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OPTIMUM DESIGN OF VISCOELASTICALLY DAMPEDSANDWICH PANELSA. K. LALL a , B. C. NAKRA a & N. T. ASNANI aa Mechanical Engineering Department , Indian Institute of Technology , Hauz Khas, NewDelhi, 110016, IndiaPublished online: 24 Oct 2007.

To cite this article: A. K. LALL , B. C. NAKRA & N. T. ASNANI (1983) OPTIMUM DESIGN OF VISCOELASTICALLY DAMPEDSANDWICH PANELS, Engineering Optimization, 6:4, 197-205

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Enyinrvriny 0plimi:otion. 1983. Vol. 6. pp. 197-205 0305-215X/83/0604-0197S18.50/0

@ 1983 Gordon and Breach Science Publirhcrs. Inc. Printed in the U.S.A.

OPTIMUM DESIGN OF VISCOELASTICALLY DAMPED SANDWICH PANELS

A. K. LALL, B. C. NAKRA and N. T. ASNANl

Mechanical Engineering Deparrmenr, Indian lnsfirufe of Technology, Haez Khas, New Delhi 110016, India

on the design variablcr and other requircrnents

NOTATION

time dependent co-efficients of the pres- sure series plate length plate width Young's modulus of skin material real part of complex Young's modulus of damping material real part of complex shear modulus of damping material complex shear modulus face plate thickness core thickness number of flexural half waves in plate length

n number of flexural half waves in plate

'lmn

'ld

width

p mass per unit area of sandwich plate I., Poisson's ratio for skin material v, Poisson's ratio for core material p, density of core p, density of skin r thickness of core/thickness of face plate

(I,,, shear parameter w,, resonant frequency of mn-th mode w, assumed resonant frequency w frequency in general p density in general

p, density at reference temperature a, shift parameter a stress E strain

INTRODUCTION external pressure on plate static stiffness In the field of structures. the present trend of using lateral semi-wave length/longitudinal light and continuous structures gives rise to easy semi-wave length, (h /n~ ) / (a /n ) transmission of vibrations. Resilient mounting of temperature the source of vibration is not totally effective reference temperature particularly when the structure is light and flex- temperature at given density p ible.' The conventional approach for controlling transverse displacement vibrations is to avoid resonance due to coinci- space coordinates dence of forcing frequency with the natural f l frequencies of the system. This approach cannot core loss factor (normalized imaginary be used in cases where vibrations occur over a part of complex stiflness) wide frequency range. In turbojets, missiles, rockets, system loss factor of the mn-th flexural etc., a number of resonances of structural elements mode may get excited. In such cases, vibrations can be extensional loss factor of damping mat- controlled by providing damping in the system erial itself.

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I98 A. K. L A L L . B. C. N A K R A A N D N. T. A S N A N I

One of the widely used configurations is a con- strained type sandwich configuration, in which one or more layers of flexible, high damping material are bonded between layers of stiff, elastic materials. On bending, shear strain is induced in the visco- elastic layer and vibratory energy is dissipated.

It is seen from review paper^^,^ on vibration control with viscoelastic materials that work on optimum design studies of the constrained type configuration is limited as it involves a large num- ber of design variables. Plunkett and Lee4 have optimized the length of the constrained visco- elastic layer to obtain maximum damping. They used the indirect method of search for single parameter optimization. Lang and DymS have carried out a study for optimal acoustic design of sandwich panels using pattern search. They have taken the masses and thicknesses of core and skin as design variables with some practical constraints. A direct search procedure was used to treat the optimization of layered structure subjected to dynamic loading for minimizing tensile interfacial s t r e s s e ~ . ~ Lunden7.' has used a sequential uncon- strained minimization procedure for finding the optimum distribution of additive damping for vibrating beams and frames. Rao9 has considered two aspects of the optimum design of sandwich beams:

a) For given geometrical parameters and visco- elastic material loss factor, finding the optimum shear parameter which will result in a maximum of the system loss factor.

b) Substitution of a homogeneous beam for an equivalent sandwich beam to obtain optimum shear.

