The vibration characteristics of a beam with an axial force

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Journal of Sound and Vibration (1978) 59(2), 237-244

THE VIBRATION CHARACTERISTICS OF A BEAM WITH

AN AXIAL FORCE R. E. D. BISHOP AND W. G. PRICE

Department of Mechanical Engineering, University College London, London WC1 E 7JE, England

(Received 30 December 1977, and in revisedform 27 February 1978)

Equations of motion are found for a non-uniform damped Timoshenko beam with a distributed axial force. Principal modes may be extracted by numerical means when the boundary conditions are specified, and the appropriate orthogonality conditions are given. The theory of linear forced vibration can thus be derived. It is an implicit requirement that all axial forces are conservative. That is to say, tangential, follower and partial follower axial forces (whether applied at an extremity or distrubuted along the beam) are excluded.

1. INTRODUCTION

The literature on the transverse vibration of beams is enormous, but only a comparatively small proportion of it admits axial tension or compression. A number of writers examine approximate methods, mostly with the lowest mode of vibration only in view; e.g., see the books by K&-man and Biot [I], Harris and Crede [2] and Timoshenko and Gere [3]. Although some results have appeared referring to centrifugal loading of helicopter blades, aeroplane propellers and turbine blades, the writers know of no general treatment of this subject. On the other hand many papers have appeared in which axial forces are ignored, as for instance with the oscillation of chimneys and other tall structures in which the axial loading is due to gravity.

The purpose of this paper is to fill this evident gap in the literature and thus to provide a means of checking the effects of axial forces. Within the limitations of linear theory, the treatment is of some generality, admitting the effects of rotary inertia, shear deflection, various combinations of ideal boundary conditions and internal damping of shear and bending. Specifically, however, this paper admits only conservative axial loading; it does not cater for tangential, follower or partial follower axial forces.

2. EQUATIONS OF MOTION

Consider axes Oxyz placed as shown in Figure 1 with the plane Oxy horizontal. A non- uniform beam, whose boundary conditions we shall leave open for the moment, occupies the region 0 < z < I when it is in its equilibrium condition. The lateral displacement of the beam at any section will be denoted by u(z, t) and will be assumed to occur in the plane Oxz. This displacement may occur in free vibration or be brought about by a transverse load X(z, t) per unit length.

Consider the motion of an element of the beam, between sections “1” and “2”, whose thickness is AZ. This element suffers a lateral displacement of its centre of mass C, the magni- tude of which is u(z, t) as shown in Figure 2(a). The element also suffers two types of distortion and it is convenient to examine these separately.

237

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238 R. E. D. BISHOP AND W. G. PRICE

I-

1

,,A3

0

,,,,,

x

Figure 1. The beam column in its equilibrium configuration. The structure may be subjected to distributed axial (vertical) forces and transverse (horizontal) forces.

A shear strain y(z, t) is imparted to the element by the shearing forces V(z, t). This is illustrated in Figure 2(b). It will be seen that, while the axial tensions Tl and Tz remain parallel to Oz, they now tend to rotate the element about an axis which passes through C and is parallel to Oy. The bending moment M(z, t) rotates the faces of this element through slightly different angles f?(z, t), as shown in Figure 2(c).

Motion of the element parallel to Ox is governed by the equation P(Z) dzii(z, t) = X(z, t) AZ + (V, + Tl e,) - (V, + T2 e,),

where p(z) is the local mass per unit axial length. If we divide by AZ and then let AZ tend to zero, we find that

arqz,t) a

p(z)

qz,

t) = X(z, 2) + -7jy + $‘(z, r) e(z,

t>l,

or, with obvious abbreviations,

,~cii = X+ V’ + (Te)‘. ₍₁₎

A prime represents differentiation with respect to z and an overdot indicates differentiation with respect to t.

