Optimum distribution of additive damping for vibrating frames

Journal of Sound and Vibration (1980) 72(3), 391-402

OPTIMUM

DISTRIBUTION

OF ADDITIVE

FOR VIBRATING

DAMPING

FRAMES

R. LUNDBN Division of Solid Mechanics, Chalmers University of Technology, S-412 96 Gothenburg, Sweden (Received 11 December 1979, and in revised form 18 April 1980)

Redistribution of an initially uniformly applied additive damping is numerically and experimentally investigated for a vibrating plane frame. It is found that an optimum redistribution can reduce amplitudes of resonant responses by up to 60% (with the cost or the weight of the damping treatment kept constant). Optimally selected non-uniform distributions of additive damping may thus be worth considering in many practical cases.

1. INTRODUCTION

Noise and vibration problems in engineering can often be successfully attacked by treating the vibrating structures with viscoelastic materials (dissipating mechanical energy into heat). The use of different such treatments has increased rapidly during the last decades. New damping materials and new design methods have been developed-sometimes causing communication problems between material physicists, structural engineers and industrial users [ 11. Jones [2] has discussed available damping treatments, considering the influence of frequency, temperature, structural configuration, etc. Design methods for various damping arrangements are described in reference [3]. A review of commercially available damping materials is given in reference [4]. Considering cost and weight aspects, the engineer should try to find damping materials and arrangements which are optimal for the situation at hand. Hitherto such efforts have been mainly aimed at finding better materials and better techniques for their attachment to vibrating structural members (see references [5-81). The importance of the spatial distribution of damping treatment has been investigated in a few papers (see references [g-11]). Mead and Pearce [12] studied beams with partial damping treatment, which they found profitable. They also concluded that the allowable operating temperature interval will be almost unaffected by a redistribution of the damping material. In the present paper, the optimum spatial distribution of damping for a harmonically vibrating plane frame (see Figure l(a)) is considered. Numerical and experimental results are presented for redistribution of an initially uniformly distributed additive damping such that minimum resonant responses are obtained. The general methods of a previous paper [lo] are applied. Possible applications of the results of the optimizations performed here can be found in references [13-151. 2. THEORY

OF VIBRATION

A brief recapitulation of the theory behind the vibration analysis and the optimizations performed in this paper is given in this and the next section. A detailed presentation is given in reference [lo]. 391 0022-460X/80/190391 + 12 $02.00/O

@ 1980 Academic Press Inc. (London) Limited

392

R. LUNDtiN

mVO

m

VO

30

db

2 m

vo

(EI), mUO

+c f

Rigid plate

k

L

(a)

base

>I

(b)

Figure 1. (a) Frame model clamped to rigid base plate. Beam length L=400 mm, thickness I = 5.00 mm and width b = 50.0 mm. Beam bending stiffness is (El),, = 109 Nm’. Mass per unit length of vertical beams is moo = l-95 kg/m. Horizontal beams (including carried load) weigh mhc = 4mUo = 7.80 kg/m. Forces PI-P5 and displacements pI-p5 are indicated. These ten quantities are complex amplitudes to be multiplied by common time factor exp iwt. (b) Frame in Figure l(a) with overall uniform damping nbu = 0.3 excited in its second resonance mode by a horizontal base displacement pt exp ior (pr real) with w = 146.7 fad/s. Real part of complex displacement is depicted for or = 37r/4. Absolute values of complex displacement amplitudes and algebraic values of phase angles (in degrees) at different locations are written in figure. Arrows indicate chosen positive directions of real part of complex displacements.

A plane frame structure composed of beam members is considered. The members are numbered i = 1,2,3, . . . , n b. Each member i of the structure is assumed to be uniform. It has length Li, bending stiffness (El)09 and mass per unit length moi (index 0 refers to the undamped frame). The members are connected to each other and to the ground at joints. The frame is excited at its joints by a set of stationary harmonic forces or/and displacements with common radian frequency w (but with different phase angles) and given amplitudes. The frequency w of the stationary excitation will later be varied. Uniform additive damping is applied to each member i. It results in a new bending stiffness (EI), a new mass per unit length mi, and a non-zero cross-sectional loss factor 17;. Only hysteretic frequency-independent damping is considered here (see reference [3]). The vibration of the frame under the described excitation is exactly analyzed, under the assumption that Euler-Bernoulli beam theory with a complex bending stiffness EI(1 + in b, is applicable. The differential equation for the transverse deflection w (n, t) thus is (a list of notation is given in the Appendix) EI(l+inb)w”“(x,

t)+mti(x,t)=O.

