An analytical solution is obtained for the problem of free and forced vibrations of a finite Euler Bernoulli beam with arbitrary (partially fixed) boundary conditions. The effects of linear viscous damping, Winkler foundation, constant axial tension, a concentrated mass, and an arbitrary forcing function are included in the analysis. No restriction is placed on the values of the parameters involved, and the solution presented here contains all cited previous solutions as special cases

Amba-Rao, C. L.

Hussaini, M. Y.

English

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https://ntrs.nasa.gov/search.jsp?R=19770019589 2020-03-22T10:06:47+00:00Z
1	 4
June 1977
NASA TECHNICAL
	 NASA TM 73,253
MEMORANDUM
(NASA-TM-Y-73253) THE GENERAL SOLUTION TO 	 N77-26533
THE CLASSICAL PROBLEM OF FINITE EULER
BERNOULLI BEAM (NASA) 13 p HC A02/MF A01
CSCL 20K	 Unclas
G3/39 31796
F-
a
z
THE GENERAL SOLUTION TO THE CLASSICAL
PROBLEM OF FINITE EULER BERNOULLI BEAM
M. Y. Hussain and C. L. Amba-Rao
Ames Research Center
Moffett Field, California 94035
n
11. Report No. 2. Government Accession No. 3. Recipient's QWW9 No.
NASA TM 73,253
4. Title and Subtitle f;. Report Data
THE GENERAL SOLUTION TO THE CLASSICAL PROBLEM
6. PrAOrming OrganizationtbdaOF FINITE EULER BERNOULLI BEAM
7. Author(s) 9. Perfamirq organization Report No.
M. Y. Hussaini and C. L. Amba-Rao A-7076
10. work Unit No.
505-06-12-079. Performing organization None and Address
Ames Research Center	 Vikram Sarabhai Space
Moffett Field, Calif.	 and	 Center
11	 contract or Grant No
94035	 Trivandrum, India
13. Type of Report and Period Covered
Technical MemorandumI!. Sponsoring Agency Name and Address
National Aeronautics and Space Administration 14. Sponsoring Agency code
Washington, D.C. 20546
15. Supplennentery Nom
16. Ai,stract
An analytical solution is obtained for the problem of free and forced
vibrations of a finite Euler Bernoulli beam with arbitrary (partially
fixed) boundary conditions.	 The effects of linear viscous damping,
Winkler foundation, constant axial tension, a concentrated mass, and an
arbitrary forcing function are included in the analysis. 	 No restriction
is placed on the values of the parameters involved, and the solution
preEented here contains all cited previous solutions as special cases.
17. Key Words (Suggested by Author(s)) 18. Distribution statement
Euler Bernoulli beam
Free and forced vibrations Unlimited
STAR Category - 39
19. Security Claaif, lof this report) 20. Security Claaif. iof this page) 21. No. of Pages 22. PNW'
Unclassified Unclassified 12 $3,25
'For sale by the National Technical Information Service, Springfield, Virginia 22161
THE GENERAL SOLUTION TO THE CLASSICAL PROBLEM OF FINITE
EULER BERNOULLI BEAM
M. Y. Hussaini and C. L. Amba-Rao*
NASA Ames Research Center, Moffett Field, California
SUMMARY
An analytical solution is obtained for the problem of free and
forced vibrations of a finite Euler Bernoulli beam with arbitrary
(partially fixed) boundary conditions. The effects of linear viscous
damping, Winkler foundation, constant axial tension, a concentrated mass,
and an arbitrary forcing function are included in the analysis. No
restriction is placed on the values of the parameters involved, and the
solution presented here contains all cited previous solutions as special
cases.
1. INTRODUCTION
Since the classical theory of beam was evolved by Euler and
Bernoulli, a large literature has accumulated on the subject, the
major part of which deals with either infinite beams or finite beams
with standard boundary conditions. The general solution for the dynamic
response of the infinite Euler-Bernoulli beam with arbitrary initial
conditions, subjected to an arbitrary load including the effects of
damping, an elastic foundation and constant axial load, was obtained
comparatively recently by Stadler and Shreeves (1). The analogous
* Vikram Sarabhai Space Center, Trivandrum, India
2problem for a finite beam does not seem to have been solved in all
generality. The first study allowing an elastically restrained edge
condition, which is of particular interest in the aerospace field, appears
to be that of Carmichael (2). Among subsequent investigations, those
of Haltbaek (3); Hess (4), Amba-Rao (5), Sharp and Cobble (6) and
Amba-Rao and Hussaini (7) are perhaps worth mentioning.
Maltbaek (3) solved the problem of a uniform beam with a rigid
mass attached at its center and with elastic supports. Hess (4) gave
a more complete solution. Amba-Rao (5) considered the free vibrations
of an elastically restrained beam carrying an arbitrary number of discrete
masses. He used the transform method, treating the added masses as impulses
(represented by Dirac delta functions) in the mass density function.
Sharp and Cobble (6) solved the problem of the arbitrarily loaded damped
beam elastically restrained against rotation. Amba-Rao and Hussaini (7)
presented a closed form solution to the problem of free vibrations of a
partially fixed, linearly damped; beam on a Winkler foundation, carrying
arbitrary masses. The present note extends the latter solution to include
the case of forced vibrations under arbitrary initial conditions, and
unifies all earlier investigations into a general theory.
2. STATEMENT OF THE PROBLEM
In addition to the usual assumptions of the Euler-Bernoulli theory,
the beam is assumed to have constant axial tensile load, with viscous
damping proportional to velocity, and to be resting on a Winkler foundation
whose resistance is proportional to the transverse displacement. The
relevant differential equation in operator form is
.s
3
.Cy(x.t) - q(x,t)
	
