Transverse vibration and buckling of a cantilevered beam with tip body under axial acceleration

Journal

of Sound and

Vibration (1985) !W(l), 43-52

TRANSVERSE

VIBRATION

A CANTILEVERED TIP BODY

UNDER

AND

BUCKLING

BEAM

AXIAL

OF

WITH

ACCELERATION

J. STORCH AND S. GATES The Charles Stark Draper Laboratory, Inc., Cambridge, Massachusetts 02139, U.S.A. (Received 1 February 1984, and in revised form 18 April 1984) The transverse vibration and buckling of a cantilevered beam subject to constant axial acceleration with rigid tip body is investigated. Two classes of tip bodies are recognized: those with mass centers located along the beam tip tangent line, and those with mass centers having an arbitrary offset with respect to the beam attachment point (but not lying along the beam tip tangent line). For the former class, the critical buckling loads and shapes as well as the natural frequencies and mode shapes are determined analytically. It is shown that for the latter class of tip bodies, steady state solutions exist except for certain critical values of acceleration. The free vibration problem for this later class of tip bodies is addressed. Numerical comparisons are made between the exact analysis and a Rayleigh-Ritz procedure for the first three natural frequencies for values of base acceleration between zero and the first critical buckling value. 1. INTRODUCTION

The influence of axial loads on the transverse bending vibrations of long slender members is a problem frequently encountered by structural dynamicists in the fields of aerospace, civil and mechanical engineering. These loads which can arise from impressed or inertial forces may be either constant or time varying, and it is well known that their presence can have a profound impact on the mechanical stability of the member. Accelerating missiles and spacecraft, components of high speed machinery, robotic members and vertically erected civil structures provide numerous situations where such problems can be of importance. Several authors have addressed problems of this nature. Seide [ 1] analyzed the transverse vibration of a free beam under constant axial thrust while Beal [2] investigated the dynamic stability of a free beam under pulsating thrusts. The natural frequencies of a cantilevered beam with a concentrated mass and moment of inertia at the tip were obtained by Durvasula [3]. Bhat et al. [4,5] treated the case of a tip body with distributed mass and inertia but restricted mass center offset. The present work is an extension of their treatment [4,5] in two respects. The base is not inertially fixed, but experiences a constant axially directed acceleration. In addition the mass center of the tip body is allowed to lie at a point off the beam tip tangent direction. With regard to buckling, Timoshenko [6] discussed the buckling of prismatic bars under the action of distributed axial loads. Specifically, the buckling of a bar under its own weight was treated. The problem of this paper exhibits the same internal tension profile but adds the complexity of a tip body which gives rise to several interesting effects. 2. PARTIAL In this governing

DIFFERENTIAL

EQUATION

section, the partial differential the planar transverse vibration

AND

BOUNDARY

CONDITIONS

equation and associated boundary conditions of a beam with tip body under the action of 43

0022-460X/85/050043 + 10 $03.00/O

@ 1985 Academic

Press Inc. (London)

Limited

.I

STORCH

? \

AND

5

GATES

VIBRATIONS OFACCELERATING BEAM

45

an axial force are derived. Figure 1 depicts the situation under consideration. At one end of the beam (x = 0) an axial force is applied imparting a velocity uO(t), along the x axis. The beam is assumed inextensible and of length 1. At the other end of the beam (x = I) a tip body is rigidly attached at point P. The rigid tip body has mass m and moment of inertia Z, about its mass center. The distance from P to the mass center of the tip body is c, and this directed line segment makes an angle, y, with the beam tip tangent at P. The base 0 of the beam moves along the x direction while the deformation u(x, t) occurs along the y direction. Note that the angle y is maintained constant. Dynamic equilibrium of a beam element yields (with rotary inertia and shear deformation neglected)

as/ax = p a2u/at2, aM/ax + S(X,t) - T(x, t) au/ax = 0,

aTlax

(192)

= pao( t),

(3)

where T, S and M denote the internal tension, shear force and bending moment respectively. The mass per unit length is p and a, is the x component of acceleration. Assuming an Euler-Bernoulli beam with bending stiffness El(x) one can write the partial differential equation for the beam deflection:

(a2/ax2)(Eza2u/ax2)- (a/ax)(T au/ax)+ p a2u/at2= 0. The translational equations one to write the first boundary mc cos y

