Journal of Mechanical Engineering Science 1978 Rao 271 82

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DOI: 10.1243/JMES_JOUR_1978_020_047_02 1978 20: 271Journal of Mechanical Engineering Science

D. K. RaoFrequency and Loss Factors of Sandwich Beams under Various Boundary Conditions

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27 1

FREQUENCY AND LOSS FACTORS OF SANDWICH BEAMS UNDER VARIOUS

BOUNDARY CONDITIONS D. K. Raot

A complete set of equations of motion and boundary conditions governing the vibration of sandwich beams are derived by using the energy approach. They are solved exactly for important boundary conditions. The computational difficulties that were encountered in previous attempts at the exact solution of these equations have been overcome by careful programming. These exact results are presented in the form of design graphs and formulae, and their usage is

illustrated by examples.

1 INTRODUCTION

The damping of vibration and noise propagated through a structural layer, by bonding a viscoelastic layer to it and covering it with a constraining layer, has become a standard practice in the last decade. During the initial stages of design of such a sandwich beam, one will normally be interested in determining its resonant frequencies and loss factors in the first few modes of vibration. Although compiled information on the exact natural frequencies of Euler beams for various boundary conditions is available (1)(2)$, similar data on sandwich beams for important boundary conditions seem to be unavailable to date because of computational difliculties.

The differential equations governing the vibration of sandwich beams are well known and have been derived by DiTaranto (3) and Mead and Markus (4), by analysing the nature of displacements and forces acting in an element of the beam. The literature based on these equations can be classified broadly as those dealing with (a) exact frequency and loss factors, (b) approximate frequency and loss factors, and (c) optimal design.

1.1 Exact frequency and loss factors The basic sandwich beam equations (4) have been solved exactly only for simple (mostly symmetric) boundary conditions so far. For example, frequency and loss factors are derived by DiTaranto (3) for simply supported beams with unrestrained ends, and are given in (5) for those with riveted ends. These factors for clamped4amped beams are given in (5) and (6). Exact fundamental frequencies of undamped clamped-free beams are analysed by Mead and Markus (7). Their modes of vibration are studied in (8).

1.2 Approximate frequency and loss factors The perturbation method is used to obtain approximate fundamental loss factors of clamped-free beams by Oravsky et al. (9). The same method is used by them for other boundary conditions in (10) and (1 1). The method consists of linearizing the damped sandwich-beam :quations (around the undamped one) with respect to core oss factor. In view of this approximation, results by this method may be inaccurate for beams with highly damped cores.

The MS. of this paper was received at the Institution on 10th May I977 and acceptedforpublication on 14th March 1978. Indian Institute of Technology, Kharagpur, West Bengal.

$ References are given in Appendix 2.

1.3 Optimal design The first step in optimal design is to find that value of optimum shear parameter, gopt, which (for a given geometric parameter, Y, and core loss factor, q,) yields maximum loss factor. Expressions and curves for gopt in terms of Y and q2 are well known for the simply supported case (12)( 13) and for the clamped4amped case (6)( 14). However, similar information for other boundary conditions seem not to be available so far.

Having found the variation of gopt with Y, the second step in optimal design consists of finding that optimal sandwich beam replacing a given homogeneous beam satisfying specified constraints on weight, height, etc. Markus and their co-workers (1 1)( 15)( 16) have outlined a nomogramic method of solving this problem. As their approach is based on the perturbation method, it may be inexact for beams with highly damped cores.

The above survey indicates that, for many non- symmetric boundary conditions, exact frequency and loss factors seem not to be available so far. This is due to the fact that their determination gives rise to basic computational daculties, as reported in (9). These difficulties have been recently overcome by the author (17). This makes it possible to present the exact results here in the form of graphs and formulae for important boundary conditions. The problem of optimal design of sandwich beams in the light of these exact results is also investigated in this paper.