A similar study was carried out by Mead" for the optimum design of sandwich plates. The criteria considered were: Constant weight, constant stiff- ness and choice of optimum core material based on constant weight theory. A trial procedure is used to find geometrical parameters for materials so as to obtain an optimum shear parameter which corresponds to the maximum system loss factor for a particular mode of vibration. Mead" has also discussed the various criteria for damping effective- ness, which include system loss factor and damping response effectiveness.

In the present work, an optimum design study is carried out on a sandwich configuration with equally thick skin faces for modal system loss factor maximization and displacement response minimization, with constraints on design variables

and other requirements such as mass, static stiff- ness ratio etc. The temperature and frequency dependent nature of the viscoelastic material is also taken into account.

OBJECTIVE FUNCTIONS

The study is divided into two main parts. In the first part, the objective function is the maximization of modal system loss factors; in the second part, it is the minimization of the displacement response at resonance. Expressions for system loss factor and resonant frequency of the simply supported sand- wich plate shown in Figure 1 for the mn-th mode are given by Meadlo and are:

where Shear parameter (I,,

and

w 2 = K,, mn

P where

2 0 . n4 K,, = -.

b4 m4. ( r 2 +

w,, being the resonant frequency for the mn-th mode

For minimization of the displacement response the additional design equations are:

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DAMPED SANDWICH PANELS 199

FIGURE I Three layer sandwich plate.

Where A,, are the co-efficients of the uniform steady fluctuating pressure distribution series, and are given by

Job J iP(x, y, r ) sin

A," = f[ sin2 ) - . s i n k r ? ) d x d y 0 0

( 4 )

DESIGN VARIABLES AND CONSTRAINTS

A sandwich plate has many physical parameters. The parameters which we want to vary for our objective functions to be extremized are termed design variables. In selecting design variables it is preferable to group them so as to reduce their total number in a problem. In the present formula- tions, five design variables were considered. These are; p,, p,r,, r , and T. Linear relationships between material density and Young's modulus developed by Krokosky13 are used to couple these two design variables.

The core material was selected in advance to take into account the effect of temperature and frequency on the core properties. The core material selected was filled silicon elastomer. As the core material properties are affected by temperature and frequency, so temperature is taken as a design variable and the effect of frequency is taken into account by the temperature-frequency dependence of the dynamic properties of filled silicon elas- tomer."

For a linear material subjected to extensional strain c; a = E,(w, T ) [ 1 + iqDjw, T ) ] E

Where ED and qD are functlon of frequency w and temperature T . According to the temperature- frequency equivalence principle, E D and qD at

different temperatures and frequencies obey the relationships:

E d w , T ) = ( T , I T o P o ) E,(w.a,)

VD(\V, T ) = q o ( f i 7 . ~ r ) where a, is known as the shift factor. The relation used in this work was'':

By proper selection of a, , the 3dimensional relationship reduces to a 2-dimensional relation- ship. For the same material, curves were fitted to both the graphs by a least squares method i.e. ( T O I T ) E D versus w . a , and q, versus w . a , yielded the following relationships.

The following are the types ofconstraints incorpor- ated in a optimal sandwich design problems.

a ) Design uariable consrrainrs:

X f < X j < X y , f o r j = 1 , 5

Where X j is the j-th design variable and Xf and X y are the lower and upper bounds respect- ively. In the case of a sandwich plate, these five variables are p,, p , , r,, r , and T.

b) Muss consrrai~lr :

where p is the mass per unit area of the sandwich and pL, pu are the lower and upper bounds respec- tively.

(c) Sraric srifi~ess consrrainr:

In some applications static stiffness ofthe sandwich is an important requirement, and forms a con- straint bounded above and below.

d ) Resonant frequency constraint:

In dynamic problems this is an important require- ment. This is incorporated to take into account the resonance phenomenon. This too is restricted with upper and lower bounds.

