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Rigid–flexible coupling dynamic analysis on a mass attached to a rotating flexible rod
Applied Mathematical Modelling xxx (2014) xxx–xxx
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Rigid–flexible coupling dynamic analysis on a mass attached to a rotating flexible rod Zhanfang Liu a,b,c,⇑, Pei Ye a, Xiaowei Guo a, Yuan Guo a a
Department of Engineering Mechanics, College of Resources and Environmental Science, Chongqing University, Chongqing 400030, PR China State Key Laboratory of Coal Mine Disaster Dynamics and Control, Chongqing University, PR China c Chongqing Key Laboratory of Heterogeneous Material Mechanics, Chongqing University, PR China b
a r t i c l e
i n f o
Article history: Received 13 October 2012 Received in revised form 20 December 2013 Accepted 25 March 2014 Available online xxxx Keywords: Rigid–flexible coupling Dynamic frequency Motion trajectory Critical angular velocity
a b s t r a c t A rigid–flexible coupling dynamic analysis is presented where a mass is attached to a massless flexible rod which rotates about an axis. The rod is limited to small deformation so that the mass is constrained to move in the plane of rotation. A strongly nonlinear model of the system is established based on the couplings between the elastic deflections of the mass and rigid rotation, in which the mass deflection and rigid rotation are both treated as unknown variables. The additional inertia forces on the mass and coupling mechanism are elucidated in the system model. In the case of varied but prescribed rigid rotation, a set of time-varying differential equations governing mass motion is obtained. The trajectories of mass motion are examined for the spin-up and spin-down rotation. Under constant rigid rotation, a set of ordinary differential equations is further attained, and the issues with dynamic frequencies and critical angular velocity of the system are analyzed. The effects of the centrifugal, Coriolis and tangential inertia forces on the dynamic responses are discussed. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction Dynamics of rotating structures has been a subject of interest because of its importance in a lot of engineering applications such as helicopter rotor blades, wind turbine blades, etc. Problems of rotating cantilever beams were especially investigated where the natural frequency, mode shape and dynamical response were involved, and many attentions were also paid to the matters with dynamic stiffening and system stability [1]. However, the rigid–flexible coupling mechanism and modeling as well as its solution strategy for rotating structures remain in extensive study, refered to Shabana [2], Wasfy and Noor [3], Schielen [4], etc. In order to get insights into the rigid–flexible coupling nature, a simple system composed of a mass fixed to a rotating flexible rod is taken into account in the paper. The rod is assumed massless and within small deformation so that two-dimensional deflection of the mass in the plane of rotation is preserved. Additional inertia forces on the mass due to the coupling between the flexible deflection and rigid rotation may be reflected in the system. It is because of the kinematical coupling that the dynamics of rotating system is different from non-rotating systems. A survey on the simple system is certainly beneficial for understanding of dynamics of general rotating structures.
⇑ Corresponding author at: Department of Engineering Mechanics, College of Resources and Environmental Science, Chongqing University, Chongqing 400030, PR China. Tel.: +86 13648354436. E-mail address: [email protected] (Z. Liu). http://dx.doi.org/10.1016/j.apm.2014.03.038 0307-904X/Ó 2014 Elsevier Inc. All rights reserved.
Please cite this article in press as: Z. Liu et al., Rigid–flexible coupling dynamic analysis on a mass attached to a rotating flexible rod, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.03.038
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Some approaches were taken to the topics that a mass is attached to a flexible rod, but the rod is fixed radially to the inner side of a rigid ring rotating about its radial symmetry axis. Weber [5] proposed the motion equations for the mass moving in the plane of rotation and orthogonally to the plane of rotation (referred to (1) and (3) in Ref. [5]) respectively, the system instability and the mass motion were discussed. Hernried and Gustafson [6] dealt with the similar problem and they built up the nonlinear differential equations governing the response and stability of the system (referred to (12) and (13) in Ref. [6]). In those two studies Coriolis effects were specially analyzed, but only constant rigid rotation and one-dimensional deflection of the mass were both postulated. Brons et al. [7] accounted for the mass to has two-dimensional deflection in the plane of rotation and a set of implicit differential equations for two deflections was developed (referred to (1.1), (1.2), (2.1) and (2.2) in Ref. [7]), an impressive result was shown that the periodic motion of the mass followed a figure-eight shaped curve. In all those works, the nonlinear differential equations governing the mass motion were given out even for constant rigid rotation and a linear elastic rod, Coriolis influence on the system response was paid to special attention. To our knowledge, the issues such as system frequencies, instability, dynamical response as well as the effects of additional inertia forces as a whole has not yet satisfactorily answered. The situation is not yet approached that arbitrary rigid rotation and deflections of the mass are both treated as the variables of system. In this paper, a system with a massless flexible rod fixed to a rotating rigid body and the mass carried by the free end of the rod is considered. The rod is initially orthogonal to the plane of rotation and arbitrary angular velocity is imposed to the rigid rotation. The mass is permitted to move in the plane of rotation as a linear elastic behavior of the rod is postulated. Rigid–flexible coupling mechanism is elucidated. A set of non-linear differential equations is produced when deflection of the mass and angular velocity are both treated as variables. For an arbitrary prescribed rotation a set of time-changing differential equations is degenerated. Dynamic frequencies are proposed under constant rigid rotation and a critical angular velocity is further derived in approaching the elastic limit of the rod. The moving trajectory of mass is studied for a given rigid rotation. 2. A nonlinear rigid–flexible coupling modeling of the system As a preliminary, let’s consider rigid body rotation about a fixed axis. Any point of the rigid body runs a circumferential trace around the axis at an angular velocity x, then the velocity of point is expressed by:
v ¼ x x;
ð1Þ
where x indicates a point position of the rigid body. A second-order anti-symmetric tensor is defined with help of the permutation tensor e as follows:
X ¼ ‰ x or Xij ¼ 2ijk xk :
ð2Þ
A mapping for any vector w with a second-order anti-symmetric tensor is satisfied by the relation:
X w ¼ x w:
ð3Þ
It is clear from (3) that an anti-symmetric tensor corresponds to the angular velocity of rotation. In order to describe a finite rigid rotation about a fixed axis, an orthogonal tensor R has to be introduced. To this end, supposing h is finite rotational angle and n is a unit vector along the rotational axis. It follows from (2) that an anti-symmetric tensor A = e n is determined. The orthogonal tensor R meets Euler–Rodrigues rotation formula [8]:
R ¼ I þ sin hA þ ð1 cos hÞA2 ;
ð4Þ
where I is the second-order unit tensor. An alternative for Euler–Rodrigues rotation formula is expressed in the form of power series as follows [8,9]:
R ¼ ehA :
ð5Þ
Now consider a flexible rod with a concentrated mass at free end is mounted to a rotating rigid disk as shown in Fig. 1. The rod is assumed to be massless and limited to linear elastic deformation, the mass is then confined to move in the plane of rotation. Two reference frames are further introduced as shown in Figs. 1 and 2, a top view is represented by Fig. 2. An inertia frame is stationary with origin at the center of rigid disk and its base vector e03 directs upwards along the rotational axis. A rotating frame runs together with the rotating rigid disk and the origin is located at the root of rod. The base vectors, e1 and e2 are respectively set in the radial and tangential directions of disk but e3 points to the tip of rod at rest. The origin of rotating frame relative to the inertia frame is located by r0, the deflection of mass is indicated by r in the rotating frame, then the position of mass in the rotating frame is given by:
r ¼ r 0 þ r:
ð6Þ
The coordinate transformation between rotating frame and inertia frame is conveyed through the orthogonal tensor R, thus, the absolute position of mass in the inertia frame is written in the form:
r0 ¼ R r ¼ R ðr 0 þ rÞ:
ð7Þ
Please cite this article in press as: Z. Liu et al., Rigid–flexible coupling dynamic analysis on a mass attached to a rotating flexible rod, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.03.038
Z. Liu et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx
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Fig. 1. A flexible rod with a mass mounted on a rotating disk.
