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Eigenvalue problem of a beam with a mass and spring at the end subjected to an axial force
Journal of Sound and Vibration (1980) 71(3), 453-457
LETTERS EIGENVALUE
TO THE EDITOR
PROBLEM OF A BEAM WITH A MASS AND SPRING SUBJECTED TO AN AXIAL FORCE 1.
AT THE END
INTRODUCTION
A beam with a mass and spring at the end subjected to an axial force can be used to model a steel pier of a highway bridge. It is necessary to know the natural frequencies and mode shapes to discuss its dynamic behavior, especially when modal analysis is employed. In this note, the frequency equation and orthogonality conditions of such a beam are derived. Eigenvalue results are presented for axial force values up to the critical value, and selected end mass and spring values. 2.
DIFFERENTIAL
EQUATION
The problem geometry is shown in Figure 1: a beam of span 1 is clamped at the end x = 0, and at x = 1 it is loaded by an end mass with a transverse spring and subjected to an axial
Figure 1. Geometry and co-ordinate system.
force P. The axial co-ordinate
is X. The strain energy of the beam is
~=~~~~~)2d~-~~(~)2d~+~~~:.*
(11
0
0
in which E is Young’s modulus, I is the moment of inertia of the cross-section, P is the axial compressive force, k is the spring constant and y is the transverse deflection. The kinetic energy expression is
(2) where w is the weight of the beam per unit length, A is the cross-sectional area, g is the acceleration due to gravity and W is the weight of the end mass. By making use of the energy expressions (1) and (2) and applying Hamilton’s principle, the following differential equation and mechanical boundary conditions are obtained: EI a4y/ax4+P
M = -EI
a*y/ax*,
Q = -EI
a*y/ax*+(wA/g)
a3y/ax3 = -(W/g)
a*y/at*
= 0,
a2y/at2 - ky + P ay/ax at x = 1.
(3) (4)
453 0022-460X/80/
150453
+ 05 $02.00 JO
@ 1980
Academic Press Inc. (London) Limited
454
LETTERS
TO THE
EDITOR
To solve equation (3), one takes natural modes of vibration in the form y =X(x) e’“‘,
(5) where w is the radian frequency, X is a function of x and i = J-1. Upon substituting equation (5) into equation (3), an ordinary differential equation for the function X is obtained: d4X/dx4+Pd2X/dx2-(wA/g)W’X=O.
(6)
This differential equation has the solution X=AcoshS&+BsinhS[+Ccosy[+Dsiny&
J$(p+x@Ti?>,
(7)
p=
s =&-p +&?z,, y= Pl’/EI = (r2/4)p, A = twA14w2/EI, ,f= x/l and p = P/P,,, in which Per is the buckling load of the cantilever (P,, = 7r2EI/4i2), and A, B, C and D are constants of integration. In the present problem, the boundary conditions at x = 0 and 1 may be expressed as
where
dX/dx=Oatx=O,
x=0,
EId3X/dx3+PdX/dx+[(W/g)w2-k]X=Oatx=l.
d2X/dx2 = 0,
After substituting equation (7) into equation produces the frequency equation
(8)
(8), expansion of the 4 x4 determinant
S{a4 + y4 + 2S2r2 cash S cos y + Sy(y2 - 6’) sinh S sin y} + (h4e - ~){(a’+ y2) sinh 6 cos y - (S/y)(S2 + r2) cash S sin y} - pS{(S2 - r’) - (S2- r2) cash S cos y - 2Sy sinh S sin y} = 0, where E = W/wAiand
K =
k13/EI.
2. PROOFS
OF ORTHOGONALITY
Orthogonality properties of the eigenfunctions and s as follows [l]: WA TtofXr=EIT+P
(9)
d4Xr
dx
d2Xr -
dx* ’
CONDITIONS
can be studied by considering modes r
WA --fX,=EI$+P$. g
(10)
Multiplying the first of equations (10) by X, and the second of by X,, integrating over the length of the beam, subtracting the second result from the first and finally adding the constant (W/g)(wP - w f )X,X, to both sides gives
0
0
(11)
LETTERS
Integration of the right-hand side of equation conditions (i.e., equations (8)) yields
(0:
455
TO THE EDITOR
(11) by parts and use of the boundary
-w:)(j-YXrXsdx +F,,=,)
(12)
= 0.
0
To satisfy equation (12) when r # s and the natural radian frequencies are distinct (w, # w,) one must have
dx +-wXrXs,=, g
:XrXs
= 0.
(13)
0
Multiplying equation (13) by W$ and using the second of equations (10) gives I
1
or
2
llirfX,xs g
dx +TXrXsx=,
Xsdx=O.
=o: 0
0
(14) Integrating the right-hand side of equation (14) and using the boundary conditions gives I
1
EI d2Xr d2Xs
(13
~~dx+kXrX~~=,=O.