1.4 Notation Area of cross-section of ith layer Central distance between face layers ith layer flexural rigidity (=EiZi) Beam flexural rigidity (=E,Z, + E3ZJ Youngs modulus of ith layer Frequency, Hz Real part of the core complex shear modulus, G: = G2( 1 + j q J Shear parameter, see equation (7) Half-thickness of ith layer Area moment of inertia of ith layer about its own midline Index for layer

Complex characteristic value Length of beam Bending moment Mass per unit length of ith layer Mass per unit length of beam

Journal Mechanical Engineering Science @ IMechE 1978 Vol20 No 5 1978

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212 D. K. RAO

N

P P*

Q f t t0 T

ui

V

x Y Y z tl2 tl tl Pi a* a0 a B

U

W

w

Horizontal shear Complex natural frequency Beam resonant frequency, rad/s Vertical shear Non-dimensional time (=tho) Time mLVD Beam kinetic energy Shear strain measure (=2hzLy2/c) Axial displacement of central line of ith layer Strain energy of beam Vertical deflection of beam Non-dimensional length co-ordinate (=x/L) Geometric parameter, see equation (7) Shear strain in core Core loss factor Loss parameter (=q/qJ Sandwich beam loss factor Mass density of ith layer Complex frequency factor Reference Euler-beam frequency factor Frequency factor Frequency parameter (=QlS& Non-dimensional time derivative of w

Non-dimensional space derivative of w (=awlat)

( = a W / a i )

2 BASIC EQUATIONS A typical sandwich beam consisting of a constraining elastic layer (1) and structural layer (3) sandwiching a viscoelastic core layer ( 2 ) is shown in Fig. 1. The core properties are described by the complex shear modulus, G: = Gz(l + jqJ, and the combined flexural rigidity of face layers is denoted by D = ElI, + EJ,.

The following assumptions are made in the analysis: (1) the beam deflection is small and uniform across any

( 2 ) the axial displacements are continuous, (3) the face layers bend as per Euler hypothesis, (4) the weak core layer deforms mainly through shear

strain, y, and does not carry much axial force (Kerwin assumption), and

( 5 ) longitudinal and rotatory inertia effects are ignor- able.

The sandwich-beam equations can be derived from Hamiltons principle by using these assumptions and noting that strain energy contributions due to bending and extension of face layers, and that due to shearing of the core only, are important. In the usual notation,

section,

(1 + dextension = f Jt ( V1 + V3)bending = f Jt Dw*.L dX

(Jshearing = f Jt G: ~1,: + E3A3 ~ 3 7 3 dX (1) 1 fi dX

where u1 and u3 denote axial displacements of particles on the face layer centre-lines, while yz denotes the shear strain in the core. From Fig. 1, it is clear that they are inter- related by

(2) Further, the Kerwin assumption of a weak core implies

U, - ~3 = CW,, - 2hz yz

Initial position

Displaced position

Fig. 1. Geometry of sandwich beam

that

(3) E l 4 ul,x + E 4 3 UJ,, = 0 Solving for ul,x and u3,, from equations (2) and (3), and substituting them in (l), the energies can be written in terms of the variables

w = w(i, i) (deflection) 2h2 L

u = - y2(3,i) (shear strain measure) C

as

1 D T = - -

where W = aw/ai, and the non-dimensional length and time variables, X and 5, are defined by 1 (6) x = XIL

i = t/to [to = ~(mL4/D)l

and the standard beam parameters are G: A, L2 (ElA,+ E 3 4 )

=g(l + jvJ = 4hi E l A l E 3 A 3 (g = shear parameter)

J (the geometric parameter)

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FREQUENCY AND LOSS FACTORS OF SANDWICH BEAMS

1

w I I = o

= 0

-

w = o

w = o

w = o

w = o

WI = 0

w'=O

213

2

wIv- p +2 w = o

wv -p*Z w' = 0 WII = 0

W"-g'(l+ Y)W"=O

w'=O

w1= 0

w'v - g' yw" -p*Z w = 0 WIII = 0

Applying Hamilton's principle to energies (5) then leads to the sandwich-beam equations:

fi+ [ 1 + Y)a;w- ya,u-

Yaiw + [g*Y- Ya;)u = 0 -OI- or, in terms of the deflection alone,

r-q + g*(l + Y ) a4, - @(a; - g*)1 w = 0 (9)

which are equivalent to those in (3) and (4). In addition, Hamilton's principle yields the following beam end conditions:

At any end

1 Q = D/L3[(l + u)a: w - Yaiul= 0 M=-D/L,[(~ + Y)a:w-Ya,ul=O or w,,=o N = y1a;l.v- a,u1 = o or u = O or w = 0 (10)

While only four basic end conditions are possible in an Euler beam, from equations (10) it is clear that there are eight basic end conditions for a sandwich beam. These are listed in Table 1, which completes the already known partial list of end conditions (10).

