Closed-form solution for the Timoshenko beam theory using a computer-based mathematical package

-

Conwrcrs

1

Pergamon

0045-7949(94loo475-7 . _

& Srrumws Vol. 55. No. 3. pp. 40542. 1995 Copyright J: 1994 Elkier Science Ltd Printed in Great Britain. All rkhts reserved w5-7949195-39.50 + 0.00

CLOSED-FORM SOLUTION FOR THE TIMOSHENKO THEORY USING A COMPUTER-BASED MATHEMATICAL PACKAGE

BEAM

A. M. Horr and L. C. Schmidt Department

of Civil and Mining

Engineering, University N.S.W., Australia

(Received 24 December

of Wollongong,

Wollongong,

1993)

Abstract-There has been considerable

research interest in applying Timoshenko beam theory to the transient response of beams as well as for free and forced vibration. Conventional finite elements treat the dynamic load induced by the mass and rotary inertia of the beam as concentrated loads and moments applied at the ends of the element. In many structures the structural joints may be far apart, and therefore many elements must be used if the inertia distributions are to be modelled accurately. The purpose of this study is to determine the influence of distributed rotary inertia and shear deformation on the motion of a mass-loaded clamped-free Timoshenko beam by means of an exact solution. The goberning differential equations are solved, and the frequency results are presented graphically and are compared with those derived for a Euler-Bernoulli beam. Mathematics. which is a computer-based mathematical package, has been used to solve the frequency equation.

NOTATION

A E

Mh M

5

cross-sectional area of beam Young’s modulus of beam material frequency beam depth shear modulus second moment of area of beam moment of inertia of the rigid body-tip cross-sectional shape factor length of the beam mass of the beam mass at the free end time eigenvalue density natural frequency

shear deformation. Thus, we have two independent motions, v(x, 1) and 0(x, t). EULER-BERNOULLI

THEORY

Consider the flexural vibration beam with an end mass, as shown Euler-Bernoulli theory for flexural perfectly elastic undamped beam kinetic and strain energy formulation

mass

pAJ* dx

s

of clamped-free in Fig. 1. The vibration of the starts with the as follows:

(kinetic energy)

(I)

(strain energy).

(2)

L

INTRODUCITON

EIy”* dx

0

The Bernoulli-Euler theory of flexural motion of elastic beams has been found to be inadequate for the prediction of higher modes of vibration, and also inadequate for those beams when the effect of crosssectional dimensions on frequencies cannot be neglected. Timoshenko beam theory takes into account the effects of rotary inertia and shear deformations during vibration of a beam, as it is easy to see that during vibration a typical element of a beam performs not only a translatory motion, but also rotates. With the introduction of shear deformation, the assumption of the elementary theory that plane sections remain plane is no longer valid. Consequently, the angle of rotation, which is equal to the slope of the deflection curve, is not simply obtained by differentiating the transverse displacement owing to the

Note that the effects of shear deformation and rotary inertia have been neglected in the calculation of the kinetic and strain energy. Using Hamilton’s principle,

Considering the beam with one end fixed, the other end free, the boundary conditions are:

405

v(0) = 0

v”(l) = 0

(4)

v’(0) = 0

V”‘(I) = 0.

(5)

A. M. Horr and L. C. Schmidt

406 Defining

the variable

a as:

(6) the differential

eqn (3) can be written

vided by nodal cross-sections into comparatively short length portions. Timoshenko [I] gave the equation of motion for the clamped free beam with an end mass as:

as:

(f)$KAG($-;)=O

(13)

(7) E,$+KAG(; For this case, the frequencies of transverse are solutions of the eigenvalue equation,

-O)-(F)$=O.

vtbratton

cos /I cash /I = - I

(8)

where K is a factor depending on the shape of the cross-section and G is the modulus of shear rigidity. The boundary conditions of the beam are:

where

\‘(O)=O

Note that the frequencies are independent of the shear modulus, as expected, as it is neglected already. When the beam carries an end-mass M at the free end, the eigenvalues are solutions of: M -zz pAL

(14)

.

.,,(,

-F)=Mv

d%(L) -I,--_, St-

8(L) EIp= ax

Q(O)=0

Huang [2] gave equation as:

the

(15)

solution

of

the

(16) differential

I + cos fi cash fi jI(sin /I cash /I - cos fl sinh p).

