VIBRATIONAL MOTION OF AN ELASTIC BEAM WITH PRISMATIC AND REVOLUTE JOINTS

Journal of Sound and Vibration (1996) 190(2), 195–206

VIBRATIONAL MOTION OF AN ELASTIC BEAM WITH PRISMATIC AND REVOLUTE JOINTS B. O. A-B  Y. A. K Department of Mechanical Engineering, King Fahd University of Petroleum and Minerals, KFUPM Box 1767, Dhahran 31261, Saudi Arabia (Received 16 December 1993, and in final form 22 February 1995) A dynamic model for the vibrational motion of an elastic beam with prismatic and revolute joints is presented. The Lagrangian approach in conjunction with the assumed modes technique is employed in deriving the equations of motion. The model developed accounts for all the dynamic coupling terms, as well as the stiffening effect due to the beam reference rotation. The tip mass dynamics is included together with the associated dynamic coupling between the modal degrees-of-freedom. In addition, the dynamic model devised takes into account the gravitational effects, thus permitting motions in either vertical or horizontal planes. Numerical simulations are performed and, whenever possible, comparisons with results of previous investigations are presented. 7 1996 Academic Press Limited

1. INTRODUCTION

The problem of modeling the dynamics of flexible beams with prismatic joints has attracted the attention of investigators in several areas of engineering applications. Examples are robotic manipulators, telescopic members of loading vehicles, and deployable space structures. It has become evident that a reliable dynamic model, for a translating and rotating beam, which accounts for the interaction between rigid and flexible body motions, is highly demanded. Such a dynamic model is crucial to the design, performance evaluation, and control of light-weight, high-speed, and high-precision applications. The vibration characteristics of a translating beam were first studied by Mote [1] in his work on band saws. In reference [1], the dependence of the natural frequencies on the velocity and initial tension in the band saw was reported. The dynamic model introduced by Tabarrok et al. [2] utilized the conservation of momenta and the continuity equations to derive the equations of motion. Their dynamic model resulted in four non-linear partial differential equations and one algebraic equation which, after the assumption of small deformation gradients and constant tension, were reduced to one partial differential equation. The solution of their simplified model was obtained in a semi-analytic form, for specific axial velocities. A dynamic model of a beam moving longitudinally at a prescribed rate over bilateral supports was introduced by Buffinton and Kane [3]. The supports were modeled as kinematical constraints imposed on the unconstrained beam motion. The beam was discretized by using the assumed modes technique. Wang and Wei [4] studied the vibrations of a translating robot arm. They used Newton’s approach in deriving the partial differential equation, and utilized Galerkin’s approximation technique in discretizing the elastic displacements. The equations of motion were presented in integro-differential form. Kalaycioglu and Misra [5, 6] reported approximate analytical solutions for the problem of vibrations in deploying appendages. The Lagrangian 195 0022–460X/96/070195+12 $12.00/0

7 1996 Academic Press Limited

196

. . -  . . 