3 EXACT SOLUTION To solve equations (9) and (10) for harmonic vibrations, assume the deflection function form

= A 4p. t ek.x = A 4Pi ek.i (1 1) where p* and Kc denote the unknown complex natural frequency and characteristic values to be determined. Substitution of equation (1 1) into (9) yields

where the complex frequency factor, P, and the associated (real) frequency factor, O, and loss factor, q, of

-Kc' + g*(1+ Y)P' + PZ(P2- g*) = 0 (12)

the beam are inter-related by

sd* =p* to = O(1 + jq)1'2 sd =pro = d(Re(O*')) q = Im(sd*2)/Re(sd*2)

For any 0*, let c, e, ..., k;: denote the six zeros of equation (1 2). Then, a general solution of equation (9) can be written as

w = (A, eklx + A, eci + A, eei + A, ek:; + A, eci + *-

+ A , ec*i) $7 (14) where A,, ..., A, are six constants which have to be determined from six boundary conditions (10).

A sixth-order characteristic determinant, which has to be zero for non-trivial solution, can then be easily formed, by using (14) with appropriate boundary conditions from (lo), by a standard procedure. Let d denote the so-formed characteristic determinant. Then, for non-trivial solution,

d(sd*,P) = 0 (15) Equations (12) and (15) are two nonlinear complex equations for two unknowns, ST and k*. The solution of these equations has been successfully programmed in (17). They will have an infinite number of solutions, which can be ordered with respect to frequency factors as (a:, k;c), (n:, q'), ... where OI < a,, < 0111 ... (roman subscripts denote mode numbers).

The numerical difficulties that were encountered earlier (9) in solving the same problem are overcome in (17) by careful programming in (complex) double precision, using improved iteration procedures for finding roots.

The procedure adopted in (17) for solving these equations is as follows. The starting values for k: and O* are taken to be those of a reference Euler beam. These are then used to compute the characteristic determinant, d, by (15). Normally, it will not be exactly zero. An improved Muller's method is then used to compute a more accurate value of c, and the corresponding value of d is again

Table 1 . Basic end conditions

No.

6

I

8

- -

End arrangement Abbreviation 7 - L Free, riveted I Fr Pinned

Pinned, riveted

Clamped, C"