(IO) Using Huang’s

It is clear that when M/PAL, becomes large, the beam with an end-mass behaves like a single degree of freedom system with a natural frequency w = Vm when k = (3EI)/I’. For completeness, the solution of eqn (7) for the transverse vibration of an undamped system can be in the form of: v = X(A cos cot + B sin cot),

(11)

non-dimensional

the boundary

‘y = c”’

A and B are arbitrary

can be rewritten

(x = 0)

(12)

V” =

as:

0

a, = 0

constants <=I

(19)

l do, / Ah? 2MK’ o,, = 0

(x=L)

L d[

TIMOSHENKO THEORY

In the previous section it was assumed that the cross-sectional dimensions of the beam were small in comparison with its length (neglecting shear and rotary inertia effects). Corrections to the theory have been given in Timoshenko for the purpose of taking into account the effects of the cross-sectional dimensions on the frequency. With the introduction of shear deformation, the assumption of the EulerBernoulli theory that plane sections remain plane is no longer valid. It means that the slope 0 of any section along the length of the beam simply cannot be obtained by differentiation of the transverse displacement V. Consequently, there are two independent motions 0, 1’. These corrections may be of considerable importance for studying the modes of vibration of higher frequencies when a vibrating beam is subdi-

(18)

KAGL’

conditions 5= 0

as:

EI

sz=_

AL’

where

and coefficients

I

y?=_

variables

ldv

__A

(j

’

Ldc

2

pAL-

M +_&h pAL

=o,

L

(20) Omitting re-written

the factor as:

en”‘, eqns (13) and (14) can be

$1;;’+ b’(r’ + .s’)v{ - by I - b’/+‘)v,,

0;; + b’(r’ + s’)Oi - h’(l - b’r’s’)0, The solutions

= 0

(21)

= 0.

(22)

of eqns (21) and (22) are:

r,, = c, cash br[ + c2 sinh br[ + c1 cos bfl[ + c4 sin b/l<

(23)

Closed-form

for the Timoshenko

407

beam theory

Mg

L

I

solution

\

Fig. I. Clamped-free

beam with an end-mass.

0, = c; sinh bar + ci cash ba[ + c; sin bB[

+ c; cos b/35,

(24)

L/H=8

Figure 2 First and Second Nat. Freq. for T. Beam

where Fig.

c; are constants. For the clampedand c’, . . car c; . free beam, there are two branches: (r’ - s2)’ +

>(r2 + s2) $ Ir2

2. First

I,2 1<(9+s2).(26)

For the first branch which considers the ratio of the shear stiffness GAK to the rotary inertia pl in the frequency equation, it can be seen that: L ~,=~[I+b~~~(a~+r~)Jc;

cz =;

cj = -@

[I - b’s’(sc’+ L

r2)]c;

[I + b2s2(p2 - r2)]c;

cd = k [I + b2s’(B2 - r’)]c;. bb’

second natural frequencies shenko beam.

(24) into the boundary the variables

I

(r2 - .Y’)~+ 5

and

for

Timo-

eqns (19) and (20) and using

a2 + s2 n=-

m=--

s2- p2 B

a

lb4 MI PZ=gpZA’

Mb2 q=pAL

(32)

k, = q cash ba + b sinh ba a

(27)

kz = q sinh ba + b cash ba a

(28)

k,=qcosbp+isinbp

(2%

k,=qsinbfl-icosb/l

(30)

k, = n(br cash ba -p

sinh ba)

k, = n(ba sinh ba - p cash ba) If the second branch lation can be used:

assumed,

the following

formuk,=n(bbcosbb-psinbfi) k,=n(b/lsinbp+pcosbb),

(33)

(31) The value of s( can then be substituted into eqns (23) and (24). Solutions of eqns (23) and (24) are the solution of the original coupled eqns (13) and (14). For simplicity, in this work we will follow the first branch. Substituting the solution of eqns (23) and

Table

E KM 200

f

4 3

first not. /

2

Ii

1

freq.

/

I

G [@aI

L [mm1

116

400

A [mm’]

I0,000

I [mm41 2.08 x 1Oh

L/H=8 Fig. 3. First natural

frequencies

for E-B beam.

