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Natural frequencies of the beam-mass system: Comparison of the two fundamental theories of beam vibrations
Journal ofSound
and Vi&ration (1991) 150(2), 330-334
NATURAL FREQUENCIES OF THE BEAM-MASS SYSTEM: COMPARISON THE TWO FUNDAMENTAL THEORIES OF BEAM VIBRATIONS M. J. MAURLZI
Department
of Engineering,
P. M.
AND
Universidad National (Received
OF
BELL!%
de1 Sur, 8000 Bahia Blanca, Argentina
17 January 1991)
1. INTRODUCTION
The natural frequencies of a beam-mass system are of practical interest in several fields of engineering applications [l-3], and it is well known that the corresponding numerical values may deviate considerably from those of the beam itself. This system is frequently chosen as a design model in engineering. Due to this fact, the solution of this problem has usually been obtained by means of an exact analysis [4-81, as well as by employing some approximate methods [9, lo]. It is the purpose of this note to present a comparison between Bernoulli-Euler and Timoshenko theories applied to the determination of the fundamental frequency of a simply supported beam with a concentrated mass attached to it. Results are presented in the form of tables, which can be used for a wide range of conditions. 2. THEORY
In Figure 1 is shown a simply supported uniform beam of length L carrying mass M at an arbitrary distance x = AL (0~ A s 1) along the static axis.
Figure
1. The beam-mass
an attached
system analyzed.
In a detailed study by Grant [6] the effect of the attached mass was expressed by the use of the Dirac delta function and the equations of motion were derived by using the Timoshenko beam theory. The coupled differential equations were conveniently solved by means of the Laplace transform method. This led to a frequency equation from which the natural frequencies can be determined [6]: sinh ba sin b/3 +-
*+t5
~[sinhbaAsinhbn(l-A)sinb~-~sinhbasinb~Asinb~(l-A)]=O,
(1)
where 4=MIW,
b’ = pAL4w2/
El,
rL = I/AL’,
s’=
EI/kAGL”,
(2-S)
330 0022-460X/91/200330+05
%03.00/O
0 1991 Academic
Press Limited
LEnERS
331
TO THE EDITOR TABLE
1
Fundamentalfrequency coeficients of a beam with a mass attached to it: (A), Bernoulli- Euler theory; (B, C), Timoshenko beam theory with r* = 0.0025 and r* = 0.005 respectively
4
0
0.2
04
0.5
0.6
0.8
1
A 0
(A) (B) (Cl
3.14159 3.06782 3.00378
3.14159 3.06782 3.00378
3.14159 3.06782 3.00378
3.14159 3.06782 3.00378
3.14159 3.06782 3.00378
3.14159 3.06782 3.00378
3.14159 3.06782 3.00378
0.1
(A)
3.14159 3.06782 3.00378
3.11187 3.03915 2.97593
3.08274 3.01058 2.94779
3.06842 2.99638 2.93367
3.05426 2.98225 2.91953
3.02646 2.95425 2.89132
2.99937 2.92668 2.86329
3.14159 3.06782 3.00378
3.04727 2.96880 2.90748
2.94810 2.87949 2.81948
2.90648 2.83834 2.77870
2.86721 2.79939 2.73999
2.79501 2.72757 2.66844
2.73023 2.66297 2.60399
3.14159 3.06782 3.00378
2.96098 2.89353 2.83449
2.81942 2.75536 2.69904
2.75934 2.69647 2.64113
2.70488 2.64303 2.58851
2.60967 2.54947 2.49632
2.52877 2.4699 1 2.41790
3.14159 3.06782 3.00378
2.90637 2.84112 2.78386
2.73761 2.67695 2.62342
2.66901 2.61001 2.55784
2.60813 2.55053 2.49953
2.50420 244888 2.39979
2.41802 2.36453 2.31699
3.14159 3.06782 3.00378
2.88726 2.82282 2.76621
2.71028 2.65083 2.59829
2.6393 1 2.58166 2.53059
2.57672 2.52057 2.47077
2.47059 2.41689 2.36914
2.38319 2.33144 2.28534
(B) (C) 0.2
(A) (B) (C)
0.3
(A) (B) (C)
0.4
(A) (B) (0
0.5
(A) (B) (C)
in which E is the modulus of elasticity, G is the modulus of elasticity in shear, I A is the cross-sectional area, p is the area moment of inertia of the cross-section, per unit volume, k is a numerical shape factor depending on the cross-section, w radian frequency, 4 is the ratio of the attached mass M to the total mass of the Mb = PAL, and (r =(l/~){-(r2+s2)+[(r2-s2)2+4/b2]“2}1’2,
(6)
p =(1/~)~+(r2+s2)+[(r2~s2)~+4/b2]“~~~’~,
(7)
~=(/32-.s2)/(~2-r2)=(a2+r2)/(a2+s2),
Y = fflP, where
is the mass is the beam
b2r2s’ < 1. If b*r*s* > 1 it is convenient
(839)
to use
(u=ja’=j(l/~){(r’+.~2)-[(r2-s2)2+4/b2]1’2}1’2
(10)
where Y’= Then the frequency
equation
sin ba' sin bp --
9a’b
a’/ P,
can be expressed
j=&i.
