Dynamic stability of a free Timoshenko beam under a controlled follower force

Journal of Sound and Vibration (1987) 113(3), 407-415

DYNAMIC STABILITY OF A FREE T I M O S H E N K O BEAM U N D E R A CONTROLLED FOLLOWER FORCE Y. P. PARK

Department of Mechanical Engineering, Yonsei University, Seoul, Korea (Received 21 August 1985, and in revisedform 25 January 1986) A uniform, free-free, Timoshenko beam is driven by a follower force with controlled direction. A finite element model of the beam transverse motion in the plane is formulated through the extended Hamilton's principle. The dynamic stability of the model is studied with respect to (i) the location of the follower force direction control sensor, (ii) the sensor gain, (iii) the magnitudes of the rotary inertia and shear deformation parameters of the beam, and (iv) the magnitude of the constant follower force. Both divergence and flutter instabilities can occur over the range of free-free Euler-Bernoulli beam models examined. The analysis shows that the effects of the rotary inertia and shear deformation parameters on the stable transverse motion of the beam are significant in certain ranges. 1. INTRODUCTION The non-conservative stability problem of a free-free beam subjected to a controlled follower force has been the subject of m a n y studies simulating the stability of flying flexible rockets and missiles. In 1965 Beal [1] introduced a direction control mechanism for the follower force to eliminate the tumbling instability of a free-free beam under a follower force. He also investigated the parametric instability of the system for periodic follower force excitation. Feodos'ev [2] investigated the dynamic stability of a slender elastic beam under uniform acceleration by an uncontrolled follower force. Matsumoto and Mote [3] examined the stability o f an Euler-Bernoulli beam under a follower force whose direction is controlled by a system with finite time delay. Wu [4-6] studied the stability of a free-free beam under a controlled follower force by using finite element discretization with an adjoint formulation. Park and Mote [7] studied the m a x i m u m controlled follower force on a free-free beam carrying a concentrated mass. They predicted the location and the magnitude of the additional concentrated mass and the location and the gain of the follower force direction control sensor that permit the follower force to be maximized for stable transverse motion of the beam. Most o f the above analyses are restricted to Euler-Bernoulli, free-free beam models where the magnitude of the rotary inertia parameter is considered as zero and the shear deformation parameter is considered as infinite. The effects of these two parameters on the stability of the b e a m become significant primarily for a stubby beam. In this study, the influence o f these parameters on the stability of a free-free Timoshenko beam with a controlled follower force is investigated. 2. FORMULATION A uniform beam of length L, bending stiffness EI, shear stiffness KAG, mass per unit length pA, and inertia per unit length pl is subjected to a constant follower force P. The 407 0022-460x/87/060407+09 $03.00/0 O 1987 Academic Press Inc. (London) Limited

408

Y. P. PARK

Figure I. Free-freeTimoshenko beam model. undeformed axis z is the centerline of the beam in a state of dynamic equilibrium, and u(z, t) denotes a small transverse disturbance from the equilibrium axis (see Figure 1). The angle between the follower force and the undeformed z axis at z = L, if(t), is assumed to be small and is controlled by proportional feedback of the rotation of the beam at z = z,:

~(t) = Kou~(z,, t)+ u~( L, t),

(1)

where Ko is the gain of the sensor. Subscripts for u denote ditferentiation with respect to the subscripted variable. The extended Hamilton's principle for the system can be expressed as ~5

I

I2

2

( T - V+ We) d t +

cSW.+dt = 0

(2)

tl

with

T=

+#

(pld~ + pAu~) dz,

wo_-

V=~ f:{Elck~+KAG(u:-ck)2}dz, 6W, c = -P~(t)6u( L, t),

Jo L

(3,4) (5, 6)

where W+ and IV.,. are the conservative and non-conservative work done by the follower force, 4' is the bending slope, and K is the shear coefficient of the cross-section of the beam model. From the variation indicated in equation (2), with equations (3)-(6), one obtains

f '~( f L{oldp',~ck + oAu,,6u + Elck/Sqb: + KAG ( u: - ch)~( uz - ck) - ( Pz/ L )ufiSu..} dz tj

"tJO

\

+ P{Kouz(z,, t)+ u+(L, t)}Su(L, t)}) dt =0.

