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Vibration frequencies for a uniform beam with one end spring-hinged and subjected to a translational restraint at the other end
Jownal of Sound and Vibration(1976) 48(4), 565-568
VIBRATION FREQUENCIES FOR A UNIFORM BEAM WITH ONE END SPRINGHINGED AND SUBJECTED TO A TRANSLATIONAL RESTRAINT AT THE OTHER END
The purpose of the present study is to deal with the free vibration of a beam hinged at one end by a rotational spring and subjected to the restraining action of a translational spring at the other end. The eigenfrequencies for the fundamental mode are presented for different values of the parameters K,LIEI and KTL3/EI, where KR and KT are the stiffness constants of the springs. 1.
INTRODUCTION
Considerable research has been devoted to the problem of free vibration of beams and the normal functions and natural frequencies can be found in various textbooks [l-3] and tables [4-6] for different combinations of end conditions. Hess [7] obtained natural frequencies for a uniform beam with a central mass and symmetrical spring-hinged ends. Chun [8] considered the free vibration of a beam hinged at one end by a rotational spring with a constant spring constant and the other end free. Goel [9] obtained the vibration frequencies for the same problem including a concentrated mass at any point of the beam. Lee [lo] deduced the vibration frequencies for a uniform beam with one end spring-hinged and carrying a mass and concentrated rotary inertia at the free end. MacBain and Genin [l l] have studied the effect of rotational and translational support flexibility on the fundamental frequency of an almost-clamped-clamped beam and a flexibly supported cantilever beam [ 121. In recent papers, Prasad and Krishnamurthy [13], and Venkateswara Rao and Kanaka Raju [14] have applied the Galerkin finite element method to vibration problems of beams. These studies have shown that the method is reliable and very accurate. This letter is concerned with the free vibration of a uniform beam subjected to a rotational restraint at one end and a translational restraint at the other end.
2.
THEORY
The differential equation for small-amplitude, free vibrations of a uniform beam is EZa*y/ax’ + pA a2y/at2 = 0,
(1) where y is the lateral deflection, EZ is the modulus of flexural rigidity of the beam, p is its density per unit volume, A is its cross-sectional area, x is the distance along measured from the spring-hinged end (Figure 1) and t is the time. The boundary conditions for the structural system under consideration are as follows : at x = 0,
Y@,1) = 0,
KRay(0, t>/ax = Eza2 y (0, t)/axz,
@a, b)
Ezasy(L, t)/axJ = K,y(L, r),
(2~ d)
atx=L,
azy(L, tyax2 = 0,
where KR is the rotational spring constant and KT is the translational spring constant. 565
566
LETTERS TO THE EDITOR
Figure 1. Structural system analysed.
Using the method of separation of variables one assumes a solution of the form J&t)
= 2 Y,(x)T(t),
(3)
ll=l
where Y,(x) is the nth mode of natural vibration and is given by Y,(x) = A, cos k. x + B,, sin k, x + C, cash k. x + 0. sinh k. x,
(4)
k; = (Y,/L)~ = o;(pA/EZ).
(5)
Substituting equation (4) in equation (3), and then in equations (2), results in a linear system of homogeneous equations in the unknowns A,, B,, C, and 0.. Since the solution must be non-trivial the vanishing of the determinant of the coefficient matrix gives the following frequency equation : (E2Zz/KR ZGL’)fi(siny,coshy, - (EZ/KTL3)$(cosy.coshy, - (siny,coshy,,
- sinhy,cosy,)
-
+ 1) - 2(EZ/K,L)y,siny,sinhy,
- sinhy,cosy,)
= 0,
(6)
where y,, = k,,L, and the vibration frequency of the nth mode is f. = 0,/27t = (k;f/2lc) (EZ/~.IA)“~.
(7)
As KR --f 0 and Kr + 0, equation (6) becomes tany, - tanhy, = 0,
(8)
which is the frequency equation for the hinged-free beam [2,5]. If KR -+ co and KT + 0, equation (6) yields the frequency equation for the clamped-free beam [4], cosy,coshy,
+ 1 = 0.
(9)
As KR + 0 and KT --t co, equation (6) tends to the frequency equation for the simply supported beam [2], sin yn = 0.
(10)
Finally, if KR -+ co and KT + co, equation (6) reduces to the frequency equation for the clamped-supported beam [4], tany,, - tanh y,, = 0.