*(I axat

of motion condition

’

for the tip body at x = 1, t)-mnoE(l,

I)-EZ$(l,

(4)

along with equation

t)+m$(l,

where EZ has been assumed uniform and non-linear terms Rotational motion of the tip body yields the second boundary

(3) allow

t)=O,

in u have been dropped. condition at x = 1:

(Z, + mc2 sin’ y) ~(Z,t)+cEZcosy~(Z,t)+EZ~(Z,r)-mcsinyn,(f)=O. ,~‘,ut

(6)

The partial differential equation governing u(x, t), equation (4), requires a specification of the internal tension T(x, t) for 0 < x G 1. Integrating equation ( 1) and using the x translational equation of the tip body as a boundary condition to specify T( Z, t) one obtains

3 r(x,t)=-[p(Z-x)+m]aO(t)+mcsin~--$$(Z,r). With a compresssive load PO(t) > 0 assumed from equation (7) that

(7)

to be applied

at x = 0 it follows

immediately

3

t),

Po(f)=(pl+m)a,(t)-mcsin~~(l, which relates the applied axial force to the base acceleration. motion can thus be written as

The governing

equation

of

(9) where the non-linear term [(du/dx)(x, t)][(a3u/dx is assumed to be cantilevered at x = 0 so that ~(0, t) = 0

and

(au/ax)(O,

at2)(1, t)] has been dropped.

t) = 0

for all t.

The beam

(10)

.I. STORCH

46

AND

S. <,Arl-.S

The partial differential equation (9) is to be solved subject to the geometric boundary conditions (10) and the natural boundary conditions of equations (5) and (6).

3. STEADY 3.1.

BUCKLING

OF CANTILEVERFD

STATE

BEAM

WITH

SOLUTIONS TIP BOLlY

UNDER

AXIAL.

‘THRCJST

(y

= 0)

Given a sufficiently large compressive axial force it is plausible that the beam in Figure 1 will buckle. To find the critical buckling loads one seeks solutions of the above boundary value problem which are independent of time. For the case y = 0 one requires y(x) f 0 such that

EZy”‘( 1) +

-p&l) =0,

(13)

(pl+m)Ezy”(l)-mcP,ry’(l)=O,

(14)

where PC, denotes those values of thrust which lead to buckling. One can immediately integrate equation ( 11) once and satisfy the boundary condition ( 13) by taking the constant of integration to be zero. Upon making the subsequent substitutions

Jgzw,

[~(P_y]“(,

z=

the differential equation transforms into d2w/dZ2+ conditions (12), and condition ( 14) require w=.

at

-25)

zw = 0, while the second

of boundary

z+)“3(!!y,

[(pl+ m)-w2”(KrP) "'~+mcP,,w=O

at

*=$[,,13;,,]“‘.

This homogeneous differential equation with the above homogeneous boundary conditions leads to an eigenvalue problem on P,, which is readily solved in terms of Bessel functions of the first kind (see reference [7] formulas 9.1.52 and 9.1.30). The critical buckling loads are determined by the transcendental equation J~,,3(~h)[~~J~2,3(~A)+f:~~~*hJ,,3(~A)] +J,,,(cwA)[&J,,,(pA)-:J’m*c*AJ_,,,(@)]=O. The corresponding

functions w

(15)

w are given by

=JZ[J,,~(LYA)J~,,~(~~~‘?)

(Of course these are determined the notation A = $J PJ2/ El, has been used. The dimensionless

-J~,:,(~~A)J,,~(~z~‘~)I.

only up to an arbitrary a=1+m*,

p = m*/JI

tip body parameters m* = mlpl,

multiplicative

c* = c/l.

(16) constant.)

+ I/m*,

Here

(17)

m* and c* are given by (18)

VIBRATIONS

OF

ACCELERATING

47

BEAM

Note that the critical buckling loads are functions of m* and c* and are independent of the tip body inertia. Figure 2 displays the variation of the first critical buckling value (A) m* and c*. The buckling with the parameters shapes can be obtained by integrating equation (16) and recalling that y(O) = 0. 1

I

I

1

1

1

I

I

I

I

/

1(

Figure 2. Parametric

For the simple

variation

of first critical

buckling

value with m* and c* ( y = 0)

case of no tip body (m* = 0), the buckling

pcvl = ix(EU 12), where j, is the nth root of the Bessel function PC,, = 7.8664( EI/12),

n=l,2,3

loads can be given directly:

,...,

of the first kind of order

l/3, viz.