(1)

For each member i, a complex 6x6 member stiffness matrix can be found by using equation (1) and including also tensional vibration (here taken as undamped). A displacement method can then be applied to analyze the frame [lo]. The cross-sectional loss factors np, i = 1,2, . . . , II b, of the beams will later be varied. The bending stiffness (El)i and the mass per unit length mi of a member i will then vary with its

OPTIMUM

cross-sectional

DAMPING

393

FOR FRAMES

loss factor 71: as [lo]

TIp/(Tfay -Vb)l,

UWi = (EOOi[l+

Wli= mci{l +[(nlay -TI~“)/n~“][nlp/(nfay

(2) - ?I!)]}.

(3)

Here the reference quantity 7:“’ has been chosen as the loss factor which (if realized) would have given the beam member a doubled weight. The loss factors n “, i = 1,2, . . . , n ‘, are collected in a column matrix nb. 3. THEORY OF OPTIMIZATION Let a response amplitude of the structure under the stationary excitation described be denoted by f(qb, w). This response f(qb, w) can be a local displacement (velocity, acceleration, stress, support reaction, etc.) or a weighted sum of such quantities. It can also be a non-linear function of these quantities. Further, let the maximum response amplitude for all w in a given frequency interval [w, w,] be denoted by F(IJ~). The objective of the present study is to reduce the response F as much as possible for a given cost (or weight) of the damping treatment. The optimization problem is formulated as minimize F(q b,

(44

subject to the constraints n,PaOVi,

? Gb($)

= CLmp.

(Jb, c)

i=l

Here Cdampdenotes the prescribed cost (or weight) of the damping treatment for the entire structure. The cost function Gb relates the cost of the damping treatment for a member to its loss factor. Two models for G” will be used in this study: namely, Gb(n”).= al;cinP,

Gb(77~)=aLicirlP/(771ay-77P).

(5a, b)

Here a is a constant relating cost to volume for a damping layer. The loss factor of the damping material is n lo’. The factor ci is defined in reference [lo] (ci = (El)i/E3h:1, see also Figure 6 of section 8). It has the same value for all i in this study. The optimization is numerically performed by using a Sequential Unconstrained Minimization Technique (SUMT). The computer program applied has been written in FORTRAN (double-precision arithmetic). It has been run on an IBM 360/65 computer. A representative optimization with 12 variables np took about 20 min of central processing time (calculations only). The core memory required was 220 kbytes. 4. PLANE FRAME The frame in Figure l(a) has been investigated numerically and experimentally. It derives its shape from the ship-building industry (modelling a deck house [15]). This symmetric frame is assumed to be excited antisymmetrically by a horizontal ground displacement pr exp iwt or a horizontal joint force P4 exp ior, or symmetrically by a force P3 exp iot or a force PS exp iwt. The response amplitudes to be studied are the displacements p2 and p4 (antisymmetric mode), and p3 and p5 (symmetric mode). As an example, Figure 1(b) shows a calculated instant picture of the stationarily vibrating damped frame in Figure l(a) subject to a horizontal ground displacement excitation.

R. LUNDeN

Figure 2. First four fixed-base undamped eigenmodes of frame in Figure l(a). Displacement and bending moment distributions are shown. Modes 1 and 2 are antisymmetric. Modes 3 and 4 are symmetric. Corresponding calculated [16] eigenfrequencies are as follows: Mode

1

2

3

4

&ad/s)

45.08 7.175

145.1 23.10

308.7 49.14

387.4 61.66

fU-W

In Figure 2 the displacement and bending moment distributions in the first four undamped eigenmodes are given for the frame in Figure l(a). They were calculated by using the program described in reference [ 16 J. Higher modes are not considered here. As stated in reference [lo] knowledge of the bending moment distribution in the frame is essential when considering optimal damping treatments. High damping values should be located where bending moment amplitudes (or, rather, bending curvature amplitudes) are large.