(11)
o-1 1 10 100 1000 co
i-01
KR LIEI \
-
KT L=/EI
1247917 1.722742 1.856787 1.873233 1.875104
0.735782
0 0.415934
0
0.416159 0.494814 0.754059 1.252003 l-724553 l-858332 l-874751 1.876619
0.01
o-739730 O-757696 0.878208 1.287038 l-740584 1a872052 1.888235 1a890076
o-1
I.309812 1.313392 1.343678 1.535805 1.8792% 1993947 2.008360 2.010003
1
2.988644 2990110 3-003006 3.108456 3.441219 3.613348 3.637722 3640542
100
and KTL3/EI
2.231325 2.232562 2.243362 2.326474 2.538831 2.626162 2.637613 2.638925
10
TABLE 1 Values of y1 = (k, L) as a function of K,L/EI
3.126081 3.127658 3.141553 3.256645 3642277 3.861458 3.894008 3.897801
1000
3.141593 3.143181 3.157181 3.273286 3.664644 3.889185 3.922695 3.926602
0)
568
LETTERS TO THE EDITOR
3. RESULTS AND DISCUSSION
The frequency equation (6) depends on the stiffness ratios K,L/EI and KTL3/EZ. Its roots have been computed numerically on a Hewlett-Packard 9810A and the numerical values of the eigenfrequencies for the fundamental mode are shown in Table 1. It is seen that the effect of both parameters is to increase the natural frequencies. For the beam without a translational spring the natural frequencies agree with Chun’s results [8]. Solid Mechanics Laboratory, Universidad National del Sur, Bahia Blanca, Argentina
M. J. MAURIZI R. E. Ross1 J. A. REYES
(Received 16 June 1976) REFERENCES
1. J. P. DEN HARTOG1956 Mechanicuf Vibrations, 4th ed. New York: McGraw-Hill Book Company, Inc. 2. L. S. JACOBSENand R. S. AYRE 1958 Engineering Vibrations. New York: McGraw-Hill Book Company, Inc. See pp. 482-496. 3. R. E. D. BISHOPand D. C. JOHNSON1960 The Mechanics of Vibration. Cambridge University Press. 4. D. YOUNG and R. P. FELGAR 1949 The University of Texas, Austin, Publication No. 4913 (Engineering Research Series No. 44). Tables of characteristic functions representing normal modes of vibration of a beam. 5. R. E. D. BISHOPand D. C. JOHNSON 1956 Vibration Analysis Tables. Cambridge University Press. 6. B. AKE.%S~N and H. H. T~~GNFORS1971 Chalmers University of Technology, Division of Solid Mechanics, Gothenburg, Publication No. 23, I-12. Tables of eigenmodes for vibrating uniform one-span beams. 7. M. S. Hass 1964 Journal of Applied Mechanics 31(3) (Trans. ASME 86, Series E), 556-557. Vibration frequencies for a uniform beam with central mass and elastic supports. 8. K. R. CHUN 1972 Journal of Applied Mechanics 39(4) (TYans. ASME 94, Series E), 1154-l 155. Free vibration of a beam with one end spring-hinged and the other free. 9. R. P. GOEL 1973 Journal of Applied Mechanics 40(3) (Trans. ASME 95, Series E), 821-822. Vibrations of a beam carrying a concentrated mass. 10. T. W. LEE 1973 Journal of Applied Mechanics 40(3) (Trans. ASME 95, Series E), 813-815. Vibration frequencies for a uniform beam with one end spring-hinged and carrying a mass at the other free end. 11. J. C. MACBAIN AND J. GENIN 1973 Journal of the Franklin Institute 296, 259-273. Effect of support flexibility on the fundamental frequency of vibrating beams. 12. P. A. A. LAURA,M. J. MAuRrzI and J. L. POMBO1975 Journalof Soundand Vibration 41,397-405. A note on the dynamic analysis of an elastically restrained-free beam with a mass at the free end. 13. K. S. R. K. PRA~ADand A. V. KRISHNAMURTHY1973 American Institute of Aeronautics and Astronautics Journal 11, 544-546. Galerkin finite element method for vibration problems. , 14. G. VENKAT~WARARAO and K. KANAKARAJU 1974 Journal of Sound and Vibration 37,567-569. A Galerkin tinite element analysis of a uniform beam carrying a concentrated mass and rotary inertia with a spring hinge.
