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A galerkin finite element analysis of a uniform beam carrying a concentrated mass and rotary inertia with a spring hinge
Journal o f Sound attd Vibration (1974) 37(4), 567-569
A GALERKIN FINITE ELEMENT ANALYSIS OF A UNIFORM BEAM CARRYING A CONCENTRATED MASS AND ROTARY INERTIA WITH A SPRING tllNGE 1. INTRODUCTION
The problems of vibrations of beams carrying concentrated masses and with spring hinges has received considei'able attention previously. Chun [1] presented the vibration frequencies for a uniform beam with one end spring hinged and the other end free. Lee [2] obtained the vibration frequencies for the same problem including a concentrated mass and concentrated rotary inertia at the free end. Goel [3] considered the case of a uniform beam with a spring hinge at one end and a concentrated mass at any point of the beam. Recently the Galerkin finite element method has been found to be a reliable and very accurate method and it has been applied to vibration problems of beams by Prasad and Krishnamurthy [4]. As a first step to apply the Galerkin finite element method to the problem of vibration of spring hinged tapered beams with concentrated masses and concentrated rotary inertias, a study is presented in this note to check the reliability and accuracy of the Galerkin finite element applied to the vibration of a spring hinged uniform beam with a concentrated mass and a concentrated rotary inertia [2]. The present study reveals that the Galerkin finite element method is very accurate and even with a one element idealization of the beam the fundamental frequencies coincide up to five significant figures with the results of reference [2]. 2. FINITE ELEMENT FORMULATION The differential equation governing the free vibrations of a uniform beam in the nondimensional form is d4w -- ).w
= 0,
(l)
dx 4
with too9 2 L 4
EI where m is the mass per unit length of the beam, o9 is the circular frequency, L the length of the beam, E is Young's modulus and I is the moment of inertia. The boundary conditions for the case of a beam with a spring hinge ofstiffness Kt at x = 0 and with a concentrated mass, M, and concentrated rotary inertia, J, at x = 1 are as follows: a t x = 0: w = o,
(2)
d2w
KaLdw
"dxa
EI d x
= 0,
(3)
atx=l: d2w --
dxa
J
dw
+ ;.----
M L ad x 567
= 0
(4)
568
LETTERS TO TIlE EDITOR
and
d3w ;.Mw=o. dxa
(5)
mL
The deflection, w, in a typical finite element is expressed as a seventh degree polynomial of the independent variable x as w = [Fl{a,)
(6)
in which [F] = [lxx2xaMxSx6xV], {0t,}T = [a~ % a3 a, a5 ~6 ~v as], {ct,} being the generalized co-ordinates. With w, w', w" and w" denoting the nodal values, the eight generalized co-ordinates can be expressed as {~.) = [ r l { a , } ,
{6,} being the column vector of nodal values and [T] a 8 x 8 transformation matrix. Hence one has w = [~]{a~),
(7)
kb] being [F][T] -t. Upon substituting equation (7) in equation (1), the residual, R, is found to be given by R = [,/,1" {6~1 - ;.[~1{6.}.
(8)
The Galerkin finite clement method requires that the residual, R, multiplied by a set of weighting functions which arc typically identical to the polynomial approximations of the dependent variable, should be minimized with respect to {fie}, to yield
0__ f wR dx = 0.
a{a,}j
(9)
Equation (9) gives rise to a set of algebraic equations: [k]{,~,) - ;4cl{a.} = 0.
(10)
The final matrix equation governing the problem is obtained by the usual assembly procedure as [K]{6} =
;.[c1{6}.
(1 l)
To satisfy the complicated boundary conditions specified in equations (2)-(5), a suitable transformation matrix is used to transform the matrices K and C, in such a way that the boundary conditions can be directly applied as in the classical finite element method. 3. RESULTS AND DISCUSSION
The non-dimensional fundamental frequencies 0 ?/4) for different values of the spring stiffness, mass and rotary inertia are presented in Table 1, for one element idealization. The results are in excellent agreement, up to five significant figures, with those presented by Lee [2].