5. OPTIMIZATION

Two mathematical models have been used in the optimization (see also reference [lo]), as follows. Case 1. The cost (or weight) of additive damping giving a beam member the crosssectional loss factor 77: is assumed to increase linearly with n! as stated in formula (5a). The increases in stiffness (elastic) and mass as caused by the damping treatment are assumed to be negligible in the vibration analysis. Case 2. The cost (or weight) of additive damping giving a beam member the crosssectional loss factor n f is assumed to depend non-linearly on q f as stated in formula (5 b) with nhy = 1.0 and with ~7:“’=O-600 and 7:“’ = O-857 (vertical and horizontal

OPTIMUM

DAMPING

FOR

395

FRAMES

members, respectively). The increases in stiffness (elastic) and mass as caused by the damping treatment are considered according to formulae (2) and (3). Case 2 is expected to give the most realistic description of the real situation behind the present study. A number of optimizations were performed to reduce maximally some selected response quantities of the structure as shown in Figure 1 I a). An initially uniform damping distribution over the damped parts of the frame was considered. The redistributions were made with the cost (or weight) of the damping treatment maintained at a given constant value. The reduced value of a response quantity f will be denoted by J$ SYM

SYM

SYM

12’ ‘O /

4

II

I

SYM

3’ 4l’

6

’

kJ3 flR 9

,Joints

6

75

4

2

6

5

3

3

4

1’2’

4

2

2

3

3

2

0

b

C

d

Figure 3. Subdivisions a, b, c and d of frame in Figure 1(a) into uniform beam members as used in optimization. Identification numbers of members are given. Joints between members are indicated. In studies with subdivision c, members 1, 2, 3, 4 are damped and members l’, 2’, 3’, 4’undamped, and vice versa.

Subdivision a in Figure 3 was used in the Main Study. The ground excitation p1 exp iot and the response amplitude lpzl (see Figure l(a)) were studied in the investigation of resonance in antisymmetric modes. For symmetric modes, the force P3 exp iwt and the response amplitude lpsl were studied. Subdivisions b, c, d in Figure 3 were used in several Comparative Studies where also other combinations of excitations and responses were investigated. 6. NUMERICAL RESULTS IN THE MAIN STUDY Optimal distributions of additive damping for modes I, 2 and 3 as calculated in the Main Study are compiled in Table 1. At the damping levels studied here, mode 4 was found to be suppressed (no separate fourth resonance peak was found) and it has therefore been omitted in Table 1. The calculated damping distributions should be compared with the bending moment distributions in Figure 2. It is observed that also the non-linear cost model in Case 2 leads to comparatively large local damping values in Table 1. This is partly due to the stiffening effect which accompanies large damping values and reduces the response. In Figure 4(a) resonant responses are presented for modes 1,2 and 3 of the uniformly damped frame in Figure l(a). A comparison of Case 1 and Case 2 shows that the stiffening effect of the damping treatment will give only a slight change of the response for modes 1 and 2 (ground displacement excitation) but a considerable reduction for mode 3 (for the force excitation studied) at high values of nbu. The reduction factor y versus the uniform loss factor nbUis displayed in Figure 4(b). For mode 1 the reduction factor increases monotonically with nbU. For modes 2 and 3 the situation is more complex. At certain levels of nbU the optimization calls for a radical

R. LUNDkN

396

TABLE

Examples

of calculated

1

optimal distributions of damping

loss factors 77: for Case 2,

subdivision a, Main Study. Resonance in modes 1, 2 and 3 is studied ; cost of damping treatment corresponds to uniform loss factors vbU= 0.1, 0.3 and 0.5 over entire structure; calculated reduction factor y is given

Mode

Uniform dam ing $”

Reduc tion 1

2

3

Loss factor 17; in member number 4 5 6 7 8 9

10

11

12

fac:or

0.00 0.10 0.23

0.52 0.74 0.82

1

0.1 0.3 0.5

0.37 0.14 O-00 O-10 0.31 O-56 0.42 O-08 O-41 0.49 O-78 O-67 0.25 0.62 0.63

O*OO 0.01 0.27 0.15 0.44 0.17

0.00 0.00 0.00

0.00 0.12 0.24

0.01 0.25 0.36

0.00 0.24 0.35

2

0.1 0.3 o-5

0.11 0.00 O-18 0.00 0.00 0.19

0.00 0.21 0.31

0.00 0.30 O-15 0.52 0.52 0.69

0.00 0.00 O-33 0.17 O-54 0.43

0.22 0.46 0.63

0.23 0.00 0.57 0.45 0.17 0.74 O-59 O-38 0.60

0.1

0.00 0.00 O-15 0.00 0.44 0.21

O-00 0.06 0.24 0.43 O-51 0.62

3

0.3 o-5

0.18 0.00 0.31 0.28 O-22 0.68

0.01 0.39 O-24 0.57 O-41 0.72

0.00 0.05 0.00

0.01 0.25 0.41

0.03 O-00 0.00 0.28 0.15 0.25 O-42 O-19 0.50

0.36 O-41 0.50 0.52 O-70 0.48

change of the damping distributions (Table 1) which creates a jump in the curves. For mode 3 the interference with mode 4 is also a source of disturbance. For the complex cases in modes 2 and 3 it can be argued that the true optima may not necessarily have been found by the computer program. However, the principal results and the levels of the y-values in Figure 4(b) should remain useful (at least, an upper bound of y has been found). For modes 2 and 3 the reduction factor y in Case 2 decreases with increasing high values of n6”. This is due to the stiffening effect coupled with high damping values-which is seen