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200 A. K. LALL. B. C. N A K R A A N D N. T. ASNANI

OPTIMIZATION STUDIES

In the first part of the study where the objective function is to maximize the system modal loss factor q,., the following six cases have been studied:

I(:I) Maximization of 11,. with constraints on p,, p,, r , , I,, T and p. I(b) Maximization of q,, with constraints on p, ,p , , r , , r , ,TandS.

I(c) Maximization of average I!,, for the first four modes with same constraints as l(a).

I(d) Maximization of q,. and minimization of p subjected to the same constraints as l(a)

I(e) Maximization of q,, with constraints on p,,p,,r,,r,, T . ~ a n d w I . , .

I(f) Same as l(e) hut with different constraint range on a',, , .

In the second part of the study the minimization ol the displacement response at the centre of a rectangular sandwich plate subjected to harmonic excitation has been carried out for the following three cases:

Il(a) Minimization of the displacement response in the first resonant mode subject to the same con- straints as in case l(a) I l(b) Minimization of the displacement response in the first resonant mode subject to the same con- straints as in case l(b)

Il(c) Minimization of the average displacement response of the first two resonant modes subject to the same constraints as in case l(a).

The maximization of an objective function is equivalent to minimization of the negative of the objective function, or minimization of the recipro- cal of the objective function. The reciprocal ap- proach is generally accepted as it flattens out the objective function, thus the application of a penalty function method remains effective in general as the constraints are violated. In the case of i(d), the objective function is coupled i.e. maxi- mization of loss factor and at the same time minimization of mass per unit area.

The flow diagram of Figure 2 shows the logical sequence of computations performed to implement the algorithm. In Block A, the input data for the problem is read (Table I). In Block B, r a n d T are calculated lor the required mode. The resonant

frequency of the sandwich structure is assumed in Block C. In Block D, the shift factor is calculated using Eq. (5). Extensional storage modulus and loss factor for the core material at the assumed frequency are calculated from Eqs. (6) and (7) in Block E.

In Block F, the complex shear modulus is calculated. The loss factors in shear and extension are treated as equal and the storage modulus in tension as three times the storage modulus in shear.

In' Block G, Young's modulus for the skin material is calculated using the linear relationship between material density and Young's modulus developed by K r o k ~ s k y . ' ~ The linear relationship was developed by curve fitting to match data for common materials ranging from polystyrene foam to aluminium alloy, and is necessary to include the effect of coupling between density and Young's modulus as well as to reduce the number of design variables. The modal stiffness of the sandwich (K, , ) , shear parameter ($,,) and mass per unit area of the sandwich plate (p) are calculated in Block H. In Block I, w,, is calculated. In Block J, a decision is made whether the calculated resonant frequency. w,,, is approximately equal to the assumed frequency. Il'u. If it is, then the modal loss factor is calculated and is treated as an objec- tive function for case ](a), l(b), i(c), I(e) and I(f). The optimization procedure is called in Block L and the search continues for a maximum system loss factor with new design variables if pre-set accuracies are not met.

If w,, is not close to u7,, the calculated resonant frequency is set equal to the assumed frequency and the procedure is repeated until convergence takes place. This complete the algorithm for the first part of the study.

For the second part of the study the algorithm continues with Block M, where the pressure series co-efficients are calculated. In Block N, the dis- placement response is calculated at the geometrical center of the sandwich olate. The first two ~ e a k s are located by an interval ialving technique in'Block 0 . The first resonant peak is treated as the objective function for cases Il(a), Il(b), but the average of the first two is used in Il(c). Finally, the optimiza- tion procedure is called in Block P and the search continues for a minimum displacement with new design variables until pre-set accuracies are met.

For the optimization study, an algorithm based on the constrained type direct method is used. The constrained problem is first converted into an

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DAMPED SANDWICH PANELS

MODAL FREQUENCY.