VIBRATION FREQUENCIES FOR A UNIFORM BEAM WITH ONE END SPRINGHINGED AND SUBJECTED TO A TRANSLATIONAL RESTRAINT AT THE OTHER END
The purpose of the present study is to deal with the free vibration of a beam hinged at one end by a rotational spring and subjected to the restraining action of a translational spring at the other end. The eigenfrequencies for the fundamental mode are presented for different values of the parameters K,LIEI and KTL3/EI, where KR and KT are the stiffness constants of the springs. 1.
INTRODUCTION
Considerable research has been devoted to the problem of free vibration of beams and the normal functions and natural frequencies can be found in various textbooks [l-3] and tables [4-6] for different combinations of end conditions. Hess [7] obtained natural frequencies for a uniform beam with a central mass and symmetrical spring-hinged ends. Chun [8] considered the free vibration of a beam hinged at one end by a rotational spring with a constant spring constant and the other end free. Goel [9] obtained the vibration frequencies for the same problem including a concentrated mass at any point of the beam. Lee [lo] deduced the vibration frequencies for a uniform beam with one end spring-hinged and carrying a mass and concentrated rotary inertia at the free end. MacBain and Genin [l l] have studied the effect of rotational and translational support flexibility on the fundamental frequency of an almost-clamped-clamped beam and a flexibly supported cantilever beam [ 121. In recent papers, Prasad and Krishnamurthy [13], and Venkateswara Rao and Kanaka Raju [14] have applied the Galerkin finite element method to vibration problems of beams. These studies have shown that the method is reliable and very accurate. This letter is concerned with the free vibration of a uniform beam subjected to a rotational restraint at one end and a translational restraint at the other end.
2.
THEORY
The differential equation for small-amplitude, free vibrations of a uniform beam is EZa*y/ax’ + pA a2y/at2 = 0,
(1) where y is the lateral deflection, EZ is the modulus of flexural rigidity of the beam, p is its density per unit volume, A is its cross-sectional area, x is the distance along measured from the spring-hinged end (Figure 1) and t is the time. The boundary conditions for the structural system under consideration are as follows : at x = 0,
Y@,1) = 0,
KRay(0, t>/ax = Eza2 y (0, t)/axz,
@a, b)
Ezasy(L, t)/axJ = K,y(L, r),
(2~ d)
atx=L,
azy(L, tyax2 = 0,
where KR is the rotational spring constant and KT is the translational spring constant. 565
566
LETTERS TO THE EDITOR
Figure 1. Structural system analysed.
Using the method of separation of variables one assumes a solution of the form J&t)
= 2 Y,(x)T(t),
(3)
ll=l
where Y,(x) is the nth mode of natural vibration and is given by Y,(x) = A, cos k. x + B,, sin k, x + C, cash k. x + 0. sinh k. x,
(4)
k; = (Y,/L)~ = o;(pA/EZ).
(5)
Substituting equation (4) in equation (3), and then in equations (2), results in a linear system of homogeneous equations in the unknowns A,, B,, C, and 0.. Since the solution must be non-trivial the vanishing of the determinant of the coefficient matrix gives the following frequency equation : (E2Zz/KR ZGL’)fi(siny,coshy, - (EZ/KTL3)$(cosy.coshy, - (siny,coshy,,
- sinhy,cosy,)
-
+ 1) - 2(EZ/K,L)y,siny,sinhy,
- sinhy,cosy,)
= 0,
(6)
where y,, = k,,L, and the vibration frequency of the nth mode is f. = 0,/27t = (k;f/2lc) (EZ/~.IA)“~.
(7)
As KR --f 0 and Kr + 0, equation (6) becomes tany, - tanhy, = 0,
(8)
which is the frequency equation for the hinged-free beam [2,5]. If KR -+ co and KT + 0, equation (6) yields the frequency equation for the clamped-free beam [4], cosy,coshy,
+ 1 = 0.
(9)
As KR + 0 and KT --t co, equation (6) tends to the frequency equation for the simply supported beam [2], sin yn = 0.
(10)
Finally, if KR -+ co and KT + co, equation (6) reduces to the frequency equation for the clamped-supported beam [4], tany,, - tanh y,, = 0.
