Applied AcouStics 14 (1981) 93-99

VIBRATIONS OF A BEAM FIXED AT ONE END AND CARRYING A GUIDED MASS AT THE OTHER

P. A. A. LAURA and P. L. VERNII~RE DE IRASSAR

Institute of Applied Mechanics, Puerto Belgrano Naval Base, 811 (Argentina)

(Received: 12 May, 1980)

S U M M A R Y

This paper presents an exact solution o f the title problem, using classical beam theory. It is also assumed that the tip mass is guided in such a manner that the end o f the beam does not rotate.

INTRODUCTION

Consider the mechanical system shown in Fig. 1. The bar is rigidly clamped at the lower end. It carries a rigid mass at x = Lwhich is

guided in such a manner that the tangent to the elastic curve remains vertical. As shown in reference 1, the critical buckling load for this structural element is

simply:

7r2 E1 P e r - L 2 (1)

On the other hand, it appears that the dynamic aspect of the behaviour of such a basic system has not been investigated in the past. (Reference is made here to the excellent survey paper by Wagner and Ramamurti . 2 Admittedly, the problem and method of solution are quite elementary but since no numerical results are available in the open literature it is hoped that the present paper will be of interest to design engineers.)

Frequency coefficients and modal shapes are determined in the present paper for a wide range of the governing mechanical parameters.

93 Applied Acoustics 0003-682X/81/0014-0093/$02.50 © Applied Science Publishers Ltd, England, 1981 Printed in Great Britain

94 P. A. A. LAURA, P. L. VERNIERE DE IRASSAR

-I × = L

!

Ix:O / I I I / 1 1 / / / , . , ' / / / / / / / /

Fig. 1. Vibrating system under study,

MATHEMATICAL STATEMENT OF THE PROBLEM AND ITS SOLUTION

Assuming classical beam theory, the behaviour of the system is governed by the partial differential equation:

0%, pA o ~2w

O.r 4 + E1 6qt 2 - - 0

subject to the boundary conditions:

~, 14'

__o x = L

~314'1 = M (32w

In the case of normal modes of vibration:

w(x, t) = W(x)e ~°"

and, substituting into eqn. (2), one obtains:

d4W . . . . k 4 W = 0

dx 4

h.~ E[ T " + T = 0

pAo

(2)

(3(a). 3(b))

(4)

(5)

(6)

(7(a))

(7(b))

95

Igl,.(Yi)[ = 0

where:

gl l (Yi) = 1

gla(Yi) = 1

g21(Yi) = 0

g 2 3 ( Y i ) = I

g31 (Yi) = -- sin Yi

g 3 3 ( Y i ) = Ca'~

M g41 ( Y i ) = sin Yl + - - Yi co s Yi

M g 4 3 ( Y i ) = ey' d- q T y i ~''

M t ,

M~ = pAoL

( l , m = 1 . . . . . 4)

g12(Yi) = 0

g14(Yi) = 1

g Z 2 ( Y i ) = 1

g E 4 ( Y i ) = -- 1

g 3 2 ( Y i ) = c o s y i

g a 4 ( Y i ) : - e -~"

M g 4 2 ( Y i ) =- - - c o s y i + - - Y i sinyi

M g44(Yi) = - e - Y ' + ~ Yi e-y'

Yi = ki L

Expanding the secular equat ion (eqn. (l 1)), one finally arrives at the expression'

M s iny i c o s h y i + cosyi sinhy~ - - = ( 1 2 )

My Yi[1 - cosy i coshy i ]

N U M E R I C A L RESULTS

In order to obtain the roots o feqn . (12) it is convenient to express it in the following form:

M v [ s i n y c o s y + c o s y s i n h y ] 1 z ( y ) = M [1 - c o s y c o s h y ] Y - 1 = 0 (13)

(11)

V I B R A T I O N S OF A FIXED BEAM C A R R Y I N G A G U l D E D MASS

The solution of eqn. (7(b)) is:

T = A costot + B s i n m t (8)

where:

tO = k 2 E p ~ ° (9)

On the other hand, the solution of eqn. (7(a)) results in:

W = C l c o s k x + C z s i n k x + C 3 e kx -k- C 4 e - k x (10)

Substituting eqn. (10) into the governing boundary condit ions (eqns. 3(a) and (b), (4) and (5)) yields the following determinantal equation:

96 P. A. A. LAURA, P. L. VERNII~RE DE IRASSAR

TABLE 1 VALUES OF (kiL) AS A FUNCTION OF M / M ,

M/M, y 0 (I.2 0'4 0"6 0"~

y~ 2"365 2"133 1"982 1"872 1"787 .v 2 5"497 5-174 5"040 4"968 4"923 Y3 8"639 8"215 8"085 8"023 7"987 .v4 11"780 1 1 " 2 9 3 11"176 11"125 14"217 .v 5 14'922 14"389 1 4 " 2 8 5 14"241 17'345

y 1 l-2 1"4 1"6 1"8

3'1 1"718 1'661 1'612 1"570 1"532 3"2 4"892 4"870 4-853 4'840 4-829 Y3 7"964 7"947 7"935 7"926 7"918 .v4 11'078 1 1 " 0 6 5 11"056 11"049 11"043 )'5 14'202 17"324 17"318 17"313 17"309

y 2 2"2 2"4 2"6 2-8

)'1 1"499 1"469 1'442 1"417 1"394 .v 2 4"820 4"813 4'806 4"801 4.796 Y3 7"912 7"907 7"903 7"899 7'896 y~ 11'038 1 1 " 0 3 5 1 1 " 0 3 1 1 1 " 0 2 9 11"026 .v~ 17'306 17"304 17-302 17"300 17"298

y 3 3.2 3.4 3.6 3.x

y~ 1.373 1.353 1.335 1.318 1.302 )'2 4.792 4.789 4.785 4.783 4.780 Y3 7.893 7'891 7.889 7.887 7.885 .v 4 11.024 11.023 1 1 . 0 2 1 11.020 11.018 Y5 17.297 17.296 17-295 17-294 17-293

y 4 4.2 4.4 4.6 4.~

y, 1.287 1.272 1.259 1.246 1.234 )'2 4.778 4.775 4.773 4.772 4.770 Y3 7.883 7.882 7.881 7.880 7.878 3'4 11.017 11.016 11.015 11.014 11.014 35 17.293 17.292 1 7 . 2 9 1 1 7 . 2 9 1 17-290

y 5 10 15 20

vl 1.222 1.037 0.939 0.876 v2 4.768 4-749 4.743 4-740 ).~ 7.877 7.865 7.861 7.859 Y4 11.013 11.004 1 1 - 0 0 1 11.000 )'5 17.290 17.284 17.282 17.281

VIBRATIONS OF A FIXED BEAM CARRYING A GUIDED MASS 97

The function z(y) shows rapid oscillations, attaining very large values between successive roots. The slope of the function at each root is, therefore, very close to the vertical.

This characteristic of the function suggests the use of the 'modified false position method' to ensure convergence in the determination of the roots. 3 In order to initiate the iterative process, the roots were first bracketed by means of a straight search process.

Table 1 shows values of Yi = (kiL) for the first five modes of vibration and as a function of M/M,,.

Figures 2, 3 and 4 show the first three modes of vibration for several values of M/M,,. It is quite easy to show that the modal shape is defined by:

+ \ c~skL~c-osshki ;~slnkL L -

It is interesting to note that, as M/M,, increases, the node situated at the left of the concentrated mass becomes closer to the end of the beam. One should remember at this point that, as M approaches infinity, eqn. (12) degenerates into the well known expression:

cos(k/L) , cosh (kiL) = 1 (15)

which corresponds to a beam fixed at both ends.

I_

M/M v : 020 M/M v = 10

Fig. 2. Firs t three moda l shapes (M/M, = 0.20; 1-0).

98 P. A. A. LAURA. P. L. VERNII~RE DE IRASSAR

L F

M/M v : 2 0 M/My: 30

Fig. 3. First three modal shapes (M/M, = 2.0; 3"0).

L

M/M v = 10 0

Fig. 4.

9 ~ ~ 200

First three moda l shapes (M/M, = 10.0; 20.0).

VIBRATIONS OF A FIXED BEAM CARRYING A GUIDED MASS 99

ACKNOWLEDGEMENTS

The authors are indebted to research engineers R. O. Grossi and L. Ercoli for their generous co-operation in the determination of the numerical results.

REFERENCES

1. S. TIMOSHENKO and J. M. GERE, Theory of elastic stability, McGraw-Hill Book Company Inc., New York, 1961.

2. H. WAGNER and V. RAMAMURTI, Beam vibrations. A review. The Shock and Vibration Digest, 9(9) (1977), pp. 17-24.

3. S. D. C o t ~ , Elementary numerical analysis. McGraw-Hill Book Company, Inc., New York, 1965.