569
LETTERS TO THE EDITOR TABLE 1
Fundamentalfrequencies (2 TM)o f a uniform cantilever beam with a spring hhzge at one end attd concentrated mass and concentrated rotary inertia at the other end M
mL J
t-
M L3 KL/EI = 0.01 KL/EI = 0.1 KL/EI = 1-0 KL/EI= 10'0 KL[EI= 1000
1"0 10-0 100-0 1-0 10-0 100.0 1"0 10'0 1000 1-0 I00 100-0 1"0 I0'0 100'0
A
9
1"0
10"0
100"0
0-25553 0"17197 0'09942 0.44931 0.30007 0-17312 0-72732 0"46460 0-26544 0'89768 0"54418 0-30850 0"92798 0"55668 0-31520
0-17219 0"14870 0'09735 0-30377 0.26113 0-16977 0"50431 0"41915 0"26217 0-65857 0-51144 0"30646 0'69374 0"52744 0"31340
0"09959 0'09748 0'08393 0.17578 0.17196 0-14736 0"29314 0"28528 0"23631 0"38766 0-37203 0"28798 0"41061 0"39173 0"29692
T h e G a l e r k i n finite element m e t h o d is thus f o u n d to be a reliable a n d accurate tool f o r investigating v i b r a t i o n p r o b l e m s a n d the a u t h o r s are presently w o r k i n g on its a p p l i c a t i o n to the case o f t a p e r e d beams.
ACKNOWLEDGMENT The a u t h o r s t h a n k D r C. L. A m b a - R a o for his c o n s t a n t e n c o u r a g e m e n t in the p r e p a r a t i o n o f this note.
Structural Engflteering Division, Space Science and Technology Centre, Trivandrum-695022, Kerala, India (Received 12 August 1974)
G. VENKATESWARARAO
K. KANAKA RAJU t
REFERENCES
1. K. R. CttUN 1972 Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics 39, 1154-1155. Free vibrations of a beam with one end spring hinged and the other free. 2. T. W. LEE 1973 Transactions of the American Society of Alechanical Engineers, Journal of Applied Mechanics 40, 813-815. Vibration frequencies for a uniform beam with one end spring-hinged and carrying a mass at the other free end. 3. R. P. GOEL 1973 Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics 40, 821-822. Vibrations of a beam carrying a concentrated mass. 4. K . S . R . K . PRASADand A. V. KRISH~AMUR'rm' 1973 American Institute of Aeronautics andAstronautics Journal 1 l, 544-546. Galerkin finite element method for vibration problems. 1"Physics and Applied Mathematics Division.
A GALERKIN FINITE ELEMENT ANALYSIS OF A UNIFORM BEAM CARRYING A CONCENTRATED MASS AND ROTARY INERTIA WITH A SPRING tllNGE 1. INTRODUCTION
The problems of vibrations of beams carrying concentrated masses and with spring hinges has received considei'able attention previously. Chun [1] presented the vibration frequencies for a uniform beam with one end spring hinged and the other end free. Lee [2] obtained the vibration frequencies for the same problem including a concentrated mass and concentrated rotary inertia at the free end. Goel [3] considered the case of a uniform beam with a spring hinge at one end and a concentrated mass at any point of the beam. Recently the Galerkin finite element method has been found to be a reliable and very accurate method and it has been applied to vibration problems of beams by Prasad and Krishnamurthy [4]. As a first step to apply the Galerkin finite element method to the problem of vibration of spring hinged tapered beams with concentrated masses and concentrated rotary inertias, a study is presented in this note to check the reliability and accuracy of the Galerkin finite element applied to the vibration of a spring hinged uniform beam with a concentrated mass and a concentrated rotary inertia [2]. The present study reveals that the Galerkin finite element method is very accurate and even with a one element idealization of the beam the fundamental frequencies coincide up to five significant figures with the results of reference [2]. 2. FINITE ELEMENT FORMULATION The differential equation governing the free vibrations of a uniform beam in the nondimensional form is d4w -- ).w
= 0,
(l)
dx 4
with too9 2 L 4
EI where m is the mass per unit length of the beam, o9 is the circular frequency, L the length of the beam, E is Young's modulus and I is the moment of inertia. The boundary conditions for the case of a beam with a spring hinge ofstiffness Kt at x = 0 and with a concentrated mass, M, and concentrated rotary inertia, J, at x = 1 are as follows: a t x = 0: w = o,
(2)
d2w
KaLdw
"dxa
EI d x
= 0,
(3)
atx=l: d2w --
dxa
J
dw
+ ;.----
M L ad x 567
= 0
(4)
568
LETTERS TO TIlE EDITOR
and
d3w ;.Mw=o. dxa
(5)
mL
The deflection, w, in a typical finite element is expressed as a seventh degree polynomial of the independent variable x as w = [Fl{a,)
(6)
in which [F] = [lxx2xaMxSx6xV], {0t,}T = [a~ % a3 a, a5 ~6 ~v as], {ct,} being the generalized co-ordinates. With w, w', w" and w" denoting the nodal values, the eight generalized co-ordinates can be expressed as {~.) = [ r l { a , } ,
{6,} being the column vector of nodal values and [T] a 8 x 8 transformation matrix. Hence one has w = [~]{a~),
(7)
kb] being [F][T] -t. Upon substituting equation (7) in equation (1), the residual, R, is found to be given by R = [,/,1" {6~1 - ;.[~1{6.}.
