By using Galerkin's variational method an approximate solution is developed for a system of two differential equations with linear partial derivatives of the second order. This system of differential equations corresponds to the physical model, known in the literature as the Timoshenko Beam. The results obtained are applied to two particular cases representing respectively: the case of a beam with a rectangular section, with a constant height and a basis with a linear variation; and the case of a beam with a constant basis and a height with cubic variation

Zainea, B.

English

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A STUDY ON THE FORCED VIBRATION 
OF A TIMOSHENKO BEAM 
Bucur Zainea 
SUMMARY 
By using Galerkin's  v a r i a t i o n a l  method w e  b u i l d  up an approximate s o l u t i o n  
f o r  a system of two d i f f e r e n t i a l  equations with l i n e a r  p a r t i a l  de r iva t ives  of 
t h e  second order. 
physical  model, known i n  t h e  l i t e r a t u r e  as the  Timoshenko Beam. The r e s u l t s  
obtained can be  f i n a l l y  applied t o  two p a r t i c u l a r  cases represent ing  respec t ive ly :  
t h e  case of a beam with a rec tangular  s ec t ion ,  w i t h  a cons tan t  height and a b a s i s  
wi th  a l inear va r i a t ion :  
t h e  case of a beam with a constant b a s i s  and a he ight  wi th  cubic  va r i a t ion .  
This system of d i f f e r e n t i a l  equations corresponds t o  the  
INTRODUCTION 
We are taking i n t o  consideration a heterogenous e l a s t i c  s t r a i g h t  beam pos- 
s e s s ing  v a r i a b l e  geometrical and mechanical c h a r a c t e r i s t i c s  a l l  along the  beam. 
We are considering the  s m a l l ,  cross-cut non-damping forced o s c i l l a t i o n s .  
The mathematical model chosen t o  be  subjected t o  ana lys i s  c o n s i s t s  i n  a system 
of two l i n e a r  equations with p a r t i a l  de r iva t ives  of second order ,  corresponding 
t o  the phys ica l  model known i n  t h e  l i t e r a t u r e  under t h e  name of Timoshenko Beam. 
This model is more exac t  than t h e  classical one usua l ly  employed i n  the  engineer- 
ing  ca l cu la t ions ,  t h a t  i s  t h e  Euler-Bernoulli model. The d i f f e rence  between them 
cons i s t s  i n  t h e  f a c t  t h a t  while f o r  t h e  Euler-Bernoulli model only the  deforma- 
t i o n s  given by t h e  bending moment o r  by t h e  t r a n s l a t i o n  i n e r t i a  are taken i n t o  
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https://ntrs.nasa.gov/search.jsp?R=19770003350 2020-03-22T12:11:11+00:00Z
account, i n  the  Timoshenko model t h e  t ransverse  shear  and the  r o t a t i o n a l  i n e r t i a  
are a l s o  taken i n t o  consideration. A s  a r e s u l t  t h e  Timoshenko model r e f l e c t s  
more exac t ly  t h e  phys ica l  r e a l i t y .  It is  well-known t h a t  (ref.  1) t h e  d i f f e r -  
ences between t h e  two theo r i e s  become s i g n i f i c a n t  i n  the  case of ( r e l a t i v e l y )  
s h o r t  beams and t h i s  cannot be neglected any longer. 
Although the  l i t e r a t u r e  r e f e r r i n g  t o  the  dynamics of t he  Timoshenko Beam 
is  abundant enough, t h e  matter of t he  non-damping beam has been i n s u f f i c i e n t l y  
t r ea t ed .  
I n  t h e  present  paper w e  t r y  t o  determine the  approximate so lu t ions  of t h e  
phenomenon by means of the  Galerkin v a r i a t i o n a l  method. W e  are of t h e  opinion 
t h a t  the  above mentioned method is  most s u i t a b l e  i n  so lv ing  t h e  sub jec t  consid- 
ered. The choosing of t he  system of coordinates required by t h e  Galerkin method 
assures  the  convergence of the  obtained so lu t ions .  
SYMBOLS 
K 
G 
P 
E 
A b )  
Dirac function 
1 
Euler 's  Beta function: B(p,g) =l xp-l(l-x)g-ldx 
c o e f f i c i e n t  of t h e  form of t h e  sec t ion  
cross-cut modulus of e l a s t i c i t y  
dens i ty  of material 
long i tud ina l  modulus of e l a s t i c i t y  (Young) 
area of cross-cut s ec t ion  
672 
c;[o,11 
moment of inertia of cross-cut section 
cross-cut displacement 
rotation angle 
1 
scalar product: (f,gj = f f(x)g(x)dx 
length of beam 
time-independent cross-cut displacement 
time-independent rotation angle 
cross-sectional area parameters 
moment of inertia parameter 
class of functions defined on 0 to 1 
THE DIFFERENTIAL EQUATIONS OF THE PHENOMENON 
The differential equations for the phenomenon are as follows: (ref, 2) 
Solutions for the differential equations are determined as follows: 
673 
f o r  boundary conditions 
and f ( x , t )  is a perturbance fo rce ,  a mobile, bu t  concentrated fo rce  f o r  a unit  
magnitude: 
LWt q@,Q = Lc?--=QL 
By considering equation (2) t h e  system of equation (1) becomes two d i f f e r -  
e n t i a l  equations of t h e  fou r th  order f o r  V(x) and U(x) as follows: 
The d i f f e r e n t i a l  equations (4) and (5) f o r  t h e  following two cases are as 
follows : 
674 

