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Vibrations of a beam clamped at one end and carrying a guided mass with an elastic support at the other
Journal of Sound and Vibration (1989) 129(2), 345-349
VIBRATIONS
OF A BEAM CLAMPED AT ONE END AND CARRYING MASS WITH AN ELASTIC SUPPORT AT THE OTHER
A GUIDED
1. INTRODUCTION
This letter deals with the dynamic behavior of the mechanical system consisting of a uniform cantilever beam with a guided tip mass. The right end of the beam does not rotate and is elastically restrained against translation. Both the problem and method of solution are quite elementary since classical beam theory is used. It is hoped, however, that design engineers will find the results useful in situations such as wind tunnel testing, piping systems with concentrated masses (e.g., valves) and elastic supports, and so on. Frequency coefficients and modal shapes are determined for a wide range of the governing mechanical parameters. 2. SOLUTION OF THE PROBLEM If one disregards shear and rotary inertia, the differential equation for small-amplitude, free vibrations of the beam shown in Figure 1 is EI a4y/aX4+pAoa2y/at2=0
(1)
where y is the lateral deflection, EI is the modulus of flexural rigidity of the beam, p is its density per unit volume, A0 is its cross-sectional area, x denotes the length co-ordinate of the beam, and t is the time.
;----------
L --;
Figure 1. Vibrating system analysed.
The boundary conditions for the structural system under consideration at x=0, at x = L,
ay/ax
Y(0, t) = 0, ay/ax( L, t) = 0,
EI a3y/ax3(L,
are as follows:
=0
(% b)
t) = M a2y/at2(L,
t) + K&L,
t).
(2c, d)
Here K, is the translational spring constant and M represents a rigid mass at x = L, which is guided in such a manner that the tangent to the elastic curve remains horizontal. In the case of normal modes of vibration one assumes a solution of the form Y(X, t) = T Y,(x)T(t), “=I
(3)
where Y,(x) is the nth mode of natural vibration and is given by Y,,(x) = A,, cos hx + B, sin k,,x + C, cash k,,x + D, sinh k,,x, k4,= (y,/ L)4 = o:( pA,/EI).
(4) (5)
345 0022-460X/89/050345+05
$03.00/0
@ 1989 Academic Press Limited
LETTERS TO THE EDITOR
346
Substituting equation (4) in equation (3), and then in the governing boundary conditions (2), results in a linear system of homogeneous equations in the unknowns A,, B,,, C, and 0,. In order to have a non-trivial solution, the vanishing of the determinant of the coefficient matrix gives the frequency equation (sin y, cash y, + cos y, sinh y,,) -
$y”
- $$;
(l-cosy,
b
coshy,)=O,
(6)
n
where y, = k,,L, and the vibration frequency of the nth mode is f. = 0,/2~
= (k’,/27r)(EI/pAO)“*.
(7)
Consider the case when K, = 0. Equation (6) then reduces to the frequency equation for a beam fixed at one end and carrying a guided mass at the other [l], (M/Mb)(l
-cos y. cash y,,) -(sin y, cash y, +cos y, sinh y.) =O.
If &+a, equation (6) tends-for clamped-clamped beam [2],
all values of M-to
(8)
the frequency equation for a
cos yn cash y, - 1 = 0.
TABLE
(9)
1
Values of the first three roots of the frequency equation (6) K,L3/ EI
0
1.0
10.0
100~0
1000~0
0
2.36502 5.49780 8.63938
2.41118 5.50078 864016
2.73320 5.52795 8.64718
3.84832 5.82222 8.72077
4.62469 7.28859 9.56627
0.5
1 a92354 4.99995 8.04992
1.96217 5dIOO52 8.04999
2.23544 5.00572 8.05064
3.31387 5.06701 8.05734
4.59375 6.25013 S-15366
1.0
1.71888 4.89277 7.96446
l-75354 4.89300 7.96448
1.99911 4.89506 7.96470
2.98765 4.91836 7.96692
4.54419 5.58377 7.99456
2.0
149954 4.82063 7.91266
1.52982 4.82071 7.91267
l-74460 4.82138 7.91273
2.61635 4.82868 7.91337
4.35334 5.02603 7.92064
4.0
1.28711 4.77804 7.88398
1.31312 4.77806 7.88398
1 a49763 4.77826 7.88400
2.24856 4.78030 7.88417
3.86906 4.81533 7.88602
6-O
1.17145 4.76269 7.87396
1.19513 4.76270 7.87396
1 a36309 4.76279 7.87397
2.04707 4.76373 7.87405
3.53873 4.77153 7.87488
8.0
1+@418 4.75477 7.86886
1.11629 4.75478 7.86886
l-27319 4.75483 7.86887
1.91223 4.75537 7.86891
3.31024 4.76240 7.86938
10-o
1.03713 4.74995 7.86578
1.05809 4.74995 7.86578
l-20681 4.74999 7.86578
1.81260 4.75034 7.86581
3.13964 4.75464 7.86611
\M/Mb
co
4.73004 7.85320 10.99561
LETTERS 2
(a)
3rd
TO THE
EDITOR
347
2nd
1
0
-1
Figure 2. Shapes of first three modes; a, = 0.8. (a) Ml Mb = 0; (b) Ml Mb = 2; (c) M/Mb = 10.