-z-S0
0
Equations (13) and (15) constitute the orthogonality relationship for transverse vibration of the beam. For the purpose of transforming the equation of motion (3) into principal co-ordinates, expansion of the transverse motion in terms of the time functions Ti and displacement functions Xi gives
i=l
Substituting equation (16) into the energy expressions orthogonality relations then yields
(1) and (2) and use of the
I
V=iif,
T;(jEf$)2dx-P/ 0
(d$)2dx+kX;_j, 0
(17) Lagrange’s equations of motion in the principal co-ordinates
e.+w;T,=O,
then are (18)
LETTERS
456
TO THE EDITOR
where
_:=[i~~~)2d~-~i(~)2d~+~X~~= dx+;X:_,),
0
0
0
being the natural radian frequency of the ith mode. If the beam were to be subjected to additional forcing, the response of the beam could be obtained by this method in a similar manner.
wi
15
I
I
I
1
7aB6m)
3.9266KS)
L
I
Ol
I
I
I
IO
00
IOooo
loo0 mtio,K
Stii
Figure 2. Eigenvalues A us stiffness ratio K for various end mass ratios K (initial axial force ratio fi = 0). CF, clamped-free; CS, clamped-simply supported.
15,
I
I
I
I
I
IO x
3-
K iii 5
0
0
Axialfora,mtiqp Figure 3. Eigenvalues A us axial force ratio p for various stiffness ratios K (end mass ratio E = 0). CF, CS as in Figure 2.
LEl-l-ERS
457
TO THE EDITOR
Figure 4. Eigenvalues A us axial force ratio & for various end mass ratios E (stiffness ratio K = 0). in Figure 2.
3. NUMERICAL
CF, CS as
RESULTS
Numerical results for the first four eigenvalues A of the beam, for various values of the non-dimensional end mass ratio E(= IV/ wAl), a’xial force ratio c.c(=P/P,) and stiffness ratio K (= k13/EI) are presented in Figures 2-4, for the beam shown in Figure 1. Eigenvalues of the beam correspond to the clamped-free condition for the critical case of E = 0, p = 0 and K = 0, and correspond to the clamped-simply supported condition for the critical case of K = cc or p = 00. Department of Civil Engineering, Nagasaki University, Nagasaki, Japan
K. TAKAHASHI
(Received 19 March 1980)
REFERENCE 1. S. TIMOSHENKO, New York: John
D. H. YOUNGand W. WEAVER,JR. 1974 Vibration Problems in Engineering. Wiley and Sons. See pp. 41.5-420.
LETTERS EIGENVALUE
TO THE EDITOR
PROBLEM OF A BEAM WITH A MASS AND SPRING SUBJECTED TO AN AXIAL FORCE 1.
AT THE END
INTRODUCTION
A beam with a mass and spring at the end subjected to an axial force can be used to model a steel pier of a highway bridge. It is necessary to know the natural frequencies and mode shapes to discuss its dynamic behavior, especially when modal analysis is employed. In this note, the frequency equation and orthogonality conditions of such a beam are derived. Eigenvalue results are presented for axial force values up to the critical value, and selected end mass and spring values. 2.
DIFFERENTIAL
EQUATION
The problem geometry is shown in Figure 1: a beam of span 1 is clamped at the end x = 0, and at x = 1 it is loaded by an end mass with a transverse spring and subjected to an axial
Figure 1. Geometry and co-ordinate system.
force P. The axial co-ordinate
is X. The strain energy of the beam is
~=~~~~~)2d~-~~(~)2d~+~~~:.*
(11
0
0
in which E is Young’s modulus, I is the moment of inertia of the cross-section, P is the axial compressive force, k is the spring constant and y is the transverse deflection. The kinetic energy expression is
(2) where w is the weight of the beam per unit length, A is the cross-sectional area, g is the acceleration due to gravity and W is the weight of the end mass. By making use of the energy expressions (1) and (2) and applying Hamilton’s principle, the following differential equation and mechanical boundary conditions are obtained: EI a4y/ax4+P
M = -EI
a*y/ax*,
Q = -EI
a*y/ax*+(wA/g)
a3y/ax3 = -(W/g)
a*y/at*
= 0,
a2y/at2 - ky + P ay/ax at x = 1.
(3) (4)
453 0022-460X/80/
150453
+ 05 $02.00 JO
@ 1980
Academic Press Inc. (London) Limited
454
LETTERS
TO THE
EDITOR
To solve equation (3), one takes natural modes of vibration in the form y =X(x) e’“‘,
(5) where w is the radian frequency, X is a function of x and i = J-1. Upon substituting equation (5) into equation (3), an ordinary differential equation for the function X is obtained: d4X/dx4+Pd2X/dx2-(wA/g)W’X=O.