THE APPROXIMATE SOLUTION 
W e  shall i n t e g r a t e  the  d i f f e r e n t i a l  equations (6), (7), ( 8 ) ,  and (9) by 
m e a n s  of the Galerkin method. 
I n  the  case of boundary conditions of equation (3) w e  shall consider Z=unit 
which is always poss ib l e  by 
-= x. x : 0&3c& => 0 5  x&\ 
-49 
Using t h e  Galerkin method, we  s h a l l  determine an approximate so lu t ion  f o r  
equation (6) as follows: 
"rl -
We choose $,(x) of t h e  form ( re f .  3) 
6 76 
The system of coordinate functions $(x) has to satisfy the boundary 
conditions of equation (3) which become equivalent with the following 
conditions: 
The approximate solution (10) becomes: 
The % constants are determined out of the following algebraic system: 
where L is the left part of equation ( 6 ) ,  and g is the right part of the 
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The system of equation (11) is a non-damped algebraic system of n equa- 
tions with n indeterminates. This system is compatible because the determi- 
nant formed with the coefficients of the undeterminants is a Gramm determinant 
of a linear independent system of functions. For the calculation of the scalar 
product (L$,$Ij) and (g,$Ij), we have kept in view the following points: 
We have used the Euler's Beta function 
We have used the following formula (ref. 4) in calculating the scalat 
product : 
then 
For equation (7) with the boundary conditions of equation (3) which mean . 
V(o)=V(l)=o we are going to give an approximate solution of the following form: 
678 
where t h e  constant 8, is drawn from the  following a lgeb ra i c  system: 
where L is  t h e  l e f t  s i d e  of equation (7) and g i s  t h e  right: s i d e  of t he  
same equation. 
Analogous t o  equation (8) we bu i ld  up an approximate s o l u t i o n  of t h e  follow- 
ing  form: 
where the yk cons tan ts  are determined from the  following a lgebra ic  system: 
where L is  t h e  l e f t  s i d e  of equation (8) and g t h e  r i g h t  s i d e  of t h e  same 
equation 
F ina l ly ,  f o r  equation (9) we bu i ld  up a s o l u t i o n  of t he  following form: 
where t h e  6, cons tan ts  are determined from t h e  following a lgeb ra i c  system: 
where L and g are the  l e f t  s i d e  and r i g h t  s i d e  of equation (9). 
679 
As a conclusion t o  case 1 t h e  approximate solutions b u i l t  up by the Galerkin 
method are t h e  following: 
and, f o r  case 2 t he  approximate so lu t ions  are t h e  following 
PECULIAR CASES 
I n  the  following l i n e s  w e  s h a l l  use  t h e  obtained so lu t ion  f o r  two pa r t i cu la r  
cases, which w i l l  be a l s o  an i n d i r e c t  checking of the  accuracy of t h e  obtained 
r e s u l t s .  
W e  bu i ld  up the  f i r s t  two approximations Ql; Q2 and r e spec t ive ly  W1; 
W2 f o r  t he  following s i t u a t i o n s :  
They represent  respec t ive ly  t h e  case of a beam wi th  a rec tangular  s ec t ion ,  hav- 
ing  a constant he ight  and a base with a l i n e a r  v a r i a t i o n ,  and t h e  case of a 
beam with a constant base and a he ight  with a cubic v a r i a t i o n  and t h i s  because, 
from an app l i ca t ive  po in t  of view t h e  beam sec t ions  are i n  many cases considered 
6 80 
rectangular. Case (a) The equation (6), if we consider (14) is reduced t o  the 
following equation 
The f i r s t  and second approximations are respectively: 
If w e  compare $J with q2 f o r  a rectangular beam made of steel we come 1 
t o  the conclusion tha t  the two approximations are comparable: 
ce r t a in  A values and f o r  ce r t a in  x values 
$, = q2 f o r  
This conclusion r e s u l t s  from the  following calculation: 
f 
The al,a2 constants are determined from the following algebraic  system:. 
681 
where 
and f o r  the steel in S.I. units 
Equation (7) then becomes : 
682 
The first and second approximations are 
here 
683 
is  made t h a t  the  two approximations are comparable f o r  c e r t a i n  X values and 
f o r  c e r t a i n  x values such as 
CONCLUSIONS TO THESE PECULIAR CASES 
For equations (8) and (9) w e  come t o  the  same r e s u l t ,  t h a t  is: the  f i r s t  
two approximate so lu t ions  are equal f o r  the given values  of 
value of x :0,5: 0 I x I 1 t h a t  is, the approximate solut ions are comparable 
among themselves i n  the  v i c i n i t y  of where the concentrated perturbance f o r c e  is  
applied: when x = < = -. 
X f o r  the  same 
1 
2 
6 84 
REFERENCES 
1. Newmann, K.: Viscous Damping in Flexural Vibrations of Bars. Journal of 
Applied Mechanics. September 1959. 
2. Sun, C. T.: On the Equations for a Tirnoshenko Beam Under Initial Stress. 
Journal of Applied Mechanics. March 1972. 
3. Zainea, Bucur : 
Buletinul Stiintific a1 Institutului de Constructii Bucuresti. Anul XVI 
Nr. 3/ 1973. 
Concernant 2 la Convergence de la M6thode de Galerkin. 
4 .  Schwarts, Laurent: Methodesmathgmatiques poursciences physiques (Chapitre 
11. Dgrivation des distributions) . 
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A study on the forced vibration of a Timoshenko beam

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