(Ila,
h)
as follows:
5
1+5
x[sinba’Asinba’(l-*)sinbp+~sinbn’sinbB*sinbp(l-A)]=O.
(12)
LETTERS
332
TO THE
EDITOR
On the other hand, the exact analysis reported in reference [8] makes it possible to obtain the fundamental frequency and the corresponding mode shape of a simply supported beam, with a concentrated mass attached to it, according to the classical onedimensional Bernoulli-Euler theory of flexural vibrations. In this case, one is able to determine the following frequency equation: &L[sin
pL sinh ApL sinh (1 - h)pL - sinh pL sin ApL sin (1 - A)pL]
+ 2 sin pL sinh pL = 0,
(13)
(PL)~ = pAL4w2/ EI. Here pL are the non-zero
solutions 3.
of the transcendental NUMERICAL
(14) equation
(13).
EXAMPLE
In order to adopt the quasi-universal notation whereby the symbol 0 denotes natural frequency coefficient, it was decided to present Tables I,2 and 3 in terms of this commonly accepted notation. In Table 1 are shown fundamental frequency coefficients determined by means of the exact expression (1) for a Timoshenko beam of uniform cross-section for r2 = O-0025 and 0.005, with s2 = 28r2/9 [ 1 I]. Numerical results are given for six different mass locations along the beam and for mass ratios 4 = 0, 0.2, 0.4, 0.5, 0.6, 0.8 and 1. Additionally, the coefficients were computed according to the Bernoulli-Euler theory by employing equation (13). The following conclusions may be of interest to a designer. (1) The relation Q,,,/fl,, (where Oibm is the fundamental frequency coefficient of the beam-mass system and Oib is the fundamental frequency coefficient of the beam) is essentially a function of the parameter 4 = M/Mb for a previously assigned value of A ; see Table 2. For instance, for A =0*5 and M/M, = 0.5, the parameter finlbrn/flnlb is approximately 0.19 for any value of r2. For A = 0.5 and M/M, = 1, the ratio Oib,,,/Oib is of the order of 0.32 for the entire range of values of r2 considered in the present study. (2) As shown in Table 3, the ratio OTb,Jnrb”m (where OTibm is the fundamental frequency coefficient of the beam-mass system determined by using Timoshenko’s theory and OFfm is the fundamental frequency parameter determined by means of the Bernoulli-Euler theory) is, essentially, a function of r2. In other words, once r2 is assigned, the ratio 0:,,,,/0:,“, remains practically constant regardless of the values of A and 4. For instance, for r2 = 0.005, the ratio 0T,,,,,/0::“, is approximately 0.96, while for r2 = 0.000625 it is of the order of 0.99. In conclusion, if one is interested in determining only the fundamental frequency of the simply supported beam-mass system one may always use, as a first order approximation, the Bernoulli-Euler theory. This conclusion will certainly not apply, in general, to higher frequencies. If one is interested in finding higher order frequencies of a mechanical system, the same procedure can be followed. ACKNOWLEDGMENTS The authors take this opportunity to thank Professor P. A. A. Laura for his encouragement during the course of this work. The present study has been sponsored by CONICET Research and Development Program (PID 3000500/88), while the second author has been supported by a grant from the Comision de Investigaciones Cientificas de la Provincia de Buenos Aires (CIC).