Fig. 2. Top view of the inertia and rotating frames.
The position of mass may be determined in two frames. In making use of (3) and in noting R_ ¼ R X, it is easy from (7) to take down the absolute velocity and absolute acceleration of mass in the inertia frame:
v0 ¼
dr 0 _ ¼ R x ðr 0 þ rÞ þ R r: dt
ð8Þ
Attention is paid to r_ in (8) stand for the relative velocity in the rotating frame, respectively. Hamilton principle is used to build up the motion equation of mass in the rotating frame. For a holonomic system with the kinetic energy T and the virtual work U by external forces, Hamilton principle takes the form:
Z
t2
d
ðT þ UÞdt ¼
t1
Z
t2
ðdT þ dUÞdt ¼ 0:
ð9Þ
t1
It is easy to approve that the kinetic energy in the inertia frame is identical with that in the rotating frame, the kinetic energy is given by
T¼
1 _ ½X ðr 0 þ rÞ þ r: _ m½X ðr0 þ rÞ þ r 2
ð10Þ
The virtual work is written out in the form:
dU ¼ F dr:
ð11Þ
Please cite this article in press as: Z. Liu et al., Rigid–flexible coupling dynamic analysis on a mass attached to a rotating flexible rod, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.03.038
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With use of (10) and (11) and through a mathematical treatment, one may write (9) into the form:
Z
Z
t2
ðdT þ dUÞdt ¼
t1
þ
t2
mdr €rdt
t1 Z t2
Z
t2
mdr X X ðr0 þ rÞdt 2
t1
Z
t2
_ mdr X rdt t1
Z
t2
_ ðr 0 þ rÞdt mdr X
t1
F drdt ¼ 0:
ð12Þ
t1
For the details to get (12), please refer to Appendix A. In noticing dr in (12) to be arbitrary, the motion equation of mass is attained as follows:
_ ðr 0 þ rÞ þ 2x r_ þ €r ¼ 0: F m½x ðx ðr 0 þ rÞÞ þ x
ð13Þ
It follows from (13) that the mass is subjected to an external force (elastic restoring force), a centrifugal inertia force, a tangential inertia force, a Coriolis inertia force and a relative inertia force. Because of the relative deflection of mass, the current position is made of an initial portion r0 and an incremental portion r. The centrifugal force is from coupling of the angular velocity with the current position of mass. The tangential inertia force represents coupling between the angular acceleration and the current position. The Coriolis force means coupling of the angular velocity with the relative velocity of mass. The deflections and external forces of mass in the plane of rotation are decomposed into two components under the rotating frame referred to Fig. 2:
r ¼ ue1 þ v e2 ;
F ¼ F 1 e1 þ F 2 e2 ;
ð14Þ
where u and v, F1 and F2 represent the components of displacement, the external force along the base e1 and e2, respectively. The displacement in relation with the force at the free-end of flexible rod is well-known due to Euler–Bernoulli beam model, 3
u¼
l F1; 3EI
3
v ¼
l F2 3EI
ð15Þ
here a cylindrical rod is assumed, l is the rod length and I is the sectional moment of inertia of rod. In noting that the angular velocity x = xe3 in which x is its absolute value and r0 = r0e1, substitution of (14) and (15) into (13) leads to a set of differential equations in the rotating frame in the form:
€ 2mxv_ þ mu
mv€ þ 2mxu_ þ
3EI l
3
3EI l
3
_ v mx2 r 0 ¼ 0 mx2 u mx
mx2
v þ mx_ u þ mx_ r0 ¼ 0
ð16Þ
ð17Þ
To next one has to account for the theorem of moment of momentum of system. The rotary inertia of system may change with the deflections of mass, as a result, the motion of mass will affect rotation of the system. The moment of momentum of system L is given by
L ¼ mr v ¼ mr ðx r Þ:
ð18Þ
One rewrites (18) in the form:
L¼J x
ð19Þ
where the rotary inertia takes the form:
J ¼ mðr r I r rÞ;
ð20Þ
where the symbol indicates a dyadic operation. The theorem of moment of momentum is then written in the form:
Q¼
dL _ _ ¼J xþJx dt
ð21Þ
where Q is the torque of system. It is easy from (20) to get the form:
_ mðr 0 r_ þ r_ r 0 þ r r_ þ r_ rÞ J_ ¼ 2mðr 0 þ rÞ rI
ð22Þ
In recalling r 0 ¼ r 0 e1 , r ¼ ue1 þ v e2 , Q ¼ Q e3 , x ¼ xe3 and in use of (22), one rewrites (21) in the form:
_ Q ¼ 2mðr 0 u_ þ uu_ þ v v_ Þx þ mðr 20 þ 2r 0 u þ u2 þ v 2 Þx
ð23Þ
Here Q is the component of the torque Q in the direction of rotation. For the rigid–flexible coupling system, (16) and (17) together with (23) make up a set of nonlinear differential equations which determines the mass motion and system rotation. Because the defections u and v as well as the angular velocity x are unknown variables, the set of differential equations is a strongly nonlinear set. Evidently, the strong nonlinearity is entirely Please cite this article in press as: Z. Liu et al., Rigid–flexible coupling dynamic analysis on a mass attached to a rotating flexible rod, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.03.038
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Z. Liu et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx
from the rigid–flexible coupling effects. But for an arbitrary prescribed rotation the system is governed by (16) and (17), which is reduced to a time-changing differential equation set.
