on the dynamic behaviour of structural elements carrying elastically mounted, concentrated masses

ON THE DYNAMIC BEHAVIOUR OF STRUCTURAL ELEMENTS CARRYING ELASTICALLY MOUNTED,

CONCENTRATED MASSES

P. A. A. LAURA, a E. A. SUSEMIHL, b J. L. POMBOf L. E. LUISONI ~ and R. GELOS e

a Director and Principal Investigator, CONICET. b Scientist, SENID. c Scientist, CON1CET. n Research Engineer, CONICET. e Research Engineer, CONICET.

Institute o f Applied Mechanics, Base Naval Puerto Belgrano (Argentina)

S U M M A R Y

The present paper deals with the analysis of certain dynamic aspects of the blhaviour of beams and plates which support elastically mounted masses.

Shear and rotatory inertia effects are not taken into account in the present investigation. An exact solution is presented in the case of a simply supported beam.

This solution can be easily extended to the problem of a simply supported rectangular plate.

It is also shown that use of a variational formulation leads to accurate and simple expressions for natural frequencies and dynamic displacements and stresses which are ideal from a designer's viewpoint. The case of supports elastically restrained against rotation is also considered.

The experimental phase of the investigation shon;ed good agreement with experimental results.

1. INTRODUCTION

Naval and mechanical engineers are very often confronted with the problem of mounting different varieties of engines and motors on structural elements. In order to avoid dangerous resonance situations, the designer must be able to predict natural frequencies of the overall mechanical system: structure-motor and its elastic mounting.

Ultimately he should also determine mode shapes and dynamic stresses induced by any dynamic disturbance generated by the engine (Fig. l).

121 Applied Acoustics (10) ( 1977 ) - -© Applied Science Publishers Ltd, England, 1977 Printed in Grea t Britain

122 P. A. A. LAURA, E. A. SUSEMIHL, J. L. POMBO, L. E. LUISONI, R. GELOS

Several papers have been written on the subject but in general it is assumed that the mass is rigidly attached to a beam or plate. 1-6

The present paper deals with the solution of a few simple but practical problems considering several types of structural elements and boundary conditions.

Motor of t Mass M (t): Dynamic

Disturbanc

Elastic , . . • .

Fig. I. Mass moto r elastically mounted on a structural element.

The investigation reported herein belongs to a broad but detailed long-range research plan aimed at the improvement of the dynamic design of structural elements. More specifically, the ultimate goal is to provide the basis for more rational design procedures.

2. THE CASE OF A SIMPLY SUPPORTED BEAM

Consider the mechanical system shown in Fig. 2. If one neglects shear deformation and rotatory inertia effects, the dynamic behaviour of the system is described by the equations:

E104w(x' t) 02w Ox 4 + p A - ~ T = F ( t ) 6 ( x -- ;~) (1)

~2 W w(x , t ) = ~x 2 = 0 (x = O , L ) (2)

where E = Young's modulus; I = moment of inertia; p and ,4 = density of cross- sectional area of the beam and 6(x - ~) = Dirac's delta function of argument ( x - ~).

DYNAMIC BEHAVIOUR OF STRUCTURAL ELEMENTS 123

The functional relation F(t) denotes the action of the spring-mass system on the beam (Fig. 2).

x:O

ILiliilll ll]l lllIllllll

jo.)t F,-e

7- f

JLlJlJJl]JlJlJJJiJ:JlilllJJIJlJlIIJJJil]l /___N

x='~

Fig. 2.

X

Simply suppor ted beam carrying a concentrated mass elastically m o u n t e d on the s tructure.

Let z I be the displacement of M at any given instant and z2 the displacement of the other spring end. One immediately obtains:

d2zl g - ~ i ~- = k(z2 - zl) + Fo . exp (icot) (3)

Making z2 = wA. exp (iogt) and replacing in eqn. (3) results in the ordinary differential equation:

d2z M-~-~ + kz = - - (o92 .g .wA + Fo)exp (iogt) (4)

where: Z ~ Z 2 - - Z 1

The particular solution of eqn. (4) is given by:

~ 2 . M . w A + Fo z = o92M - k exp (Rot) (5)

The action of the spring-mass system then results:

~o2.M.w~ + Fo F( t ) = - k . z = 1 - o92M/k exp (imt) (6)

Taking now: w(x, t) = W(x) exp (icot) (7)

124 P. A. A. L A U R A , E. A. SUSEMIHL, J. L. POMBO, L. E. LUISONI, R. GELOS

and replacing eqns. (6) and (7) in eqn. (1) one obtains:

d 4 W 0)2 M . W~ + Fo 6(x 7) (8) E I ~ - p A . w 2 W = "

dx '~ 1 - 0)2 M / k

Let W(x) and 3(x - 7) be expanded in terms of the infinite set of modal functions of the structure sin (nnx /L) :

W(x ) = b . . s in ~ (9(a)) n = l

6 (x - ~,) = x ~ c . . nT[x

s i n (9(b)) L

n = l

where: 2 f L nzrx 2 n t c ,

c. = 3(x -- y) sin - - dx = - sin - - (9(c)) o L L L

Substituting eqns. (9) into eqn. (8) results in the expression:

E1 - p . A 0 ) 2 b. sin ~.

oc

Fo + eo2M E bi s i n J__~__ ~

J=| L 2 n ~ l n r c ) ' . s i n n n X = 1 ~-~-Y'M~" = sin L ~ (10)

From the analysis of eqn. (I0) one concludes immediately that the b.'s are solutions of an infinite number of equations since, equating like terms, one obtains:

00

--~ - p . A 0 ) 2 b. = 1 - 0 )2M/k L T (11)

It is convenient to introduce the following definitions:

0)0 = 17 :=: (/)1

[EI] '~ ( L ) 2 a)l Wn = \ p A ] = 0)1n 2 6 = Wo

(12)

0)I = ~ ' = -- L

M p A L


where:

From eqns. (12) one can easily show that:

62 = 6ol 2 E I n 4 M /(M/pAL)] n* = m_ n , (002 = p . A . L 4 . k = [(k/EI/L3)] r

k r =

EI/L 3

Substituting eqns. (12) and (13) into eqn. (11) results in the expression:

,~, (n 4 - 02)(1 - r/262) sin (nn~') s i n (nnr') /21/b'j sin (jnT') - 2mq2 b'. = 2mr/2

Defining:

where: pAre 12L pAEIn4L Eln 4

b'i = 2F-"-'~ bl = pAL4 .2Fo bl = 2FoL 3 bi

7' = 7/L

] .j a l l a 1 2 • . . a l n 1

A = { a i f l = a.21 a22 . . . a2,

k an 1 an2 • " • ann

V2 V ~ {/3i} ~--- --n

Lb.J

(13)

(14)

(IS)

f sin ( ina') sin (jrca') i ~ j

a l i l s in2 ( jna') - ( j4 _ r/2)(1 _ r/262) ~m--~m ~ ~ i = j

sin (in0~') vl = 2mr/2

One can express eqn. (14) in matrix notat ion: Ab' = v (16)

The solution of the free vibrations problem is attained making Yo = 0 in eqn. (l 1). The natural frequency coefficients r h are roots of the secular determinant:

det A(r/i ) = 0 (17) The amplitudes of motion in the case of forced vibrations are given by:


t l n

Making:

W*(x) = EIW(x). x' = x _ Fo L3 ' L

and substituting into eqn. 08) one obtains:

(18)

W*(x) = ~ b', sin (19) n

Bending moment and shear force amplitudes are calculated using well known expressions from the strength of materials theory. For instance, the amplitudes of bending moments are given by:

d2W [M(x)l = - E1 dx---- T (20)

Substitution of eqn. (18)into eqn. (20)yields:

(7 ) 2 IM(x)l = E1 ~ b,. sin (nnx') n

Elrt2 2FoL 3 Eln4 - L2 Eln4 ~ n 2 2~oL 3 b, sin (nnx')

n

2 = FoL -~ ~ , nZb', sin (nnx')

n

Defining a dimensionless parameter:

M*(x) =

one finally obtains:

(21)

IM(x)l (22) FoL

2 M*(x) = 7T ~.._, nZb', sin (nnx') (23)

2.1. Numerical results Tables 1 through 7 depict values of amplitudes of displacement and bending

moments for several values of the parameters M/My; ~/L; k/EI/L 3 as the frequency ratio ~o/¢oo varies.

Reference 7 contains numerical information on the subject when the exciting force is applied directly to the beam. The equations derived in the present investiga-

TABLE 1 COMPARISON OF AMPLITUDES OF DYNAMIC DISPLACEMENTS IN THE CENTRE OF

A SIMPLY SUPPORTED BEAM

to/coX~Xk ~,' = ~,/L 0-1 0.3 0"5

0'1 0-006230* 0.016667 0.021041 0-006236"I 0 " 0 1 6 6 6 7 0"021026

0-3 0.006794 0-018143 0-022864 0-006800 0-018142 0.022849

0.5 0"008281 0.022037 0-027678 0.008287 0.022036 0-027663

0.7 0'012261 0.032459 0"040562 0.012267 0.032458 0"040547

0.9 0.033213 0.087313 0.108367 0'033219 0.087313 0.108352

* Reference 7, p. 103. ~" Values obtained in the present investigation:

M/M,, = 10-1o; k/EI/L 3 = 107.

TABLE 2 DYNAMIC AMPLITUDES AS A FUNCTION OF x/L FOR SEVERAL VALUES OF (D/tO 1 AND y/L (M/My = 0.20;

k/EI/L3 = 1)

M/My = 0'20 k/EI/L 3 = I x/L to/o91 = 0.10 0.30 0.50 0.70 0-90

0 0 0 0 0 0 0-10 0-00335 0.00379 0.00086 0'00053 0.00074 0.20 0.00587 0.00667 0.00153 0.00097 0-00138 0'30 0-00732 0.00837 0.00194 0.00126 0.00186 0.40 0.00789 0-00908 0.00214 0 - 0 0 1 4 1 0.00214 0"50 y/L = 0.1 0.00775 0.00897 0.00213 0-00143 0.00222 0'60 0.00696 0.00809 0.00194 0.00132 0.00209 0.70 0.00565 0-00659 0.00159 0.00109 0.00176 0'80 0.00398 0-00465 0-00113 0-00078 0.00127 0.90 0.00207 0.00242 0.00059 0.00041 0.00067 ! .00 0 .00000 0 0 0 0

0 0 0 0 0 0 0.10 0.00734 0.00812 0.00191 0-00122 0-00174 0.20 0.01380 0.01528 0.00359 0 " 0 0 2 3 1 0.00331 0.30 0-01847 0.02050 0-00484 0-00314 0.00453 0.40 0.02078 0-023 ! 4 0.00550 0.00360 0.00527 0.50 0-02077 0-02322 0.00556 0-00368 0.00549 0-60 y/L = 0"3 0.01888 0'02118 0-0051 ! 0"00342 0'00517 0'70 0"01551 0-01744 0 " 0 0 4 2 3 0'00286 0.00437 0-80 0-01097 0'01237 0 - 0 0 3 0 1 0 " 0 0 2 0 5 0.00316 0-90 0.00566 0.00639 0-00156 0'00107 0.00165 1.00 0 .00000 0 0 0 0

0 0 0 0 0 0 0.10 0.00779 0.00857 0.00207 0.00137 0.00201 0.20 0.01492 0-01641 0-00396 0-00262 0-00384 0'30 0.02081 0.02287 0.00550 0'00363 0.00529 0.40 0.02481 0.02723 0.00654 0.00430 0.00624 0"50 )'/L = 0'5 0"02625 0-02880 0'00691 0 " 0 0 4 5 4 0.00657 0"60 0'0248 ! 0'02723 0'00654 0"00430 0'00624 0"70 0-02081 0"02287 0;00550 0 " 0 0 3 6 3 0.00529 0.80 0"01492 0'01641 0.00396 0 " 0 0 2 6 2 0.00384 0.90 0'00779 0.00857 0.00207 0 " 0 0 1 3 7 0.00201 1.00 0 0 0 0 0