LETTERS
TO THE EDITOR TABLE
333
2
Values of 0, of the system shown in Figure 1 and analysis of the variation of the parameter as a (fllb,,,/LJnlb); (a) values of 0, as a function of A, C$ and r2; (b) values of 0,,,L?,, function of A, 4 and r2 0.02
0.005
0.0025
0.000625
0.00025
0
2.73460 2.30807 2.08255
3.00378 2.53059 2.28534
3.06782 2.58166 2.33144
3.12207 2.62419 2.36965
3.13369 2.63320 2.37772
3.14159 2.6393 1 2.38319
2.73460 2.40267 2.19281
3.00378 2.64113 2.41790
3.06782 2.69647 2.46991
3.12207 2.74281 2.51332
3.13369 2.75265 2.52253
3.14159 2.75934 2.52877
2.73460 2.52440 2.35231
3.00378 2.77870 2.60399
3.06782 2.83834 2.66297
3.12207 2.88852 2.71251
3.13369 2.89921 2.72306
3.14159 2.90648 2.73023
1 0.185 0,313
1 0.187 0.314
1 0.188 0.316
1 0.190 0.318
1 0.190 0.318
1 0.190 0.318
1 0.138 0.247
1 0.137 0.242
1 0.138 0.242
1 0.138 0.242
1 0.138 0.242
1 0.139 0.242
1 0.083 0.163
1 0.081 0.154
1 0.081 0.152
1 0.081 0.151
1 0.081 0.151
1 0.080 0.150
A
(a)
0
0.5
0.5 1 0
0.3
0.5 1 0
0.2
0.5 1
(b)
0
0.5
0.5 1
0.3
0
0.2
0
0.5 1
0.5 1
TABLE
3
,bm as a function of A, 4 and r2, where a:,,,, and a:&, are the fundamental Values of 0 Tb,L? nE frequency coejicients of the beam-mass system determined by using Timoshenko’s theory and Bernoulli- Euler theory respectively 0.02
0.005
0.0025
0.000625
0.00025
0
1
0.870 0.875 0.874
0.956 0.959 0.959
0.976 0.978 0.978
0.994 0.994 0.994
0.997 0.998 0.998
1 1 1
A 0.5
0
0.5 0.3
0.2
0
0.870
0.956
0.977
0.994
0.997
0.5 1
0.871 O-867
0.957 0.956
0.977 0.977
0.994 0.994
0.998 0.998
1 1 1
0 0.5 1
0.870 0.869 0.862
0.956 0.956 0.954
0.977 0.977 0.975
0.994 0.994 0.994
0.997 0.998 0.997
1 1 1
334
LETTERS TO THE EDITOR REFERENCES
1. P. A. A. LAURA,J. L. POMBO and E. A. SUSEMIHL 1974 JournalofSound and Vibration 37, 161-168. A note on the vibrations of a clamped-free beam with a mass at the free end. 2. M. SWAMINADHAM and A. MICHAEL 1979 Journal ofSound and Vibration 66, 144-147. A note on frequencies of a beam with a heavy tip mass. 3. C. N. BAPAT and C. BAPAT 1987 Journal of Sound and Vibration 112, 177-182. Natural frequencies of a beam with non-classical boundary conditions and concentrated masses. 4. L. S. SRINATH and Y. C. DAS 1967 Journal of Applied Mechanics, Transactions of the American Society of Mechanical Engineers, Series E, 784-785. Vibrations of beams carrying mass. 5. R. P. GOEL 1976 Journal of Sound and Vibration 47, 9-14. Free vibrations of a beam-mass system with elastically restrained ends. 6. D. A. GRANT 1978 Journal of Sound and Vibration 57, 357-365. The effect of rotary inertia and shear deformation on the frequency and normal mode equations of uniform beams carrying a concentrated mass. 7. H. SAITO and K. OTOMI 1979 Journal of Sound and Vibration 62, 257-266. Vibration and stability of elastically supported beams carrying an attached mass under axial and tangential loads. frequency of a 8. J. H. LAU 1981 Journal of Sound and Vibration 78, 154-157. Fundamental constrained beam. 9. P. A. A. LAURA, C. FILIPICH and V. H. CORTINEZ 1987 Journal of Sound and Vibration 117, 459-465. Vibrations of beams and plates carrying concentrated masses. 10. W. H. LIU and F.-H.YEH 1987 Journal of Sound and Vibration 117, 555-570. Free vibration of a restrained non-uniform beam with intermediate masses. 11. T. M. WANG and J.E. STEPHENS 1977 Journal of Sound and Vibration 51, 149-155. Natural frequencies of Timoshenko beams on Pasternak foundations.