3. Dynamical response of mass for arbitrary prescribed rotation For an arbitrary prescribed rotation or given time-changing angular velocity, the motion equations (16) and (17) are left and only two variables have to be treated. One rewrites (16) and (17) in a familiar matrix form:
m
0
€ u
0
m
v€
þ
0
2mx
u_
2mx
0
v_
" þ
3
3EI=l mx2
_ mx
_ mx
3EI=l mx2
3
# u
v
( ¼
mr 0 x2 _ mr 0 x
) ð24Þ
A set of time-varying differential equations (24) determines the motion of mass when rigid rotation is known. For comparison with general structure dynamics, it is necessary to revise (24) into a matrix form:
½Mf€r g þ ½C C fr_ g þ ð½K ½K C þ ½K T Þfrg ¼ fF 0 g
ð25Þ
Here column matrix {r} represents unknown deflections of mass in (25). Mass matrix [M] is a diagonal matrix. Coriolis matrix [CC] is an anti-symmetric matrix, but it does not bring about any damping effect. Structure stiffness matrix 3 3 ½K ¼ ½3EI=l 0; 0 3EI=l depends on material property and structure geometry. Centrifugal matrix ½K C ¼ ½mx2 0; 0 mx2 is _ ; mx _ 0 is another anti-symmetric a diagonal matrix but related with angular velocity. Tangential matrix ½K T ¼ ½0 mx matrix but dependent of angular acceleration. Inhomogeneous terms in the right hand of (25) rely on the initial centrifugal _. force mr0x2 and initial tangential force mr 0 x In order to examine the deflection of mass, a prescribed rigid rotation is designed by a specific form:
8t 2-p sin 2Tp t; 0tT > T > > < -; T t 43 T 4 x¼ 7 2 p 7 > t 3 T þ 2p sin T t 3 T ; 3 T t 73 T > > : T 0; t 73 T
ð26Þ
Parameters in (26) are settled such as T = 15 s and - ¼ 2 rad=s, four time intervals are divided. The system is initially at rest and a spin-up rotation is carried out in 15 s. A constant rotation with 2 radians per second is kept from 15 s to 20 s and follows a spin-down rotation from 20 s to 35 s, finally the rigid rotation stops after 35 s. The rigid rotation is illustrated in Fig. 3, and the mass, material and geometrical parameters are given in Table 1. A numerical solution is carried out with use of MATLAB for the set of time-varying differential equations (Appendix B). The motion trajectory of mass is examined in the rotating frame as shown in Fig. 4. During the first time interval, the mass moves radially outwards and tangentially backwards (v is negative in sign), referred to Fig. 5. The tangential deflection increases in the early stage and then converges to small amplitude because the centrifugal forces increases but the tangential inertia force undertakes an up-down change. The radial deflection is more or less ten times than the tangential deflection in magnitude. For the second interval from 15 s to 20 s, the constant rotation is followed (Fig. 3), a motion trajectory is shown in Fig. 6, in which a petal pattern is formed since each elliptical trace conducts rotation. The radial deflection is about three orders in magnitude more than the tangential deflection which is jointly produced by the centrifugal force and elastic restoring force.
Fig. 3. Rotary angular velocity versus time.
Please cite this article in press as: Z. Liu et al., Rigid–flexible coupling dynamic analysis on a mass attached to a rotating flexible rod, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.03.038
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Z. Liu et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx
Table 1 The system parameters for dynamic simulation. Mass
Rod length
Rod diameter
Elastic modulus
Initial eccentricity of mass
m = 0.3 kg
l = 0.3 m
d = 0.003 m
E = 100 GPa
r0 = 0.1 m
Fig. 4. Trajectory of the mass motion in the rotating frame.
Fig. 5. Trajectory of the mass motion in the rotating frame from start to 15 s.
Fig. 6. Trajectory of the mass motion in the rotating frame from 15 s to 20 s.
Please cite this article in press as: Z. Liu et al., Rigid–flexible coupling dynamic analysis on a mass attached to a rotating flexible rod, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.03.038
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Fig. 7. Trajectory of the mass motion in the rotating frame from 20 s to 35 s.
Fig. 8. Trajectory of the mass motion in the rotating frame from 35 s to 40 s.
In the stage the tangential inertia force vanishes due to constant rotation, the petal pattern is only influenced by Coriolis force. In the third stage from 20 s to 35 s, the trajectory of mass is depicted in Fig. 7. The centrifugal force decreases with spindown rotation and the tangential inertia force changes its sign, then the radial deflection decreases and the tangential deflection moves forward (v is positive in sign). In the final stage after 35 s, there is no rigid rotation so that there is no rigid–flexible coupling effect. A stable elliptical motion trajectory of mass is held as shown in Fig. 8. In the stage only the elastic restoring force and relative inertia force control the mass motion. The stable oscillation in Fig. 8 agrees very well with conventional vibration of mass with two degrees of freedom. 4. Dynamical characteristics of mass motion for constant rotation In the case of constant angular velocity, its dynamical characteristic of system may be explored. It follows from (24) that a set of inhomogeneous ordinary differential equations is turned out in the form:
m
0
€ u
0
m
v€
þ
0
2mx
u_
2mx
0
v_
" þ
3EI=l mx2
0
# u
0
3EI=l mx2
v
3
3
( ¼
mx2 r 0
)
0
ð27Þ
Then a linear dynamic system is reached. It is easy to have two order eigen-frequencies from (27) in the form:
x1 1 f1 ¼ ¼ 2p 2p
sffiffiffiffiffiffiffiffi 3EI
!
x2 1 ¼ x ; f2 ¼ 3 2p 2p ml
sffiffiffiffiffiffiffiffi 3EI ml
3
! þx ;
ð28Þ
where f1 and f2 are both termed by the dynamic frequencies as change with angular velocities. With use of the parameters in Table 1, the changes of dynamic frequencies with discrete angular velocities are depicted in Fig. 9. Without rigid rotation, both of the dynamic frequencies return to the unique natural frequency of the linear system, Please cite this article in press as: Z. Liu et al., Rigid–flexible coupling dynamic analysis on a mass attached to a rotating flexible rod, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.03.038
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Z. Liu et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx
Fig. 9. Two order dynamic frequencies of the system vary with discrete constant angular velocities.
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 i.e., its circular frequency is x0 ¼ 3EI=ðml Þ. With constant rotation, two dynamic frequencies are extended from the same natural frequency. The second-order frequency grows up with increasing angular velocity, but the first-order frequency goes down until vanishing, similar results are pointed out by some authors [10,11]. It is noticed that the vanishing dynamic frequency leads to a maximal angular velocity that is identical to the natural frequency. Once the maximal angular velocity is approached, stability of the linear system will be definitely lost from (27). However, it is insufficient to determine the critical angular velocity of system only up on vanishing dynamic frequency because the deflections of mass have to be judged whether its value exceeds its elastic tolerance. The deflections of mass are related to the initial conditions of deflections and velocities, the initial eccentricity r0 as well as the current angular velocity. With leaving out a lengthy derivation, the deflections of mass for any current angular velocity are written out in the form [12]:
uðtÞ
v ðtÞ
¼ A1
cosðx1 t /1 Þ sinðx1 t /1 Þ
þ A2
8
9
1cos x t 1cos x t x2 r0 < x1 1 þ x2 2 = þ 2x0 : sin x1 t þ sin x2 t ; cosðx2 t /2 Þ x1 x2
sinðx2 t /2 Þ
ð29Þ
where the coefficients A1, A2, /1 and /2 are dependent of initial conditions of deflections and velocities of the mass. The deflections in (29) are composed of the periodic and eccentric portions. For given initial conditions of deflections and velocities for the mass, the dynamical deflections are entirely decided by the angular velocity and initial eccentricity. However, the deflection of mass is in fact limited to a certain finite value to insure linear elasticity of the bent rod. Therefore, the bigger is the initial eccentricity, the less is the angular velocity. As a result, a critical angular velocity is extended for the rotating linear system. Suppose a limited deflection is set 3.0 mm, the relationship of critical angular velocity with initial eccentricity is shown in Fig. 10. Without eccentricity (r0 ¼ 0), the critical angular velocity is equal to the natural frequency. With increasing initial eccentricity, the critical angular velocity decreases until vanishing, namely, the linear system is more
Fig. 10. An illustrative for critical angular velocity versus initial eccentricity.