128 P. A. A. LAURA, K. A. SUSKMIHL, J. L. POMBO, I,. E. I_UISONI. R. GELOS

TABLE 3 DYNAMIC AMPLITUDES AS A FUNCTION OF x/L (M/M,. - I '0; k/EI/L 3 :- 1,0)

M/M,, = 1.0 k/El/L3 : I x lL <o/co i = 0-I 0 0.30 0.50 0.70 0-90

0 0 0 0 0 0 0-10 0-11576 0-00037 0.00014 0-00001 0.00014 0.20 0.20292 0.00065 0-00025 0-00018 0-00026 0.30 0-25290 0.00081 0.00032 0.00023 0.00035 0-40 0.27254 000088 0.00035 0.00026 0.00041 0"50 7/L = 0.1 0.26785 0.00087 0.00035 0.00026 0.00042 0.60 0.24066 0.00079 0.00032 0.00024 0.00040 0.70 0.19509 0.00064 0.00026 0.00020 0.00033 0.80 0-13741 0.00045 0-00019 0-00014 0.00024 0.90 0.07143 0.00024 0.00001 0.00007 0-00013 1 0 0 0 0 0 0 0

0 0 0 0 0 0 0-10 0-51324 0.00080 0.00032 0.00022 0.00033 0.20 0-96493 0.00151 0.00060 0.00042 0.00063 0.30 1.29178 0.00203 0.00081 0.00056 0.00086 0.40 1.45334 0.00229 0.00091 0-00066 0-00100 0.50 v/L = 0.3 1.45277 0-00229 0.00093 0-00068 0-00104 0.60 1.32040 0.00209 0.00085 0-00063 0.00098 0.70 1.08448 0.00172 0.00070 0.00052 0.00083 0.80 0-76729 0.00122 0.00050 0-00038 0.00060 0.90 0.39593 0.00063 0.00026 0.00020 0-00032 1.00 0 0 0 0 0

0 0 0 0 0 0 0.10 1.14894 0.00085 0.00034 0.00025 0.00038 0.20 2.20135 0"00163 0.00066 0.00048 0.00073 0.30 3-07054 0.00228 0.00092 0.00067 0.00101 0.40 3.66117 0.00271 0.00110 0.00079 0.00119 0.50 ~,/L = 0.5 3.87348 0.00287 0.00115 0.00083 0.00125 0.60 3.66117 0.00271 0.00110 0-00079 0.00119 0.70 3.07054 0.00228 0.00092 0.00067 0.00101 0-80 2-20135 0-00163 0.00066 0.00048 0.00073 0.90 1-14894 0.00085 0.00034 0.00025 0-00038 1.00 0 0 0 0 0

TABLE 4 COMPARISON OF DYNAMIC BENDING MOMENT

AMPLITUDES M/FoL (y/L = 0.5)

co log i Reference 21 M/My = 10- l o k/EI/L 3 = 108

0'1 0"252 0.250 0-3 0.270 0.268 0'5 0.318 0.316


T A B L E 5 DYNAMIC BENDING MOMENT AMPLITUDES: IM[/FoL (7/L = 0'1; M/Mo = 1 0 - 1 ° ; k/EI /L3 = 108)

CO/to I x/L 0"10 0 '30 0"50 0.70 0.90

0 '10 0"08819 0.09001 0.09477 0.10724 0-17116 0"30 0"07053 0-07517 0.08734 0"11958 0-28750 0"50 0"05063 0"05617 0.07082 0.11006 0"31678 0.70 0"03049 0-03486 0'04646 0.07782 0-24453 0-90 0"01019 0-01183 0"01621 0-02811 0"09178

T A B L E 6 DYNAMIC BENDING MOMENT AMPLITUDES: IMI/FoL (7/L = 0"30; M / M v = 10-1o ; k /El /L3 = 108)

tO/tO I x / L 0.10 0 '30 0.50 0.70 0-90

0"I0 0"07053 0-07517 0"08734 0-11958 0"28750 0"30 0-20934 0.22136 0"25293 0"33688 0-77589 0"50 0'15165 0"16620 0-20462 0"30746 0"84882 0 '70 0'09131 0"10286 0'13349 0"21599 0'65300 0 '90 0"03049 0"03486 0.04646 0'07782 0"24453

T A B L E 7 DYNAMIC BENDING MOMENT AMPLITUDES: IMI/FoL (7/L = 0"10; M/M~, = 1 ; k /EI /L 3 = I)

(O/tO 1 x/L 0.10 0.30 0.50 0.70 ' 0.90

0- I 0 3 "39142 0.01155 0.00404 0-00228 0"00218 0"30 3"03217 0-00965 0.00373 0.00255 0"00365 0-50 2' 17674 0-00721 0"00302 0.00234 0"00402 0"70 i '31107 0'00447 0-00198 0.00166 0'00310 0-90 0'43793 0-00152 0 '00069 0.00060 0"00116

tion should degenerate properly into those obtained in reference 7 as M / M v

approaches zero and k / E I / L 3 approaches infinity. In order to carry out this comparison it was decided to take numerical values

for the parameters which would give an idea of the trend of the present analytical solution. Accordingly, Tables l and 4 show a comparison of results when M / M o =

10- io; k / E I / L s = l0 s. The agreement between numerical results is quite good. One can see in all cases the influence of all the parameters involved. Tables 5 and

6 illustrate very clearly the fact that the maximum bending moment amplitude displaces towards the centre of the beam as the exciting frequency increases.

The extremely high values of ] M I / F o . L corresponding to ~o/o91 = 0-10 in the case of Table 7 indicate'the proximity of a resonance situation.

Fifty terms of the expansion defined in eqns. (9) and (1 l) have been taken in all cases.