When discussing rotation of the element, it is convenient to refer to Figure 2(d). Moments taken about an axis parallel to Oy and passing through C give

Z,(z) AZ&Z, t) = Ml - Mz - (T, - I’, e,)$(y + 8,)

-(T2- v,tl*)$(Y+ e,)+(T,e, + W$+(T,B, + vg, where Z,,(z) is the moment of inertia per unit length. It follows that

and, since

z,B=iw-T(y+e)+Te+

v

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VIBRATION OF A BEAM WITH AXIAL FORCES 239

L-_-

0 x (b)

L--_

L---

0 Y 0 x (c) id)

Figure 2. An element of the beam bounded by faces 1 and 2. The deformation consists of (a) a displacement, (b) shearing and (c) bending. Diagram (d) shows the actions on the slice.

this becomes

I,,~=M’-Tu’+TO+ V. (.I!)

The equation of motion parallel to Oz is essentially one of static equilibrium since it is assumed that, to the first order, the element has no acceleration in the axial direction. That

is to say

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240 R. E. D. BISHOP AND W. G. PRICE

where Z(z, t) is the upward force per unit length. From this equation it is found that, to the first order,

T’+Z=O. ₍₄₎

The foregoing equations of motion must be supplemented by shearing force and bending moment relations. These are

V(z, t) = kAG(z) [yb t) + a(z) $(z, r)l, (5) MC-5 t) = Wz) [W, f> + B(z) &z, t>l, (6) w’.ere a(z) and /I(z) represent the distributed structural damping and kAG(z), EZ(z) are the sl ear and flexural rigidity, respectively.

3. SOME COMMENTS ON THE EQUATIONS OF MOTION

The relationships (l)-(6) are the equations of motion. They represent an adaptation of the Timoshenko beam theory but it has to be accepted that the present use of that theory introduces an added element of arbitrariness. The basic assumptions of beam theory have not been discarded-e.g., plane sections are still assumed to remain plane-and we have made further demands. At best we can achieve only a logical development of this semi- empirical theory, such as it is.

Under the circumstances it is as well to check that the foregoing equations conform to simpler existing theory for special cases.

(i) If Z(z, t) = 0, T is constant and so equations (l)-(3) become

@=X+ I”+ T6”, u’ = 8 + y, I,t?=M’-Tu’+TB+ V. (7)

When used with equations (5) and (6), these equations govern the motion of a damped Timoshenko beam under simple tension or compression.

(ii) If T(z, t) = 0, equation (7) further reduces to

@= X+ V’, UT =

e + 7,

r,e=w+

L ₍₈₎

With equations (5) and (6), these equations govern the motion of a damped Timo- shenko beam without axial loading.

(iii) If m(z) = 0 = p(z) the equations govern the motion of undamped beams.

(iv) If, in addition, Z,, = 0, y(z, t) -+ 0 and kAG(z) 4 m, equations (6) and (8) reduce further to

pti=

x+

V’, O=M’+

v,

M(z, f) = E/(z)

uyz,

t). (9)

These results form the basis of the Bernoulli-Euler beam theory.

(v) If Z(z) = 0 as in (i) and, in addition, El(z) = 0 = kAG(z) = y(z, t) = Z,(z), equations (7) reduce further to

pii = X+ Tu”. ₍₁₀₎

This is the equation of motion of a taut string.

(vi) If T(z,t) = T(t), and the conditions appropriate to the Bernoulli-Euler beam theory as stated in (iv) are valid, equations (7) reduce to

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VIBRATION OF A BEAM WITH AXIAL FORCES 241

Bolotin [4] discusses these equations in great detail in the dynamic stability of elastic systems, as, for example, when a simply supported beam is loaded by a periodic longitudinal force of the form T(t) = To + Tcoswt.

4. PRINCIPAL MODES

If all structural damping is ignored and there is no lateral excitation X(z, t), free vibration is possible in the principal modes. The equations of motion are found from equations (l)-(6):

pii = V’ + (To)‘, u’ = e + y, I,di= M’-Tu’+ TO+ V,

T’ + Z = 0, V = kAGy, M = EI%‘. (12)

Motion in the rth principal mode, performed at the rth natural frequency w,, will be of the form

u(z, t) = u,(z) sin 0, t, &z, t) = 8,(z) sin w, t, The first of equations (12) shows that

y(z, t) = r&5-) sin w, t. (13)

-w; pu, = V: + (To,)‘, (14)

where T is independent of time, V,(z) being the distribution of shearing force in the rth mode. Multiply this result throughout by u,(z) and integrate with respect to z over the length of the beam; it is then found that

1 1 1

-of

i pu, u,dz= s V: u, dz f I (To,)’ u, dz.