Please cite this article in press as: Z. Liu et al., Rigid–flexible coupling dynamic analysis on a mass attached to a rotating flexible rod, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.03.038
Z. Liu et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx
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and more sensitive to be violated since the eccentric portion of deflection is larger and larger to approach the tolerance of deflection. 5. Concluding remarks Dynamics of a rotating flexible rod with a mass is studied in the paper where the rod is assumed massless and limited within small deformation. Coupling between the elastic deflection of mass and rigid rotation produces the additional inertia forces on the mass such as Coriolis force, tangential inertia force and centrifugal forces, which lead to complex dynamical characteristics and response for the rigid-flexible system. A rigid–flexible coupling model is established with a set of three strongly nonlinear differential equations when the rotary angular velocity and the deflections of mass are all treated as unknown variables. The strong nonlinearity of system is motivated by coupling of elastic deformation with rigid rotation. Under the condition of given rigid rotation, the rigid-flexible system is governed by a set of two time-varying differential equations. The dynamical deflections of mass are numerically examined for prescribed rigid rotation and the dynamical trajectory of mass is traced. It is attractive to view the petal pattern for constant angular velocity and the elliptical pattern for no rotation. In the case of constant rotation, there are two order dynamic frequencies but the first one tends to vanishing with growth of angular velocity. Since the deflections of mass are limited to assure the linear elastic system, the critical angular velocity is shown to decrease with increasing initial eccentricity of the mass. In spite of a simple system treated here, the results would be beneficial for understanding dynamics of general rotating structures. Acknowledgements The research is supported by NASF (Grant No. 11176035). The authors would like to acknowledge the financial support by the National Science Foundations of China (Nos. 11072276; 11372365) and National Basic Research Program of China (No. 2011CB612211). Appendix A. The motion equation of mass with use of Hamilton principle The kinetic energy and virtual work done by external forces are given by
T¼
1 1 1 _ fR½X ðr 0 þ rÞ þ rg _ ¼ m½X ðr 0 þ rÞ þ r _ ½X ðr0 þ rÞ þ r _ mv 0 v 0 ¼ mfR½X ðr 0 þ rÞ þ rg 2 2 2
dU ¼ F dr
ðA:1Þ ðA:2Þ
With replacement of ^r ¼ Xðr0 þ rÞ, the variational function of kinetic energy is written out in the following:
1 1 _ ð^r þ rÞ _ _ ð^r þ rÞ _ ¼ m½dr_ r_ þ d^r ^r þ dr_ ^r þ d^r r _ ¼ md½ð^r þ rÞ dT ¼ d m½ð^r þ rÞ 2 2 d d dr r_ þ md^r ^r þ m dr ^r þ md^r r_ ¼m dt dt ¼m
d d _ mdr €r þ md^r ^r þ m ðdr ^r Þ mdr ^r_ þ md^r r_ ðdr rÞ dt dt
ðA:3Þ
For a holonomic system, Hamilton principle may be taken in the form:
Z
t2
ðdT þ dUÞdt ¼
t1
¼
Z
t2
Z t2 Z t2 Z t2 Z t2 d d _ ðdr rÞdt mdr €r dt þ md^r ^rdt þ m ðdr ^r Þdt mdr ^r_ dt dt dt t1 t1 t1 t1 t1 Z t2 Z t2 _ þ þ md^r rdt ðF drÞdt
Z
t2
m
t1
t1
_ ðmdr €r þ md^r ^r mdr ^r_ þ md^r rÞdt þ
t1
Z
t2
ðF drÞdt
ðA:4Þ
t1
Attention is paid to the following relationships:
Z
t2
m
t1
Z
d t ðdr r_ Þdt ¼ mdr r_ jt21 ¼ 0; dt
t2
t1
md^r ^r dt ¼
Z
Z
t2
t1
m
d ðdr ^r Þdt ¼ mdr ^r jtt21 ¼ 0; dt
t2
mdðX rÞ X ðr 0 þ rÞdt ¼ t1
Z
ðA:5Þ
t2
mdr X X ðr0 þ rÞdt;
ðA:6Þ
t1
Please cite this article in press as: Z. Liu et al., Rigid–flexible coupling dynamic analysis on a mass attached to a rotating flexible rod, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.03.038
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Z. Liu et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx
Z
t2 t1
mdr ^r_ dt þ
Z
t2
_ ¼ md^r rdt
Z
t1
t2
_ ðr 0 þ rÞ þ dr X r_ dðX rÞ rÞdt _ m½dr X
t1
¼ 2
Z
t2
_ mdr X rdt
t1
Z
t2
_ ðr0 þ rÞdt: mdr X
ðA:7Þ
t1
With use of (A.5)–(A.7) Hamilton principle becomes
Z
t2
ðdT þ dUÞdt ¼
t1
þ
Z
t2
t1 Z t2
mdr €rdt
Z
t2
mdr X X ðr0 þ rÞdt 2
t1
F drdt ¼ 0:
Z
t2
t1
_ mdr X rdt
Z
t2
_ ðr0 þ rÞdt mdr X
t1
ðA:8Þ
t1
In noticing an arbitrary dr, the motion equation of mass takes the form:
_ ðr 0 þ rÞ þ 2x r_ þ €r ¼ 0; F m½x ðx ðr 0 þ rÞÞ þ x
ðA:9Þ
Appendix B. A brief description of subroutine with Matlab The subroutine can be separated into three parts. The first part is assign values to parameters, such as Young’s elastic modulus, cross-sectional area moment of inertia, rod length and mass. The second part is selective statement by using a keyword called ‘if-elseif-else’. With the selective statement, the different angular velocity can be used for calculation correctly. The last part is function call. The ODE function is called to solve the set of the ordinary differential equations, whereas a PLOT function is further called to plot the figures. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
T.R. Kane, R.R. Ryan, A.K. Banerjee, Dynamics of a cantilever beam attached to a moving base, AIAA J. Guid. Control Dyn. 10 (2) (1987) 139–151. A.A. Shabana, Flexible multibody dynamics: review of past and recent developments, Multibody Sys.Dyn. 1 (1997) 189–222. T.M. Wasfy, A.K. Noor, Computational strategies for flexible multibody systems, Appl. Mech. Rev. 56 (6) (2003) 553–613. W. Schielen, Computational dynamics: theory and applications of multibody systems, Eur. J. Mech. A Solids 25 (2006) 566–594. H.I. Weber, A note on the stability of a rod subjected to compression by centrifugal force, J. Sound Vib. 46 (1) (1976) 105–111. A.G. Hernried, G.B. Gustafson, On the dynamic response of a single-degree-of-freedom structure attached to the interior of a rotating rigid ring, J. Appl. Mech. 55 (1988) 201–205. M. Brons, P.G. Hjorth, W. Kliem, Coriolis effects in the dynamics of a rotating elastic structure, Acta Mech. 116 (1996) 223–227. B. Palais, R. Palais, Euler’s fixed point theorem: the axis of a rotation, J. Fixed Point Theory Appl. 2 (2007) 215–220. J. Gallier, D. Xu, Computing exponentials of skew-symmetric matrices and logarithms of orthogonal matrices, Int. J. Rob. Autom. 17 (4) (2002) 1–11. W. Kim, J. Chung, Free non-linear vibration of a rotating thin ring with the in-plane and out-of-plane motions, J. Sound Vib. 258 (1) (2002) 167–178. W.S. Shum, Z. Lin, Dynamic motion of whirling rods with Coriolis effect, Appl. Math. Modell. 341 (2010) 203–1216. Z.F. Liu, X.W. Guo, Dynamical analysis on a mass at the tip of a flexible rod on a rotating base, Appl. Mech. Mater. 105–107 (2012) 536–540.