PC,.2= 55.977( EI/l’).

As noted in the introduction, Timoshenko [6] treated the buckling of a vertical cantilever with free tip under the action of its own weight. The estimates of the first critical buckling load given there agree favorably with the value given above for the simple degenerate case. 3.2.

STEADY

STATE

SOLUTIONS

(y

f

0)

As will be seen, the permissible values of axial thrust for which a steady state solution exists in the case y # 0 are quite different from those of the previous section ( y = 0). In fact, no eigenvalue problem arises and the steady state solution, when it exists, is unique. The steady state solution y(x) must now satisfy equations (1 1)-( 13). Equation (14) is replaced by (pl+ m) Ely”( I) - mc cos yP,,y’( I) = mc sin yPc,..

(19)

.1. STOR(‘H

48 Making

the same transformations

AND

as before

y(x) =,

1

CrAlkS

one finds

i [c,J,,dl)+ I <,A

s = -( I//)(ih’/cr)“” The constants

S.

and

c,J , 7(t)l dr,

(20)

< -= cuA( I -s/cu~)~ ‘.

(21)

c, and c7 must satisfy the following

J,,daAk,

+J-,,,(d

= -Jm*c*[(sin

system:

1c.z= 0,

+[;JY m c * A cos YJJ,/~(~A)

m*c*A cos rJ,&3h)]c,

[JiJJZ13(@h)+;J

non-homogeneous

-~‘;J,,,(@h)]c:

-y)//3”‘](3A/2)“‘.

(22)

In general these equations will have a unique solution. If A (axial thrust) assumes a value such that the coefficient matrix in equations (22) is singular, it can be shown that the equations are inconsistent. Note that the equation obtained by setting the determinant I

I

I

I

I

I

I

I

I

I

I

1

I

I

I

I

I

12 -

I

I

x,/2

-

(x,+x2)/2

-

6-

0

g -75 z ‘;

I

II

I

III

I

III

1

I

III

I

II

40 20

0 l"""""""l""1

-16 (x3+x,)/2 0.00

13.40

26.80

40.20

53.60

67.00

80.40

93.80

107.20

120.60

134.00

x (ft) Figure 3. Steady state solutions when y = 45.0” for A lying between the first four critical values. Parameters: m*=2.778; c*=O.5; I= 134.Oft; y=45.0”. First four critical values of A: A, =0.89: A,=2.74; A,=4,81; h,=691.

VIBRATIONS

OF

ACCELERATING

49

BEAM

of the system (22) to zero is exactly the same as equation ( 15) except for the replacement c* + c* cos y. Hence the system can become singular for an infinite number of real values of A. Figure 3 illustrates the steady state solutions given by equation (20) at values of A lying between the first four critical values. Numerical experiments revealed that the overall shapes of these curves are insensitive to m*, c*, 1, and y within the respective ranges of A. It thus has been shown that for the case in which the mass center of the tip body lies off the beam tip tangent direction, a unique steady state solution to the vibration equation exists, except when the axial thrust assumes certain critical values (such that the linear system (22) becomes singular) for which no solution exists. 4. VIBRATION

4.1.

NATURAL

FREQUENCIES

AND

MODE

SOLUTIONS

SHAPES

(7

=

0)

The eigenvalue problem for the transverse vibration of a cantilevered body under the action of axial thrust can be formulated as follows. The is assumed constant and y = 0. Using equations (5), (6) and (8)-( 10) one has a homogeneous boundary on u(x, t). Assuming a solution of the form e”“‘4(x) yields an eigenvalue with eigenvalues w2. This eigenvalue problem is conveniently scaled by the 5 = x/ I and y( 5) = C$(15). Appropriate dimensionless parameters are a2 = Po12/EI,

b4 = (p14/ EI)w’,

problem

value problem problem on 4 transformation

m* = m/pi,

J* = (I, + mc2)/p13.

C*=C/l,

The eigenvalue

beam with tip axial thrust PO

then transforms

(23)

into

(24)

‘(l)+m*b4y(l)=0,

(27) Here (‘) denotes differentiation with respect to 5. The general solution of equation (24) can be written linearly independent solutions, Y(5)

For convenience, are specified as

the initial yj’-“(0)

= ,i,

combination

of four

ciYi(5)

values of these four functions

= 6,,

as a linear

and their first three derivatives

i, j = 1,2, 3,4 (~5,~ = Kronecker

delta).