(2.1)
where
	
4 
	
2
-EJ a4 -P-L2- +f+da +p
aty
	
ax	 ax
Here y(x,t) is the lateral deflection, EJ the flexural rigidity
of the beam, p the mass density of the beam and concentrated was",
x the space coordinate measured along the length of the beam,
P the tensile axial force on the beam, f the Winkler foundation
constant, d the damping constant, t the time, and q the forcing
function. The initial and boundary conditions are, respectively:
y - u(x) , 
at - v(x)	 at t - 0
	
(2.2)
1 - Y
	T^ ax3	
at x - 0 for all t	 (2.3)
- 
-R1 a
1 a3y_
T2 ax3 Y
at x = L for all t	 (2.4)
= R
	
axe 	 2 ax	 +r
where L is the length of the beam an^ T i ' T2 , Ri . and R2 are spring
constants (partial fixity factors. ` Thlese boundary conditions are
unmixed. The method of solution 41 en here, is applicable even if the
conditions are mixed.	 I'
3. SOLUTION	 t
The general theory of linear idp"Ators suggests two fundaaental
IN
methods of solving equatiun (2.1). Une method is to find the inverse
Aw
.f
4
of the operator 1, that is, in effect to obtain the Green's.function.
The operator -C is not self-adjoint with respect to either x or t.
However, if the Green's function G is evaluated satisfying the homogeneous
initial and boundary conditions, then the problem can be solved in
principle by an extended definition of l as given in equation (3.1)
below:
.Cy - q + [pu(x)6'(t) + pv(x)d(t) + d6(01
+ terms due to inhomogeneous boundary conditions
Q (say)	 (3.1)
where 6(t) is the delta function. The solution takes the form
f
Lfm
Y(x.t) - J G(x,t: E,T)Q(&,T)dg dT
0 0
(3.2)
In the.present paper, the solution is obtained by the second method
using spectral representation of -C in terms of x-eigenfunctions.
First, the solution is sought as a sum of two functions, one of which is
the solution of a homogeneous problem with inhomogeneous boundary conditions,
while the other is the solution of an inhomogeneous equation with homo-
geneous boundary conditions. Thus, let
Y(x . t ) - y l (x , t ) + y2(x,0
	
(3.3)
where
tyi - 0
	
(3.4)
(3.5)
(3.6)
(3.7)
5
with the boundary conditions
yi"(O,t) - -Tlyi(O,t)
y'l
"(A,t) - T2yi(X't)
yl(O,t) - -Riyi(O,t)
y'i (7,t) - R2 i (n.t)
and
Xy2 - q
with the boundary conditions
y2 (O,t) - y2 (*., t) 	 0
y2(0 , t ) - y2(A,t) - 0
and the initial conditions
y2 (x,0) - u(x) - y i (x,0)
 