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Contents lists available at ScienceDirect
Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Rigid–flexible coupling dynamic analysis on a mass attached to a rotating flexible rod Zhanfang Liu a,b,c,⇑, Pei Ye a, Xiaowei Guo a, Yuan Guo a a
Department of Engineering Mechanics, College of Resources and Environmental Science, Chongqing University, Chongqing 400030, PR China State Key Laboratory of Coal Mine Disaster Dynamics and Control, Chongqing University, PR China c Chongqing Key Laboratory of Heterogeneous Material Mechanics, Chongqing University, PR China b
a r t i c l e
i n f o
Article history: Received 13 October 2012 Received in revised form 20 December 2013 Accepted 25 March 2014 Available online xxxx Keywords: Rigid–flexible coupling Dynamic frequency Motion trajectory Critical angular velocity
a b s t r a c t A rigid–flexible coupling dynamic analysis is presented where a mass is attached to a massless flexible rod which rotates about an axis. The rod is limited to small deformation so that the mass is constrained to move in the plane of rotation. A strongly nonlinear model of the system is established based on the couplings between the elastic deflections of the mass and rigid rotation, in which the mass deflection and rigid rotation are both treated as unknown variables. The additional inertia forces on the mass and coupling mechanism are elucidated in the system model. In the case of varied but prescribed rigid rotation, a set of time-varying differential equations governing mass motion is obtained. The trajectories of mass motion are examined for the spin-up and spin-down rotation. Under constant rigid rotation, a set of ordinary differential equations is further attained, and the issues with dynamic frequencies and critical angular velocity of the system are analyzed. The effects of the centrifugal, Coriolis and tangential inertia forces on the dynamic responses are discussed. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction Dynamics of rotating structures has been a subject of interest because of its importance in a lot of engineering applications such as helicopter rotor blades, wind turbine blades, etc. Problems of rotating cantilever beams were especially investigated where the natural frequency, mode shape and dynamical response were involved, and many attentions were also paid to the matters with dynamic stiffening and system stability [1]. However, the rigid–flexible coupling mechanism and modeling as well as its solution strategy for rotating structures remain in extensive study, refered to Shabana [2], Wasfy and Noor [3], Schielen [4], etc. In order to get insights into the rigid–flexible coupling nature, a simple system composed of a mass fixed to a rotating flexible rod is taken into account in the paper. The rod is assumed massless and within small deformation so that two-dimensional deflection of the mass in the plane of rotation is preserved. Additional inertia forces on the mass due to the coupling between the flexible deflection and rigid rotation may be reflected in the system. It is because of the kinematical coupling that the dynamics of rotating system is different from non-rotating systems. A survey on the simple system is certainly beneficial for understanding of dynamics of general rotating structures.
⇑ Corresponding author at: Department of Engineering Mechanics, College of Resources and Environmental Science, Chongqing University, Chongqing 400030, PR China. Tel.: +86 13648354436. E-mail address: [email protected] (Z. Liu). http://dx.doi.org/10.1016/j.apm.2014.03.038 0307-904X/Ó 2014 Elsevier Inc. All rights reserved.
Please cite this article in press as: Z. Liu et al., Rigid–flexible coupling dynamic analysis on a mass attached to a rotating flexible rod, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.03.038
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Z. Liu et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx
Some approaches were taken to the topics that a mass is attached to a flexible rod, but the rod is fixed radially to the inner side of a rigid ring rotating about its radial symmetry axis. Weber [5] proposed the motion equations for the mass moving in the plane of rotation and orthogonally to the plane of rotation (referred to (1) and (3) in Ref. [5]) respectively, the system instability and the mass motion were discussed. Hernried and Gustafson [6] dealt with the similar problem and they built up the nonlinear differential equations governing the response and stability of the system (referred to (12) and (13) in Ref. [6]). In those two studies Coriolis effects were specially analyzed, but only constant rigid rotation and one-dimensional deflection of the mass were both postulated. Brons et al. [7] accounted for the mass to has two-dimensional deflection in the plane of rotation and a set of implicit differential equations for two deflections was developed (referred to (1.1), (1.2), (2.1) and (2.2) in Ref. [7]), an impressive result was shown that the periodic motion of the mass followed a figure-eight shaped curve. In all those works, the nonlinear differential equations governing the mass motion were given out even for constant rigid rotation and a linear elastic rod, Coriolis influence on the system response was paid to special attention. To our knowledge, the issues such as system frequencies, instability, dynamical response as well as the effects of additional inertia forces as a whole has not yet satisfactorily answered. The situation is not yet approached that arbitrary rigid rotation and deflections of the mass are both treated as the variables of system. In this paper, a system with a massless flexible rod fixed to a rotating rigid body and the mass carried by the free end of the rod is considered. The rod is initially orthogonal to the plane of rotation and arbitrary angular velocity is imposed to the rigid rotation. The mass is permitted to move in the plane of rotation as a linear elastic behavior of the rod is postulated. Rigid–flexible coupling mechanism is elucidated. A set of non-linear differential equations is produced when deflection of the mass and angular velocity are both treated as variables. For an arbitrary prescribed rotation a set of time-changing differential equations is degenerated. Dynamic frequencies are proposed under constant rigid rotation and a critical angular velocity is further derived in approaching the elastic limit of the rod. The moving trajectory of mass is studied for a given rigid rotation. 2. A nonlinear rigid–flexible coupling modeling of the system As a preliminary, let’s consider rigid body rotation about a fixed axis. Any point of the rigid body runs a circumferential trace around the axis at an angular velocity x, then the velocity of point is expressed by:
v ¼ x x;
ð1Þ
where x indicates a point position of the rigid body. A second-order anti-symmetric tensor is defined with help of the permutation tensor e as follows:
X ¼ ‰ x or Xij ¼ 2ijk xk :
ð2Þ
A mapping for any vector w with a second-order anti-symmetric tensor is satisfied by the relation:
X w ¼ x w:
ð3Þ
It is clear from (3) that an anti-symmetric tensor corresponds to the angular velocity of rotation. In order to describe a finite rigid rotation about a fixed axis, an orthogonal tensor R has to be introduced. To this end, supposing h is finite rotational angle and n is a unit vector along the rotational axis. It follows from (2) that an anti-symmetric tensor A = e n is determined. The orthogonal tensor R meets Euler–Rodrigues rotation formula [8]:
R ¼ I þ sin hA þ ð1 cos hÞA2 ;
ð4Þ
where I is the second-order unit tensor. An alternative for Euler–Rodrigues rotation formula is expressed in the form of power series as follows [8,9]:
R ¼ ehA :
ð5Þ
Now consider a flexible rod with a concentrated mass at free end is mounted to a rotating rigid disk as shown in Fig. 1. The rod is assumed to be massless and limited to linear elastic deformation, the mass is then confined to move in the plane of rotation. Two reference frames are further introduced as shown in Figs. 1 and 2, a top view is represented by Fig. 2. An inertia frame is stationary with origin at the center of rigid disk and its base vector e03 directs upwards along the rotational axis. A rotating frame runs together with the rotating rigid disk and the origin is located at the root of rod. The base vectors, e1 and e2 are respectively set in the radial and tangential directions of disk but e3 points to the tip of rod at rest. The origin of rotating frame relative to the inertia frame is located by r0, the deflection of mass is indicated by r in the rotating frame, then the position of mass in the rotating frame is given by:
r ¼ r 0 þ r:
ð6Þ
The coordinate transformation between rotating frame and inertia frame is conveyed through the orthogonal tensor R, thus, the absolute position of mass in the inertia frame is written in the form:
r0 ¼ R r ¼ R ðr 0 þ rÞ:
ð7Þ
Please cite this article in press as: Z. Liu et al., Rigid–flexible coupling dynamic analysis on a mass attached to a rotating flexible rod, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.03.038
Z. Liu et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx
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Fig. 1. A flexible rod with a mass mounted on a rotating disk.