0 0 0

On integrating the terms on the right-hand side by parts we find that

-0: jpu,u,dr= ,V,u,,:- jV,u:dz+ ire,u,[,$-jT6’,u;dz.

0 0 0

The integrated termSin this last equation vanish provided that, at both ends of the beam, (a) shearing force or displacement vanish, and (b) tension, slope due to bending or dis placement vanish. These conditions are met at a pinned support (u, = 0), a clamped support (u, = 0), a free end with zero tension (V, = 0 = T), a sliding end with zero tension (V, = 0 = T). Under these conditions,

1 1 1

w:

i PU, u, dz = I V, u:dz + s Ttl, u; dz.

0 0 0

This must remain true if the subscripts are reversed, and so

1 1 1

03

I pu, us dz = s V, u: dz + s TB, u:dz.

0 0 0

Subtraction thus reveals that

ku:-o,j 1 1

jm, u, dz =

I (V, u; - V, u;) dz + I T(B, u: - 8, u;) dz.

0 0 0

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242 R. E. D. BISHOP AND W. G. PRICE

Turning now to the third of equations (12), we see that -of I, 0, = M,! - Tu: + Te, + V,.

Multiply by 13, and integrate as before; this gives

-o+,J,&dz= IM.O,l:,-jM~O:dz-JTu:B,dz+jT~,8,dz+jV~0,dz.

0 0 0 0 0

The integrated term now requires that both ends of the beam shall have either zero bending moment or zero slope due to bending. This condition is met by a pinned end or a free end

(Mr = 0) and by a clamped support or a sliding end (0, = 0) and so these requirements place no further restrictions on the theory and we may ignore the integrated term.

By reversal of the subscripts we find that

1 1 1 1 1

-of I,B,t?,dz=- M,t?;dz-

I I I Tu,‘B,dz+ I Tt$t&dz+ I V,t$dz.

0 0 0 0 0

If this equation is now subtracted from the last one and it is remembered that M, = El(z)&,

it is found that

1 1 1

(wf - CD:)

I Z, 19,0, dz = I T(u; 0, - u; 0,) dz + I ( V, 0, - V, f?,) dz. (16)

0 0 0

Equations (15) and (16) provide orthogonality conditions for the principal modes. By adding them together we find that

1 1

(4 - 4) (,w~s + h&Qdz =

I I

(V,Y, - vsyr)dz

0 0

and, since V, = kAGy,, the integral on the right-hand side vanishes. We may thus write

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where a,, is the Kronecker delta function.

A second orthogonality condition is found by returning to the equations containing only of and adding them together. This gives

which simplifies to I

I [ER?; 0; + T(u; 8, + u; 8, - S, 0,) + kAGy, yJ dz = of a,, 6,,, (18)

0

where T is a function of z alone.

It will be understood that we have chosen to define T(z) as positive when tensile. Thus if T(z) represents the excess of buoyancy over weight in a marine structure (such as a single point mooring of the tower buoy type), then

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VIBRATION OF A BEAM WITH AXIAL FORCES 243 where Z,(z) is the buoyancy force per unit length of the column. But if, due to the fluid particle velocity in waves, there is in addition a time dependent component of Z(z), equal to Z,(z,f) say, this will produce a time dependent component of the axial tension. To the first order, we then have

W,

t) = j

P=,(q) - dq)gl dq

-+dq,

1) dq.

2 L

The first component is independent of time and partially determines the principal modes as before; the second integral represents an “excitation” and plays a part in producing a forced oscillation.

5. FORCED OSCILLATION

Once the principal modes have been found, by adaptation of the Prohl-Myklestad method or otherwise, and their orthogonality established, a modal analysis of general forced oscillation becomes possible. The approach is quite well known and we need do no more, here, than briefly outline the method.