130 P . A . A . LAURA, E. A. SUSEM1HL, J. L. POMBO, L. E. LUISONI, R. GELOS

3. BEAMS WITH ENDS ELASTICALLY RESTRAINED AGAINST ROTATION

In this case, the governing differential system is defined by eqn. (8) and the boundary conditions (Fig. 3):

/ lJllJllllfJllJllIJIJJJJJJJ

I M ] []] l l l l l l ] l l l l l l l l l

× ' - ' ~

Fig. 3. Beam with ends elastically restrained against rotat ion.

W(L/2) = 0 (24(a))

~ (L/2) d 2 W = - tpE. l.--d~x2 (L/Z) (24(b))

where ~0 is the flexibility coefficient of the supports (in the case of a rigid clamp q~ = 0 and for a simply supported beam ~o = oo).

Since the modal functions involve non-trigonometric and hyperbolic functions, it will be shown that it is more convenient to make use of a variational formulation. The solution of the differential system is expressed in terms o f . a polynomial expansion where each co-ordinate function satisfies identically the boundary conditions. Substitution of the polynomial expression in eqn. (8) results in an error or residual function. The Galerkin method is then used to minimise the error expression. It is quite convenient to express W in the form:

N

W "~ E Aj(otyx'* + fli x2 -t- l)x J (25) jffi0

where ~tj and fly are obtained substituting each co-ordinate function in eqn. (24). For instance, as a first approximation one can take the first term of eqn. (25). The approximate solution is then given by:

W a ---- Ao(0~o X4 "4" flo X2 d- 1) (26) For the free vibrations problem one makes F o -- 0 and, substituting eqn. (26),

one obtains the error function:


e(x) = 2 4 ~ o E I - pAo92(tXo X4 + flo x2 4- I)

to2M 1 - o92M/k

The coefficients ct o and flo are given by:

• 1 ~ o ' + 1 % = (L/2) 4 5¢p' + 1

where:

(CtoXa 4- flox 2 4- 1)3(x - 7) (27)

flo = - 1 + a (Lf2)2 (28)

~pE.I.

U2

Galerkin 's method requires that the error funct ion be or thogonal with respect to each co-ordinate function. Accordingly one sets:

f LI2 ~(X)(~O X4 4- flO X2 4- l ) d x = 0 (29) -LI2

The following frequency equation is then obtained:

O94pAL N 1 -- o92 E . I . L . N 2 + p A . L . N ~ + ~ M N 3 + E . I . L . N 2 = 0 (30)

= ~z L 8 ~fl -6 f12 L'* N, + +-WO

3 ~2L4 N 2 = " ~ + otflL 2 + 12ct

N3 = (0~),4 + fl)~2 + 1)2

The roots o f eqn. (30) are then given by:

L 4 l k l k

00 2 pAL'* --f N: 4- 2M/M"'----~ EI['-~ N, 4- -~ EI/L------ $ Na

"El = N 1

L 4 f l 4- ~ (31(a)) + ~ + LZ 1

-T-

(31(b))

N2 + 2M/M-------~EI/LaN! + 4 MFMvEI/L 3 IN2 NI (32)

Equat ion (32) yields the two lowest natural frequencies o f the coupled system beam-spring-concentrated mass.

Clearly, the approximation defined by eqn. (26) is poor in the case of a forced vibration problem if one is interested in predicting displacements and stresses. Taking a two-term approximation:

132 P . A . A . LAURA, E.. A. SUSEMIHL J. L. POMBO, L. E. LUISONI, R. GELOS

W a = Ao(C(o x4 + flo x2 + 1) + A l ( ~ l x 4 + /61 x2 + ] )x

a n d a p p l y i n g the p r o c e d u r e p r e v i o u s l y de sc r ibed , o n e o b t a i n s " (33)

w ( x , t ) ~ - t A o [ 1 6 ~ ' O ( L ) 4 - 8 ~ ' O ( L ) e + 1]

[ (;)' (;)]) + L . A j 16~'1 - 8fl ' 1 + e x p ( i ~ t ) (34)

w h e r e : (R1 /R3) - [(RIR2 + R1R42 + R:Rs2)/(R2R3)](¢o/(Oo) 2

Ao = Folk

Folk RIRa L . A j = (R1/R3) _ [(R1R2 + RjR42 + R2R32)/(R2R3)](oa/Oao) 2 'R2R3

R 3 ~ N 3

R , I " ~ N 4

NI = 76"8e 'o 2 - 256CC'o/~'o + 384~ 'o N2 = 0 " l l l l l ~ ' o 2 - 0"571428~ 'of l 'o + 0"8fl 'o 2 + 0"4C~'o - 1"3333/?'o + 1

N 3 = 16~'oO' /L) 4 - 8/~'o('Y/L) 2 + 1 N,, = 16~'~(~//L) 5 - 8~'t(~/L) 3 + (?,/L) N5 = 68"571428a ' j 2 - 192c~'1/~'1 + 160a ' j

N 6 = 0"0227272~ '12 _ 0"11111~'1/T1 + 0"1428571TI 2 + 0"0714286~'z - 0"2,B'~ + 0"083333

0C O 161 + 3 q ) '

L 4 I + 7q;

Ct'o -- ~oL '* /16; f l 'o = floL2/8

16 8 1 + 5tp' 8 L4 c~ o, flo L 2 1 + 7~o' L : 'g '°

k r = Ei/L 3

M m = M,,' M v = p A . L


The amplitude of the bending moment is then given by:

[M(x,t)[ ~- L 2 I, o

+ A , L [ 3 2 0 c ( I ( L ) a - 48fl'J ( L ) ] ) (35)

3.1. Numerical results Tables 8 through 11 contain numerical values for the two lowest frequencies of

the coupled mechanical system (~o' = 0, 1, 5 and oo) for different values of the parameters M/My; k/E1/L 3 and y/L.

Figures 4 and 5 depict the variation of frequencies for a simply supported and a rigidly clamped beam, respectively. The variation of the frequencies follows the trend of variation corresponding to a two degree of freedom, discrete system (see Fig. 6).