and Vi&ration (1991) 150(2), 330-334
NATURAL FREQUENCIES OF THE BEAM-MASS SYSTEM: COMPARISON THE TWO FUNDAMENTAL THEORIES OF BEAM VIBRATIONS M. J. MAURLZI
Department
of Engineering,
P. M.
AND
Universidad National (Received
OF
BELL!%
de1 Sur, 8000 Bahia Blanca, Argentina
17 January 1991)
1. INTRODUCTION
The natural frequencies of a beam-mass system are of practical interest in several fields of engineering applications [l-3], and it is well known that the corresponding numerical values may deviate considerably from those of the beam itself. This system is frequently chosen as a design model in engineering. Due to this fact, the solution of this problem has usually been obtained by means of an exact analysis [4-81, as well as by employing some approximate methods [9, lo]. It is the purpose of this note to present a comparison between Bernoulli-Euler and Timoshenko theories applied to the determination of the fundamental frequency of a simply supported beam with a concentrated mass attached to it. Results are presented in the form of tables, which can be used for a wide range of conditions. 2. THEORY
In Figure 1 is shown a simply supported uniform beam of length L carrying mass M at an arbitrary distance x = AL (0~ A s 1) along the static axis.
Figure
1. The beam-mass
an attached
system analyzed.
In a detailed study by Grant [6] the effect of the attached mass was expressed by the use of the Dirac delta function and the equations of motion were derived by using the Timoshenko beam theory. The coupled differential equations were conveniently solved by means of the Laplace transform method. This led to a frequency equation from which the natural frequencies can be determined [6]: sinh ba sin b/3 +-
*+t5
~[sinhbaAsinhbn(l-A)sinb~-~sinhbasinb~Asinb~(l-A)]=O,
(1)
where 4=MIW,
b’ = pAL4w2/
El,
rL = I/AL’,
s’=
EI/kAGL”,
(2-S)
330 0022-460X/91/200330+05
%03.00/O
0 1991 Academic
Press Limited
LEnERS
331
TO THE EDITOR TABLE
1
Fundamentalfrequency coeficients of a beam with a mass attached to it: (A), Bernoulli- Euler theory; (B, C), Timoshenko beam theory with r* = 0.0025 and r* = 0.005 respectively
4
0
0.2
04
0.5
0.6
0.8
1
A 0
(A) (B) (Cl
3.14159 3.06782 3.00378
3.14159 3.06782 3.00378
3.14159 3.06782 3.00378
3.14159 3.06782 3.00378
3.14159 3.06782 3.00378
3.14159 3.06782 3.00378
3.14159 3.06782 3.00378
0.1
(A)
3.14159 3.06782 3.00378
3.11187 3.03915 2.97593
3.08274 3.01058 2.94779
3.06842 2.99638 2.93367
3.05426 2.98225 2.91953
3.02646 2.95425 2.89132
2.99937 2.92668 2.86329
3.14159 3.06782 3.00378
3.04727 2.96880 2.90748
2.94810 2.87949 2.81948
2.90648 2.83834 2.77870
2.86721 2.79939 2.73999
2.79501 2.72757 2.66844
2.73023 2.66297 2.60399
3.14159 3.06782 3.00378
2.96098 2.89353 2.83449
2.81942 2.75536 2.69904
2.75934 2.69647 2.64113
2.70488 2.64303 2.58851
2.60967 2.54947 2.49632
2.52877 2.4699 1 2.41790
3.14159 3.06782 3.00378
2.90637 2.84112 2.78386
2.73761 2.67695 2.62342
2.66901 2.61001 2.55784
2.60813 2.55053 2.49953
2.50420 244888 2.39979
2.41802 2.36453 2.31699
3.14159 3.06782 3.00378
2.88726 2.82282 2.76621
2.71028 2.65083 2.59829
2.6393 1 2.58166 2.53059
2.57672 2.52057 2.47077
2.47059 2.41689 2.36914
2.38319 2.33144 2.28534
(B) (C) 0.2
(A) (B) (C)
0.3
(A) (B) (C)
0.4
(A) (B) (0
0.5
(A) (B) (C)
in which E is the modulus of elasticity, G is the modulus of elasticity in shear, I A is the cross-sectional area, p is the area moment of inertia of the cross-section, per unit volume, k is a numerical shape factor depending on the cross-section, w radian frequency, 4 is the ratio of the attached mass M to the total mass of the Mb = PAL, and (r =(l/~){-(r2+s2)+[(r2-s2)2+4/b2]“2}1’2,
(6)
p =(1/~)~+(r2+s2)+[(r2~s2)~+4/b2]“~~~’~,
(7)
~=(/32-.s2)/(~2-r2)=(a2+r2)/(a2+s2),
Y = fflP, where
is the mass is the beam
b2r2s’ < 1. If b*r*s* > 1 it is convenient
(839)
to use
(u=ja’=j(l/~){(r’+.~2)-[(r2-s2)2+4/b2]1’2}1’2
(10)
where Y’= Then the frequency
equation
sin ba' sin bp --
9a’b
a’/ P,
can be expressed
j=&i.