Clamped C

Sliding S

unrestrained

1 Sliding, riveted s,

In terms of ofwandu

0 3

w'V-g'(l+ Y)W"-p*2w=O

w v = o



214

~~~

Clamped-pinned c-P 15.418 49.965 104-25

Clamped-clamped C - C 22-373 61.673 120.9

D. K. RAO

Boundary conditions

Table 2. Reference Euler-beam frequency factors, 9,

Abbreviation No. -

1

2

3

- -

5 -

Clamped-free I C-F I 3.516 I 22.034 I 61.697

Pinned-pinned I P-P I n2 I 4x2 I 9n* Freefree I F-F I 22-373 I 61.673 I 120.9

computed. The procedure is iterated until Id I is smaller than a specified tolerance.

follows. From equation (5), it is clear that the sandwich- beam equations reduce to those of a reference Euler beam (which is defined as a homogeneous beam of mass per unit

parameter, fi, which are defined by

The starting values for k: and lP are determined as - = - D = resonant frequency of sandwich beam .rS, resonant frequency of reference Euler beam

- '1 loss factor of sandwich beam '1=- = '12

length, m, and flexural rigidity, D) when Y = g" Y = 0. Then, equation (12) reduces to loss factor of core

g 4 = a; (16) where zero subscripts denote reference Euler-beam values. The reference M~-beam frequency factors, OW are known (2), 8s listed in Table 2. Knowing DO from this table, equation (16) yields k: = d(Do). k: and Do thus found are the required starting values.

They depend on five parameters: g, Y, mode, boundary conditions and q2. The effects of the first four parameters are displayed in the design Curves of Figs 2 to 6, while the effect of q2 is shown in Fig. 7. Since q2 is assumed to be small in drawing Figs 2-6, they are accurate only for q2 < 0.3.

Major conclusions that may be drawn from Figs 2-6 are :

4 NUMERICAL RESULTS Eflect of shear Parameter As can be expected, there exists an optimum shear

The computed exact results are displayed as graphs and formulae for the frequency parameter, 8, and loss maximum.

parameter, go,, for which the loss parameter reaches a

u1

3

-1 mode

10-1 100 10' 10' 10-2 Shear parameter, g

Fig. 2. Frequency and loss parameters of clamped-free beam

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FREQUENCY AND LOSS FACTORS OF SANDWICH BEAMS 215

1 00

lo-'

10-2

lo-'

10-4

10-2 10-1 lo0 10' Shear parameter, g

Fig. 3. Frequency and loss parameters of clamped-pinned beam

10-2

100

10-1

i 3

s 10-2 u2

5 P u1 s 4

10-3

10-4

10-2 10-1 100 101 102 Shear parameter, g

Fig. 4. Frequency and loss Parameters of clamped-clamped beam

Effect of geometric parameter An increase in Y increases the loss and frequency parameters. shear parameter.

Eflect of mode The mode-I1 loss parameter is greater than the mode-I value only when the shear parameter is very large.

Effect of boundary conditions Boundary conditions do influence the value of optimum

Effect of loss factor of core Fig. 7 shows that the inaccuracy (of ignoring the effect of q& increases with g and Y. It also shows that an increase

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276 D. K. RAO

c) d E

i Y B : P 111

100

10-1

10-2

10-3

10-4

10

5

1 10-1 10-1 100 10' 102

Shear parameters, g Fig. 5. Frequency and loss parameters of pinned-pinned beam

100 I I 1 ~ 1 l 1 1 I 1 I I I 1 1 1 I I I , , , I , 10 - __--- 5 ------------- --___. --_

lo-' L 5 @*

10-2

10-3

10-4

1 10-2 10-1 100 10' 102

Shear parameter, g

Fig. 6. Frequency and loss parameters of free-free beam

in q2 reduces the loss parameter. For example, for a beam with g = 10 and Y = 1, as the core loss factor increases from 0.1 to 1, the loss parameter reducesfrom 0.0953 to0.0609 by 36 per cent. In other words, a ten-fold increase in tf2 has resulted in only a 6.4-fold increase in beam loss factor. However, the corresponding increase in frequency factor is computed to be only 2.14 per cent, and is negligible.

5 APPROXIMATE FORMULAE Exact frequency and loss factors computed as above are

used to generate formulae in terms of g and Y. As in design curves, they are computed to first-order approximation relative to tf2. They are accurate to within 5 per cent of exact values and are given in Appendix 1.