Fig. 2. Top view of the inertia and rotating frames.
The position of mass may be determined in two frames. In making use of (3) and in noting R_ ¼ R X, it is easy from (7) to take down the absolute velocity and absolute acceleration of mass in the inertia frame:
v0 ¼
dr 0 _ ¼ R x ðr 0 þ rÞ þ R r: dt
ð8Þ
Attention is paid to r_ in (8) stand for the relative velocity in the rotating frame, respectively. Hamilton principle is used to build up the motion equation of mass in the rotating frame. For a holonomic system with the kinetic energy T and the virtual work U by external forces, Hamilton principle takes the form:
Z
t2
d
ðT þ UÞdt ¼
t1
Z
t2
ðdT þ dUÞdt ¼ 0:
ð9Þ
t1
It is easy to approve that the kinetic energy in the inertia frame is identical with that in the rotating frame, the kinetic energy is given by
T¼
1 _ ½X ðr 0 þ rÞ þ r: _ m½X ðr0 þ rÞ þ r 2
ð10Þ
The virtual work is written out in the form:
dU ¼ F dr:
ð11Þ
Please cite this article in press as: Z. Liu et al., Rigid–flexible coupling dynamic analysis on a mass attached to a rotating flexible rod, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.03.038
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Z. Liu et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx
With use of (10) and (11) and through a mathematical treatment, one may write (9) into the form:
Z
Z
t2
ðdT þ dUÞdt ¼
t1
þ
t2
mdr €rdt
t1 Z t2
Z
t2
mdr X X ðr0 þ rÞdt 2
t1
Z
t2
_ mdr X rdt t1
Z
t2
_ ðr 0 þ rÞdt mdr X
t1
F drdt ¼ 0:
ð12Þ
t1
For the details to get (12), please refer to Appendix A. In noticing dr in (12) to be arbitrary, the motion equation of mass is attained as follows:
_ ðr 0 þ rÞ þ 2x r_ þ €r ¼ 0: F m½x ðx ðr 0 þ rÞÞ þ x
ð13Þ
It follows from (13) that the mass is subjected to an external force (elastic restoring force), a centrifugal inertia force, a tangential inertia force, a Coriolis inertia force and a relative inertia force. Because of the relative deflection of mass, the current position is made of an initial portion r0 and an incremental portion r. The centrifugal force is from coupling of the angular velocity with the current position of mass. The tangential inertia force represents coupling between the angular acceleration and the current position. The Coriolis force means coupling of the angular velocity with the relative velocity of mass. The deflections and external forces of mass in the plane of rotation are decomposed into two components under the rotating frame referred to Fig. 2:
r ¼ ue1 þ v e2 ;
F ¼ F 1 e1 þ F 2 e2 ;
ð14Þ
where u and v, F1 and F2 represent the components of displacement, the external force along the base e1 and e2, respectively. The displacement in relation with the force at the free-end of flexible rod is well-known due to Euler–Bernoulli beam model, 3
u¼
l F1; 3EI
3
v ¼
l F2 3EI
ð15Þ
here a cylindrical rod is assumed, l is the rod length and I is the sectional moment of inertia of rod. In noting that the angular velocity x = xe3 in which x is its absolute value and r0 = r0e1, substitution of (14) and (15) into (13) leads to a set of differential equations in the rotating frame in the form:
€ 2mxv_ þ mu
mv€ þ 2mxu_ þ
3EI l
3
3EI l
3
_ v mx2 r 0 ¼ 0 mx2 u mx
mx2
v þ mx_ u þ mx_ r0 ¼ 0
ð16Þ
ð17Þ
To next one has to account for the theorem of moment of momentum of system. The rotary inertia of system may change with the deflections of mass, as a result, the motion of mass will affect rotation of the system. The moment of momentum of system L is given by
L ¼ mr v ¼ mr ðx r Þ:
ð18Þ
One rewrites (18) in the form:
L¼J x
ð19Þ
where the rotary inertia takes the form:
J ¼ mðr r I r rÞ;
ð20Þ
where the symbol indicates a dyadic operation. The theorem of moment of momentum is then written in the form:
Q¼
dL _ _ ¼J xþJx dt
ð21Þ
where Q is the torque of system. It is easy from (20) to get the form:
_ mðr 0 r_ þ r_ r 0 þ r r_ þ r_ rÞ J_ ¼ 2mðr 0 þ rÞ rI
ð22Þ
In recalling r 0 ¼ r 0 e1 , r ¼ ue1 þ v e2 , Q ¼ Q e3 , x ¼ xe3 and in use of (22), one rewrites (21) in the form:
_ Q ¼ 2mðr 0 u_ þ uu_ þ v v_ Þx þ mðr 20 þ 2r 0 u þ u2 þ v 2 Þx
ð23Þ
Here Q is the component of the torque Q in the direction of rotation. For the rigid–flexible coupling system, (16) and (17) together with (23) make up a set of nonlinear differential equations which determines the mass motion and system rotation. Because the defections u and v as well as the angular velocity x are unknown variables, the set of differential equations is a strongly nonlinear set. Evidently, the strong nonlinearity is entirely Please cite this article in press as: Z. Liu et al., Rigid–flexible coupling dynamic analysis on a mass attached to a rotating flexible rod, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.03.038
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Z. Liu et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx
from the rigid–flexible coupling effects. But for an arbitrary prescribed rotation the system is governed by (16) and (17), which is reduced to a time-changing differential equation set.