(7)

The beam is subdivided into N finite elements o f equal length ! (see Figure 2). The weak form of equation (7) can be obtained as follows: ! i=l fO

{plop,, Sdpc"+pAu(,~>fu<"+ Eldp~"Sd)~"+KAGtu,"('> - ~b~'J)S(u~') -~b ~ "--('1

- ( Pz/ L)u~') ~Su~''} d z + P{ Kou~(z,, t) + u,( L, t)lSu( L, t) =0.

(8)

FREE

BEAM

Element Ilo

Noel no.

WITH

1

2

I

1

CONTROLLEI) 3

I

2

FOLLOWER

I

3

N

I

4

409

FORCE

I

I

i

I

i.1

I

N N~'I

Figure 2. Finite element and local co-ordinate. For convenience, by using the local co-ordin~tes z ' = z - ( i - l ) ! of the ith elements, respectively, the following non-dimensional parameters are introduced:

~= z'/I,

"11= u/I,

Q = PL2/EI,

S= KAGL2/EI,

R = I / A L 2.

(9)

Here S is the shear deformation parameter and R is the rotary inertia parameter. The finite element model for Timoshenko beams, developed by Thomas and Abbas [8, 9], is used, with the solution assumed to be o f the form ~b('~(~, t) = A{i~(,~)e ~

"0(~ :, t) = B"~(~:) e"'.

(10)

The stability and characteristic frequency dependence of the beam model upon the force Q. the shear deformation parameter S, and the rotary inertia parameter R can then be investigated by examining the behaviors of the eigenvalues (A 2= pAL4a2/EI) of the global discretized eigenvalue equations as calculated for the system. 3. RESULTS AND DISCUSSION The eigensolutions were calculated for a beam model composed of 20 uniform finite elements by using a Cyber 170-825 computer with Subroutine RGG of the EISPACK package. 70

-

60

~....-.--6030 .,...,...,..

K8=00 !

I',t

"~ 5 0 - . ,

O

"-.

,

~,,

"',, - .

~o

o>

"-.'7"--, \..\ \

zz37

$

~__-zz 23

1ok

ol

--

.

0

,

2

,

,

,

\

I

ul5!

i

,

,

4 6 8 Fotlower force (QlrtZ}

10

12

Figure 3. Eigenvaluedependence upon shear deformation parameter without direction control. 5 = 1 0 6 , - - - , R = 0 , S = I 0 ~ ; - - - , R = 0 , S = I 0 ~.

.R=0.

410

v.P.

PARK

3.1. FORCE Q WITHOUT DIRECTION CONTROL

Excitation of the beam by a constant follower force Q without directional control always leads to a system model with two zero eigenvalues associated with rigid body translation and rotation of the model. In the discussion of instability here it is assumed that rigid body instability does not occur and that the mechanism perventing instability does not otherwise influence the model dynamics. The dependence of the first two eigenvalues upon the magnitudes Of shear deformation parameter S and rotary inertia parameter R is illustrated in Figures 3-6. For R = 0 and 15 Ke=O0

0

I

1113. , ~ . . . . . . . . . .

-

~'.T,,Io13 / i 8 fJ

~-103 2

I

f

3 4 ,5 Sheor deformolion poromeler (log S)

6

Figure 4. Flutter force dependence upon shear deformation parameter without direction control.

, R=0;

9 R = 0-001.

70

K=0 0

0

,-----6175 60 " " - - . . .

V ~40

'~

--~

.22-37

xI

I \z164

~,,,~/i

~

10~-

lal 11.13,,I'~

1085 kLi ol

0

,

,

2

,

,

,

,

,

,

4 6 8 Follomerforce (O/rrZ)

,

,

10

,~,,

12

Figure 5. Eigenvalue dependence upon rotary inertia parameter without direction control. - - - , R =0; , S = 106, R =0.001.

S = 106,

151

FREE

BEAM WITH CONTROLLED

1Ke=O0

11"13 , , I 11-I 2 .

FOLLOWER

411

FORCE

l-~--a

~

1(?