(8)
The Galerkin finite clement method requires that the residual, R, multiplied by a set of weighting functions which arc typically identical to the polynomial approximations of the dependent variable, should be minimized with respect to {fie}, to yield
0__ f wR dx = 0.
a{a,}j
(9)
Equation (9) gives rise to a set of algebraic equations: [k]{,~,) - ;4cl{a.} = 0.
(10)
The final matrix equation governing the problem is obtained by the usual assembly procedure as [K]{6} =
;.[c1{6}.
(1 l)
To satisfy the complicated boundary conditions specified in equations (2)-(5), a suitable transformation matrix is used to transform the matrices K and C, in such a way that the boundary conditions can be directly applied as in the classical finite element method. 3. RESULTS AND DISCUSSION
The non-dimensional fundamental frequencies 0 ?/4) for different values of the spring stiffness, mass and rotary inertia are presented in Table 1, for one element idealization. The results are in excellent agreement, up to five significant figures, with those presented by Lee [2].
569
LETTERS TO THE EDITOR TABLE 1
Fundamentalfrequencies (2 TM)o f a uniform cantilever beam with a spring hhzge at one end attd concentrated mass and concentrated rotary inertia at the other end M
mL J
t-
M L3 KL/EI = 0.01 KL/EI = 0.1 KL/EI = 1-0 KL/EI= 10'0 KL[EI= 1000
1"0 10-0 100-0 1-0 10-0 100.0 1"0 10'0 1000 1-0 I00 100-0 1"0 I0'0 100'0
A
9
1"0
10"0
100"0
0-25553 0"17197 0'09942 0.44931 0.30007 0-17312 0-72732 0"46460 0-26544 0'89768 0"54418 0-30850 0"92798 0"55668 0-31520
0-17219 0"14870 0'09735 0-30377 0.26113 0-16977 0"50431 0"41915 0"26217 0-65857 0-51144 0"30646 0'69374 0"52744 0"31340
0"09959 0'09748 0'08393 0.17578 0.17196 0-14736 0"29314 0"28528 0"23631 0"38766 0-37203 0"28798 0"41061 0"39173 0"29692
T h e G a l e r k i n finite element m e t h o d is thus f o u n d to be a reliable a n d accurate tool f o r investigating v i b r a t i o n p r o b l e m s a n d the a u t h o r s are presently w o r k i n g on its a p p l i c a t i o n to the case o f t a p e r e d beams.
ACKNOWLEDGMENT The a u t h o r s t h a n k D r C. L. A m b a - R a o for his c o n s t a n t e n c o u r a g e m e n t in the p r e p a r a t i o n o f this note.
Structural Engflteering Division, Space Science and Technology Centre, Trivandrum-695022, Kerala, India (Received 12 August 1974)
G. VENKATESWARARAO
K. KANAKA RAJU t
REFERENCES
1. K. R. CttUN 1972 Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics 39, 1154-1155. Free vibrations of a beam with one end spring hinged and the other free. 2. T. W. LEE 1973 Transactions of the American Society of Alechanical Engineers, Journal of Applied Mechanics 40, 813-815. Vibration frequencies for a uniform beam with one end spring-hinged and carrying a mass at the other free end. 3. R. P. GOEL 1973 Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics 40, 821-822. Vibrations of a beam carrying a concentrated mass. 4. K . S . R . K . PRASADand A. V. KRISH~AMUR'rm' 1973 American Institute of Aeronautics andAstronautics Journal 1 l, 544-546. Galerkin finite element method for vibration problems. 1"Physics and Applied Mathematics Division.