(b) 0.6 -

0.6

-

h

Figure 4. (a) Response ratio Ipl/pd IX. loss factor n bufor modes 1, 2 and 3 of structurein Figure l(a) with uniform damping q bu. Modes 1 and 2: p =pz, per= [pII.Mode 3: p =ps, p”= 5pp1 where pp’ is static deflection as caused by a static force P3 on undamped frame. (b) Reduction factor y us. loss factor qbu for Case 1; ----, Case 2. subdivision a. Modes 1.2 and 3 are studied. For both figures -,

397

OPTIMUMDAMPING FOR FRAMES

to be less significant for mode 1. The low values of y for mode 3 means an important reduction of amplitudes caused by the redistribution of damping. In Table 2 the calculated resonant frequencies for uniformly and optimally distributed damping are given. The redistribution of uniformly applied damping is seen to increase resonant frequencies in most cases; see also the eigenfrequencies of the undamped frame in Figure 2. Note that mass, stiffness and damping parameters all are involved in the redistribution (Case 2). TABLE 2 Resonant frequencies

(Hz) for frame with uniform and optimal distributions of damping for Case 2

Loss factor n bu Uniform damping Optimal damping

2

1

Mode

0.1 743 8.04

0.3 7.93 8.72

0.1 0.5 8.60 24.0 9.68 25.3

0.3 26.4 28.0

3

0.5 29.9 29.6

0.1 51.7 56.0

0.3 60.4 64.9

0.5 70.0 744

7. NUMERICAL RESULTS IN COMPARATIVE STUDIES For modes 1 and 2 and Case 2, the two sets of values nirn = O-300 and ni”’ = 0.632, and 77tm=0-800 and vim = O-941, were also tried. Only very small changes in response amplitudes, damping distributions and reduction factors were observed as compared to those obtained in the Main Study (77:“’= 0.600, r7irn = 0.857). The additional mass imposed by the damping treatment thus has a relatively small effect (at these low frequencies). Optimizations were performed in Case 2 with subdivision b in Figure 3. The -y-value rose by about O-2 for modes 1,2 and 3 compared with that in the Main Study. With subdivision d the increase was less important. The prescribed partial treatment for subdivision c gave somewhat differing results. For low values of nbU the reduction was significant (y = O-6) which indicates that a nonuniform distribution can be worth considering also for the damped parts of a partially damped structure. A replacement of the response quantities lpzl and lpsl studied previously by 1p4(and 1~~1 (see Figure l(a)) did affect the optimal distributions and reduction factors y obtained. However, this effect was small except for high values of n bu. 8. EXPERIMENTS Laboratory experiments were performed to corroborate the calculated theoretical results. A plane frame with geometry as in Figure l(a) was constructed in steel (SIS 14 1410). The two upper corners were bent to the required shape (see Figure 5(b)). The two middle joints were welded (see Figure 5(a)). The frame was rigidly attached to a heavy steel plate (see Figure 5(a)). The spacers were fabricated from a 1.5 mm aluminium sheet and screwed to the base structure (see Figure S(b)). A viscoelastic layer of type LD-400 from Lord Kinematics [17,18] was attached to the outside of the spacers (see Figure 5(b)) by using a two-component epoxy adhesive. The LD-400 material was selected since it combines a high loss factor with a high elastic modulus. The excitations and responses studied were chosen to be in accord with those of the Main Study as described in section 5. A horizontal harmonic displacement ( pl) of the base plate was generated by using a programmable hydraulic testing machine (MTS). A

398

R. LUNDeN

Figure 5. (a) Photograph of built-up structure with geometry according to Figure l(a). Aluminium spacers with damping layer are attached to base steel structure. Additional masses are attached to horizontal beams. Force exciter and accelerometer can be seen. (b) Detail from the upper right part of frame.