approach with the assumed modes technique was used in deriving the equations of motion, and the discretized differential equations were solved analytically. For specified deployment schemes, solutions were obtained in the form of Bessel functions. Recently, Tadikonda and Baruh [7] introduced a dynamic model for a translating flexible beam with a prismatic joint. The equations of motion were derived by using the Lagrangian approach with time-dependent eigenfunctions of the fixed-free beam. The dynamic coupling between the translational motion and the elastic displacements was included in the equations of motion. In all of the above mentioned investigations, only the translational motion of an elastic beam with a prismatic joint was considered. Chalhoub and Ulsoy [8, 9] investigated the effect of the structural flexibility on the control of a robotic arm with one prismatic and two revolute joints. The equations were derived by using the Lagrangian approach in conjunction with the assumed displacement approximation. Their model, however, did not account for the inertia coupling between the rigid body motion and the elastic deformations. Wang and Wei [10] proposed a feedback control law to minimize the vibrations of a flexible robot arm with revolute and prismatic joints. They utilized their earlier model [4] in addition to a prescribed rotational motion. Consequently, the effect of the elastic motion on the reference rotational motion did not appear in the equations. Krishnamurthy [11] extended the work of reference [10] by incorporating the dynamics of the beam tail. The choice of the body axes, however, resulted in a dynamic model wherein the inertia coupling between the translational motion and the elastic deformations was ignored. That is, if one eliminates the rotational-degree-of-freedom from the equations of motion, the model of reference [11] is rendered uncoupled. Recently, Yuh and Young [12] developed a dynamic model of an axially moving and rotating beam, wherein, the motion is confined to the horizontal plane. Newton’s approach was employed in developing the equation of motion in the form of one partial differential equation. By using the assumed modes technique, a system of linear differential equations with time varying coefficients was obtained. Following the same methodology of reference [3], Buffinton [13] modeled the dynamics of an elastic manipulator with a prismatic joint, wherein the prismatic joint was modeled as a two-point support. Some other investigators [14, 15] modeled the translating and rotating beam by a set of elastically connected massless rigid links to obtain simplified models. A displacement based finite element formulation has been also addressed by Pan et al. [16].

Figure 1. Schematic diagram of the rotating and deploying beam.

     

Figure 2. Tip deflection of a deploying beam. ——, Fast deployment (L=1·8+0·5t m); . . ... , slow deployment (L=1·8+0·1t m).

197

Figure 3. Tip deflection of a deploying beam. ——, With end mass; . . ... , without end mass.

The objective of the described work in what follows was to develop a general dynamic model for an elastic beam with prismatic and revolute joints. The model proposed accounts for all the dynamic coupling terms, as well as the stiffening effect due to the beam reference rotation. The tip mass dynamics is included together with the associated dynamic coupling between the modal-degrees-of-freedom. In addition, the model devised takes into account the gravitational effects, thus permitting motions in either vertical or horizontal planes. The Lagrangian approach in conjunction with the assumed modes technique is employed in deriving the equations of motion. Simulations of various motion strategies are conducted and, whenever possible, comparisons with results of previous investigations are presented.

2. MATHEMATICAL MODEL

The beam configuration shown in Figure 1 is considered with the following features and assumptions: (1) the reference axes XY represent the inertial frame, and the coordinate system xy is fixed to the hub which is rotating about a stationary revolute joint O as shown in Figure 1; (2) the base, or the hub, is rigid with mass moment of inertia Jh ; the motion

Figure 4. Tip deflection of a retracted beam. ——, Fast retraction (L=3−0·2t m); . . ... , slow retraction (L=3−0.1t m).

198

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Figure 5. Tip deflection of a deploying beam (L=1·8+0·1t m). ——, Fast rotation (u =0·785 rad/s); . . . . . , slow rotation (u =0·157 rad/s).

of the beam inside the hub is frictionless; (3) the beam is uniform and slender at all times, and thus the Euler–Bernoulli beam theory is applicable with small deformations; the beam is inextensible; (4) the gravitational potential energy due to the elastic deformations is neglected compared to the gravitional potential energy resulting from the rigid body motions; (5) a tip mass is considered to be concentrated at the free end of the beam; the rotary inertia of the tip mass is neglected. 2.1.     The global position of an arbitary material point p on the beam can be expressed as Rp=Arp ,

(1)

where the matrix A is the rotational transformation matrix from the moving coordinate system xy to the fixed reference frame XY, and rp is the location of the point p in the rotating coordinate system xy. As shown in Figure 1, the location of the point p in

Figure 6. Tip deflection of a retracted beam (L=3−0·2t m). ——, Fast rotation (u =0·785 rad/s); . . . . . , slow rotation (u =0·157 rad/s); – – –, no rotation.

