 350 m 58 m 10.4 m 0.875 355600 metric tons

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6.3. Details of rotor shaft, bearing, finite element discretization and electromagnetic actuator for simulation

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In order to simulate the characteristics of a ship rotor, an overhung rotor, which represents the propeller shaft of a ship, is chosen for analysis. The details of the overhung rotor are taken from Friswell et al. [25] and a schematic of the same is shown in Figure 8. In order to choose the number of elements with which the rotor shaft system has to be discretized, and to check the grid sensitivity, the Campbell diagram for the overhung rotor shaft system is plotted for two different cases of finite element grid size. Firstly the number of finite elements are taken as 6 (thereby, the length of the Rayleigh beam element becomes 0.25 m) and then the number of finite elements are taken as 12 (thereby, the length of the Rayleigh beam element becomes 0.125 m).

Figure 8: Schematic of overhung rotor shaft system with finite element discretization.

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Figure 9: Campbell diagram for overhung rotor. The Campbell diagram plot for two cases with different number of finite elements (namely, 6 and 12) are plotted in Figure 9. It can be seen from Figure 9 that there is a close match between the natural frequency map for both the cases of discretization. Based on this convergence of the Campbell diagram, six finite elements (Rayleigh beam elements) are used to discretize the rotor shaft continua. The frequency response of the un-controller rotor-shaft-bearing system with unbalance excitation of 10 gm-mm at the rotor disk is shown in Figure 10. As expected, a peak at the first critical speed found using the Campbell diagram is also observed in the frequency response. Physical details of the system, like dimension, stiffness and damping constants of bearings, details of the actuator as well as the details of discretization of the shaft continuum used for simulation are given in Table II.

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6.4.

333 334 335 336 337 338 339 340 341 342 343

The equations of motion of a ship under the sea-keeping condition (Eq. (1) to (6)) are numerically integrated using ODE45 function of MATLAB. The force term $ is the excitation force which is sum of diffraction force $L # and Froude-Krylov force $‘•  acting on the ship hull. For the present case of a  VLCC ship the same has been found as – = [10 0 0 25 J 10 cos S… 7 J 10› cos S… 0]œ . The time response of ship motion namely, surge, sway, heave, roll, pitch, yaw are plotted in Figure 11. It is apparent from Figure 11 that the pitch, heave and roll motion are the dominant motions of the ship as compared to yaw and sway motion. The reason for this type of motion of ship is that bow water waves were considered to impinge upon the ship hull. The frequency of these motion parameters, as seen in Figure 11, are expectedly different for pitch and roll motion because of directional nature of the hydrodynamic coefficients of the ship hull. These ship motion parameters form the base motion excitation to the rotor shaft system mounted on the ship.

Max. Rotor Disk Displacement (m)

Ship motion under sea-keeping condition.

344 345

Figure 10: Frequency response for an un-controlled rotor-bearing system with unbalance excitation.

346

Table II: Rotor shaft bearing, finite element discretization and actuator parameter details Parameter Name Shaft diameter Shaft length Disk diameter Disk thickness Rotor Shaft Bearing Bearing stiffness Young’s modulus Torsional rigidity Density Rotor spin speed No. of finite elements Finite Element Discretization No. of nodes Length of each finite element No. of coil turns Pole face area Electromagnetic Actuator Bias current Current stiffness

Value 50 mm 1.5 m 350 mm 70 mm 10 MN/m 211 GN/m2 81.2 GN/m2 7810 kg/m3 1000 rpm 6 7 250 mm 800 5 cm2 5A 355.43 N/A 13

Displacement stiffness

0.89 MN/m

347

6.5.

348 349 350

In order to fairly compare the performance of the four control actions, namely, PD, PID, TE/MRPD and FE/MRPID control actions, it is necessary to find the optimum controller gains for the same. The performance of a controller or control action can be judged from two points of view

351 352 353 354 355 356 357

Multi-objective optimization of control gains

• •

The control action should yield the minimum vibration amplitude at the disk and The cost of the controller should be the minimum in terms of energy input to the actuator. Higher the control current requirement would result in higher weight and the cost of the control hardware.

Since the above two performance parameters are conflicting with each other, non-linear multi-objective optimization of the control gains is carried out using the MATLAB Optimization Toolbox. To this end, two objective functions are defined:

q€l + q,€  ž (26) Xn nd o€l + o,€ 2 objective function  ž on where, q€l + q,€ is the sum of the maximum value and root-mean-square (RMS) value of the displacement of the rotor disk, o€l + o,€ is the sum of the maximum value and RMS value of the control current of the actuator. Constraints imposed are: 1st objective function

358 359 360

ql{ €l Ÿ 0.1 Xn zp , zP   0 N/ , N , NL   0 z , z , †   0 z , z , † , †   0

PD control PID control TE/MRPD control FE/MRPID control

361

(a)

(27)

(b)

14

(c)

(d)

(e)

(f)

362

Figure 11: Ship motion parameters versus time.