An idea of the accuracy of the variational approach proposed for calculating dynamic displacements and bending moments may be inferred from Table 12. The approximate results are compared with the exact values in the case of a simply supported beam for several values of M/Mv; k/EI/L a and y/L. The frequency ratios selected for this comparison are 0.70 and 0.90. Displacement and bending moment amplitudes have been evaluated at cross-sections located at x/L = 0.20 and 0.30.

4. RECTANGULAR PLATES WITH EDGES ELASTICALLY RESTRAINED AGAINST ROTATION

For normal modes the free vibrations problem is governed by the differential equation (see Fig. 7):

t o 2 M D v 4 W - Ph' t°2W = l - co2(M/k) W.6(x - Xo) 6(y - Yo) (36)

where V 4 = biharmonic operator; D -- flexural rigidity; p = density of the plate material and h = thickness.

The boundary conditions are defined by:

w(x, y, t)lx= _+o/2 = w(x, y, t)ly= _+b/2 = 0 (37(a))

\OX 2 + t't Oy 2] Ix=a/2 (37(b))

?2w o2w ~ y=b/2 = -O.tpy \OY 2 -t- 12~xx2 ] y=b/2 (37(C))

134 P. A. A. LAURA, E. A. SUSEMIHL~ J. L. POMBO, L. E. LUISONI~ R. GELOS

TABLE 8 FREQUENCY COEFFICIENTS FOR A RIGIDLY CLAMPED BEAM (~p' ~ 0)

co 1. LZ(pA/EI) ~ M/My k/El/L3 y = 0 ~, = (1/3) L y = (1/6) L

0.2 18.17 87-38 21.89 72-53 19-44 81-66 0.5 14.56 68.94 20.93 47.98 16-33 61.48 1.0 1000 11.47 61.91 19.16 37.05 13-25 53.58 2.0 8.62 58.23 16.00 31-37 10.16 49.43 3.0 7.19 56.98 13.79 29.72 8.53 48.03

0.2 15.89 31.58 20.11 24.97 17.04 29.46 0.5 11.03 28.78 ! 3.64 23.27 11.85 26.79 1-0 100 8.01 28.04 9.72 23-09 8.58 26.17 2.0 5-73 27.70 6.90 23-02 6.13 25.90 3.0 4.70 27.59 5.63 23.00 5-02 25-81

0.2 6.89 23.05 7.05 22.51 6.95 22.83 0.5 4.36 23.01 4.46 22.50 4.40 22.80 1.0 10 3.09 23.00 3.15 22.50 3.11 22.80 2.0 2-18 23.00 2.23 22-50 2.20 22.79 3.0 1.78 23.00 1-82 22.50 1.80 22.79

0-2 2.23 22.50 2.23 22.46 2-23 22.48 0.5 1.41 22.50 1.41 22.46 1.4t 22-48 t .0 1 1.00 22.50 1-00 22.46 1.00 22.48 2.0 0.71 22.50 0-71 22.46 0-71 22.48 3.0 0.58 22.50 0.58 22.46 0-58 22-48

TABLE 9 FREQUENCY COEFFICIENTS FOR A BEAM WITH SUPPORTS ELASTICALLY RESTRAINED AGAINST ROTATION

( ~ , ' = 1)

091. L2(pA/E1) ½ M/My k/EI/L 3 y = 0 7 = (1/3) L ~, = (I/6) L

0"2 10"74 84-31 12.24 74"01 11.19 80.95 0'5 8"90 64'38 I 1 '47 49'96 9'57 59.88 1"0 1000 7'19 56"33 10.39 38-98 7"92 51'14 2'0 5"52 51-90 8'84 32"39 6'19 46"25 3'0 4.64 50-36 7.80 30"00 5-25 44.53

0"2 10"37 27"61 12-05 23-78 10'84 26.42 0"5 8'06 22"48 10.43 17"37 8.63 21.00 1"0 100 6'16 20'79 8-22 15"58 6-65 19.27 2'0 4'53 19'99 6'06 14"96 4"90 18.48 3"0 3"75 19'73 5.00 14-79 4.06 18.23

0"2 6-54 13"85 6"93 13'06 6.66 13.60 0"5 4"19 13"68 4.40 13"01 4"26 13"46 1"0 10 2'97 13"63 3-12 13'00 3.02 13.42 2"0 2-10 13-61 2-20 12"99 2.14 13.40 3"0 1 '72 13'60 1-80 12"99 1.75 13.40

0"2 2"22 12"89 2-23 12"83 2-23 12.87 0"5 1'40 12'89 1.41 12"83 1.41 12"87 1 '0 1 0.99 12'89 1.00 12-83 1-00 12.87 2'0 0.70 12"89 0.71 12"83 0.70 12.87 3"0 0.57 12"89 0.58 12'83 0.57 12.87


T A B L E 10 FREQUENCY COEFFICIENTS FOR A BEAM WITH SUPPORTS ELASTICALLY RESTRAINED AGAINST ROTATION

(~' = 5)

to I . L2(pA/EI) ½ M/My k /E l /L 3 y = 0 y = (I/3) L 7 = (1/6) L

0 '2 8"95 83'87 10"12 74'23 9"29 80-82 0"5 7'45 63'71 9.46 50-23 7'97 59'62 I '0 1000 6"05 55"48 8"56 39'23 6'62 50'74 2"0 4"66 50"92 7"30 32.54 5'19 45"73 3"0 3"93 49"32 6"45 30.07 4'41 43"95

0 '2 8'75 27'14 10"01 23'72 9"10 26"09 0"5 6"97 21'55 8'93 16.82 7'44 20 '20 1"0 100 5"42 19"59 7"35 14'45 5"86 18"14 2"0 4-03 18'62 5"57 13"48 4'38 17"15 3"0 3"35 18'31 4'64 13'21 3"64 16"83

0 '2 6'27 11"98 6-82 11 '01 6"43 11 "69 0"5 4"07 11 '67 4"36 10'90 4.16 11 '43 1 '0 10 2 '90 11-59 3-09 10.87 2"96 11"36 2"0 2'05 11"56 2"19 10"86 2 '10 11"33 3 '0 1 '68 11"55 1-79 10"86 i.71 11 "33