3. Dynamical response of mass for arbitrary prescribed rotation For an arbitrary prescribed rotation or given time-changing angular velocity, the motion equations (16) and (17) are left and only two variables have to be treated. One rewrites (16) and (17) in a familiar matrix form:
m
0
€ u
0
m
v€
þ
0
2mx
u_
2mx
0
v_
" þ
3
3EI=l mx2
_ mx
_ mx
3EI=l mx2
3
# u
v
( ¼
mr 0 x2 _ mr 0 x
) ð24Þ
A set of time-varying differential equations (24) determines the motion of mass when rigid rotation is known. For comparison with general structure dynamics, it is necessary to revise (24) into a matrix form:
½Mf€r g þ ½C C fr_ g þ ð½K ½K C þ ½K T Þfrg ¼ fF 0 g
ð25Þ
Here column matrix {r} represents unknown deflections of mass in (25). Mass matrix [M] is a diagonal matrix. Coriolis matrix [CC] is an anti-symmetric matrix, but it does not bring about any damping effect. Structure stiffness matrix 3 3 ½K ¼ ½3EI=l 0; 0 3EI=l depends on material property and structure geometry. Centrifugal matrix ½K C ¼ ½mx2 0; 0 mx2 is _ ; mx _ 0 is another anti-symmetric a diagonal matrix but related with angular velocity. Tangential matrix ½K T ¼ ½0 mx matrix but dependent of angular acceleration. Inhomogeneous terms in the right hand of (25) rely on the initial centrifugal _. force mr0x2 and initial tangential force mr 0 x In order to examine the deflection of mass, a prescribed rigid rotation is designed by a specific form:
8t 2-p sin 2Tp t; 0tT > T > > < -; T t 43 T 4 x¼ 7 2 p 7 > t 3 T þ 2p sin T t 3 T ; 3 T t 73 T > > : T 0; t 73 T
ð26Þ
Parameters in (26) are settled such as T = 15 s and - ¼ 2 rad=s, four time intervals are divided. The system is initially at rest and a spin-up rotation is carried out in 15 s. A constant rotation with 2 radians per second is kept from 15 s to 20 s and follows a spin-down rotation from 20 s to 35 s, finally the rigid rotation stops after 35 s. The rigid rotation is illustrated in Fig. 3, and the mass, material and geometrical parameters are given in Table 1. A numerical solution is carried out with use of MATLAB for the set of time-varying differential equations (Appendix B). The motion trajectory of mass is examined in the rotating frame as shown in Fig. 4. During the first time interval, the mass moves radially outwards and tangentially backwards (v is negative in sign), referred to Fig. 5. The tangential deflection increases in the early stage and then converges to small amplitude because the centrifugal forces increases but the tangential inertia force undertakes an up-down change. The radial deflection is more or less ten times than the tangential deflection in magnitude. For the second interval from 15 s to 20 s, the constant rotation is followed (Fig. 3), a motion trajectory is shown in Fig. 6, in which a petal pattern is formed since each elliptical trace conducts rotation. The radial deflection is about three orders in magnitude more than the tangential deflection which is jointly produced by the centrifugal force and elastic restoring force.
Fig. 3. Rotary angular velocity versus time.
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Z. Liu et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx
Table 1 The system parameters for dynamic simulation. Mass
Rod length
Rod diameter
Elastic modulus
Initial eccentricity of mass
m = 0.3 kg
l = 0.3 m
d = 0.003 m
E = 100 GPa
r0 = 0.1 m
Fig. 4. Trajectory of the mass motion in the rotating frame.
Fig. 5. Trajectory of the mass motion in the rotating frame from start to 15 s.
Fig. 6. Trajectory of the mass motion in the rotating frame from 15 s to 20 s.
Please cite this article in press as: Z. Liu et al., Rigid–flexible coupling dynamic analysis on a mass attached to a rotating flexible rod, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.03.038
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Z. Liu et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx
Fig. 7. Trajectory of the mass motion in the rotating frame from 20 s to 35 s.
Fig. 8. Trajectory of the mass motion in the rotating frame from 35 s to 40 s.
In the stage the tangential inertia force vanishes due to constant rotation, the petal pattern is only influenced by Coriolis force. In the third stage from 20 s to 35 s, the trajectory of mass is depicted in Fig. 7. The centrifugal force decreases with spindown rotation and the tangential inertia force changes its sign, then the radial deflection decreases and the tangential deflection moves forward (v is positive in sign). In the final stage after 35 s, there is no rigid rotation so that there is no rigid–flexible coupling effect. A stable elliptical motion trajectory of mass is held as shown in Fig. 8. In the stage only the elastic restoring force and relative inertia force control the mass motion. The stable oscillation in Fig. 8 agrees very well with conventional vibration of mass with two degrees of freedom. 4. Dynamical characteristics of mass motion for constant rotation In the case of constant angular velocity, its dynamical characteristic of system may be explored. It follows from (24) that a set of inhomogeneous ordinary differential equations is turned out in the form:
m
0
€ u
0
m
v€
þ
0
2mx
u_
2mx
0
v_
" þ
3EI=l mx2
0
# u
0
3EI=l mx2
v
3
3
( ¼
mx2 r 0
)
0
ð27Þ
Then a linear dynamic system is reached. It is easy to have two order eigen-frequencies from (27) in the form:
x1 1 f1 ¼ ¼ 2p 2p
sffiffiffiffiffiffiffiffi 3EI
!
x2 1 ¼ x ; f2 ¼ 3 2p 2p ml
sffiffiffiffiffiffiffiffi 3EI ml
3
! þx ;
ð28Þ
where f1 and f2 are both termed by the dynamic frequencies as change with angular velocities. With use of the parameters in Table 1, the changes of dynamic frequencies with discrete angular velocities are depicted in Fig. 9. Without rigid rotation, both of the dynamic frequencies return to the unique natural frequency of the linear system, Please cite this article in press as: Z. Liu et al., Rigid–flexible coupling dynamic analysis on a mass attached to a rotating flexible rod, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.03.038
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Z. Liu et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx
Fig. 9. Two order dynamic frequencies of the system vary with discrete constant angular velocities.