363 364 365 366 367 368 369

In the case of a multi-objective optimization problem, where the objective functions are conflicting each other, it is usually not possible to find an optimal solution that can simultaneously optimize each objective function. A solution is called Pareto optimal, if none of the objective functions can be improved in value without affecting the value of the other objective function values. A plot of all such Pareto optimal points, comprises of the Pareto front [26]. The results of the constrained multi-objective optimization, resulting in the Pareto fronts for PD, PID, TE/MRPD and FE/MRPID control laws are shown in Figure 12. Results in tabular form are shown in Table III - VI for PD, PID, TE/MRPD and FE/MRPID control respectively.

370

6.5.1.

371 372 373

The first objective function defined above in Eq. (26) is a representative of the rotor shaft vibration response and the second objective function is a representative of the control current used by the electromagnetic actuator. Since both the objective functions must be ideally minimized, therefore a

Choosing the optimal control parameters

375

decision variable, Ξ is chosen as the product of the two objective functions, Ξ =

376 377 378 379

minimum. Therefore, for PD control action the value corresponding to serial number 3 in Table III, for PID control action the value corresponding to serial number 3 in Table IV, for TE/MRPD control action the value of gains corresponding to serial number 3 in Table V and for FE/MRPID control action the value corresponding to serial number 8 in

374

s

J

. Now, a set of values of control gains is chosen for which the value of decision variable, Ξ is

$¢£¤ ¥$¦¢U  $

P¢£¤ ¥P¦¢U 

15

380 381

Table VI is chosen for further simulation and comparison. The chosen values of the control gains are given in Table VII.

382

Table III: Optimization result for PD control law.

S. No. 1 2 3 4 5 6 7 8 9 10

383 S. No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

q€l + q,€ o€l + o,€ Ξ on Xn 9398.7 7190.7 1.543 0.91916 1.4183 7720.1 2957 1.5518 0.8719 1.3530 2891 2659 1.5773 0.79928 1.2607 3211 2799.4 1.5749 0.80781 1.2722 5452.5 2713.2 1.5622 0.84049 1.3130 6034.4 2852.9 1.5591 0.8507 1.3264 6436.6 2967.5 1.5571 0.85763 1.3354 8868 3299.2 1.5471 0.8884 1.3744 3831.7 3001.3 1.5706 0.82187 1.2908 7851.3 3331.9 1.5507 0.87884 1.3628 Table IV: Optimization results for PID control law. zp

zP

N/

N

NL

9368.2 8407.8 2005.7 2078.4 8273.5 7657.1 8209.6 8454.3 9368.2 7489.1 8188.1 7018.9 8535.1 3361.9 4430.8 3358.6

9464 6634.9 1.3724 89.042 2408.3 1596.6 3977.8 4941 9464 603 6001.1 2311.1 7916.7 1233 1237.1 56.915

259.5 173.4 18.112 46.542 183.53 133.92 164.89 246.19 259.5 176.81 207.84 219.57 211.33 174.99 173.25 95.817

q€l + q,€ Xn 0.3590 0.3649 0.4084 0.4072 0.3768 0.3807 0.3720 0.3691 0.3590 0.3848 0.3667 0.3794 0.3622 0.3938 0.3903 0.4013

o€l + o,€ on 0.2229 0.2029 0.0438 0.0482 0.1563 0.1416 0.1758 0.1868 0.2229 0.1254 0.1970 0.1485 0.2129 0.1014 0.1109 0.0681

Ξ

0.080031 0.074076 0.017893 0.019631 0.058881 0.053916 0.065415 0.068966 0.080031 0.048247 0.072231 0.056334 0.077131 0.039926 0.043319 0.027317

384

16

385 386

Figure 12: Pareto fronts for PD, PID, TE/MRPD and FE/MRPID control laws.

387

Table V: Optimization results for TE/MRPD control. S. No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

z

3594.0 3127.0 2010.8 3047.2 2229.7 2477.6 3819.6 2347.1 2554.0 231.1 3374.9 3002.3 2933.2 2106.0 2294.0

z

6.6755 7.1541 7.8112 6.7349 7.0532 7.3098 5.9784 7.7622 7.0625 6.8472 6.5034 7.1427 6.9542 7.7308 6.8512

†

7607.9 7792.4 7862.1 7632.7 7789.6 7725.3 7554.3 7631.6 7651.8 7586.5 7650.1 7700.9 7632.7 7642.2 7727.9

q€l + q,€ Xn 0.3813 0.3833 0.3889 0.3837 0.3876 0.3865 0.3803 0.3873 0.3861 0.3828 0.3822 0.3839 0.3842 0.3885 0.3873

o€l + o,€ on 0.0534 0.0480 0.0335 0.0470 0.0365 0.0399 0.0559 0.0382 0.0409 0.0492 0.0509 0.0465 0.0456 0.0349 0.0374

ql{ €l Xn 0.0358 0.0690 0.0400 0.0372 0.0394 0.0387 0.0353 0.0390 0.0385 0.0367 0.0364 0.0373 0.0375 0.0397 0.0392

Ξ

0.0204 0.0184 0.0131 0.0181 0.0142 0.0154 0.0213 0.0148 0.0158 0.0189 0.0195 0.0179 0.0176 0.0136 0.0145

388 389 17

390

Table VI: Optimization result for FE/MRPID control law.