Figure 3. Shapes of first three modes; $, = 80. (a) M/M,
= 0; (b) M/M,
= 2; (c) M/M,
= 10.
348
LETTERS
Figure 4. Shapes of first three modes;
TO THE
Z?, = 800. (a)
EDITOR
M/Mb = 0; (b) M/Mb = 2;
(c)
M/Mb =
10.
Finally, if K, = 0 and M = 0, equation (6) becomes tan y, + tanh y, = 0, which is the frequency equation for a clamped-guided 3.
NUMERICAL
(10) beam [2].
RESULTS
3.1. Determination of eigenvalues The frequency equation (6) depends on the ratios M/Mb and K,L3/ EZ. Its roots have been computed numerically on a Texas PC and the eigenfrequencies y, = k,L for the first three modes of vibration are presented in Table 1 for different values of the cited parameters. 3.2. Determination of modal shapes It is easy to show that the modal shapes are given by the expression
Figures 2, 3 and 4 show the first three modes of vibration for $( K,L3/EZ) = O-8,80, 800, and in each case for M/Mb = 0, 2 and 10. Department of Engineering, Universidad National del Sur, Au. Alem 1253-8000 Bahia Blanca, Argentina
(Received 20 June 1988)
M. J. MAURIZI P. BELLES
LETTERS TO THE EDITOR
349
REFERENCES 1. P. A. A. LAURA and P. L. VERNIERE DE IRASSAR 1981 Applied Acoustics 14,93-99. Vibrations of a beam fixed at one end and carrying a guided mass at the other. 2. B. AKESSON and H. T.&GNFORS 1971 Chalmers University of Technology, Division of Solid Mechanics, Gothenburg, Publication No. 23, 1-12. Tables of eigenmodes for vibrating uniform one-span beams.
VIBRATIONS
OF A BEAM CLAMPED AT ONE END AND CARRYING MASS WITH AN ELASTIC SUPPORT AT THE OTHER
A GUIDED
1. INTRODUCTION
This letter deals with the dynamic behavior of the mechanical system consisting of a uniform cantilever beam with a guided tip mass. The right end of the beam does not rotate and is elastically restrained against translation. Both the problem and method of solution are quite elementary since classical beam theory is used. It is hoped, however, that design engineers will find the results useful in situations such as wind tunnel testing, piping systems with concentrated masses (e.g., valves) and elastic supports, and so on. Frequency coefficients and modal shapes are determined for a wide range of the governing mechanical parameters. 2. SOLUTION OF THE PROBLEM If one disregards shear and rotary inertia, the differential equation for small-amplitude, free vibrations of the beam shown in Figure 1 is EI a4y/aX4+pAoa2y/at2=0
(1)
where y is the lateral deflection, EI is the modulus of flexural rigidity of the beam, p is its density per unit volume, A0 is its cross-sectional area, x denotes the length co-ordinate of the beam, and t is the time.
;----------
L --;
Figure 1. Vibrating system analysed.
The boundary conditions for the structural system under consideration at x=0, at x = L,
ay/ax
Y(0, t) = 0, ay/ax( L, t) = 0,
EI a3y/ax3(L,
are as follows:
=0
(% b)
t) = M a2y/at2(L,
t) + K&L,
t).
(2c, d)
Here K, is the translational spring constant and M represents a rigid mass at x = L, which is guided in such a manner that the tangent to the elastic curve remains horizontal. In the case of normal modes of vibration one assumes a solution of the form Y(X, t) = T Y,(x)T(t), “=I
(3)
where Y,(x) is the nth mode of natural vibration and is given by Y,,(x) = A,, cos hx + B, sin k,,x + C, cash k,,x + D, sinh k,,x, k4,= (y,/ L)4 = o:( pA,/EI).