6:2_2.. . . . . . . . . . . . . . . . . . . . G

- " ~ ' " " "~.~,..., ~ 9

4.e8 4.8".'

\.

'~-.-4.73

3d 03

0 -co

0

I -4 Rotory inerfio

("

}'

1

86

I

-3

-2 (log R}

porometer

-1

Figure 6. Flutter force dependence upon shear deformation and rotary inertia parameters without direction control. , S = 106; ~ - ~ , S = 10~; - - - , S = 104; - - , S = 103. - - - - - , S = 10z; - - - , S = 10.

S = 1 0 6 (a case which can be considered as a Euler-Bernoulli beam), Q / - a ' 2 > 11-13 produces flutter, this limiting value being 0.5% of the flutter force for the Euler-Bernouili beam case as found in references [1-7]. The results in Figure 3 show that the force at flutter (flutter force: Qf), with R = 0 and S = 102 is approximately 44% smaller than that of an Euler-Bernoulli beam. Flutter force dependence upon S is shown in Figure 4 for R = 0 and 10 -3. The results show that the Qf dependence upon S is negligible for S > 10~. For S< 103, however, the effects of S on Qf become significant and Of decreases as S

301

Ke=I 0 z ~ = 0

I

~ ,22.37 .~2'23

.

2110

"~ "1

,+_o

~

~

o

...,,b-"r~ "- - " - - " : , ~

_.---

E "~'

-/

"~

,

,

',

''

..;\ ,,2"16. 'L',~257~ '~ "~,\

1.701.71

~.

. '~1'

"E

0

,

~

I

06

,

~

I

,

12 Fotlower

force

,

I

18 (O/Tr z)

a

~

'~t,~

'-.

I

24

"~'~-

30

Figure 7. Eigenvalue dependence upon shear deformation parameter with direction control. Euler beam; . . . . , R e 0 , $ = 103; - - - , R = 0 , S = 102 .

z,/L=0.0.)--,

( K ~ = 1.0,

Y . P . PARK

412

decreases. Qr dependence upon R is shown in Figure 5 for S = 10 6. As can be seen in Figure 5, the effects of R on Q.f are negligible for the cases of both R = 0 and R = 10 -3. Figure 6 gives the total Qf dependence upon S ( 1 0 < S < 10 6) and R ( 0 < R < 10-1). The two parameters have negligible effects on the Q / f o r S > 104 and R < 10 -3. Outside the

501

Ko=I O, z=/L =O 5

]

{

f

i'~

\\I , ,

O

2

4 6 8 Follower force (O/rr 2)

10

12

Figure 8. Eigenvalue dependence upon shear deformation parameter with direction control. K e = 1.0, 0.5. Key as Figure 7.

501

z,/L=

K0= l O, zs/L = l O

401

I~--0

i

30522-37

I/~2-23

-..,, - ~ . \\ '\\

o 2o1-.2. 0 'o ~

~o

/

I

_e

2oI-I 0

zh~"

.

1

2

,

!1

i

".".._

,

,

,

,

,

4 6 8 Follower force (Q/Tr2)

,

,

10

12

Figure 9. Eigenvalue dependence upon shear deformation parameter with direction control. 1.0. Key as Figure 7.

Ke =

1-0, z , / L =

FREE BEAM WITH CONTROLLED FOLLOWER FORCE

413

above ranges, the effects of the two parameters on Of become significant. However, in many applications, R is relatively small, say R < 10 -3, and so the contribution of R to the stability appears to be negligible. 3.2. FORCE Q WITH DIRECTION CONTROL With control of the follower force direction one can control the rigid body rotation or tumbling of the model. Either divergence and flutter is predicted at sufficiently large Q depending upon R and S, upon the location of the sensor z, and upon the beam rotation feedback gain KA. With Ko = 1 and z, = 0 , flutter is initiated when Q/Tr 2= i'70 for R = 0 and S = 103, and when Q/r: 2= 1.56 for R = 0 and S = 102 (see Figure 7). In both cases, a stable response theoretically occurs when Q is further increased. With Ko= I and z , / L = 0.5, 1.0, the initial instability type becomes divergence for R = 0 and S = l0 s, 103 (see Figures 8 and 9). From Figures 7-9, one can see the dependences of the critical force (Q,,, flutter or divergence) upon S. The dependence o f the initial instability type encountered with increasing Q, called the primary instability, upon z, and Q with R = 0 is shown in Figure 10. The boundary 1.0 I

w

IB

IG

0

Divergence

~ 3'25

0.14 0

I

2

Flutter I

1

3 4 5 Shear deformat,on pmam~er (loq S)