0"2 2'22 10'72 2'23 10-65 2.22 10"69 0"5 1'40 10.72 i"41 10-65 1"40 10'69 1-0 0 '99 10"72 1-00 10'64 0-99 10"69 2"0 1 0-70 10'72 0"71 10'64 0"70 10'69 3"0 0"57 10"72 0"58 10-64 0.57 10"69

T A B L E I l FREQUENCY COEFFICIENTS FOR A SIMPLY SUPPORTED BEAM (tO' ~ CO)

w 1- L2(pA/EI) ½ M/M,, k /EI /L 3 ~, = 0 7 = (1/3) L ), = (1/6) L

0"2 8.34 83"74 9-40 74.29 8"65 80'78 0'5 6"95 63"51 8"78 50"31 7"42 59"54 1 "0 1000 5"66 55-23 7-95 39.30 6"17 50"62 2 '0 4.36 50'63 6.78 32'59 4.85 45'57 3-0 3"68 49'01 5'99 30-09 4-12 43'78

0"2 8'18 27.00 9"32 23-70 8"50 25"99 0.5 6.56 21'28 8.37 16.69 6.99 19"97 I'O 100 5.14 19'23 6'99 14.13 5"55 17.81 2 '0 3.84 18"20 5.36 13'03 4.17 16-75 3"0 3-19 17"86 4-48 12.72 3.48 16.40

0-2 6" ! 3 I I "39 6'75 ! 0.35 6-30 l 1.09 0.5 4.01 l l '01 4-33 10,19 4.11 10.76 1.0 10 2.86 10"92 3"07 10.16 2"92 10"68 2'0 2.03 10"88 2"18 10"14 2"08 10.64 3-0 1.66 10'86 1 '78 10'14 1.70 10-63

0.2 2.21 9"98 2.23 9.90 2"22 9"96 0-5 1-40 9"98 1'41 9-90 1 '40 9'95 1.0 1 0'99 9-98 1.00 9.90 0.99 9"95 2-0 0.70 9-98 0-71 9.90 0.70 9,95 3.0 0.57 9.98 0"58 9-90 0.57 9,95

36 P . A . A . LAURA, E. A. SUSEMIHL~ J. L. POMBO~ L. E. LUISONI~ R. GELOS

N - d

I t

80

6 0

z,O

2~

0.2 0.5 1

L~ =~o ; ? =o I M

M v

000

K/El/ L 3

lO0

1 0 0 0 / - 1 0 0 /

3

Fig. 4. Lowest f requency coefficients for a simply suppor ted beam which carries an elastically moun t ed concentra ted mass (Ca" = oo).

D Y N A M I C B E H A V I O U R O F S T R U C T U R A L E L E M E N T S 137

BO

70

50

j 50 o,~,..j

I !

,.o

2C

.1C

L I I 02 0.5 1

My

1000

t L L 1

I

t I L

I00

I0

- - )00

100

3

Fig. 5. Case of a beam with rigidly c l am ped ends (tp' = 0).

] 38 P . A . A . LAURA, E. A. SUSEMIHL, J. L. POMBO, L. E. LUISONI, R. GELOS

!

4.

4.

. - I o,, 14"

4.

4'

I I

= 10

k 2

kl

i ~ k2= oO

k2 : 10

kl

0 i I I i I I 0.2 0.5 0.'/5 1 2

o.os M_3.Z M.f

Fig. 6. Discrete, two degree of freedom vibrating system.


TABLE 12 COMPARISON OF DISPLACEMENT AND BENDING MOMENT AMPLITUDES FOR A SIMPLY SUPPORTED BEAM

M k 09 ~, x Iw(x, t ) l IM(x, t)l ~o' Mv EI/L3 col ~" ~ FoLa/EI FoL

Approximate Approximate solution Exact solution Exact

oo 0'20 oo 0.5 oo 0'5 oo 0.2 oo 1 oo 1

1 0.9 0"2 0-3 0.003306 0.003308 0.036473 0'03477 2 0.9 0.2 0.3 0.002441 0.00244 0.026929 0.025657

10 0,7 0'4 0'2 0'007319 0.00721 0.086158 0.07991 lO00 0.9 0 0"2 0.12447 0.12030 1.234420 1.16942

l 0.9 0'4 0"2 0.0003546 0.0003523 0.0037796 0.0036495 1 0.7 0.4 0'2 0"0002327 0-0002298 0-0027388 0"0025467

~ ~ x ~ II11

l M

1111111111111111111111111111111111111111 I1 []111

-~ (Xo, yo) ×

b

Fig. 7. Case of a rectangular plate with edges elastically restrained against rotation.


Similar relations to eqns. (37(b)) and (37(c)) hold for x = - a / 2 and y = - b / 2 ,

respectively. Following references 5 and 6 one takes as a first approximation:

W ( x , y ) ~- W 1 = A , , (o t , x 4 + ,8,x z + l)(yty 4 + 61y z + 1) = A , , X ( x ) Y ( y ) (38) The ~,, fl,, 7~ and 8, coefficients are obtained substituting eqn. (38) into eqn. (37).

Accordingly one has:

where:

1' I = = - ' a

l 1 ° _ _ ~ 6 r

(b') 2 ttr) ~

1 + (p'~ ~'1 = 1 + 5~'----~' fl'l = - ( 1 + e ' , )

t

l + ~ r Y'1 = - - " 6' 1 = - (1 + Y'i)

1 + 5~o'y'

a' = a/2; b' = b/2

D(Ko, + r/2Ko2 + Koa) - - -

where:

Following Galerkin's criterion one determines the following frequency equation:

t .o2M a 2 Pht°2 a4Ko,~ = X~m Y~yo) ~l (39)

4 1 - ( t o2M/k ) "-4

q = a/b

Ko, 96a', + + 1 7 2 2y ' ,6 ' , + 2y', + 6', + + 1 = - - + 7 5 - 5 -

Ko= 3 2 ( ~ , z 7 6ct', + f l ' , 2 + f l , ) = , + ~ ' , f l ' l + 3

(~ 7 6)"' + a " = a ' ) x y, 2 + ~y ' , 6 ' , + 3 + 1

( 9 6'1 ) ( V 2 2~" +f l"2 2 ) Ko3 = 96y', + --3- + 1 + ~ a ' , f l ' l + 5 + 3 fl ' ' + 1