     

199

Figure 7. Tip deflection of a deploying beam (L=1·8+0·1t m, u =0·785 rad/s); ——, Without end mass; . . . . . , with end mass.

the rotating coordinate system xy can be expressed as (a list of nomenclature is given in the Appendix) rp={x, u(x, t)}T.

(2)

To approximate the elastic motion, the assumed modes method is used. It has been shown in a previous investigation [12] that the assumed displacement technique can be applied to the problem of axially moving beams, with the assumed shape functions being time-dependent. The elastic displacements are represented in the form k

u(x, L, t)= s Yi (x, L)ui (t),

(3)

i=1

where Yi (x, L) are the assumed shape functions, ui are the modal coordinates, and k is the number of modes. In equation (3), the beam length L is a function of time. The eigenfunctions of a stationary cantilever beam will be used as the assumed shape functions. The cantilever eigenfunctions, after being normalized with respect to the mass per unit length r, can be represented in the form Yi (x, L)=

1

$

0

x x x x sin ei −sinh ei −ai cos ei −cosh ei L L L L zrL

1%

,

(4)

where ai=(sin ei+sinh ei )/(cos ei+cosh ei ), ei=bi L and bi4=vi2 r/EI. The values of ei are defined by the transcendental relation 1+cos ei cosh ei=0.

(5)

2.2.   The velocity of an arbitrary material point p can be obtained by differentiating equation (1) with respect to time, as R p=Ar˙ p+Au u rp ,

(6)

. . -  . . 

200 where Au=

$

−sin u 1 A= cos u 1u

−cos u −sin u

%

r˙ p={x˙ , u˙ (x, t)}T.

and

(7, 8)

The velocity of the point p can be written in the inertial reference frame as R p=

6

−sin u(xu +u˙ )−cos u(uu −x˙ ) +cos u(xu +u˙ )−sin u(uu −x˙ )

7

.

(9)

By differentiating equation (3), the time derivative of the elastic displacement can be written as k

$

u˙ (x, L, t)= s Yi u˙i+ i=1

%

dYi u . dt i

(10)

Since Yi is function of x(t) and L(t), its time derivative is dYi /dt=(L /L)[−12 Yi+Y'i (L−x)].

(11)

Equation (10) can now be written in the form

$

%

k L u˙ (x, L, t)= s Yi u˙i+ (−12 Yi+Y'i (L−x))ui . L i=1

(12)

The velocity vector obtained is used to obtain the kinetic energy of the beam as Ub=

1 2

g

R pT R p dm.

(13)

By substituting for the value of R p from equation (9) and making use of some tigonometric properties, the kinetic energy of the beam can be written as Ub=

1 2

g

[(u˙+xu )2+(uu −x˙ )2 ] dm.

(14)

Next, the contribution of the end mass to the kinetic energy of the system can be evaluated. The location of the end mass in the fixed reference frame can be written as Re=Are ,

(15)

re={L(t), u(L(t))}T.

(16)

where

The velocity vector of the end mass can be written in the fixed reference frame as R e=A0 u re+Ar˙ e .

(17)

By substituting for A0 and A from equations (7) and (15), the global velocity vector of the end mass can be established as R e=

−sin u(Lu +u˙ (L))+cos u(L −u(L)u )

6

+sin u(L −u(L)u )+cos u(u˙ (L)+Lu )

7

.

(18)

The kinetic energy of the end mass is expressed in the form Ue=12 me R eT R e ,

(19)

     

201

which, after substituting for the velocity vector from equation (18), can be expressed in the form Ue=12 me [(L 2+u 2(L))u 2+L 2+u˙ 2(L)]+me Lu u˙ (L)−me L u u (L).

(20)

The total kinetic energy is given by U=Ub+Ue .

(21)

The potential energy of the system is composed of three parts: the elastic strain energy stored in the flexible system, the gravitational potential energy, and the strain energy resulting from the centrifugal force field due to the beam rotation. First, the strain energy for small deformations may be written as Vs=

1 2

g

L

EI(x)

0

$ % 1 2u 1x 2

2

dx.