CALCULATE STORAGE MODULUS 1 AND LOSS FACTOR FOR CORE

t CALCULATE COMPLEX MODULUS 1 IN SHEAR FOR CORE

CALCULATE \Ymn.Kmn &JJ

I CALCULATE SYSTEM

K MODAL LOSS FACTOR

7 0 1 LOCATE DISPLACEMENT PEAKS

P I CALL OPT. PROCEDURE I

IABLES ACCURACY

ST0 P

FIGURE 2 Flow diagram for algorithms

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202 A. K. LALL. B. C. NAKRA A N D N. T. ASNANI

unconstrained type by the application of an ex- ternal parabolic penalty. First the multi-dimen- sional space is randomly scanned to select a good starting point, then a sequential interval halving search is carried out to find a minimum along the steepest descent d i r e ~ t i o n . l ~ - ' ~ The search is con- tinued from Block B with new values of the design variables in the steepest descent direction with appropriate increments in the design variables, unless the pre-set accuracy on the design variables and objective function is met. For global optimum design, different starting points or more random search points have to be considered which will lead to the same solution.

NUMERICAL RESULTS

FORTRAN Computer programs have been de- veloped, based upon the flow diagram of Figure 2. Some modifications are made for different cases to meet specific requirements. Input data for the problem is given in the Table I. The results of problems ](a), I(b) and I(c) are tabulated in Table 11. In Table 111 results for problems I(d), I(e) and

1(f) are given. The results of problems Il(a). II(b) and Il(c) are given in Table IV.

In all the cases the results are within the con- straints set up initially, except in problems I(e) and I(f) for constraints on 1, and T. In [(a) a low temperature is selected from the broad range so that the core is operative in its glassy region area. The core material loss factor was toward its upper limit. There is not much difference in the system modal loss factor whether the first or the second modal loss factor is maximized.

In I(b), the presence of a static stiffness constraint in place ora mass per unit area constraint changed the design variables selection considerably. The maximum modal loss factor decreased to satisfy the constraint put on the static stiffness ratio. The static stiffness ratio was defined as the ratio of the static stiffness of the sandwich to the static stiffness of the skin faces alone, i.e. without core.

In I(c), the average of the first four modal loss factors is maximized. Results clearly indicate that it is immaterial whether the material selection is based on the first or the second resonant mode.

In I(d), a coupled objective function was used i.e. maximization of modal loss factor with mini- mum mass per unit area of the sandwich plate. Comparing the results with I(a) it is seen that there

TABLE I

Values of parameters and ranges for the constraints

Plole poromerer~ Length af the plate = 400 mm Width of the plate = 300 mm Poisson ratiu far the core = 0.5 Poisson ratio for the skin = 0 .3 Pressure = 0.1 N/mm2 for I@), Il(b) and ll(c) only Reference temperature = 30'C

0.0666 s r, s 20.0 mm 0.2559 5 I, 5 7.64 mm

- 12.3 5 7 5 80°C 6.688 x 5 p S 2.0 x 10.' Kg/mm2, all cases except i(b) and ll(b)

Static stiffness of sandwich 5 8 for l(b) and ll(b) only

Static stiffness of facer without core 275 5 H,,,, 5 350 for l(e) only

I50 5 w,,,, 5 200 for l(f) only

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D A M P E D SANDWICH PANELS 203

is decrease in mass per unit area and a slight loss in loss factor.

Cases I(e) and I(f) indicate that the frequency constraint effects the selection of design variables for the sandwich configuration. In all the above cases. the temperature constraint was towards its lower limit.

In summary, the temperature selected was to- ward the lower limit and i t was found that the core material loss factor always approaches the maxi- mum value available. Moreover the material selection based on the fundamental or the second resonant frequency does not give any significant difference in the values of the modal loss factor.

In the second part of the study where the objec- tive function was minimization ofthe displacement

response, the results obtained were diflerent from those for the first part for the same configuration. In II(a), the thickness of the skin was toward its maximum value but its density is quite low. More- over, the core material loss factor did not tend towards the maximum available. The mass per unit area was towards the upper limit.