Ko4 = + 2~',fl', + + + 1

(~__~ 2y',&', 2 y ' , + 6 ' j 2 ~ ) x + ~ + 5 + 6 ' , + 1


The roots o f eqn. (39) yield the two lowest frequencies o f the coupled sys tem:

ph. 4 ~'~ 2 = (02 - - f f a

(r/m)(Ko4/4) + gi + (qr/4)X~o)Y~yo) = +- {[(r/m)(Ko,/4) + gl + (qr/4)X~o)Y~yo)] 2 - (gt . r /m)Ko,} ~ (40)

Ko412

where: M M

r = ka2 /D; m . . . . ; gl = Kol + r/2Ko2 + go3 ph. a. b M plate

4.1. Numerical results Tables 13 th rough 16 depic t numerical in format ion for (a) a s imply suppor t ed

pla te ; (b) a r igidly c lamped pla te ; (c) (P'x = 9'y = 1 and (d) (p'x = (p'y = 2. Several values o f t h e p a r a m e t e r s M / M p and k . a2/D have been selected. I t has been assumed in all cases tha t the spr ing-mass system is p laced in the centre o f a square plate.

TABLE 13 FREQUENCY COEFFICIENTS to,(ph/D)½a 2 FOR A SIMPLY SUPPORTED SQUARE PLATE

ka2 /D M/Mp 0"2 0"5 1"0 2"0 5 100 oo

0 19.75 19.75 19-75 19.75 19.75 19.75 19.75 19-77 19.80 19.85 19-96 20.31 39.10 co

0. I 1.41 2.23 3.15 4.43 6-88 15.97 16.73 19.77 19.80 19-85 19.95 20.27 32.14 oo

0.25 0.89 1.41 1-99 2.80 4.36 12.29 14.02 19.77 19.80 19.85 19.95 20.25 29.95

0.50 0.63 1-00 1.41 1.98 3.08 9.33 11.46 19.77 19.80 19-85 19-95 20-25 28-94 oo

1.0 0-45 0.71 1.00 1.40 2,18 6.83 8-89 19.77 19-80 19.85 19.95 20,24 28-46 oo

2.0 0.32 0.50 0.70 0.99 1,54 4.91 6.63 19.77 19-80 19.85 19.95 20.24 28.18 oo

5.0 0.20 0.32 0.45 0.63 0.98 3.13 4.34

It is i m p o r t a n t to poin t out that it has a l ready been shown 5"6 tha t the accuracy of the p rocedure is excellent (a) when M = 0 for any combina t i on o f flexibili ty coefficients tp~ and (py and (b) when M ~ 0; and the pla te is s imply suppor t ed a long the four edges. In case (b) the results are in very good agreement with exper imenta l results previous ly publ ished in the l i terature. 4


T A B L E 14 FREQUENCY COEFFICIENTS ogt(ph/D)½a 2 FOR A RIGIDLY CLAMPED SQUARE PLATE

ka2 /D M/Mp 0'2 0"5 1 '0 2"0 5 co

0 36.00 36.00 36.00 36.00 36-00 36.00 36.02 36.04 36.08 36.17 36.43

0.1 1.41 2.23 3.16 4.45 6-99 28.14 36.02 36.04 36.08 36-17 36.42

0.25 0.89 1.41 2.00 2-82 4.42 22.71 36.02 36.04 36.08 36.17 36.36 oo

0.50 0.63 ! .00 1.41 1-99 4.23 ! 7.94 36-02 36.04 36.08 36.17 36-42

1.0 0-45 0.71 1.00 1.41 2.21 13-55 36.02 36.04 36.08 36.17 36-42

2-0 0.32 0-50 0.71 1.00 1-56 9.94 36.02 36.04 36-08 36.17 36.42

5.0 0.20 0.32 0.45 0-63 0.99 6-44

T A B L E 15 FREQUENCY'COEFFICIENTS tot(ph/D)½a 2 FOR A SQUARE PLATE WITH ELASTICALLY RESTRAINED EDGES

(~,'~, = ~ ' , = 1)

ka 2/D M / M o 0.2 0.5 1-0 2.0 5-0 oo

0 22"90 22.90 22.90 22.90 22.90 22-90 22'92 22"95 23.00 23.09 23'41 oo

0-1 1"41 2'23 3'15 4.44 6-92 19.19 22'92 22-95 22"99 23'09 23"38

0'25 0 '89 1 '41 1 "99 2"81 4"38 15"95 22'92 22"95 22-99 23.09 23"37

0"50 0'63 1 '00 1 "41 1 "98 3-10 12'96 22"92 22"95 22'99 23 '09 23 "36 oo

I "0 0"45 0.71 1-00 1-40 2" 19 i 0"00 22"92 22'95 22-99 23'09 23 "36

2"0 0'32 0 '50 0"70 0-99 l "55 7'43 22"92 22"95 22'99 23"09 23.36

5'0 0 '20 0"32 0-45 0"63 0'98 4'86

T A B L E 16 FREQUENCY COEFFICIENTS FOR A SQUARE PLATE WITH ELASTICALLY RESTRAINED EDGES (~'x ~ ~o'y = 2)

M / M r 0"2 0"5 1 "0 2 '0 5"0

0 21 '51 21-51 21-51 21-51 21 "51 21 "51 21 '53 21 '56 21 "60 21.71 22"03 oo

0'1 1 '41 2"23 3"15 4"43 6-90 18'11 0 '25 21 '53 21 "56 21 "60 21 '70 22.00

0"89 I '41 1.99 2.80 4.37 15.11 21.53 21.55 21-60 21.70 21-99 oo

0"50 0.63 1.00 1 "41 1.98 3"09 12.31 21 '53 21 "55 21 "60 21.70 21 "98

I "0 0"45 0"71 1-00 1-40 2'19 9'52 21 "53 21 '55 21 '60 21 "70 21 "98

2"0 0 '32 0"50 0"70 0'99 1 '55 7.09 21 "53 21 "55 21 "60 21.70 21 '98 oo

5"0 0 '20 0'32 0'45 0.63 0'98 4.64


The fact that the present approach does not present any formal difficulties in the case of rib-reinforced plates should also be emphasised (a well accepted mathe- matical model is that of the orthotropic biharmonic operator

04 (~4 04 D:, 8-x, + 2H &¥2 c~),----- ~ + Dy~y4

instead of V 4 in eqn. (36)).