(4) (5)
345 0022-460X/89/050345+05
$03.00/0
@ 1989 Academic Press Limited
LETTERS TO THE EDITOR
346
Substituting equation (4) in equation (3), and then in the governing boundary conditions (2), results in a linear system of homogeneous equations in the unknowns A,, B,,, C, and 0,. In order to have a non-trivial solution, the vanishing of the determinant of the coefficient matrix gives the frequency equation (sin y, cash y, + cos y, sinh y,,) -
$y”
- $$;
(l-cosy,
b
coshy,)=O,
(6)
n
where y, = k,,L, and the vibration frequency of the nth mode is f. = 0,/2~
= (k’,/27r)(EI/pAO)“*.
(7)
Consider the case when K, = 0. Equation (6) then reduces to the frequency equation for a beam fixed at one end and carrying a guided mass at the other [l], (M/Mb)(l
-cos y. cash y,,) -(sin y, cash y, +cos y, sinh y.) =O.
If &+a, equation (6) tends-for clamped-clamped beam [2],
all values of M-to
(8)
the frequency equation for a
cos yn cash y, - 1 = 0.
TABLE
(9)
1
Values of the first three roots of the frequency equation (6) K,L3/ EI
0
1.0
10.0
100~0
1000~0
0
2.36502 5.49780 8.63938
2.41118 5.50078 864016
2.73320 5.52795 8.64718
3.84832 5.82222 8.72077
4.62469 7.28859 9.56627
0.5
1 a92354 4.99995 8.04992
1.96217 5dIOO52 8.04999
2.23544 5.00572 8.05064
3.31387 5.06701 8.05734
4.59375 6.25013 S-15366
1.0
1.71888 4.89277 7.96446
l-75354 4.89300 7.96448
1.99911 4.89506 7.96470
2.98765 4.91836 7.96692
4.54419 5.58377 7.99456
2.0
149954 4.82063 7.91266
1.52982 4.82071 7.91267
l-74460 4.82138 7.91273
2.61635 4.82868 7.91337
4.35334 5.02603 7.92064
4.0
1.28711 4.77804 7.88398
1.31312 4.77806 7.88398
1 a49763 4.77826 7.88400
2.24856 4.78030 7.88417
3.86906 4.81533 7.88602
6-O
1.17145 4.76269 7.87396
1.19513 4.76270 7.87396
1 a36309 4.76279 7.87397
2.04707 4.76373 7.87405
3.53873 4.77153 7.87488
8.0
1+@418 4.75477 7.86886
1.11629 4.75478 7.86886
l-27319 4.75483 7.86887
1.91223 4.75537 7.86891
3.31024 4.76240 7.86938
10-o
1.03713 4.74995 7.86578
1.05809 4.74995 7.86578
l-20681 4.74999 7.86578
1.81260 4.75034 7.86581
3.13964 4.75464 7.86611
\M/Mb
co
4.73004 7.85320 10.99561
LETTERS 2
(a)
3rd
TO THE
EDITOR
347
2nd
1
0
-1
Figure 2. Shapes of first three modes; a, = 0.8. (a) Ml Mb = 0; (b) Ml Mb = 2; (c) M/Mb = 10.
Figure 3. Shapes of first three modes; $, = 80. (a) M/M,
= 0; (b) M/M,
= 2; (c) M/M,
= 10.
348
LETTERS
Figure 4. Shapes of first three modes;
TO THE
Z?, = 800. (a)
EDITOR
M/Mb = 0; (b) M/Mb = 2;
(c)
M/Mb =
10.
Finally, if K, = 0 and M = 0, equation (6) becomes tan y, + tanh y, = 0, which is the frequency equation for a clamped-guided 3.
NUMERICAL
(10) beam [2].
RESULTS
3.1. Determination of eigenvalues The frequency equation (6) depends on the ratios M/Mb and K,L3/ EZ. Its roots have been computed numerically on a Texas PC and the eigenfrequencies y, = k,L for the first three modes of vibration are presented in Table 1 for different values of the cited parameters. 3.2. Determination of modal shapes It is easy to show that the modal shapes are given by the expression
Figures 2, 3 and 4 show the first three modes of vibration for $( K,L3/EZ) = O-8,80, 800, and in each case for M/Mb = 0, 2 and 10. Department of Engineering, Universidad National del Sur, Au. Alem 1253-8000 Bahia Blanca, Argentina
(Received 20 June 1988)
M. J. MAURIZI P. BELLES
LETTERS TO THE EDITOR
349
REFERENCES 1. P. A. A. LAURA and P. L. VERNIERE DE IRASSAR 1981 Applied Acoustics 14,93-99. Vibrations of a beam fixed at one end and carrying a guided mass at the other. 2. B. AKESSON and H. T.&GNFORS 1971 Chalmers University of Technology, Division of Solid Mechanics, Gothenburg, Publication No. 23, 1-12. Tables of eigenmodes for vibrating uniform one-span beams.