Applying the geometric boundary conditions (25) one finds c, = 0 and c2 = 0. Application of the natural boundary conditions (26) and (27) yields the homogeneous system of

50

I

SIORC’H

ANI)

5

(,AllS

(28) (i = I, 2). In order for this system to have a non-trivial

solution

one requires

~(w~)~u,,a,,-u,~u~,-o,

(29)

which determines the natural frequencies w,. It still remains to find the functions y, and y4. A purely numerical technique which is easy to apply consists of integrating two initial value problems and iterating on w until values are found which satisfy equation (29). The well-known natural frequencies of a cantilevered-free beam serve to bracket the frequencies in the present case of a cantilevered beam with a tip body on an accelerating base. In what follows here the problem is treated analytically by solving the differential equation (24) in the form of a power series, thus obtaining the eigenfunctions in functional form. Note that for P,= 0, the differential equation has constant coefficients and can therefore be solved in closed form in terms of elementary functions (cf. [8]). For the case PC,f 0, the equation is linear with analytic coefficients and has a solution which is regular at 5 = 0. The series representing the solution will converge for all 5 (see reference [9]). Of course one need only be concerned with convergence on 0 S 5 C I. If one assumes a solution of equation (24) in the form

and inserts this expansion and its corresponding derivatives obtains the following recursion formula for the coefficients: A h+J=

b“A,+[u’/(l+m*)](k+1)~A,+,-~‘(k+I)(k+2)A~+~ (k+l)(k+2)(k+3)(k+4)

into equation

(24), then one

k=0,1,2

,....

(30)

These conditions determine A,, A,, A,, . . once values for A,, A,, AI and A, have been prescribed. To generate yi one must have A,, = 0, A, = 0, A, = 1. A, = 0. To generate y, one must have A0 = 0, A, = 0, A2 = 0, A, = k. Hence the eigenfunction Y,(t) corresponding to the eigenvalue w, is given by

.

I Given the infinite series I,‘_,, A&‘, then by the ratio test the series converges for those values of 5 for which

(31) (absolutely)

VIBRATIONS

OF ACCELERATING

BEAM

51

Dividing both sides of the recursion formula (30) by Ak and taking the limit (assuming limk_co IAk+,/Akl exists) one finds limk,,lAk+,/Akl = 0. H ence the series representing the solution of the differential equation (24) converges for all 5 in agreement with the general theory. 4.2.

FREE

VIBRATION

(y # 0)

The free vibrational response of the accelerating beam with y Z 0 is now investigated. By “free” one means that no external forces are acting other than PO which is producing the axial acceleration. It is important to note that in all cases considered thus far, y = 0 or steady state response, a constant axial force implied a constant base acceleration. In the present case this is no longer true (see equation (8)). It is assumed here that a base force PO(t) is applied such that the base acceleration a, is constant. With non-linear terms in u neglected, the equation of motion is

1

4

&-$+a,;

(pz+m-px)g

2

+,$=o.

[

(32)

The boundary conditions are given by equations (5), (6) and ( 10). The boundary condition (6) is inhomogeneous. The solution u may be written as u(x, t) = U(X, t) +f(x), withf(x) chosen so that the boundary conditions on v are rendered homogeneous. One then has the following requirements on f(x): f(0) = 0,

f’(0) = 0,

Elf”‘(I)

+ ma,f(

I) = 0,

c EI cos rf”‘( I) + Elf”(Z) = mc sin ya,. If, in addition

to satisfying

the conditions

(33), one requries

(33)

that f(x)

satisfies

EZ$+u,-& (pZ+m-px)+i =O, [ 1 then the differential homogeneous. The conditions equation (32) with 3.2 that f(x) exists

equation

on v(x, t), as well as the boundary

(34) conditions,

are rendered

on f(x) are precisely those governing the steady state solution of the boundary conditions (5), (6) and (10). It was shown in section for all values of a,, except when the system (22) becomes singular.

f(x) is given by _equation (20) with A = :J( 12/El)(pl+ m)a,. It is seen that V(X, t) is governed by the same equations previously encountered when considering the natural frequencies in the case y = 0, with the formal replacements, c+ccos ‘y, Z,+I,+mc2sinzy and P,+aO(pl+m); thus u(x, t) = F (Ak cos wkf+ Bk sin C+J) Yk(t), k=l

where the frequencies wk are solutions of equation (29) with c* + c* cos ‘y, PO+ a&l+ m), and the functions Yk([) are given by equation (31). All frequencies are positive for sufficiently small values of a,. When a, assumes a value such that equation (15) is satisfied (with c* + c* cos y), a frequency wk goes to zero. This corresponds to the situation wheref(x) fails to exist.