(Ila,
h)
as follows:
5
1+5
x[sinba’Asinba’(l-*)sinbp+~sinbn’sinbB*sinbp(l-A)]=O.
(12)
LETTERS
332
TO THE
EDITOR
On the other hand, the exact analysis reported in reference [8] makes it possible to obtain the fundamental frequency and the corresponding mode shape of a simply supported beam, with a concentrated mass attached to it, according to the classical onedimensional Bernoulli-Euler theory of flexural vibrations. In this case, one is able to determine the following frequency equation: &L[sin
pL sinh ApL sinh (1 - h)pL - sinh pL sin ApL sin (1 - A)pL]
+ 2 sin pL sinh pL = 0,
(13)
(PL)~ = pAL4w2/ EI. Here pL are the non-zero
solutions 3.
of the transcendental NUMERICAL
(14) equation
(13).
EXAMPLE
In order to adopt the quasi-universal notation whereby the symbol 0 denotes natural frequency coefficient, it was decided to present Tables I,2 and 3 in terms of this commonly accepted notation. In Table 1 are shown fundamental frequency coefficients determined by means of the exact expression (1) for a Timoshenko beam of uniform cross-section for r2 = O-0025 and 0.005, with s2 = 28r2/9 [ 1 I]. Numerical results are given for six different mass locations along the beam and for mass ratios 4 = 0, 0.2, 0.4, 0.5, 0.6, 0.8 and 1. Additionally, the coefficients were computed according to the Bernoulli-Euler theory by employing equation (13). The following conclusions may be of interest to a designer. (1) The relation Q,,,/fl,, (where Oibm is the fundamental frequency coefficient of the beam-mass system and Oib is the fundamental frequency coefficient of the beam) is essentially a function of the parameter 4 = M/Mb for a previously assigned value of A ; see Table 2. For instance, for A =0*5 and M/M, = 0.5, the parameter finlbrn/flnlb is approximately 0.19 for any value of r2. For A = 0.5 and M/M, = 1, the ratio Oib,,,/Oib is of the order of 0.32 for the entire range of values of r2 considered in the present study. (2) As shown in Table 3, the ratio OTb,Jnrb”m (where OTibm is the fundamental frequency coefficient of the beam-mass system determined by using Timoshenko’s theory and OFfm is the fundamental frequency parameter determined by means of the Bernoulli-Euler theory) is, essentially, a function of r2. In other words, once r2 is assigned, the ratio 0:,,,,/0:,“, remains practically constant regardless of the values of A and 4. For instance, for r2 = 0.005, the ratio 0T,,,,,/0::“, is approximately 0.96, while for r2 = 0.000625 it is of the order of 0.99. In conclusion, if one is interested in determining only the fundamental frequency of the simply supported beam-mass system one may always use, as a first order approximation, the Bernoulli-Euler theory. This conclusion will certainly not apply, in general, to higher frequencies. If one is interested in finding higher order frequencies of a mechanical system, the same procedure can be followed. ACKNOWLEDGMENTS The authors take this opportunity to thank Professor P. A. A. Laura for his encouragement during the course of this work. The present study has been sponsored by CONICET Research and Development Program (PID 3000500/88), while the second author has been supported by a grant from the Comision de Investigaciones Cientificas de la Provincia de Buenos Aires (CIC).