S. No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14

z

z

†

†

5927.6 5939.6 5947 5890.5 2667.7 5932.6 5512.8 255.96 3258.6 1547.6 5947 688.11 3814.6 4921.3

8.9204 10.432 12.97 6.4373 1.0815 10.362 4.248 0.46703 1.7291 0.86717 12.97 1.3142 2.3451 1.2776

32515 32518 32519 32512 31314 32516 32382 30540 31942 32031 32519 31999 32060 31246

12021 15476 18779 8684.2 1243.7 14342 6019.6 661.06 2363.7 2106.5 18779 1824.4 3390.9 2790.1

q€l + q,€ Xn 0.34867 0.34864 0.34863 0.3487 0.35221 0.34865 0.34903 0.35486 0.35107 0.35289 0.34863 0.35328 0.35051 0.35046

o€l + o,€ on 0.07854 0.08982 0.10106 0.06844 0.02899 0.08603 0.05938 0.00482 0.03561 0.02056 0.10106 0.01262 0.04201 0.04962

Ξ 0.02739 0.03132 0.03523 0.02386 0.01021 0.02999 0.02073 0.00171 0.01250 0.00726 0.03523 0.00446 0.01473 0.01739

391 392

Table VII: Value of optimum control gains selected for simulation. PD zp = 2891 zP = 2659

PID N/ = 2005.7 N = 1.3; NL = 18.1

z

TE/MRPD z = 2010.8 = 7.8; † = 7862.1

FE/MRPID z = 255.96; z = 0.46 † = 30540; † = 661

393

6.6.

394 395 396 397 398 399 400 401 402 403 404 405 406 407

Dynamic simulation of the finite element model of the overhung rotor mounted on a ship under seakeeping condition is carried out using the Newmark-
method in MATLAB. The vertical vibration response of the rotor disk (>P ) normalized with rotor stator air gap (Xn), without any control action and with PD control (with optimum controller gain values) is plotted in Figure 13. The temporal response comparison between PID, TE/MRPD and FE/MRPID control law with optimum controller gain values is plotted in Figure 14. It is evident that the maximum amplitude of vibration is minimum for the case of FE/MRPID control law. PD control action reduces the rotor vibration by an appreciable amount as compared to uncontrolled response, but the maximum amplitude of vibration in the transient range is much higher than that obtained with PID, TE/MRPD and FE/MRPID control action. The temporal response corresponding to PID, TE/MRPD and FE/MRPID control laws (see Figure 14), looks similar and may seem to perform similarly, however, as will be seen in the next section, that the control effort required by FE/MRPID control law is much lower than that for PID control law. Of the four control laws, the vibration levels for the case of FE/MRPID control action is the minimum. Detailed comparison of performance of the four control laws are given in Table VIII.

Rotor vibration response

18

408 409

Figure 13: Time response; without control and with PD control law.

410 411

Figure 14: Time response comparison for PID, TE/MRPD and FE/MRPID control law.

412

6.7.

413 414 415 416 417 418 419 420 421

Control current is a direct measure of control effort required by the controller. Figure 15(a) shows the plot of control current for the case when PD control law is used for simulation. It is evident from the plot that the values of control current with optimum control parameters is not in the acceptable range, as the maximum value of control current is as high as 3.6 Ampere (i.e. more than 70% of bias current value); which does not allow linearization or the use of constant current stiffness. Therefore, in a such a situation, to use the PD control law, one needs to increase the number of coil turns in the electromagnetic actuator, to contain the maximum value of the control current. However, increasing the number of coil turns would result in a heavier and costlier controller hardware. Plot of control current for PD control law with higher number of coil turns (N =3000) is given in Figure 15(b).

422 423 424 425 426

Comparison of control current values for PID, TE/MRPD and FE/MRPID control laws is shown in Figure 16. It is evident from the figure that the FE/MRPID control law utilizes much less control effort as compared to both TE/MRPD and PID control laws. The value of maximum control current (the factor which would decide the number of coil turns) and the RMS value of control current (the factor which would decide the control effort) are lesser for the FE/MRPID control law as compared to TE/MRPD and

Control current

19

427 428

PID control law. Detailed comparison of the control current requirements for the three control laws is given in Table VIII.

Figure 15: Plot of control current for PD control law.

Figure 16: Plot of control current comparison between PID, TE/MRPD and FE/MRPID control laws.

429

6.8.

430 431 432 433 434 435 436 437

A comparison of the four control laws, namely, PD, PID, TE/MRPD and the FE/MRPID control laws with optimum controller gains, based on certain performance criteria are given in Table VIII. It can be seen that the maximum and RMS values of rotor disk vibration are minimum, when the FE/MRPID control law is applied, in comparison with PD and PID and TE/MRPD control laws. Moreover, in the process of reducing vibration the FE/MRPID control law is seen to require the considerably lesser amount of control effort as compared to PD, PID and TE/MRPD control laws, the same is evident from the values of maximum and RMS values of control current. A bar plot comparing the different values of performance parameters is also shown in Figure 17.