5. NUMERICAL

RESULTS-NATURAL

FREQUENCIES

Table 1 gives the squares of the first three natural frequencies of a cantilevered with tip body under constant axial acceleration (y=O). The base acceleration

beam ranges

1 STORCH

52

TABLE Squares

S (,AThS

AND

1

of the first three natural frequencies oj‘ a cantilevered constant axial acceleration ( y = 0)

0.0 3.3470 6.6941 10.0411 13.3881 16.7352 20.0822 23.4292 26.7762 30.1233 33.4703

Power series

RayleighRitz 0.8556 0.7705 0.6852 0.5999 0.5145 0.4290 0.3434 0.2577 0.1719 0.0860 0.000 1

RayleighRitz

0.8552 0.7701 0.6849 0.5996 0.5142 0.4287 0.3432 0.2575 0.1718 0.0859 0~0000

_

Beam parameters: EI = 2,582 937 x 10’ N m’; p = 19.97716 l.O= m/p/; J*=O.1025 = (I,+ mc2)/pl”; ~~0.05 = c/l.

kg/m:

I = 40.8432

Power series

RayleighRitz

Power series

304.97 303.23 301.49 299.75 298.0 I 296.26 294.52 292.78 291.04 289.30 287.56

19.442 19.244 19.045 18.845 18.644 18.443 18.240 18.037 17.833 17.628 17.423

20.007 19.799 19.591 19.382 19.172 18.962 18.750 18.538 18.325 18.112 17.897

with tip body under

Third natural frequency (rad/s)’

Second natural frequency (rad/s)’

First natural frequency (rad/s)’ Base acceleration (m/s2)

beam

m.

Tip

292.36 290.67 288.98 287.28 285.59 283.89 282.19 280.50 278.80 277. I 1 275.41

body parameters:

m* =

between zero and the first critical buckling value (as determined by equation (15)). Two methods have been used. The first is a Rayleigh-Ritz procedure with use of the first 40 eigenfunctions of a clamped-free beam. The second method is to use the analytical solution of section 4.1, with equation (29) solved by a bisection algorithm. Six terms are used in each power series. Note that the results given for zero base acceleration are in agreement with those obtainable from reference [4].

REFERENCES 1. P.

SEIDE

longitudinal

1963 Aerospace Corporation Report TDI?- 169 (3560-3O)TN-6.

acceleration

Effect

of constant

on the transverse vibration of uniform beams.

2. T. R. BEAL 1965 American Institute of Aeronautics and Astronautics Journal 3,486-494. 3.

4. 5. 6. 7. 8. 9.

Dynamic stability of a flexible missile under constant and pulsating thrusts. S. DURVASULA 1965 Department of Aeronautical Engineering, Indian Institute of Science, Bangalore Report AE 1338. Vibrations of a uniform cantilever beam carrying a concentrated mass and moment of inertia at the tip. B. RAMA BHAT and H. WAGNER 1976 Journal of Sound and Vibration 45, 304-307. Natural frequencies of a uniform cantilever with a tip mass slender in the axial direction. B. RAMA BHAT and M. AVINASH KULKARNI 1976 American Institute of Aeronautics and Astronautics Journal 14, 536-537. Natural frequencies of a cantilever with slender tip mass. S. P. TIMOSHENKO 1936 Theory of Elastic Stability. New York: McGraw-Hill. See p. 115. M. ABRAMOWITZ and I. A. STEGUN 1968 Handbook of Mathematical Functions. Washington, D.C.: National Bureau of Standards. J. STORCH and S. GATES 1983 Charles Stark Draper Laboratory Report CSDL-R- 1629. Planar dynamics of a uniform beam with rigid bodies affixed to the ends. E. L. INCE 1956 Ordinary Differential Equations. New York: Dover.