- U(x)
ay e
	ayi
at (x, 0) = v 	 - at (x,0) - V(x)
where, by a proper choice of the origin and unit of length, the end
points of the beam are at x - 0 and x - A, and primes denote
differentiation with respect to x.
(3.8)
a) EVALUATION OF yl
The solution y l
 of equation (3.4) is given in reference 7. A
slightly different approach is presented here which yields results
identical to those of reference 7. The approach is expected to be
useful particularly if the series representing the inverse of the finite
Fourier transform is not summable in closed form•.
6Write
yl = *(x)exp(-at)cos(wt) 	 (3.9)
Substituting this in equation (3.4), we find that *(x) satisfies the
ordinary differential equation
EJ dam-P d l + fey - P0 2 + w2 )* = 0	 (3.10)dx4 	8x2
where Zap d, and with concentrated masses being represented by delta
function p m + Mj d(x - xj ). Thus, the concentrated mass is supposed to be
situated at the point x = n j ; and d is assumed proportional to p.
The boundary conditions are
	
* (3)
(0) _ -T ,*(0)	 (3)(W) = T2^(X)	 (3.11)
( 2 )	 _ _	 ( 1 )	 (2)	 _	 (1)^	 (0)	 Rl^y	 (0)	 and	 ^	 (^)	 R2^	 (1()
where * (i) denotes ith derivative of * with reopcct to X.
Define
i
* coo nx dx
o 8x
Integrating by parts, we establish the recurrence relation
2 r(_1)n,(i+l)M - (1+1) (0)]
11
-	
-1)n,(i+3)(,) _ (1+3) (0)] + n gi+4^	 (3.12)
Z(n)	 eo	 2 [ ( -1)n^y(1)(]i) - (3) (0)]n
_ n 
^(_1)n^(3)(^) - x (3) (0)] + n g4Z 	 (3.13)
The expression for $4^ is obtained from the finite cosine transform of
equation (3.4):
784 [1 + EJ n 2] • ;(n) LEJ ( a2 + W2 ) - fEJ
+ EJJ n r_1) n^(3)M 
-
x, (3) (0)] + Sj	 (3.14)
where
sj - 02 +EJ W2 Mj *(xj )coo nxj
Finally,
^V(n)	 1	 r(n2 + 2L 2 ) {(- 1 ) n, (1) (E ) - V► (1) (0)}(n2 + £ 2 ) 2 - 0
	
1(_1)n,(3)M 
_x (3) (0)} + sj ]	 (3.15)
where
P	 ( P \`	 f	 02 + WZ
RZ • EJ ' -k4 • - \ 2EJ / + EJ - m	 EJ
Then
m
^ (x) • ^ (0) +	 ^ (n)cos nx
m• 1
or
*(x) .1	 1	 Ca2EJ W2 m *(xj ) - {^4 (3) M -x(3)(0)}A j 4 _ k4
+ 2k2{x,(1)(A) _ x(1)(0)!] + a2AEJW2 Mj *(xj )[I 3 (x + xj)
+ I 3 (Ix - xj I)] - 1( [x(1)(0){I1(x) + (R2 + k2)I3(x)}
+ (1) W {I 2 (x) + (Z 2 + k 2 )I 4 (x)} + to (3) (0) I 3 (x) - V(3)(X)I4(4]
(3.16)
r
	
^_	 Ijp_
-	 a
8
where
I (x)	 -	 1	 +	 n	 cos h (n - x) A2
 + k2
i	 2(R2 + k2) 2	 sinhll Aft+P
I (x)	 1	 _ l(	 cosh n - x)
3	 2(R" - k")
	 4k2 [ k + k Z sink X
cocth (X - x)/1 2
 - k2 1
k2 s inh A i- J
I 2 (x) - I 1 (A - x) and I " (x) - I 3 (X - x)
b) SOLUTION OF ty 2 = q(n,t)
The operator L 2 is a sum of two commutative operators Lx and Lt
defined as
a2
L • EJ a"-- P --- + f	 ( 3 .17)
x	 ax"	 3x2
Lt - p 2-2  +d o•	 (3.18)
at
2
The inverse of -C2 may be obtained by considering L x as a constant.
The results will be a function of the operator L x and should be interpreted
by using the spectral representation of Lx. Consider Lx : Its domain
D  is the set of all functions in S (which is a linear vector'space of
real-valued square integrable functions over (0,A)) and they have piecewise
continuous fourth derivatives for 0 <_ x < x  and for x  < x _< 71, whic'i
satisfy the conditions
*(0) s ^ M - *11(0) - V ,M - 0	 (3.19)
and are such that Lx* belongs to S.
If * i , ^ i are in Dx , it can he easily shown following standard
pri-secdurr
y
9(^J I LX*,) - 
\yj.*i)
	