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 i.e., its circular frequency is x0 ¼ 3EI=ðml Þ. With constant rotation, two dynamic frequencies are extended from the same natural frequency. The second-order frequency grows up with increasing angular velocity, but the first-order frequency goes down until vanishing, similar results are pointed out by some authors [10,11]. It is noticed that the vanishing dynamic frequency leads to a maximal angular velocity that is identical to the natural frequency. Once the maximal angular velocity is approached, stability of the linear system will be definitely lost from (27). However, it is insufficient to determine the critical angular velocity of system only up on vanishing dynamic frequency because the deflections of mass have to be judged whether its value exceeds its elastic tolerance. The deflections of mass are related to the initial conditions of deflections and velocities, the initial eccentricity r0 as well as the current angular velocity. With leaving out a lengthy derivation, the deflections of mass for any current angular velocity are written out in the form [12]:
uðtÞ
v ðtÞ
¼ A1
cosðx1 t /1 Þ sinðx1 t /1 Þ
þ A2
8
9
1cos x t 1cos x t x2 r0 < x1 1 þ x2 2 = þ 2x0 : sin x1 t þ sin x2 t ; cosðx2 t /2 Þ x1 x2
sinðx2 t /2 Þ
ð29Þ
where the coefficients A1, A2, /1 and /2 are dependent of initial conditions of deflections and velocities of the mass. The deflections in (29) are composed of the periodic and eccentric portions. For given initial conditions of deflections and velocities for the mass, the dynamical deflections are entirely decided by the angular velocity and initial eccentricity. However, the deflection of mass is in fact limited to a certain finite value to insure linear elasticity of the bent rod. Therefore, the bigger is the initial eccentricity, the less is the angular velocity. As a result, a critical angular velocity is extended for the rotating linear system. Suppose a limited deflection is set 3.0 mm, the relationship of critical angular velocity with initial eccentricity is shown in Fig. 10. Without eccentricity (r0 ¼ 0), the critical angular velocity is equal to the natural frequency. With increasing initial eccentricity, the critical angular velocity decreases until vanishing, namely, the linear system is more
Fig. 10. An illustrative for critical angular velocity versus initial eccentricity.
Please cite this article in press as: Z. Liu et al., Rigid–flexible coupling dynamic analysis on a mass attached to a rotating flexible rod, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.03.038
Z. Liu et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx
9
and more sensitive to be violated since the eccentric portion of deflection is larger and larger to approach the tolerance of deflection. 5. Concluding remarks Dynamics of a rotating flexible rod with a mass is studied in the paper where the rod is assumed massless and limited within small deformation. Coupling between the elastic deflection of mass and rigid rotation produces the additional inertia forces on the mass such as Coriolis force, tangential inertia force and centrifugal forces, which lead to complex dynamical characteristics and response for the rigid-flexible system. A rigid–flexible coupling model is established with a set of three strongly nonlinear differential equations when the rotary angular velocity and the deflections of mass are all treated as unknown variables. The strong nonlinearity of system is motivated by coupling of elastic deformation with rigid rotation. Under the condition of given rigid rotation, the rigid-flexible system is governed by a set of two time-varying differential equations. The dynamical deflections of mass are numerically examined for prescribed rigid rotation and the dynamical trajectory of mass is traced. It is attractive to view the petal pattern for constant angular velocity and the elliptical pattern for no rotation. In the case of constant rotation, there are two order dynamic frequencies but the first one tends to vanishing with growth of angular velocity. Since the deflections of mass are limited to assure the linear elastic system, the critical angular velocity is shown to decrease with increasing initial eccentricity of the mass. In spite of a simple system treated here, the results would be beneficial for understanding dynamics of general rotating structures. Acknowledgements The research is supported by NASF (Grant No. 11176035). The authors would like to acknowledge the financial support by the National Science Foundations of China (Nos. 11072276; 11372365) and National Basic Research Program of China (No. 2011CB612211). Appendix A. The motion equation of mass with use of Hamilton principle The kinetic energy and virtual work done by external forces are given by
T¼
1 1 1 _ fR½X ðr 0 þ rÞ þ rg _ ¼ m½X ðr 0 þ rÞ þ r _ ½X ðr0 þ rÞ þ r _ mv 0 v 0 ¼ mfR½X ðr 0 þ rÞ þ rg 2 2 2
dU ¼ F dr
ðA:1Þ ðA:2Þ
With replacement of ^r ¼ Xðr0 þ rÞ, the variational function of kinetic energy is written out in the following:
1 1 _ ð^r þ rÞ _ _ ð^r þ rÞ _ ¼ m½dr_ r_ þ d^r ^r þ dr_ ^r þ d^r r _ ¼ md½ð^r þ rÞ dT ¼ d m½ð^r þ rÞ 2 2 d d dr r_ þ md^r ^r þ m dr ^r þ md^r r_ ¼m dt dt ¼m
d d _ mdr €r þ md^r ^r þ m ðdr ^r Þ mdr ^r_ þ md^r r_ ðdr rÞ dt dt
ðA:3Þ
For a holonomic system, Hamilton principle may be taken in the form:
Z
t2
ðdT þ dUÞdt ¼
t1
¼
Z
t2
Z t2 Z t2 Z t2 Z t2 d d _ ðdr rÞdt mdr €r dt þ md^r ^rdt þ m ðdr ^r Þdt mdr ^r_ dt dt dt t1 t1 t1 t1 t1 Z t2 Z t2 _ þ þ md^r rdt ðF drÞdt
Z
t2
m
t1
t1
_ ðmdr €r þ md^r ^r mdr ^r_ þ md^r rÞdt þ
t1
Z
t2
ðF drÞdt
ðA:4Þ
t1
Attention is paid to the following relationships:
Z
t2
m
t1
Z
d t ðdr r_ Þdt ¼ mdr r_ jt21 ¼ 0; dt
t2
t1
md^r ^r dt ¼
Z
Z
t2
t1
m
d ðdr ^r Þdt ¼ mdr ^r jtt21 ¼ 0; dt
t2
mdðX rÞ X ðr 0 þ rÞdt ¼ t1
Z
ðA:5Þ
t2
mdr X X ðr0 þ rÞdt;
ðA:6Þ
t1
Please cite this article in press as: Z. Liu et al., Rigid–flexible coupling dynamic analysis on a mass attached to a rotating flexible rod, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.03.038
10
Z. Liu et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx
Z
t2 t1
mdr ^r_ dt þ
Z
t2
_ ¼ md^r rdt
Z
t1
t2
_ ðr 0 þ rÞ þ dr X r_ dðX rÞ rÞdt _ m½dr X
t1
¼ 2
Z
t2
_ mdr X rdt
t1
Z
t2
_ ðr0 þ rÞdt: mdr X
ðA:7Þ
t1
With use of (A.5)–(A.7) Hamilton principle becomes
Z
t2
ðdT þ dUÞdt ¼
t1
þ
Z
t2
t1 Z t2
mdr €rdt
Z
t2
mdr X X ðr0 þ rÞdt 2
t1
F drdt ¼ 0:
Z
t2
t1
_ mdr X rdt
Z
t2
_ ðr0 þ rÞdt mdr X
t1
ðA:8Þ
t1
In noticing an arbitrary dr, the motion equation of mass takes the form:
_ ðr 0 þ rÞ þ 2x r_ þ €r ¼ 0; F m½x ðx ðr 0 þ rÞÞ þ x
ðA:9Þ
Appendix B. A brief description of subroutine with Matlab The subroutine can be separated into three parts. The first part is assign values to parameters, such as Young’s elastic modulus, cross-sectional area moment of inertia, rod length and mass. The second part is selective statement by using a keyword called ‘if-elseif-else’. With the selective statement, the different angular velocity can be used for calculation correctly. The last part is function call. The ODE function is called to solve the set of the ordinary differential equations, whereas a PLOT function is further called to plot the figures. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
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Please cite this article in press as: Z. Liu et al., Rigid–flexible coupling dynamic analysis on a mass attached to a rotating flexible rod, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.03.038