408

A. M. Herr

and L. C. Schmidt

the following

equations

can be derived: (‘, + (‘j = 0

k, c~,+ k2cz + k,c, + k,c, = 0 k,c, + k,,cz + k,c, + k,c, = 0. Fig. 4. Second

natural

frequencies

(34)

for E-B beam

Consider the coefficients of the four equations as matrix C. In order that solutions other than zero may exist, the determinant of the C matrix must be equal to zero. This leads to the frequency equation 1.5

ICI

= O+m(kzk;

- k,k, + k,k, - k2k,)

1. +n(k,k,-k,k,+k,k,-k,k,)=O

5.

(35)

or ,j’(o. E, G, GEOMETRY)

Fig. 5. Fourth

and higher natural frequencies shenko beam

for Timo-

Solving for W. the frequency mode can be derived.

= 0.

of vibration

(36)

for each

NUMERICAL RESULTS AND DISCUSSION

-0

30

5

5

w

-1 -1

5

-2 I -2 5

-

-.

-3 -3

5I L/H=??

Fig. 6. Second and higher natural

frequency

for E-B beam.

Fig. 7. The effect of normalized

In this study. the clamped free beam carrying a spherical shaped mass at the free end is considered (Fig. I). It is made up of either an isotropic material or a transversely isotropic material with the axis of isotropy coinciding with the beam centroidal axis. For the isotropic case. there are two elastic constants that govern the beam response: E and G. Solving the frequency equation for the numerical data given in Table I. and using the Mathematics computer package. the frequency function can be plotted against the frequency. Appendix A shows Mathematics’s log file which can be used to solve the frequency function.

mass on frcquencq

equation

of Tlmwhcnko

beam

Closed-form

solution

for the Timoshenko

beam theory

409

Fig. 8. The effect of normalized mass on frequency equation of Timoshenko beam over a very wide range

Considering the spherical-shaped normalized mass, c, can be assumed

end-mass, as:

the

(37) The normalized end-rotary inertia, t, can be assumed as the rotary inertia of the sphere about x = L divided by the rotary inertia of the beam about x = 0:

(! MR2) where His the depth of the beam and R is the radius of gyration for the end-mass.

Fig. 9. The eflect of normalized

For the brevity, the details of the long and tedious calculations are omitted. However, some of the interesting numerical results of these calculations are illustrated in Figs 2-6, where the frequency equation fis plotted against w for the case where 0 = 2, and t =O.l. The first six natural frequencies of a Timoshenko and an Euler-Bernoulli beam, clamped at one end and carrying an end-mass at the free end, have been determined (Figs 2-6). The natural frequency decreases as the radius of the sphere shape mass increases (Figs 9 and 10). For the first natural frequency, as the tip mass increases, we have a degenerated mode (w = 0), as expected (Fig. 10). However, for the third mode it is observed that, with increasing tip mass, the natural

mass on frequency

equation

of E B beam

A. M. Horr

410

10

0

20

30

and L. C. Schmidt

40

60

50

70

90

80

100

Normalised Mass o Fig. IO. First natural

frequency

against

normalized

frequency converges to the second fixed-free mode, and the fourth mode converges to the third (Figs I1 and 12). Figures 7 and 8 show a three-dimensional plot of normalized mass, natural frequency and frequency function. By comparing Figs 8 and 9, it can be seen that for Timoshenko theory, with increasing normalized mass, the frequency function shows an unexpected behaviour. We expect the same plot with shifting to the left for all f-o planes, as in the Euler-Bernoulli theory (Fig. 9), while in the Timoshenko beam theory (Fig. 8), with increasing normalized mass, the slope of the frequency function

mass for Timoshenko

beam

changes so that the third and fourth natural frequencies approach to the second and third natural frequencies in the zero normalized mass plane. Note that in Fig. 8, for the purpose of showing this behaviour, the frequency function is plotted over a very wide range. By looking at the two frequency functions for the two theories, it can be seen that the normalized mass is an individual term in the E-B frequency function. while in the Timoshenko frequency function some major terms are multiplied by the normalized mass. The difference between the two models is due to the shear and inertia effects of the Timoshenko beam model.