(11)
o-1 1 10 100 1000 co
i-01
KR LIEI \
-
KT L=/EI
1247917 1.722742 1.856787 1.873233 1.875104
0.735782
0 0.415934
0
0.416159 0.494814 0.754059 1.252003 l-724553 l-858332 l-874751 1.876619
0.01
o-739730 O-757696 0.878208 1.287038 l-740584 1a872052 1.888235 1a890076
o-1
I.309812 1.313392 1.343678 1.535805 1.8792% 1993947 2.008360 2.010003
1
2.988644 2990110 3-003006 3.108456 3.441219 3.613348 3.637722 3640542
100
and KTL3/EI
2.231325 2.232562 2.243362 2.326474 2.538831 2.626162 2.637613 2.638925
10
TABLE 1 Values of y1 = (k, L) as a function of K,L/EI
3.126081 3.127658 3.141553 3.256645 3642277 3.861458 3.894008 3.897801
1000
3.141593 3.143181 3.157181 3.273286 3.664644 3.889185 3.922695 3.926602
0)
568
LETTERS TO THE EDITOR
3. RESULTS AND DISCUSSION
The frequency equation (6) depends on the stiffness ratios K,L/EI and KTL3/EZ. Its roots have been computed numerically on a Hewlett-Packard 9810A and the numerical values of the eigenfrequencies for the fundamental mode are shown in Table 1. It is seen that the effect of both parameters is to increase the natural frequencies. For the beam without a translational spring the natural frequencies agree with Chun’s results [8]. Solid Mechanics Laboratory, Universidad National del Sur, Bahia Blanca, Argentina
M. J. MAURIZI R. E. Ross1 J. A. REYES
(Received 16 June 1976) REFERENCES
1. J. P. DEN HARTOG1956 Mechanicuf Vibrations, 4th ed. New York: McGraw-Hill Book Company, Inc. 2. L. S. JACOBSENand R. S. AYRE 1958 Engineering Vibrations. New York: McGraw-Hill Book Company, Inc. See pp. 482-496. 3. R. E. D. BISHOPand D. C. JOHNSON1960 The Mechanics of Vibration. Cambridge University Press. 4. D. YOUNG and R. P. FELGAR 1949 The University of Texas, Austin, Publication No. 4913 (Engineering Research Series No. 44). Tables of characteristic functions representing normal modes of vibration of a beam. 5. R. E. D. BISHOPand D. C. JOHNSON 1956 Vibration Analysis Tables. Cambridge University Press. 6. B. AKE.%S~N and H. H. T~~GNFORS1971 Chalmers University of Technology, Division of Solid Mechanics, Gothenburg, Publication No. 23, I-12. Tables of eigenmodes for vibrating uniform one-span beams. 7. M. S. Hass 1964 Journal of Applied Mechanics 31(3) (Trans. ASME 86, Series E), 556-557. Vibration frequencies for a uniform beam with central mass and elastic supports. 8. K. R. CHUN 1972 Journal of Applied Mechanics 39(4) (TYans. ASME 94, Series E), 1154-l 155. Free vibration of a beam with one end spring-hinged and the other free. 9. R. P. GOEL 1973 Journal of Applied Mechanics 40(3) (Trans. ASME 95, Series E), 821-822. Vibrations of a beam carrying a concentrated mass. 10. T. W. LEE 1973 Journal of Applied Mechanics 40(3) (Trans. ASME 95, Series E), 813-815. Vibration frequencies for a uniform beam with one end spring-hinged and carrying a mass at the other free end. 11. J. C. MACBAIN AND J. GENIN 1973 Journal of the Franklin Institute 296, 259-273. Effect of support flexibility on the fundamental frequency of vibrating beams. 12. P. A. A. LAURA,M. J. MAuRrzI and J. L. POMBO1975 Journalof Soundand Vibration 41,397-405. A note on the dynamic analysis of an elastically restrained-free beam with a mass at the free end. 13. K. S. R. K. PRA~ADand A. V. KRISHNAMURTHY1973 American Institute of Aeronautics and Astronautics Journal 11, 544-546. Galerkin finite element method for vibration problems. , 14. G. VENKAT~WARARAO and K. KANAKARAJU 1974 Journal of Sound and Vibration 37,567-569. A Galerkin tinite element analysis of a uniform beam carrying a concentrated mass and rotary inertia with a spring hinge.