LETTERS
TO THE EDITOR TABLE
333
2
Values of 0, of the system shown in Figure 1 and analysis of the variation of the parameter as a (fllb,,,/LJnlb); (a) values of 0, as a function of A, C$ and r2; (b) values of 0,,,L?,, function of A, 4 and r2 0.02
0.005
0.0025
0.000625
0.00025
0
2.73460 2.30807 2.08255
3.00378 2.53059 2.28534
3.06782 2.58166 2.33144
3.12207 2.62419 2.36965
3.13369 2.63320 2.37772
3.14159 2.6393 1 2.38319
2.73460 2.40267 2.19281
3.00378 2.64113 2.41790
3.06782 2.69647 2.46991
3.12207 2.74281 2.51332
3.13369 2.75265 2.52253
3.14159 2.75934 2.52877
2.73460 2.52440 2.35231
3.00378 2.77870 2.60399
3.06782 2.83834 2.66297
3.12207 2.88852 2.71251
3.13369 2.89921 2.72306
3.14159 2.90648 2.73023
1 0.185 0,313
1 0.187 0.314
1 0.188 0.316
1 0.190 0.318
1 0.190 0.318
1 0.190 0.318
1 0.138 0.247
1 0.137 0.242
1 0.138 0.242
1 0.138 0.242
1 0.138 0.242
1 0.139 0.242
1 0.083 0.163
1 0.081 0.154
1 0.081 0.152
1 0.081 0.151
1 0.081 0.151
1 0.080 0.150
A
(a)
0
0.5
0.5 1 0
0.3
0.5 1 0
0.2
0.5 1
(b)
0
0.5
0.5 1
0.3
0
0.2
0
0.5 1
0.5 1
TABLE
3
,bm as a function of A, 4 and r2, where a:,,,, and a:&, are the fundamental Values of 0 Tb,L? nE frequency coejicients of the beam-mass system determined by using Timoshenko’s theory and Bernoulli- Euler theory respectively 0.02
0.005
0.0025
0.000625
0.00025
0
1
0.870 0.875 0.874
0.956 0.959 0.959
0.976 0.978 0.978
0.994 0.994 0.994
0.997 0.998 0.998
1 1 1
A 0.5
0
0.5 0.3
0.2
0
0.870
0.956
0.977
0.994
0.997
0.5 1
0.871 O-867
0.957 0.956
0.977 0.977
0.994 0.994
0.998 0.998
1 1 1
0 0.5 1
0.870 0.869 0.862
0.956 0.956 0.954
0.977 0.977 0.975
0.994 0.994 0.994
0.997 0.998 0.997
1 1 1
334
LETTERS TO THE EDITOR REFERENCES
1. P. A. A. LAURA,J. L. POMBO and E. A. SUSEMIHL 1974 JournalofSound and Vibration 37, 161-168. A note on the vibrations of a clamped-free beam with a mass at the free end. 2. M. SWAMINADHAM and A. MICHAEL 1979 Journal ofSound and Vibration 66, 144-147. A note on frequencies of a beam with a heavy tip mass. 3. C. N. BAPAT and C. BAPAT 1987 Journal of Sound and Vibration 112, 177-182. Natural frequencies of a beam with non-classical boundary conditions and concentrated masses. 4. L. S. SRINATH and Y. C. DAS 1967 Journal of Applied Mechanics, Transactions of the American Society of Mechanical Engineers, Series E, 784-785. Vibrations of beams carrying mass. 5. R. P. GOEL 1976 Journal of Sound and Vibration 47, 9-14. Free vibrations of a beam-mass system with elastically restrained ends. 6. D. A. GRANT 1978 Journal of Sound and Vibration 57, 357-365. The effect of rotary inertia and shear deformation on the frequency and normal mode equations of uniform beams carrying a concentrated mass. 7. H. SAITO and K. OTOMI 1979 Journal of Sound and Vibration 62, 257-266. Vibration and stability of elastically supported beams carrying an attached mass under axial and tangential loads. frequency of a 8. J. H. LAU 1981 Journal of Sound and Vibration 78, 154-157. Fundamental constrained beam. 9. P. A. A. LAURA, C. FILIPICH and V. H. CORTINEZ 1987 Journal of Sound and Vibration 117, 459-465. Vibrations of beams and plates carrying concentrated masses. 10. W. H. LIU and F.-H.YEH 1987 Journal of Sound and Vibration 117, 555-570. Free vibration of a restrained non-uniform beam with intermediate masses. 11. T. M. WANG and J.E. STEPHENS 1977 Journal of Sound and Vibration 51, 149-155. Natural frequencies of Timoshenko beams on Pasternak foundations.