harmonic vertical force (P3) was generated by employing an electromagnetic exciter (type 8200, Briiel & Kjaer). Responses were measured by an accelerometer placed at different locations. Resonance amplitudes and pertinent frequencies were found visually by studying the maxima of responses (observed on an oscilloscope). For uniformly distributed damping, the cross-sectional loss factor qbU was first numerically estimated from the geometry chosen (see Figure 6) and from the material data

3 2 I

Figure 6. Sketch of beam member with base structure 1, spacer 2 (symbolically shown, compare with Figure 5(b)), and damping layer 3 effective in bending vibration about y-axis. Elastic moduli El = 210 GPa and E3 = 0.02Er. Loss factor n lo’= 0.5. Thicknesses ht = 5 mm and hs = 1.6 mm (for uniformly applied damping). Widths bl = 50 mm and bf = 30 mm. Distance between neutral axes is h 31= 13.3 mm. Spacer length is L, = 20 mm and gap length L, = 5 mm.

(according to reference [17] LD-400 has a loss factor T’~’= O-5 and an elastic modulus E3 = 0*02& at the temperature 24°C and frequency 100 Hz). The frame was then excited and the resonant responses and frequencies compared to computed results giving new and more accurate values of nbU and 7”‘. Finally, numerical optimizations were performed for these values of q bu and v lay according to the specifications for the Main Study, Case 2, and subdivision a. Modes 1, 2 and 3 were studied.

OPTIMUM

DAMPING

399

FOR FRAMES

The local loss factors as obtained from the numerical optimizations were physically realized in the model by redistributing the damping material (see Figure 5) while maintaining the total amount of this material. The height of the spacers was kept constant (for practical reasons). This led to a slight increase of h31 (see Figure 6) in those locations where a thicker layer was required. This fact was deemed not to affect the total results very much in the cases studied. TABLE

3

Comparison between numerical and experimental results. Loss factors 7bU and 71ay as obtained experimentally and used in numerical optimization. Reduction factor y and resonant frequency f for numerical and experimental cases are given (frequencies for uniform damping within parentheses) Mode

1

77lay 77bu

“Ion Y ;z:

f

CXP

2

3

0.90

0.80

0.67

0.10 0.56 o-55 (6.60) 6.96

0.14 0.70 0.76 (22.5) 24-l

0.17 O-56 0.42 (52.6) 55.8

(6.60) 7.16

(22.5) 23.7

(52.6) 60.2

Results from the experiments are presented in Table 3. The agreement between theory and experiments is found acceptable overall. For mode 3, however, the results are somewhat uncertain depending on interference with mode 4 and on the flatness of the response curves. Some problems were encountered in the experimental study. Firstly, the damping parameters of the damping material were uncertain as they showed a strong temperature dependence. Secondly, some spacers carrying thick layers were not able to transmit the desired shear forces betweeen the base structure and the damping layer. This last problem was successfully solved by introducing massive aluminium spacers (instead of those in Figure 5(b)) at critical locations. The arrangement of spacers as in Figure 5(b) causes the normal strain in the damping layer to be concentrated in the gaps between the individual spacers. Strain gauge measurements showed that the strain amplitude on the upper side of a 1.6 mm thick damping layer was 25% larger over a gap than over the middle of the spacers. This comparatively small difference means that the strain concentrations at gaps do not play as important a role as expected in the damping mechanism of the structure.

9. DISCUSSION

Redistribution of a uniformly distributed additive damping was found to reduce resonant responses in mode 1 by up to about 40% (see Figure 4(b)). The corresponding reduction for modes 2 and 3 is 60%. These results agree with those reported in reference [lo]. A non-uniform distribution of damping can thus reduce the cost of a damping

400

R.

LUNDtiN

treatment by about 40-60% (compared with a uniform distribution) when response amplitudes under a given excitation are to be kept below a fixed level. The crucial point in a practical situation is whether it will be technically and economically worth considering a non-uniform distribution of damping. Several aspects other than those dealt with here may make non-uniform damping treatments more expensive than specified in equations @a, b). Varying excitation frequency w and complex mode patterns in a structure may complicate the choice of optimal distributions. The additional stiffness imposed on the structure by a damping treatment with high values of its local cross-sectional loss factors can in itself have a positive effect; see Figure 3 for TbU= 0.5. An increase in loss factor then seems to be less important. In such cases (for low frequencies where stiffness is important) it might be better to apply a local stiffening by cheaper means. High frequency excitation has not been investigated in the present study. The tendency shown in Figure 4(b) is, however, that the efficiency of a redistribution increases with the mode number. In many practical situations the excitation may have a wide range of frequencies. Several modes of the structure will then be involved and an optimization cannot be limited just to one resonance peak. However, there may be one separate mode which is dominating in duration or amplitude and to which special attention should be paid. Some care must then be exercised. For one example studied (Main Study, Case 2, qbU = 0.1) the optimum damping distribution for mode 1 resulted in increased amplitudes for the higher modes, as shown in Figure 7. For modes 2 and 3 the amplitudes increased by the factors 1.21 and 6.50 compared with those for a uniformly distributed damping.