5. EXPERIMENTAL INVESTIGATION

This part of the study dealt with the determination of the lowest natural frequencies of a simply supported steel bar of the following characteristics:

L = 59.85 cm b = 5.08 cm h = 0.635 cm

W~ = 1.590 kg E = 2. 100.000 kg/cm 2

bh 3 I = - - = 0.1083936 cm 4

12

(span) (width of the bar) (thickness) (total weight of the beam) (Young's modulus)

(moment of inertia)

= 7 = 8 x 10 -6 kg seg2 (density of steel) P g cm 4

A block diagram of the experimental set-up is shown in Fig. 8. Figure 9 shows a view of the structural system investigated.

[ O=cillator [

L~[Trar~s d u cer J~u

T,.o,0oc,,r Fig. 8. Block diagram of the experimental set-up.

The fundamental frequency of the simply supported beam (no mass attached) is given by:

rt [ E. I ~ ~' Fo = ~kp-A..L 4] = 41.18 Hz

and the measured frequency is Fo = 39 Hz.

144 P . A . A . LAURA, [ . A. SUSEMIHL, J. L. POMBO, L, E. LUISONI, R. GELOS

Fig. 9. View of the structural system studied in the experimental phase of the investigation.

Different values of M, k and 7 were considered in the investigation. A comparison of analytical and experimental results is shown in Tables 17 and 18.

It can be concluded that the agreeement is, in general, quite good.

6. CONCLUSIONS

The methodology presented in this paper is quite simple and applicable to an important class of structural dynamics problems.

T A B L E 17 COMPARISON OF EXPERIMENTAL AND ANALYTICAL RESULTS (7/L = 0; CO-ORDINATE SYSTEbl AS SHOWN

IN FIG. 3)

Springs kl = 0"956kg/cm k 2 = 8"605 kg/cm k3 = 25"853 kg/cm k4 = oc Analytic Experim. Analytic E.werim. Analytic Experim. Analytic Experirn.

M 1 = 0 .000969 k g . s e g 2

c m

M e = 0"002243 kg . seg 2

c m

4 '95 5 13-77 10 20"36 20 27.52 25

41 '59 38"5 44"88 42 '5 52 '62 52

k I = 2"686 kg/cm k2 = 8'881 kg/cm k3 = 28 '774 kg/cm k4 ~ -J3 Analytic Experhn. Analytic Experim. Analytic Experhn. Analytic Experirn.

5.37 5.3 9.22 10 14-14 14 20.91 19

42-28 40 44.75 42.5 52.54 50


TABLE 18 COMPARISON OF EXPERIMENTAL AND ANALYTICAL RESULTS () ' / t = 0 " 2 5 ; CO-ORDINATE SYSTEM AS

SHOWN'IN FIG. 3)

Springs k j = 0.956kg/cm k2 = 8.605kg/cm k3 = 25"853kg/cm k4 = c~ Analytic Experim. Analytic Experim. Analytic Experim. Analytic Experim.

M 1 = 0"000969 kg.seg 2

cm

M2 =0.002243 kg .seg2

c m

4.98 5.1 14.33 14 22.33 20 32.26 28.5

41.41 37 43.13 38 47.99 46

kl = 2.686kg/cm k2 = 8.881 kg/cm k3 = 28.774kg/cm k4 = oo Analytic Experim. Analytic Experim. Analytic Experim. Analytic Experim.

5.44 5.3 9.89 10 15.65 15 26-26 22.3

41.76 37 43.056 43 47-48 45

It is hoped that naval and mechanical engineers will find it useful in their design work.

7. ACKNOWLEDGEMENTS

The first au thor wishes to acknowledge with deep grati tude the generous advice and support o f Capta in Dr Frank Andrews (US Navy), now at The Catholic

University of America, and Mr Marvin Lasky (ONR). Both helped to get h im started in research on vibrat ions and supported his work from 1966 to 1970 at The Catholic University. (P.A.A.L. returned to Argent ina in 1970.)

8. REFERENCES

I. R. E. ROBERSON, Transverse vibrations of a free circular plate carrying concentrated mass, J. Appl. Mech., 18(3) (September 1951 ) pp. 280-2.

2. W. F. STOKEY and C. F. ZOROWSKI, Normal vibrations of a uniform plate carrying any number of finite masses, J. Appl. Mech., 26(2) (June, 1959) pp. 210-16.

3. R. SOLECKI, Vibrations of straight bars and plates with concentrated masses, Rozprawy/nzh. CC II, 9(3) (1961) pp. 497-51 I. (In Polish.)

4. A. W. LEISSA, Vibration q/plates, NASA SP 160, 1969. 5. P. A. A. LAURA and E. ROMANELLI, Vibrations of rectangular plates elastically restrained against

rotation and subjected to a bi-axial state of stress, Journal of Sound and Vibration, 37 (1974). 6. P. A. A. LAURA, E. ROMANELLI and M. J. MAURIZl, Frequency coefficients for rectangular

plates with symmetrical slope restraints carrying concentrated masses. Paper presented to VIII International Congress on Acoustics, London, Great Britain, July, 1974.

7. P. A. A. LAURA el al., Concepts and applications of structural dynamics, Solid Mechanics Laboratory Publication 74-10, Bahia Blanca, Argentina, 1974. (In Spanish.)

on the dynamic behaviour of structural elements carrying elastically mounted, concentrated masses

Documents