Horizontal distortion of the beam shown in Figure 1 in the Oxz plane may be represented as the sum of distortions in its principal modes. That is, we may write

u(z. f) =

,$

U,(z)Pr(t),

ez,

r>

=

$

U,(z)p,(t), Y(Z, t) = t L(Z)Pr(f),

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r=O r=O r-0

where p,(t) is the rth principal co-ordinate which can conveniently be scaled so that unit deflection at the end z = I corresponds to p? = 1. (We shall disregard artificial problems raised by rigid body modes in this context.)

The modal index r can conveniently be identified with the number ofnodes r in the principal modes; in particular the lowest mode of a clamped-free or a pinned-free beam will have the index r = 0 and in neither case does this imply that o. = 0. Notice that some scaling other than the type mentioned will be needed if the end z = I does not move laterally; we shall here assume that such motion is possible.

Substitution of the equations (21) into equations (1) and (5) shows that

p

2 u,p, -

kAG

5 [yrpp

+

ayr/$]'

-

5

(TU,)‘p, = X(z, 1)

?=O r-0 r=O

If the equation is multiplied by u,(z) and integrated with respect to z it is found that

joj.jpu. u, dz - ?0 p,lkAGy, 4:, + ?zO p,/kAGy, 4 dz - t

r-0 d,lkAGay, 4b 0 0 + jjjkAGay,u:dz - 5 r-0 Pr17-K41:, +?_ &/TO 1 ,u:dz= X(z,r)~,(z)dz. i 0 0 0

Under conditions that are quite commonly met, the integrated terms all vanish. Suppose, for instance, that the beam represents a chimney; we then have u,(O) = 0 = V,.(I) = T(I). We are thus left with

I

$

ii,(llu,u,dz+ z pI ‘(kAGy?+ T&)(0,+ ys)dz

r=O 0 r=O I

0

1 I

+ ~~o+~G~~,(A,f ys)dz = j-X(2, t) u,@)dz,

0 0

where the result (2) has been used.

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244 R. E. D. BISHOP AND W. G. PRICE

Similarly if equations (21) are substituted into equation (3) and the result of so doing is multiplied throughout by 0,(z) and integrated, it is found that

(23) Again the integrated terms can usually be left out; thus, for the chimney, Q,(O) = 0 = Z%;(l).

When equation (23), without its integrated terms, is added to equation (22), it is found that

(s=O,1,2 ,... ), where

I u,, =

I akAGy, ys dz, /5&s = j$?Zf% 0,’ dz. (25)

0 0

Equation (24) is, of course, very familiar in linear dynamics and standard matrix techniques are readily available for its solution. As usual, the orthogonality conditions play a crucial role in arrival at the equations of motion (24).

The quantity on the right-hand side of equation (24) is the generalized applied force at the 8th principal co-ordinate ps. In particular, if X(z, t) is constant (or contains a constant component) such that X(z,t) = r(z), then the corresponding modal deflection will be constant so that p,(t) = j$. With these values we find that

1 jj,=

1

2

I

%,<z,

WI dz, s = 2,3, ,... Q, ass 0 (26) 6. CONCLUSIONS

A linear theory of beam-column vibration has been presented, it is believed for the first time. It is well known that difficult stability problems can arise from the static loading of such structures when there is an end loading or distributed axial loading which includes a follower force: e.g., see the book by Bolotin 141. The treatment presented here does not enter that particular field of speculation, however.

Within its limitations, the theory that has been given is complete and forced motions can be investigated. This is made possible by the orthogonality relationships that are established.

REFERENCES

T. VON BARMAN and M. A. BIOT 1940 Mathematical Methods in Engineering. New York: Mc- Graw-Hill.

C. M. HARRIS and C. E. CREDE 1961 Shock and Vibration Handbook. New York: McGraw-Hill. See Chapter 7, p. 18.

S. P. TIMOSHENKO and J. M. GERE 1961 Theory of Elastic Stability. New York: McGraw-Hill, second edition.