The utility of these formulae to a designer is quite obvious. For example, for a clamped-free beam with g = 10, Y = 1, and tf2 = 0-1 in fundamental mode, the very first formula in Table 3 yields a beam loss factor of 0.0098 compared with its exact value of 0.00953 (error 2.8 per cent), while the second formula yields a frequency factor



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FREQUENCY AND LOSS FACTORS OF SANDWICH BEAMS 211

1 00

10-1

be

u c * 2 10-2 E a * -1

lo-'

10-4

10-2 10-1 1 00 10' Shear parameter, g

Fig. 7 . Effect of core loss factor on beam loss factor for clamped-free beam

.... ~ - _ _ Ref. (16), approximate

Present exact theory 20 & L"

2 15 .- 2 5 5 10

a

.a 2 5 5

0

Geometric parameter, Y Fig. 8. Optimum shear parameter curves

of 4.64264 compared with its exact value of 4.64833 (error 0- 12 per cent).

6 OPTIMAL DESIGN

As mentioned earlier, there are two aspects of optimal design of sandwich beams: (a) given Y and q,, to find that optimum shear parameter, gopt, at which the loss parameter attains a maximum; and (b) given a homogeneous beam, to find an optimal sandwich conrguration under specified constraints on weight, height, etc. that achieves this optimum shear parameter.

6.1 Optimum shear parameter These values can be read off for a given boundary condition (in mode I) from Fig. 8. Corresponding

1 0 2

100

10-1

10-2

10-3

10-4

maximum loss parameters can be read off Fig. 9, where three curves are drawn for each boundary condition, the top, middle and bottom curves corresponding to qz = 0.1, 0-5 and 1, respectively. Fig. 8 shows that go, does not change much for Y > 15. Fig. 9 indicates that it is advisable to have Y > 15 to achieve a good loss parameter.

Comparison of exact go,, with approximate gopt obtained by Oravsky et al. (16) is given in Fig. 8 for a clamped-pinned beam. For example, for Y = 5 and v2 = 1, this figure shows the exact gopt to be 6.7 compared with the approximate go, (16) of 10.

6.2 Optimal sandwich codiguration A homogeneous beam with specified half-depth h, length L, elastic modulus E and mass density is given. It is



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278 D. K. RAO

Description

1 c c

From curve

Value Error Value Error

From formula - Exact

0 5 10 15 20 Geometric parameter, Y

Fig. 9. Maximum loss parameter curves

Resonant frequency (mode I), Hz

required to replace it by a sandwich beam having maximum damping. In this case, the selection of the optimal sandwich configuration consists of finding seven quantities, h, , h,, h, , E l , G,, p, and p,, in addition to three assumed ones (L, p,, E,-same as those for the homogeneous beam) that satisfy specified weight, height and other constraints, and maximize the loss parameter simultaneously. The constraint equations required for evaluating these seven quantities are described below.

~

1309 1318 0.7 per cent 1296 0.97 per cent

Weight constraint The ratio of weights of the sandwich and the homogeneous beam that it replaces should be equal to a specified constant, called the weight ratio, r,. The constraint equation is

Pi hi + P2h2 P3h3 - rwph = 0

Height constraint The ratio of thicknesses of the sandwich and the homogeneous beam should be equal to a given constant, called the height ratio, r,. The constraint equation is

h, + h , + h , - r h h = O Geometric parameter constraint Since the damping emciency increases with Y, for a good design, the geometric parameter should be greater than 15 :

Y > 15

Optimum shear parameter Once Y is prescribed, gopt at which the beam attains maximum damping can be found from Fig. 8. The so-

G, = 0.98 x 1O1O N/m2

E l = E , = 20.6 x 1010N/rn2y pI = p, = 7850 kg/m3

Fig. 10. Sandwich beam used in the example of Section 7.1

found gopt is related to optimal sandwich beam dimensions through the equation

G,A, L2 El A , + E, A , = 7 El A , E, A,