(6)
This differential equation has the solution X=AcoshS&+BsinhS[+Ccosy[+Dsiny&
J$(p+x@Ti?>,
(7)
p=
s =&-p +&?z,, y= Pl’/EI = (r2/4)p, A = twA14w2/EI, ,f= x/l and p = P/P,,, in which Per is the buckling load of the cantilever (P,, = 7r2EI/4i2), and A, B, C and D are constants of integration. In the present problem, the boundary conditions at x = 0 and 1 may be expressed as
where
dX/dx=Oatx=O,
x=0,
EId3X/dx3+PdX/dx+[(W/g)w2-k]X=Oatx=l.
d2X/dx2 = 0,
After substituting equation (7) into equation produces the frequency equation
(8)
(8), expansion of the 4 x4 determinant
S{a4 + y4 + 2S2r2 cash S cos y + Sy(y2 - 6’) sinh S sin y} + (h4e - ~){(a’+ y2) sinh 6 cos y - (S/y)(S2 + r2) cash S sin y} - pS{(S2 - r’) - (S2- r2) cash S cos y - 2Sy sinh S sin y} = 0, where E = W/wAiand
K =
k13/EI.
2. PROOFS
OF ORTHOGONALITY
Orthogonality properties of the eigenfunctions and s as follows [l]: WA TtofXr=EIT+P
(9)
d4Xr
dx
d2Xr -
dx* ’
CONDITIONS
can be studied by considering modes r
WA --fX,=EI$+P$. g
(10)
Multiplying the first of equations (10) by X, and the second of by X,, integrating over the length of the beam, subtracting the second result from the first and finally adding the constant (W/g)(wP - w f )X,X, to both sides gives
0
0
(11)
LETTERS
Integration of the right-hand side of equation conditions (i.e., equations (8)) yields
(0:
455
TO THE EDITOR
(11) by parts and use of the boundary
-w:)(j-YXrXsdx +F,,=,)
(12)
= 0.
0
To satisfy equation (12) when r # s and the natural radian frequencies are distinct (w, # w,) one must have
dx +-wXrXs,=, g
:XrXs
= 0.
(13)
0
Multiplying equation (13) by W$ and using the second of equations (10) gives I
1
or
2
llirfX,xs g
dx +TXrXsx=,
Xsdx=O.
=o: 0
0
(14) Integrating the right-hand side of equation (14) and using the boundary conditions gives I
1
EI d2Xr d2Xs
(13
~~dx+kXrX~~=,=O.
-z-S0
0
Equations (13) and (15) constitute the orthogonality relationship for transverse vibration of the beam. For the purpose of transforming the equation of motion (3) into principal co-ordinates, expansion of the transverse motion in terms of the time functions Ti and displacement functions Xi gives
i=l
Substituting equation (16) into the energy expressions orthogonality relations then yields
(1) and (2) and use of the
I
V=iif,
T;(jEf$)2dx-P/ 0
(d$)2dx+kX;_j, 0
(17) Lagrange’s equations of motion in the principal co-ordinates
e.+w;T,=O,
then are (18)
LETTERS
456
TO THE EDITOR
where
_:=[i~~~)2d~-~i(~)2d~+~X~~= dx+;X:_,),
0
0
0
being the natural radian frequency of the ith mode. If the beam were to be subjected to additional forcing, the response of the beam could be obtained by this method in a similar manner.
wi
15
I
I
I
1
7aB6m)
3.9266KS)
L
I
Ol
I
I
I
IO
00
IOooo
loo0 mtio,K
Stii
Figure 2. Eigenvalues A us stiffness ratio K for various end mass ratios K (initial axial force ratio fi = 0). CF, clamped-free; CS, clamped-simply supported.
15,
I
I
I
I
I
IO x
3-
K iii 5
0
0
Axialfora,mtiqp Figure 3. Eigenvalues A us axial force ratio p for various stiffness ratios K (end mass ratio E = 0). CF, CS as in Figure 2.
LEl-l-ERS
457
TO THE EDITOR
Figure 4. Eigenvalues A us axial force ratio & for various end mass ratios E (stiffness ratio K = 0). in Figure 2.
3. NUMERICAL
CF, CS as
RESULTS
Numerical results for the first four eigenvalues A of the beam, for various values of the non-dimensional end mass ratio E(= IV/ wAl), a’xial force ratio c.c(=P/P,) and stiffness ratio K (= k13/EI) are presented in Figures 2-4, for the beam shown in Figure 1. Eigenvalues of the beam correspond to the clamped-free condition for the critical case of E = 0, p = 0 and K = 0, and correspond to the clamped-simply supported condition for the critical case of K = cc or p = 00. Department of Civil Engineering, Nagasaki University, Nagasaki, Japan
K. TAKAHASHI
(Received 19 March 1980)
REFERENCE 1. S. TIMOSHENKO, New York: John
D. H. YOUNGand W. WEAVER,JR. 1974 Vibration Problems in Engineering. Wiley and Sons. See pp. 41.5-420.