20 + I 15

i

3rd Nat. Freq for 1

wth hp rnos~

I 10

c . /

_ 2nd Nat

Freq. for Clamped-freei.

8. wIthout tip mass

5+

0

Fig.

I I. Natural

2 Nommlised

I

05

0

frequency

against

normalized

5 Mass D

mass for Timoshcnko

beam.

IO

Closed-form

solution

for the Timoshenko

41 I

beam theory

W A0

35

30 .\

0

\

4th. Nat

Freq. 101 T

B wth

tip moss

1 0

05

1

2 Normaliscd Mass

Fig.

12. Natural

frequency

against

normalized

5

10

0

mass for Timoshenko beam.

Fig. 13. Natural frequency against normalized end-rotary inertia for Timoshenko beam. CONCLUSIONS

The use of different beam theories shows that, even though we have not much difl‘erence in the fundamental frequency for both models, as we move towards higher modes, differences between the natural frequencies in the Timoshenko beam and the EulerBernoulli beam increase, Consequently, the frequency results using the Euler-Bernoulli theory for a beam which is not slender (L/H i:25)areinaccurate and unreliable for modes higher than the fundamental mode. It IS interesting to note that. for the Timoshenko theory. the third natural frequency in a beam with a tip mass converges to that of the second mode frequency of a fixed-free Timoshenko beam as the tip mass increases, and also the fourth natural frequency

converges to the third fixed-free mode. In general, the frequency results show that, as the normalized mass incrcascs, cvcn for a fixed end-rotary inertia, the frequency decreases; also, the same is true when the normalized mass is fixed and the end-rotary inertia increases (Fig. 13). Thus in the design of flexible structures, it is necessary to consider the lower frequencies that may occur when changing the tip mass properties. HEFERENCES S. P. Timoshcnko. Yihration Problems in Enginerring, 3rd edn. Van Nostrand, New York (1955). 2. T. C. I luang, The effect of rotatory irtertia and of shear deformation OII the ftequency and normal mode equation beams with simple end conditions. Tr~mr ASME d. AP/If. MfYlr 28, 579-584 (1961)

I.

412

A. M. Horr APPENDIX

To solve the frequency ml = s = b = r = alf = bet = rl = r2 = r3 = r4 = r5 = 1-6= r7 = r8 = f=

function

with Mathematics

and L. C. Schmidt

A: MATHEMATICA

computer

LOG FILE

package,

the following

m/(ro*a*l) Sqrt[(e*i)/(Q/3)*a*g*1”2)] Sqrt[(l/(e*i))*(ro*a/9.81)*(w”2)*(lA4)] Sqrt[i/(a*l”2)] (I/Sqrt[2])*Sqrt[Sqrt[(r^s - s”2)“2 + 4/b”2] - (r”2 + s”2)] (l/Sqrt[2])*Sqrt[Sqrt[(r”2 - s”2)“2 + 4/b”2] + (rn2 + s”2)] (b/alf)*Sinh[b*alfl+ ml*b “2*Cosh[b*alfl (b/alf)*Cosh[b*alfl + ml*b”2*Sinh[b*alfl (b/bet)*Sin[b*bet] + ml*b”2*Cos[b*bet] -(b/bet)*Cos[b*lxt] + ml*b”2*Sin[b*bet] ((alf”2 + sn 2)/alf)*(b*alf*Cosh[b*alfj - (1/2)*ml*((sig)“2)*b”2*Sinh[b*alfl) ((alf”2 + sn 2)/alf)*(b*alf*Sinh[b*alfl(1/2)*ml*((sig)“2)*b”2*Cosh[b*alfl) -((bet”2 - sA2)/bet)*(b*bet*Cos[b*bet] - (1/2)*ml*((sig)“2)*bA2*Sin[b*betJ) ((bet”2 - s^2)/bet)*( - b*bet*Sin[b*bet] - (l/2)*ml*((sig)“2)*bA2*Cos[b*bet]) ((alf”2 + s”2)/alf)(r3*r8 - r7*r4 + r4*r5 - rl*r8) + ((bet”2 - sA2)/lxt)(r2*r7

log file can be used:

- r6*r3 + rl*r6 - r2*r5)