438 439 440 441 442 443

In order to check the relative performance of the control laws, with respect to different actuator location, a plot of the variation of the maximum rotor disk displacement and the maximum control current required by the actuator with the three possible actuator location (i.e. at node 2, node 3 and node 4) is shown in Figure 18. For details regarding the node locations, refer Figure 8. It can be concluded from Figure 18, that although the actuator node location affects the amplitude of rotor vibration and the control effort required by the actuator, the relative performance of the control law remains unaltered.

444

Table VIII: Comparison of control law performance.

Comparison of controller performance

Control law PD (N=800) PD (N=3000) PID (N=800) TE/MRPD (N=800) FE/MRPID (N=800)

Maximum response (mm) 1.98 1.98 0.60 0.59 0.55

Maximum control current (A) 3.630 0.570 0.160 0.130 0.018

RMS response (mm) 1.17 1.22 0.18 0.17 0.16

RMS control current (A) 0.36 0.43 0.05 0.03 0.005

20

PD (N=3000) vs. PID (N=800) PID (N=800) vs. TE/MRPD (N=800) PID (N=800) vs. FE/MRPID (N=800)

69.55% reduced

71.93% reduced

≈1 % reduced

≈21 % reduced

8.33% reduced

88.75% reduced

85.25% reduced 11.11% reduced

88.37% reduced ≈18 % reduced

90% reduced

445

446 447

Figure 17: Performance comparison of four control laws.

(a) (b) Figure 18: Actuator location effect on (a) max. response and (b) control current.

448 449

6.9

450 451 452 453 454 455 456

There could be various sources of excitation in a rotating machinery, namely, unbalance in the disc, misalignment, rotor bow, defect in bearings, looseness in the system, rubs in rotors, rotor-cracks etc. may act on the rotor bearing system to cause rotor vibration. It may not be possible to analyze the effect of all the sources of excitation in a rotor shaft as the system model will become complicated and moreover it will be difficult to decipher the effect of each excitation on the rotor shaft vibration characteristic. However, of all the possible sources of excitation in a rotating machinery, the presence of unbalance in a rotor disk is almost unavoidable even after careful balancing of the rotor disk. Therefore, it

Effect of Combined Unbalance and Base Motion Excitation

21

457 458

may be important to analyze the effect of combined base motion and unbalance excitation, particularly, on the vibration characteristics of the overhung rotor.

459 460 461 462 463 464 465 466 467 468 469 470 471 472 473

To this end, an unbalance of 10 gm-mm (following [8]) is considered on the overhung rotor disk along with the base motion excitation due to seakeeping motion of the ship. The first critical speed of the overhung rotor is close to 950 rpm (from Campbell diagram in Figure 9), therefore in order to analyze unbalance response of the rotor, close to its first critical speed, rotor spin speed is chosen to be 1000 rpm. The resulting temporal response is plotted in Figure 19, where it can be seen that the predominant response is due to the base motion of the rotor which results in a low frequency high amplitude response, whereas, the unbalance excitation results in high frequency but low amplitude response, as seen in the first 5 seconds of the response shown inside the inset in Figure 19. The response shown in inset is due to unbalance excitation only, because the base motion commences from 5 seconds of simulation time onwards. Hence it can be seen that the influence of parametric excitation due to base motion overrides the influence of unbalance excitation, even in the vicinity of the rotor critical speed. To further validate this claim for a wide range of spin speed of the rotor, frequency response for overhung rotor under three control laws with only base motion and combined excitation (both base motion and unbalance excitation) is plotted in Figure 20, where it can be seen that unbalance excitation causes only a marginal change in the frequency response of the system (see zoomed portion of Figure 20) over limited frequency band.

474 475 476 477

Figure 19: Combined unbalance and base motion temporal response for overhung rotor. The zoomed figure corresponds to only unbalance response (the base motion commences from 5 seconds onwards) is shown in the inset. The base motion response is seen to be predominant over the unbalance response.

22

478 479 480

Figure 20: Effect of unbalance excitation on frequency response of overhung rotor subject to base excitation with three control laws.

481

6.10.

482

6.10.1 Transient response for parametrically excited single degree of freedom (SDOF) case

483 484 485 486 487

488

489

490

Performance comparison between PID and FE/MRPID control law for other cases

In this subsection a parametrically excited single degree of freedom discrete system is considered with mass m, time varying stiffness z †«'S- … and time varying damping † †«'S- …. The mass is considered to be also acted upon by the control force applied through an electromagnetic actuator. Then the equations of motion of mass  can be written as, 6 + z †«'S- …6 + † †«'S- …6 = z$ o − z 6 Now, for the PID control law this equation becomes,

(28)

N (29) 6 + NL 6  + z 6 = 0 ‡ˆp Multiplying the above equation on both sides by operator ‡ˆp , one gets a third order differential equation, 6 + z †«'S- …6 + † †«'S- …6 + z$ ŒN/ 6 +