(22)

Second, the gravitational potential energy for the beam, including the end mass, can be written in the form Vg=

g

L

rgx sin u dx+me gL sin u,

(23)

L−Li

which, upon integration, yields the explicit expression Vg=rg sin u(LLt−(Lt2 /2))+me gL sin u.

(24)

Third, the strain energy due to the centrifugal force field that results from the beam rotation can be expressed as [17] Vr=

1 2

g 0 1 L

Fp

0

1u 1x

0 b1

2

dx+Fe L

1u 1x

2

,

(25)

L

where the axial forces Fp and Fe can be written as Fp=

g

L

ru 2x dx=

x

ru 2 2 (L −x 2 ), 2

Fe=me u 2L.

(26, 27)

By utilizing equations (26) and (27), the strain energy due to the beam rotation can be represented as Vr=

ru 2 4

g

L

0

(L 2−x 2 )

0 1 1u 1x

2

dx+me u 2L 2

0 b1 1u 1x

2

.

(28)

L

The total potential energy of the system is then given by V=Vs+Vg+Vr .

(29)

2.3.    In order to write the equations of motion in a more compact form, one can make the following definitions: Mij=

g

L

0

(−12 Yi+(L−x)Y'i )(−12 Yj+(L−x)Y'j )r dx,

Sij=

g

L

(L 2−x 2 )Y'i Y'j r dx,

0

(30, 31)

. . -  . . 

202

Qij=rL 4Y'i (L)Y'j (L),

Nij=

g

L

Yi (−12 Yj+(L−x)Y'j )r dx,

(32, 33)

0

ai=

1 zrL 3

g

L

rxYi dx,

bi =

0

1 zrL 3

g

L

rx(L−x)Y'i dx,

ci =

0

1 zrL

g

L

rYi dx,

0

(34–36) di=−(ai /2)+bi−ci ,

ei=(ai /2)−bi ,

fi=di−32 ai .

(37–39)

By utilizing equations (30)–(39), the total kinetic energy given by equation (21) can be written as U=12 (me+mb )L 2+

0

1

1 rL 3 r +Jh+ (Lt−L)3 u 2 me L 2+ 3 2 3

0

1

0

1

0

1

0

1

1 k k 4m (−1)i+j 1 L 2 k k m (−1)i+j + s s dij+ e u˙i u˙j+ 2 s s Mij+ e ui uj 2 i=1 j=1 rL 2 L i=1 j=1 rL

+

k L k k 2me (−1)i+j 2m (−1)i+1 s s Nij− u˙i+zrLLu s ai+ e u˙i L i=1 j=1 rL rL i=1

0

1

k a 3m (−1)i+1 +zrLL u s bi− i−ci− e ui , 2 rL i=1

(40)

where mb is the total mass of the beam. Similarly, the total potential energy of the system can be written as V=

+

0

1

1 k 2 2 Lt2 u 2 k k sin u+me gL sin u+ s s Sij ui uj 4 s Di ui +rg LLt− 2L i=1 2 4 i=1 j=1

me u 2 k k s s Q uu, rL 2 i=1 j=1 ij i j

(41)

where Di2 is obtained by solving a first order differential equation that relates the natural frequency of the beam to its length. The first order differential equation can be obtained by differentiating equation (5) with respect to L: i.e., dv/dL=−2v/L.