In Il(b), the static stiffness ratio changed the design variables. Core thickness increases towards the maximum available.

In Il(c), the design variables are close to Il(a). The solulion converges in lI(c) with the first modal response a little higher than that in Il(a). In all the cases of second part, temperature is toward the higher side of the range available.

TABLE I1

Optimum solutions far [(a). I(b) and I(=)

Density of core-kg/mm3 Density of skin-kglmm' Thickness of core-mm Thickness of skin-mm Temoeraturc-'C

Caw [(a) Caw Kb) Caw Kc)

Far For For For optimum q , , optimum q l l optimum q , , optimum q , ,

For For optimum q,. optimum q,.

as; Der unit area-kg/mm2 1.401 x lo - ' 1.215 x 10.' 3306 6 10.' 3.352 x 10-' 1.271 x lo- ' 1.228 x l o - I Resonant modal frequency-rad/sec 298.95 599.31 558.14 899.45 366.95 597.96 Young's modulus for core-N/mm2 9.872 19.75 9.376 10.308 16.143 19.59 Loss factor for core 0.1957 0.1958 0.1913 0.1924 0.1950 0.1958 Shear parameter 37.502 31.638 75.244 146.73 20.649 32.429 Stiffness ratio 19.573 16.006 1.887 1.042 26.964 16.018 System loss factor r n = I , " = I 0.1880 0.1318 0.1637 0.1614 0.1840 0.1821 System loss factor m = I . " = 2 0.1869 0.1871 0.1445 0.1452 0.1868 0.1873 Resonant frequency rn = 1.n = l 298.95 407.45 558.14 581.74 366.95 406.68 Resonant frequency m = l . n = 2 436.89 599.31 862.77 899.45 538.02 597.96 Static stiffness ratio - - 8.01 1 8.W26 - -

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DAMPED SANDWICH PANELS 2115

CONCLUSIONS

A procedure for carrying out an optimum design study of sandwich plates with viscoelastic cores has been outlined, with a view to maximizing the system loss factor or minimizing the resonant response, subject to various constraints imposed on the design variables. Numerical results have shown that the two objective functions give different results for design variables.

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I . P. Grootenhuir. "The control of vibrations with virco- elastic materialr." J . Sound& V i b , 11, 421-433 (1970).

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3. R . C. Nakra. "Vibration control with viscoelastic materials-It." Shock ond Vibrorion Digrsr. 13, (I), 17-20. January (1981).

4. R. Plunkett and L. C. Lee. "Length optimizalion for con- strained vircoelastic layer damping." J . Arousr. Soc. Amer.. 48. 150-161 (1970).

5. M. A. Lang and L. C. Dym, "Oplimal acoustic design of sandwich panels," I . Acousr. Soc. Amer., 57. 1481-1487. June 0975).

6. Y. S. Lai and 1. D. Archcnbach. "Optimal design of a layered structure undcr dynamic louding." Compurers ond Slrucrures, 3. 559-572 (1973).

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10. D. J. Mead, "The double skin configuration.'' Univ. Southampton, ASSU Rep. No. 160 (1962).

11. D. 1. Mcad, "Criteria for comparing the eKectivencss of damping treatments," Noirr Conrrol, 7, (3), 27-38 (1961).

12. D. 1. 1. Jones, "Temperature frequency dependence of dynamc properties of damping materials." J. Sound ond I,'ib., 33, (4). 451-470 (1974).

13. E. M. Krokosky. "The ideal multifunctional conaruclural material," J. S w r r . Diu., ASCE, 94. (ST4), 959-981 (1968).

14. R. L. Fox, Oprimizorion Merhods for Engineering Design. Addison-Wesley (1971).

15. R. C. Johnson, Optimum Design of Mechonicoi Elementr. John Wiley & Sons. Inc. (1980).

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