Since only four equations are available to determine the seven quantities, it is clear that no unique optimal configuration exists. However, if it is desired that the sandwich configuration be symmetric, an optimal solution can be arrived at, as illustrated in the next section.

7 EXAMPLES

Solutions of the different types of problem that may arise in the design of sandwich beams using the data presented here is best illustrated by the following examples.

7.1

When G, and q2 are constant, the procedure is as follows: (1) From the given data, L, h,, h,, h,, E , , G,, E,,

p,, pz, p, and q, for the sandwich beam, compute the parameters to, g and Y from equations (6) and (7).

(2 ) Use the computed g and Y to read off a and ij from Figs 2-6. With 0, chosen from Table 2, compute then the frequency factor, $2 = Oofi, and loss factor, q = q2*. (Alternatively, these factors can be computed directly from formulae in Table 3.)

(3) Calculate the required resonant frequency,f, and beam loss factor, q, from the following formulae:

To find resonant frequency and loss factor when core properties are constant

f = $2/(2nt&

Example As an example, the application of this procedure to the

sandwich beam shown in Fig. 10 yields results shown below.

I 0.006965 I 0.006816 12.1 per cent] 0.00'13 i 4.8percent

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FREQUENCY AND LOSS FACTORS OF SANDWICH BEAMS

G2,N/cm2

t12

279

60 65.4 68.8 71.3 73.3 76.5 I 80 0-22 0.275 0-313 0.342 0.368 0.411 0.46

In this case, the results are within 5 per cent of the exact values.

7.2 To find resonant frequency and loss factor when core properties vary with frequency

For some core materials, tabulated or graphical data for G2(f) and tl,(f) are available. The procedure in such cases is as follows:

(1) From given data, compute to and Y and choose 0, as described in the previous example. In addition, find the scale factors g relating g with G,, and 8) relating f with fi from the formulae:

g A,L2 ( E l 4 + E344,) EIA 1 E3A3

g = G , = q

- d 2nt0 f *o

a= - =- (2) Use these scale factors on Figs 2-6 to convert the g-

scale into a G,-scale and the a-scale into anfscale. With the newly constructedf-scale as a basis, superimpose the given G2(f) and qz ( f ) curves on the same figure. Proceed as in the following example.

Example The sandwich beam shown in Fig. 11 has Thiokol RD as the core material. Its shear modulus and loss factor vary with frequency (18) at 35OC as follows:

f , H z I10 120 130 I 4 0 ( 5 0 I 70 (100

El = E, = 7 x 1010N/m2 1 pI = p, = 2500 kg/m

Fig. 1 1. Sandwich beam used in the example of Section 7.2

For this beam, computed values are to = 0.0558 s, Y = 1.55, g = 0.32 cm2/N and 0 = 0.0975 s. Using the computed scale factors and tabulated data given above, the G 2 ( f ) and q,(f) curves are superimposed on the appropriate design figure (Fig. 2), as shown by AB and CD, respectively, in Fig. 12. Then, point E, which intersects the frequency parameter curve at computed Y = 1.55 with the G,(f) curve AB, is located. A horizontal line, EF, drawn from this point towards thef-scale then yields a resonant frequency of 15 Hz, while the corresponding core loss factor can be read off by its extension, EGH, as 0.25. Point E also determines the value of g at resonance. Hence, through lines EJK (where El is vertical), one can read off the corresponding loss parameter as 0.07. Here, J is the point of intersection of the loss parameter curves at Y = 1.55, corresponding to g at resonance. From the read-off values of loss parameter and core loss factor at resonance, one can easily compute the beam loss factor as 0.0175.

I I I I I I I100 lo- 0.25

H j core loss factor, v ~ ( ~ J

N/cm2 Fig. 12. Method of finding resonant frequency and loss factor of beam in the example of Section 7.2

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280

. - - - - d - - -. -. -. __ - - - -. 412

D. K. RAO

Eg

Height constraint h , + h 2 + h 3 - 1 . 3 h = 0

I

E = 20.6 x 1O1O N/m2 p = 7850 kg/m3

Geometric parameter constraint