(3.20)
where the inner product is defined as
14j-
 fo
dn	 (3.21)
In other words, Lx is self-adjoint and its eigenfunctions are *i.
These 
i 
can be obtained from the analysis of section (a) by letting
T 1 , T2 -► - and R l , R2 - 0. The mi defined by the relation
® i -	 ^,2 2 i T7T	 (3.22)
C
form the orthonormal basis for S. Hence q, U, V can be represented
as
q/P - !.r gi(t)4i(x)
-
U - 	 Uimi(x)
V -
	 Vimi(x)
1
where
gi - jo Q(E.t) mi(E)dE
P
('A
Ui 
- J U(E)mi(&)dE
0
x
Vi 
- f
o 
V( E) 0iM dIC
(3.23)
(3.24)
The equation (3.6) is written as
2	 ay
\P ate 
+ d at + Lx y2 - q(x,t) , y2 (x.o) - "(x) ,	 at (x,0) - V(x)
(3.25)
y h-_ - 1	 1
r
10
Treating Lx as a constant, we find the solution of (3.25) (by the
method of Laplace transforms or by variation of parameters) as
	
Y	
fo
t exp(-o(t - T)j sinh B(t - T)	 T x dT
	
2	 s	 P
+ exp(-at) s=nh Ot +
0
 cosh at 
U(x)
+ exp( , at) Binh St VW	 (3.26)
^f
where 8 - 42 - Lx/p and 2 ap - d., It is to be noted that
the function of the operator is written before the function on which
it acts and it is interpreted accordingly. For instance,
f(Lx)U(x)	 f(An)Ummn(x)
where An
 are the eigenvalues of Lx end n 	 the associated normlised
eigenfunctions. Thus,
Y2 
- j [Fn(t)*qn(t) + Fn (t)(aUn + Vn)
n-1
exp(-at)cosh 8 n t Uj mn (x)	 (3.27)
t
where Fn (t)*qn (t) - f Fn (t - T)gn(T)dT
0
Binh 8 t
	--AFn (t) - exp(-at)	
S 
n	
, and 
Sn - 
a3 	 n
n
T T^
11
DISCUSSION
Substitution of (3.16) in (3.11) yields four equations for five
unknowns	 (1)(0), *0) (7),  * (3) (o), * (3) (70 and *(x j ). Evaluation of
at x - xj from (3.16) provides the necessary fifth equation. The determinant
of this system set equal to zero is the equation for natural frequencies.
In the above Procedure, if equation (3.19) is used instead of (3.11), we
finally obtain the equation for the eigenvalues a n of the operator Lx.
From equation ( 3.10) it is obvious that a n have the form
X  - p02 + wn2)
and then
Bn	 ^'n
where w  are the natural frequencies of the beam simply supported at
both ends. The corresponding eigenfunctions are obtained from
equation ( 3.16). In summary, a general solution has been presented
for the free and forced vibrations of a finite Euler -Bernoulli beam.
REFERENCES
1. W. Stadler and R. W. Shreeves, Quart. J. Mech. Appl. Math. 23 (1970),
197-208.
2. T. E. Carmichael, Quart. J. Mech. Appl. Math. 12 (1959) 29-42.
3. J. C. Maltbaek, J. Mech. Sci. 3 (1961) 197-218.
4. M. S. Hess, J. Appl. Mech. 31 (1964) 556-558.
5. C. L. Amba-Rao, J. Acous. Soc. America 40 (1966) 1 67-371.
6. G. R. Sharp and M. H. Cobble, J. Acous. Soc. America 45 (1969) 654-660.
7. C. L. Amba-Rao and M. Y. Hussaini, J. Appl., Mech. 42 (1975) 735-36.
8. B. Friedman, Principles and Techniques of Applied Mathematics, Wiley,
New Yuck, 1950.


The general solution to the classical problem of finite Euler Bernoulli beam

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