(42)

The solution of equation (42) can be obtained, in closed form, as vi (L)=Di /L 2,

(43)

where Di2=ei4 EI/r. Substituting the expressions for the total kinetic and potential energies into the variational from of Lagrange’s equation of motion, one obtains (k+2) coupled non-linear ordinary differential equations. Note that the resulting dynamic model possesses (2+k) degrees of freedom. These are L, u and ur , where r=1, . . . , k. Upon performing the

     

203

necessary differentials and algebraic manipulations, the equations of motion are written in the following final forms: for the translational motion,

$

me+mb+

0

1 % $ 0 0 0

1%

k 1 k k m (−1)i+j 3m (−1)i+j Mij+ e ui uj L+ s zrLdi− e ui u 2 s s L i=1 j=1 rL i=1 zrL

0

1 1

1 1

2m (−1)i+j L 2 k k 3m (−1)i+j 1 k k u¨i uj− 3 s s Mij+ e ui uj + s s Nij− e rL L i=1 j=1 2rL L i=1 j=1 +

0

k k 2u 2 Q 2L k k m (−1)i+1 (−1)i+j− ij ui uj+ 2 s s Mij+ e u˙i uj 2 me s s rL L L i=1 j=1 rL i=1 j=1 k

me (−1)i+1

−4u s

zrL

i=1

−r(Lt−L)2u 2−

u˙i+

X

k k r L u s ei ui+zrLu s fi u˙i−(L(rL+me )u 2 L i=1 i=1

2 k 2 2 s D u +rgLt sin u+me g sin u=F; L 5 i=1 i i

(44)

for the rotation motion,

$

0

1 % 1

r rL 3 k k 4m (−1)i+j m + s s dij+ e −Sij− e Qij ui uj u me L 2+Jh+ (Lt−L)3+ 3 rL rL 3 i=1 j=1

$ 0 1% 0 $ 0 1 % $ 0 1 % 0X 0 1 0 1 k

+ s zrLdi− i=1

3me (−1)i+1 zrL

k

ui L+ s zrL 3ai+

2Lme (−1)i+1

i=1

zrL

2

u¨i

k k 1 4m (−1)i+j +2u meLL + rL 2L + s s dij+ e u˙i uj −r(Lt−L)2L u rL 2 i=1 j=1

k

k

−u s s Sij+ i=1 j=1

−4u

1

r di 2me (−1)i+1 + ui L2 LzrL

k k 3 3m (−1)i+1 u˙i+L 2 s +L s zrL di+ ai − e 2 i=1 i=1 zrL

2me L m u k k u˙i uj+ 2 e s s Qij ui uj rL L r i=1 j=1

L k k me (−1)i+j L2 s s ui uj+rg LLt− t cos u+me gL cos u=T; L i=1 j=1 rL 2

(45)

for the elastic degrees of freedom, k

0

s drj+ j=1

1

0

1

4me (−1)r+j L k 2m (−1)r+j u¨j+ s Nrj− e uj rL L j=1 rL

0

+ zrL 3ar+ k

+u 2 s j=1

0

2Lme (−1)r+1 zrL

0

1

u −

0 1 1 1

1

L 2 k 2m (−1)r+j Nrj− e u˙j 2 s L j=1 rL

Sjr 2me Qjr + −2(−1)r+j −drj uj 2 rL L

0

+zrLL u 2ar−br+cr−

4me (−1)r+1 1 + 4 Dr2 ur=0, rL L

r=1, . . . , k.

(46)

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. . -  . . 

Note that the fifth term of equation (46) accounts for both the stiffening effect due to the beam rotation, and the centripetal acceleration effect. The centripetal acceleration is known to produce a small softening effect during the lead-lag motion of a spinning beam [18, 19]. The dynamic model, as represented by equations (44)–(46), reduces to that of reference [7] if the rotational degree of freedom is eliminated, provided that the effect of shortening due to axial deformations is neglected. For the case of both translation and rotation, equation (46) reduces to the model obtained in reference [12] if the effect of the end mass is eliminated from equation (46). However, the softening effect due to the beam reference rotation was not present in the final equation of motion in reference [12]. 3. SIMULATION AND DISCUSSION OF RESULTS