Optimum shear parameter constraint G,A2LZ(E,A, + E A )

4h: E , A , E3 A , =gop,=5*1

Symmetry constraint

go,, is scaled off as 5 .1 from Fig. 8. The solution of these constraint equations results in an optimal sandwich beam with G , = 0.42 x lo6 N/cmZ (compared with 0.52 x lo6 N/cmZ of (1 1)). The dimensions of this optimal sandwich configuration are shown in Fig. 13b.

h, = h3

8 CONCLUSIONS The basic equations governing the vibration of sandwich beams are presented and solved exactly for important boundary conditions. Resulting frequency and loss factors in important modes are presented in the form of graphs and formulae. The numerical difficulties involved in programming the exact solution have been successfully overcome by using a carefully developed iterative approach. The application of graphs and formulae presented herein to typical problems is illustrated by examples.

The work reported in this paper has been done at Technische Universitat, Berlin with the assistance of a Fellowship grant from the Alexander von Humboldt- Stiftung, West Germany.

9 ACKNOWLEDGEMENT

APPENDIX 1 FORMULAE FOR FREQUENCY AND LOSS FACTORS

x = log,, g y = log,, Y Clamped-free (C-F) beam

Mode -

1

11

111

Formula

q = 0- 142675 x I .76507 x 0.38 1063 x 0-958039 x 1 .O654gx4 x 5.78986 x 0-539055 x 1*0878$ x 1*08745 x 0.976757x4Yx 0.586965Y x 1.O224lxY x 1.15050X2y2 x 0~97021oXY x 0.983707x4Y2 x T,+

a= 3.97720 + 0.593861~ + 0 ~ 1 5 1 6 4 1 ~ ~ - 0 * 0 6 1 3 3 1 ~ ~ - 0*022252~+~(0.154016 + 0.553244~ + 0*144153~-0*075502~ - 0*014758X) + ~(2.42584 + 2.14246~ + 0.294401~ - 0.194389~ - 0.0483459)

q = 0.045809 x 5.23955 x O.66021lX2 x 0.897479 x 0-992201* x 9 .96967~ 0.7086WY x O.73731Sx2Yx 1-01858cY

x 1.04428x4Y x 0.632614 x 0.782958Y x 1.16677x2y2 x 1~03702xY x 0.976208x4y2 x q2

22.6437 + 1.53915~ + 1 . 1 6 9 0 9 ~ ~ + 0.088024~ -0.096522~ +~(-1.37115 + 0.330011~ + 1.61566x2+ 0.233375~ - 0.169197~) +y2(6.24946 + 8.70502~ + 3.57741~- 0.22708~ - 0.3063089)

q=0-018430 x 8.13049 x 0.803410X2 x 0.890123 x 0-976944 x 10.9134x 0.90008 x 0*789523 x 0.964562

x 1.02404x4Y x 0.763190Y x 0.73016OrY x 0.974320X2Y x 1.05259x32 x 140884x42 x q2

a = 62.2273 + 1.82075~ + 1 . 9 6 0 3 4 ~ ~ + 0.598420~ - 0.000547~ + y(-3.00169 - 2.56321~ + 1 .23125~~ + 1.22688~ + 0.158933~) +~(8.08929 + 16.5705~ + 10.5650~ + 0.634568~ - 0.7681919)

Journal Mechanical Engineering Science @ IMcchE 1978 Vol20 No 5 1978


FREQUENCY AND LOSS FACTORS OF SANDWICH BEAMS 28 1

Mode Formula

I

Mode

q = 0440681 x 6.61077 x 0.67147OX x 0.86014OX x 0.983856d x 10-452oY x 0.739451 x 0~70669lxZy x lW221x3y x 1.05007Y x 0.662933Y x 0.74738PY x 1~11045x2y2 x 1*05O6lx3Y2 x 0.986177xY2 x qz

R = 15.7484 + 1.06717~ + 0*913755~+ 0.073022~ - 0-08182W + Y(--1.14737 - 0.048593~ + 1.22596~ + 0.246788~ - 0.126150~) + ~(4.04096 + 6.65816~ + 3.14506~- 0.271991~ - 0.3420039)

Exact formula

I11

= 0.016269 x 8.88545 x 0.84412xz x 0.881654 x 0.968293x4 x 10.8945Y x 0.938944 x 0.816825x2Y x 0.952257xY

x 1.01419x4Y x 0.792846Y x 0-722358Y x 0~928255x2y2 x 1-05499Y x 1~01888x4y x qz R = 50.3163 + 1 .3210~ + 1.53476~+ 0.530222~ + 0.0213589 +~(-2*39312 - 2.71641~ + 0.461029~ + 1.15645~

+ 0.2640899) + ~(5.85246 + 13.6697~ + 9.51636~ + 0.528498~ - 0.7870849)

q = 0.008407 x 9.69867 x 0.929659. x 0.909518 x 0.968921 x 10-6141Y x 1.04444Y x 0.934567Y x 0.939467Y x 0.987858Y x 0.878939Y x 0.752321Y x 0-86079lX2J x 1.03891x2y2 x 1.02975Y x q2

R = 104.642 + 1.02666~+ 1.57212~+ 1~10017~ +0~2611209+~( -2~71121 -6-34427~-2.63637~+ 2.01631~ + 1.05319~) + ~(6.17567 + 18.1866~ + 15.8423~ + 2.563839 - 0.8252459)

Clamped4amped (C-C) beam

Mode I Formula

1

q = 0.023039 x 8.39163 x 0.787746x2 x 0.864577x3 x 0.969520X4 x 10.843oY x 0.857714Y x 0.755173Y x 0.969394xY x 1.03218x4Y~ 0.739590Y x 0-724177Y x 0*99569lxY2 x 1-05721x x 1*00635x4Yz x t12

R = 22-6008 + 0.906251~ + 0-951003~ + 0.209149~ - 0.0368359 + Y(--1.30613 - 0.798394~ + 0.861367~ + 0.478106~ - 0.002469~) ty(3.54426 + 7.31833~ + 4.51526~ + 0.047526~- 0.4169139)