6¬ + † cosS- … + z$ NL 6 + z cosS- … − †S- sinS- … + z + z$ N/ 6 + z$ N − zS- sinS- …6 =0

Similarly, for the case of FE control law, the governing equation for the SDOF system is given as, 6 + z †«'S- …6 + † †«'S- …6  z z + [† + † z + † z ]‡ˆp + † † ‡ˆp + z$ Œ  6 + z 6 = 0 z + † ‡ˆp

491

Multiplying the equation on both sides by z + † ‡ˆp , one gets a third order differential equation,

492

Now, choosing a test case with following parameters,

(30)

† 6¬ + [z + †† cosS- … + z$ † † ]6 + [z† cosS- … + †z cosS- … − †† S- sinS- … + z$ [† + † z + † z ] + z † ]6 + [zz cosS- … − z† S- sinS- … + z$ z z + z z]6 = 0

23

“ “'  '  ; † = 0 ; N/ = 10 ; NL = 500 ; N = 100 ;     ' N/ “ “ z = N/ ; z = ; † = 500; † = 500; z = −1.252 J 10 ; z$ = 177 100    The plot of free vibration for the above SDOF system with S- = 1 rad/s with initial conditions 6n = 0.1, obtained using ode45 function in MATLAB is shown in Figure 21. The test results for the simple SDOF system shown in Figure 21 reveal that the transient response decays quickly for FE/MRPID control law as compared to PID control law, therefore shows the supremacy of the proposed FE/MRPID control law over PID control law in terms of free vibration response.  = 5 zX; z = 10

493 494 495 496 497

498 499 500

Figure 21: Transient response of parametrically excited SDOF system with PID and FE/MRPID control law.

501

6.10.2 Overhung rotor subject to periodic base motion

502 503 504 505 506 507 508 509 510

In this subsection, comparison between the PID and FE/MRPID control law is carried out for the case of overhung rotor being subject to periodic base pitching and rolling motion. Optimized control gain values for the case of seakeeping base motion given in Table VII are considered. The vertical rotor disk response and the control current plot for periodic base pitching motion with a frequency of 5 rad/s and an angular amplitude 0.1 rad is given in Figure 22 and Figure 23 respectively. Similarly, the corresponding plots for the case of periodic base roll motion with frequency of 2 rad/s and an angular amplitude of 0.1 rad is given Figure 24 and Figure 25 respectively. The FE/MRPID control law is seen to have much better control over response at the expense of lesser control current (performance characteristics) than the conventional PID control law.

511

6.11.

512 513 514 515 516 517 518 519 520 521

The comparison of the controller performance in the previous section is done based on optimum control gains. However, it is well known that it is difficult to specify and maintain the exact value of controller gains because of external or internal uncertainties in the system. Therefore, it is important to analyze the sensitivity of the controller with regards to variation in optimum values of controller gains. To this end, a study of sensitivity of the three controllers is done. Sensitivity of response and of control current is defined by the percentage change in the maximum response and the maximum control current for 1 % change in the optimum control gain. Response and current sensitivities for the four control laws are given in Table IX. The performance parameters chosen for the sensitivity analysis are maximum value of response, maximum value of control current, RMS value of response and RMS value of control current. It can be seen from Table IX, that the maximum absolute sensitivities for PD, PID, TE/MRPD and

Sensitivity analysis

24

522 523 524

FE/MRPID control are 2.63%, 0.85%, 0.84% and 0.63% respectively. These values show that, as compared to the FE/MRPID control law, the PD, PID and TE/MRPD control laws are more sensitive to change in the values of optimum control gains for the case of vibration control of ship mounted rotor. 0.02

PID

FE

0 -0.02 -0.04 -0.06 -0.08 -0.1 -0.12 0

5

10

15

20

Time (s)

Figure 22: Rotor disk response for periodic pitching of the rotor base with PID and FE/MRPID control laws.

Figure 23: Control current for the two control laws for periodic pitching of the rotor base.

Figure 24: Rotor disk response for periodic rolling of the rotor base with PID and FE/MRPID control laws

525

Figure 25: Control current plot for the two control laws for periodic rolling motion of the rotor base.

Table IX: Sensitivity analysis of PD, PID, TE/MRPD and FE/MRPID control laws. zp 0.05 2.01 -0.71 2.63

PD

q€l o€l q,€ o,€ Max. Absolute Sensitivity

zP ≈0 ≈0 ≈0 0.02

2.63

N/ -0.02 0.85 0.85 0.02

PID N NL ≈0 ≈0 0.01 ≈0 ≈0 ≈0 0.01 ≈0 0.85

TE/MRPD z z † 0.02 0.01 -0.01 -0.84 -0.01 -0.01 0.02 -0.08 0.08 -0.84 -0.08 0.08 0.85

z ≈0 0.46 ≈0 0.36

FE/MRPID z † -0.01 -0.07 -0.01 -0.53 0.04 -0.07 0.03 -0.63

† 0.01 0.53 -0.04 0.58

0.63

526 25

527

7.