Figure 10. Initial instability types for shear deformation parameter and sensor location. Ko = 1.0.

o

2

183 Initial flutler

"8

force~

I '72

~t 42 O

I

0 2

I 3

I 4

I 5

6

Shear deforrnohOn parameter (log S)

Figure II. Critical force (flutter instability) dependence upon shear deformation parameter. Ko = 1.0. --, z,/ L =.O;- - - - - , zJ L =O.125.

414

v.P.

PARK

O< Ke 4; 3 ,zs = zs(div) 260

Divergence &-f 227

O

2 o

Stable o .'2'

o

'J' I

l

I

I

4 5 Shear deformation parameter (log S') 3

Figure 12. Critical force (divergenceinstability) dependence upon shear deformation parameter. of z J L between the two types of primary instability regions moves down stream along the beam as S increases, converging to z J L = O . 1 3 9 . The dependence of Qc, upon S is shown in Figures 11 and 12. In both cases, the Q , increases as S increases, but the rate of variation o f Q , is negligible for S > 10a. If the primary instability is of divergence type, then the force at instability is independent o f K0 and z~. 4. CONCLUSIONS 4.1. F O L L O W E R F O R C E W I T H O U T D I R E C T I O N C O N T R O L (1) The instability at the critical force is of flutter type. The critical force depends upon the magnitudes o f the rotary inertia parameter R and the shear deformation parameter S. (2) The critical force is increased as S is increased, but the rate o f increase is negligible for S > 104. (3) R contributes insignificantly to the critical force when R < 10 -3. 4.2. F O L L O W E R F O R C E W I T H D I R E C T I O N C O N T R O L (1) The primary instability is of either flutter or divergence type depending upon the rotation sensor location zs and the magnitude of the sensor gain Ko. (2) The magnitude ofthe critical force depends mainly upon S for both flutter and divergence type instabilities. (3) The critical force increases as S increases, but the rate of increase is negligible for S > 104. (4) The initial instability depends on zs. The zs boundary between the two types of instability moves downstream along the beam as S increases. REFERENCES 1. R. R. BEAL 1965 American Institute of Aeronautics and Astronautics Journal 3, 486-494. Dynamic stability of a flexible missile under constant and pulsating thrusts. 2. V. I. FEODOS'EV 1965 Prikladnaya Matematika i Mekhanika 29, 391-392. On a stability problem. 3. G. Y. MATSUMOTOand C. D. MOTE, JR. 1972 Transactions of the American Society of Mechanical Engineers, Journal of Dynamic System, Measurement and Control, 330-334. Time delay instabilities in large order systems with controlled follower force. 4. J. J. Wu 1975 Journal of Sound and Vibration 43, 45-52. On the stability of a free-free beam under axial thrust subjected to directional control. 5. J. J. WU 1976 American Institute of Aeronautics and Astronautics Journal 14, 313-319. Missile stability using finite elements--an unconstrained variational approach.

FREE BEAM WITH CONTROLLED FOLLOWER FORCE

415

6. J. J. W o 1976 Journal of Sound and Vibration 49, 141-147. On missile stability. 7. Y. P. PARK and C. D. MOTE, JR. 1985 Journal of Sound and Vibration 98, 247-256. The maximum controlled follower force on a free-free beam carrying a concentrated mass. 8. J. THOMAS and B. A. H. ABBAS 1975 Journal of Sound and Vibration 41,291-299. Finite element model for dynamic analysis of Timoshenko beam. 9. J. THOMAS and B. A. H. ABBAS 1976 Transactions of the American Society of Mechanical Engineers, Journal of Engineering for Industry 1145-1151. Dynamic stability of Timoshenko beams by finite elelnent method.