~~ ~~~ ~

q = 0.011480 x 9.13923 x 0.886609 x 0.901283x2 x 0.970332x x 10.7746Y x 0.998121Y x 0.8875832y x 0.94437OXY x 0.999922Y x 0.836086Y x 0.734605 x 0~890741xzy2 x 1*04832xY2 x 1*02535Y2 x t12

R = 61-9907 + 0.980155~+ 1 . 2 7 7 2 1 ~ ~ + 0.663769~+ 0.1140339 +~( -2-18860-3 .48325~-0 .519345~ + 1.29480~ + 0.476231~) +y2(5-16093 + 12.9228~ + 9.90417~ + 1~18020~-0~609689X)

I11

q=0.006602 x 9.75759x x 0.941078 x 0.921716 x 0.972932 x 10.4967Y x 1*06583xy x 0-969662x2y x 0*940708xY

x 0.982363Y x 0.903227Y x 0.76970lxY x 0.852076x2Yz X 1~0315lX~ X 1*03004x4Y2 X qz

R = 121.283 + 0.765274~+ 1.32566~+ 1*16482X3+O*325159X4 +~(-2*40440-6*79007~-3*52460~ + 1*98512x- + 1.190539) + ~(5.57449 + 17.0361~ + 15.6782~ + 3.21381~- 0.5633099)

i

(in)gYtl, (in) + (in)(2 + Y)g + (1 + $)(1 + Y)g tl=

(in) + (in)@ + Y)g + (1 + &(I + Y)g + (in) + 2(in)g + (1 + $)g (in) 3



282 D. K. RAO

Freefree (F-F) beam

Formula

t] = 0.075403 x 4.69314x x 0.509122x2 x 0.862207$ x 1.01833x4 x 9.16243Y x 0.605483 X 0-713556x2Y x 1-05442x3Y

x 1.05569x4Y x 0.595434Y x 0-820556xY2 x 1.24622x2Y2 x 1.02599x3Y x 0-964048x42 X tfz R = 23.4275 + 2.54004~ + 1.65291~- 0.096295~- 0.21311W + y(-1.51635 + 1.40446~ + 2.54268~ + 0.030539~ - 0.362220~) + ~(9.33873 + 12.2033~ + 3.93686~ - 0.920585~ - 0.4915049)

t] = 0.022902 x 8.3877lX x 0-787927x2 x 0.86548oX x 0-96991P4 x 10.876W x 0.855323 x 0.755639x2Y x 0.971236x3Y

x 1-03273Y x 0.7394657 x 0.729053x 1~00117xzy x 1.0543+Y2 x 1*00444x4y2 x t]

f2 = 62.2957 + 2.47790~ + 2.60370~ + 0.579376~- 0.097816x4 + y(-3.50899 - 2-27008~ + 2.15589~ + 1.29628~ I1

+ 0.024284~) + ~(9.62087 + 20.0572~ + 12.5740~ + 0.217371~ - 1.144459)

t] = 0.01 1994 x 8.90854x x 0.87333oX x 0.903888x3 x 0.972967x4 + 10.8132Y x 0.985307 X 0.864767x2Y X 0.948384xy x 1.00399Xy x 0.824826 x 0.738425xY2 x 0.909043x2Y2 x 1.046672y2 x 1*02068x4y2 x t]

a = 121.596 + 2.00753~ + 2.54472~ + 1.28896~ + 0.2151489 +y(-4-32946 - 6*44049~-0.735310~ + 2.35698~ 111

+ 0.821448~) + ~(10.5496 + 25.1796~ + 18.9689~ + 2.52084~- 1.021309)

APPENDIX 2

REFERENCES

(I) YOUNG, D. and FELGAR, R. P. Tables of characteristic functions representing normal modes of vibration of a beam 1949 (University of Texas Publication No. 49 13).

(2) BISHOP, R. E. D. and JOHNSON, D. C. The mechanics of vibration 1960 (Cambridge University Press).

(3) DiTARANTO, R. A. J. appl. Mech. 1965 3 2 881. (4) MEAD, D. J. and MARKUS, S . J. Sound. Vibr. 1969 10,163. ( 5 ) MEAD, D. J.Strojnicb Casopsis 1971 2253. (6) MEAD, D. J . and MARKUS, S . J. Sound Vibr. 1970 12,99. (7) MEAD, D. J . and MARKUS, S . Strojnicb Casopsis 1970 21,2. (8) MARKUS, S. and VALASKOVA, 0. J. Sound Vibr. 1972 23,

(9) ORAVSKY, V., MARKUS, S. and SIMKOVA, 0. J. Sound

(10) MARKUS, S., ORAVSKY, V. and SIMKOVA, 0. Acustica

423.

Vibr. 1974 33,335.

1974 31,132.

(11) MARKUS, S., ORAVSKY, V. and SIMKOVA, 0. Acta Technica 1974 78,647.

(12) ROSS, D., UNGAR, E. E. and KERWIN, E. M. Structural ahp ing (Ed. J . E. Ruzica) 1959 (American Society of Mechanical Engineers).

(13) BERANEK, L. L. (Ed.) Noise and oibration control 1971 McGraw-Hill Book Company).

(14) MEAD, D. J . Seminar on the vibration of damped structures proceedings (Ed. J . C. Snowdon) 1975 (Pennsylvania State University).

(15) ORAVSKY, V. and MARKUS, S . Zbornik Vystureneplasty 1975 (Karlovy Vary).

(16) ORAVSKY, V. and MARKUS, S . Vilyane Vibrachi no organism cheloveka iproblemi vibrazatchit, Symposium 1977 Elka, USSR.

(17) RAO, D. K. Computer programs for determining exact frequency and loss factors of sandwich beams with arbitrary boundary conditions 1977 (Internal Report, Institut f i r Mechanische Schwingungslehre und Machinen Dynamik, Technische Univer- sitiit Berlin, West Germany).

(18) SNOWDON, J. C. Vibration and shock in damped machanical systems 1968 (John Wiley & Sons).

Journal Mechanical Engineering Science @ IMechE 1978 VoI 20 No 5 1Y78


Journal of Mechanical Engineering Science 1978 Rao 271 82

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