528

Following conclusions are drawn from this work:

Conclusions

529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550

1. Active control of lateral vibrations using electromagnetic actuator mitigates vibration caused by ship motion in a rotor shaft system, mounted on a ship, which is under seakeeping condition. 2. In this work, for the first time, a realistic ship motion is considered and its effect on the lateral vibration of the rotor shaft system mounted on the ship is studied. 3. Two novel control strategies, namely, Three Element Control (TE/MRPD) and Four Element Control (FE/MRPID) have been developed and their relative performance with respect to PD and PID control laws have been compared for mitigating vibrations effectively. 4. For a fair comparison among the control laws, the control gain parameters for all the four control laws are optimized using the multi-objective optimization process. In this, minimization of both response and control cost are considered. The performance of the optimum FE/MRPID control law is found to be better as compared to TE/MRPD, PD and PID control laws. The FE/MRPID control law also ensures higher robustness with regards to a change in controller gains from their optimum values. 5. Effect of location of actuator on the controller performance is also carried out. The actuator location is seen to influence the disc response amplitude and the maximum control current required, however, the relative performance of the control laws is found to be unaltered. 6. In order to further validate the efficacy of the proposed control laws, the performance of the control laws is compared for different cases, such as, (a) combined unbalance-base motion excitation, (b) for parametrically excited single degree of freedom system and (c) for overhung rotor subject to periodic pitching and rolling of the rotor base. In these cases, the proposed FE/MRPID control law is seen to have better performance characteristics as compared to the conventional PID control law.

551 552 553

The finite element code developed and used in this work has been verified with results present in the literature, however, verification of the results with experimental investigation on an actual ship-mounted rotor has not been carried out and therefore the same is proposed as a future work.

554

Acknowledgement

555 556 557 558

The authors humbly acknowledge the grant given under the grant-in-aid project titled "Investigation of the Performance of a Rotor-Shaft System Supported on Magnetic-Bearings mounted on a Maneuvering Marine Vessel based on the Stability and Vibration Response" (Project no. NRB-347/MAR/14- 15), under which this work was performed.

559

Appendix A: System Matrices

560

1. Shape Function Matrix [Ψ]

562

® [®¯] = ° ` i

563

2. Shaft finite element matrices

561

 

i i ®` −®b  

®b ®± i i

 

i i ®± −®²

 

®²     ³, where, ´ = 1 − 3 µ m ¶ + 2 µ m ¶ ; ´ = i  

 

· °µ m ¶ − 2 µ m ¶ + µ m ¶ ³; ´ = 3 µ m ¶ − 2 µ m ¶ ; ´ = · °− µ m ¶ + µ m ¶ ³ 

(A.1)

26

564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583

584 585 586

587 588 589

Inertia (A.2)

matrix:[y]]g = [yd ]]g + [y7 ]]g = ¸i [®¯]d [®¯] q6 + ¸i oP [®& ¯]d [®& ¯] q6 ¹

¹

m i −` i i (A.3) j [®& ]q6 ; [»]]g = ¸n op [®& ]d h j [®& ]q6 ` i ` i m (A.4) Bending stiffness matrix: [w¼ ]]g = ¸n O[®&& ]d [®&& ]q6 m i −` (A.5) Circulatory matrix: [w½ ]]g = ¸n O[®&& ]d h j [®&& ]q6 ` i Coriolis matrix: m m i −` i −` [½]]g = [½d ]]g + [½7]]g = ¸n [®¯]d h j [®¯]q6 + ¸n oP [®& ¯]d h j [®& ¯]q6 ` i ` i (A.6) Parametric stiffness matrix due to base motion: ] m m ` i i i ¾w¿`` À = ¸n [®¯]d h (A.7) j [®¯]q6 + ¸n op [®& ¯]d h j [®& ¯]q6 g i i i ` ] m m i i ` i ¾w¿bb À = ¸n [®¯]d h j [®¯]q6 + ¸n op [®& ¯]d h j [®& ¯]q6 (A.8) g i ` i i ] m m i ` i ` ¾w¿`b À = ¸n [®¯]d h j [®¯]q6 + ¸n op − oP [®& ¯]d h j [®& ¯]q6 g ` i ` i (A.9) Force vectors due to base motion: ] ] m m m m ` ` i i Ágb_  = ¸n 6 [®]d Ã Ä q6 + ¸n oP [®& ]d Ã Ä q6, Ágb¿  = ¸n 6 [®]d Ã Ä q6 − ¸n oP [®& ]d Ã Ä q6 g g i i ` ` (A.10) ] ] m m m ` i ` Ág`_  = ¸n op [®]d Ã Ä q6, Ág`¿  = ¸n op [®]d Ã Ä q6, {g` }] = ¸n  [®]d Ã Ä q6, {gb }] = g g i ` i m i d (A.11) ¸n  [®] Ã Ä q6 ` 3. Finite element matrices pertaining to rotor disk Æv i i i i Æv i i Inertia matrix [y]v = Å É (A.12) i i ÇÈ i i i i ÇÈ i i i i i −Æv i i Ìi i i i Ï Æv i i i Gyroscopic matrix; [º]v = Ëi i i −Ç Î ; Coriolis matrix [½]v = Å É i i i −ÇÈ ¿Î Ë i Í i i ÇÈ i Êi i Ç¿ (A.13) i i i i Æv i i i i Æv i i i i i i Parametric stiffness matrix ¾w¿`` À = Å É ; ¾w¿bb À = Åi i i i É; ¾w¿`b À = v v v i i Ç¿ i i i i Ç ¿ i i i i i Æv i i ÌÆ i i i Ï Ë v Î i i Ç¿ − ÇÈ Î Ë i i Ç¿ − ÇÈ i Í Ê i (A.14) Force vectors on disk due to base motion:

Gyroscopic matrix: [º]]g = ¸n op [®& ]d h m

27

590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611

612 613 614

615

d

Ágb_ Âv = [Æv ¯v i i ÇÈ ]d ; Ágb¿ Âv = ¾i Æv ¯v – ÇÈ iÀ ; {g`Ñ }v = ¾i i i − Ç¿ À ; Ág`¿ Âv = d

¾i i − Ç¿ iÀ ; {g` }v = [Æv i i i]d ; {gb }v = [i Æv i i]d (A.15) 4. Global matrices and vectors: [y] = ∑v[y]v + ∑][y]]g [v] = ΩT- Ô + Õ #∑L[º]v + ∑[º]]g  + 2 ΩT- Ô ∑L[½]v + ∑[½]]g  + [v]=
(A.16) (A.17)

ƒ [w] = − à Õ + ΩT- Ô #{∑L[»]v + ∑[»]]g } − ΩT- Ô {∑L[½]v + ∑[½]]g }Ä − ÖΩT- Ô {∑L[y]v + ∑[y]]g } +

Ω-WÔ Ã∑L¾w¿`` À + ∑¾w¿`` À Ä + Ω-1Ô Ã∑L¾w¿bb À + ∑¾w¿bb À Ä − Ω-1Ô Ω-WÔ Ã∑L¾w¿`b À + ƒ

]

v

g

∑¾w¿`b À Ä× + + Á[w¼ ]]g + [w]=
g

ƒ

]

v

g

v

{Ø} = − Ω-WÔ + ΩT- Ô Ω-WÔ #[∑{gbÑ }]g + ∑L{gbÑ }v ] + Ω-1Ô − ΩT- Ô Ω-WÔ #[∑{gb9 }]g + ∑L{gb9 }v ] − Õ + ΩT- #ÁΩ-W [∑{g`9 }]g + ∑L{g`9 }v ] − Ω-1 [∑{g!Ñ }]g + ∑L{g`Ñ }v ]Â − V1- + VT- Ω-W − Ô

Ô

Ô

Ô

Ô

(A.18)

Ô

ÛW-Ô ΩT- Ô #[∑{g` }]g + ∑L{g` }v ] − VW-Ô − VT-Ô Ω-1Ô + V1-Ô ΩT- Ô #[∑{gb }]g + ∑L{gb }v ] − Xcos ./ sin 23 + sin ./ sin
1 cos 23 [∑{g` }]g + ∑L{g` }v ] − Xcos ./ cos 23 − sin ./ sin
1 sin 23 [∑{gb }]g + ∑L{gb }v ] (A.19) Appendix B: Derivation of constitutive relationship of a four element viscoelastic model Consider the four element model shown in Figure 5(b), the expression for the force applied to the viscoelastic model can be written as, (B.1) „… = z ‰… + † ‰ …+z ‰… − Ü… where, „… is the time varying force input to the viscoelastic material, ‰… is the resulting displacement of the viscoelastic material, Ü… is the displacement across the damper as shown in Figure 5(b). Since, the force across the spring with stiffness z and the damper † has to be equal, the following equation is written using the differential time operator ‡ˆp (equal to

P ), P

(B.2) z ‰… − Ü…# = † Ü … Using the above equation and the differentiation operator, it is possible to eliminate Ü… from the Eq. (B.2) and then we get the required constitutive relationship for the four element model for the viscoelastic material model, as given below:

 (B.3) z z + [† + † z + † z ]‡ˆp + † † ‡ˆp ‰… z + † ‡ˆp A similar expression for the constitutive relationship for the three element model can be written as,

„… =

„… =

616

References

617 618 619 620

[1] [2]

z z + z + z † ‡ˆp ‰… z + † ‡ˆp

(B.4)

M. A. Brown and A. A. Shabana, “Application of Multibody Methodology to Rotating Shaft Problems,” Journal of Sound and Vibration, vol. 204, no. 3, pp. 439–457, 1997. F. Lin and G. Meng, “Study on the Dynamics of a Rotor in a Maneuvering Aircraft,” ASME Journal of Vibration and Acoustic, vol. 125, no. 3, pp. 324–327, 2003. 28

621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670

[3] [4]

[5]

[6]

[7]

[8] [9] [10]

[11] [12] [13]

[14]