A uniform beam similar to that of reference [7] is used in the simulation of the present work. The beam is uniform with total length Lt=3·657 m, cross-section of 15·24 cm×0·952 cm, mass per unit length r=4·015 kg/m and flexural rigidity EI= 756·65 Nm2. The translational motion of the beam will be simulated first. It consists of either axial deployment with uniform extension rate, or axial retraction with uniform retraction rate. In addition, the translational motions of the beam are simulated when it carries a pay load at its tip. In the second part of the simulation both the translational and the rotational motions of the beam are considered simultaneously. The pay load effect is also examined in the latter case. In order to activate the elastic motions, an initial tip displacement is given to the beam. The elastic deflection curve of the beam can be written as v(x)=x 2(3L−x)D/2L 3,

(47)

where D is the initial tip deflection. By utilizing the orthogonality condition of the shape functions, the initial modal displacements can be found in the form ur (L, 0)=(D/2L 3 )

0g

L

0

x 2(3L−x)Yr (x) dx

>g

L

0

1

Yr2 (x) dx .

(48)

The tip vibratory motion of the axially moving beam is shown in Figure 2, for two uniform deployment rates. As the deployment rate is increased, the frequency of oscillation decreases and the amplitude increases. This behavior is attributed to the decreasing beam stiffness as the free length of the beam increases. The same behavior was reported by some previous investigators [5–7, 10]. The end mass effect on the vibratory response of a deploying beam is shown in Figure 3. A 2 kg end mass is shown to increase both the period and the amplitude of oscillation. Figure 4 shows the tip deflection of a retracted beam with two retraction velocities. The deflection of the rotating beam during deployment is shown in Figure 5. This shows how the effect of the reference rotation becomes more pronounced as the rate of rotation increases. The tip deflections of a retracted beam that rotates with different rotational rates are shown in Figure 6. The results show that, for the same rate of retraction, the vibrational amplitude increases as the rotational rate increases. The observed behavior can be attributed to the coupling between the elastic deformations and the reference rotational motion. This result was not reported in the previous investigations where such inertia coupling terms were neglected. The end mass effect on the deploying beam that rotates at a fast rate (u =0·785 rad/s) is shown in Figure 7. The effect has mainly increased the period of oscillation. The simulation results for a rotating and

     

205

translating beam could not be compared to those of reference [12]. This is due to the inclusion of damping in reference [12], which was shown to have suppressed all the major vibrational interactions in the beam motion. In all the simulation results, three elastic modes were used. However, the responses were shown to be dominated by the first mode. The reason for this behavior is that the initial tip deflection given to the beam was picked up mainly by the first mode deflection. For instance, in this numerical simulation, an initial tip deflection of −5 mm was given to the free end of the beam in the case of deployment. The corresponding initial modal displacements were obtained by solving equation (48) for the initial beam length of 1·8 m as u1=−6·5×10−3 m, u2=1·66×10−4 m and u3=−3×10−5 m. 4. CONCLUSIONS

The dynamic model developed in this paper provides a complete representation of the vibrational motion due to small deformations of an elastic beam with prismatic and revolute joints. All the possible dynamic coupling terms between the translational motion, rotational motion and elastic deformations were included in the model. In addition, the model accounts for the stiffening effect due to the centrifugal force field resulting from the beam reference rotation. Moreover, the softening effect due to the centripetal acceleration is also included. Such an effect has not been tackled in previous investigations. This model is valid for the cases where the beam is either kinematically or inertially driven, thus lending itself to control applications. A direct extension of this work is to consider large elastic deformations together with the associated effect of shortening. ACKNOWLEDGMENT

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APPENDIX: NOMENCLATURE A EI F g Jh k L Lt mb me T U

transformation matrix flexural rigidity axial force gravitional constant mass moment of inertia total number of modes instantaneous beam length total beam length total mass of the beam end mass torque kinematic energy

u ui Yi V v D dij r v b u

elastic displacement modal coordinates shape function potential energy static deflection initial tip deflection Kronecker delta mass per unit length natural frequency frequency parameter, bi4=vi2 r/EI reference rotation of the beam