[15] [16] [17] [18] [19] [20] [21]

[22] [23]

M. Duchemin, A. Berlioz, and G. Ferraris, “Dynamic Behaviour and Stability of a Rotor Under Base Excitation,” ASME Journal of Vibration and Acoustics, vol. 128, no. 5, pp. 576–585, 2006. N. Driot, C. H. Lamarque, and A. Berlioz, “Theoretical and Experimental Analysis of a Base Excited Rotor,” ASME Journal of Computational Nonlinear Dynamics, vol. 1, no. 3, pp. 257–263, 2006. A. S. Lee, B. O. Kim, and Y. Kim, “A finite element transient response analysis method of a rotorbearing system to base shock excitations using the state-space Newmark scheme and comparisons with experiments,” Journal of Sound and Vibration, vol. 297, pp. 595–615, 2006. L. Bachelet, N. Driot, G. Ferraris, and F. Poirion, “Dynamical behavior of a rotor under rotational random base excitation,” in Proceedings of the ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, DETC2007, Las Vegas, NV, United States, 2007, vol. 1, pp. 1251–1260. F. M. A. El-Saeidy and F. Sticher, “Dynamics of a rigid rotor linear/nonlinear bearings system subject to rotating unbalance and base excitations,” Journal of Vibration and Control, vol. 16, no. 3, pp. 403–438, 2010. M. Dakel, S. Baguet, and R. Dufour, “Nonlinear dynamics of a support-excited flexible rotor with hydrodynamic journal bearings,” Journal of Sound and Vibration, vol. 333, pp. 2744–2799, 2014. M. Dakel, S. Baguet, and R. Dufour, “Steady-state dynamic behavior of an on-board rotor under combined base motions,” Journal of Vibration and Control, vol. 20, pp. 2254–2287, 2014. Z. Liu, Z. Liu, Y. Li, and G. Zhang, “Dynamics response of an on-board rotor supported on modified oil-film force considering base motion,” Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, vol. 232, pp. 245–259, 2016. Y. Yi, Z. Qiu, and Q. Han, “The effect of time-periodic base angular motions upon dynamic response of asymmetric rotor system,” Advances in Mechanical Engineering, 2018. Q. Xinghui, H. Qinkai, and C. Fulei, “Dynamic modeling and analysis of the planetary gear under pitching base motion,” International Journal of Mechanical Sciences, vol. 141, pp. 31–45, 2018. A. S. Das, J. K. Dutt, and K. Ray, “Active Vibration Control of Flexible Rotors on Maneuvering Vehicles,” American Institute of Aeronautics and Astronautics (AIAA), vol. 48, no. 2, pp. 340–353, 2010. A. S. Das, J. K. Dutt, and K. Ray, “Active vibration control of unbalanced flexible rotor–shaft systems parametrically excited due to base motion,” Applied Mathematical Modelling, vol. 34, pp. 2353–2369, 2010. G. Schweitzer et al., Magnetic Bearings: Theory, Design and Application to Rotating Machinery. Springer-Verlag, 2009. D. R. Bland, The theory of linear viscoelasticity. Pergamon Press, 1960. J. K. Dutt and B. C. Nakra, “Stability of Rotor Systems with Visco-elastic Supports,” Journal of Sound and Vibration, vol. 153, pp. 89–93, 1992. J. K. Dutt and T. Toi, “Rotor vibration reduction with polymeric sectors,” Journal of Sound and Vibration, vol. 262, no. 4, pp. 769–793, May 2003. H. K. Roy, A. S. Das, and J. K. Dutt, “An efficient rotor suspension with active magnetic bearings having viscoelastic control law,” Mechanism and Machine Theory, vol. 98, pp. 48–63, 2016. E. M. Lewandowski, The Dynamics of Marine Craft:Maneuvering and Seakeeping. World Scientific Publishing Co. Pte. Ltd., 2004. Z. Thein, “Practical Source Code for Ship Motions Time Domain Numerical Analysis and Its Mobile Device Application,” Master’s Thesis, Department of Shipping and Marine Technology Chalmers University of Technology, Goteborg, Sweden, 2011. H. D. Nelson and J. M. McVaugh, “The Dynamics of Rotor Bearing System using Finite Elements,” ASME J Eng Ind, vol. 98, no. 2, pp. 593–600, 1976. H. D. Nelson, “A finite rotating shaft element using Timoshenko Beam Theory,” ASME J. Mech. Des, vol. 102(4), pp. 793–803, 1980.

29

671 672 673 674 675 676

[24] J. K. Dutt and B. C. Nakra, “Vibration Response Reduction of a Rotor Shaft System Using Viscoelastic Polymeric Supports,” ASME Jouranal of Vibration and Acoustics, vol. 115, 1993. [25] M. I. Friswell, J. E. T. Penny, S. D. Garvey, and A. W. Lees, Dynamics of Rotating Machines. Cambridge University Press, 2010. [26] S. Dehuri, A. K. Jagadev, and M. Panda, Multi-objective Swarm Intelligence: Theoretical Advances and Applications. Springer, 2015.

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