6.0 -

5.0

Qt- 4.0 \ T 3.0 2.0 I.0 0.0

IJJ 1’ ;

e /

,’

\ \ \ \ \

1-i /

‘, \ \ \

\

\

‘.

f(Hz)

Figure 7. Response ratio Ip&Ip 11us.frequency f (Hz) for structure in Figure 1(a) (Case 2, subdivision a, Main Study) with ground excitation p1 exp iwt and with 11bu = 0.1. Frequency range covers modes 1 and 2. -, Uniform damping distribution; - - - - , optimum damping distribution for mode 1.

It should also be noted that the choice of objective function can be crucial for high values of 7 bu.If a reduction of one response quantity by, say, 50% is obtained, this does not mean that other important response quantities will be reduced by the same amount. From the studies performed with different subdivisions it can be concluded that subdivision a (the finer one) in Figure 3 should be utilized fo? these lower modes to make reductions as high as 30-60% possible. The results obtained are applicable to structures with constrained as well as free layer damping treatments. The locations of different types of constrained layers can be optimized with the method given in this paper by introducing equivalent free layers in the calculation model; see references [5, 191.

OPTIMUM

DAMPING

FOR FRAMES

401

When considering a complicated real structure an automatic computer optimization, as used here, may be too expensive or cumbersome to perform. It would then be more practicable to calculate distributions of bending moment (or bending curvature) and from such data estimate effective distributions of damping. The results of this paper may be helpful in such estimations. A direct calculation of response data will then show which of the chosen distributions will result in the best performance of the structure.

10. CONCLUDING REMARKS

A concept of optimal damping with cost and weight constraints has been presented and an application of the method has been described. Theoretical, numerical and experimental studies of a harmonically vibrating frame (Figure l(a)) indicate that representative vibration response quantities at resonance as obtained for a uniform additive damping can be significantly reduced through a redistribution of this damping. It is concluded that non-uniform distributions of additive damping are worth considering in many practical situations. Applications of the method of this paper to damping problems in the high frequency domain are planned.

ACKNOWLEDGMENT This work was supervised by Professor Bengt Akesson, Division of Solid Mechanics, Chalmers University of Technology. Mr Stefan Jansson performed the experiments as his Diploma work. Their help is gratefully acknowledged.

REFERENCES 1. E. E. UNGAR 1977 Sound and Vibration 11,ll. A decade of damping. 2. D. I.G. JONES 1972 Sound and Vibration 6, 25-31. Damping treatments for noise and vibration control, 3. L. CREMER,M.HECKL and E.E.UNGAR 1973 Structure-Borne Sound.Berlin: Springer. fornoise and vibration control. 4. W. E.PURCELL 1977Soundand Vibration 11,4-29. Materials 5. R. PLUNKETT and C.T. LEE 1970Journal ofrhe Acoustical Society of America 48,150-161. Length optimization for constrained viscoelastic layer damping. 6. A. D. NASHIF and T. NICHOLAS 1970 Shockand Vibration Bulletin 41, 121-131. Vibration tontrol by multiple layered damping tr$atment. 7. S. MARKUS, V. ORAVSKP and 0. SIMKOVA 1974 Acra Technica &AV 19, 647-662. Philosophy of optimum design of damped sandwich beams. 8. B.M. PATEL,G.E.WARNAKA~~~D.J.MEAD~~~~ ShockandVibrarionBulletin 48,39-52. New structural damping technique for vibration control. 9. R. PLUNKED 1972 Shock and Vibrarion Bulletin 42, 57-64. Optimum damping distribution for structural vibration. 10. R. LUND&N 1979Journal ofSound and Vibration 66,25-37. Optimum distribution of additive damping for vibrating beams. 11. D. K. RAO 1978 Acusrica 39,264-269. Vibration damping of tapered unconstrained beams. 12. D. J. MEAD and T. G. PEARCE 1961 University of Sourhampron, Department of Aeronautics and Astronaurics, Report 126. The optimum use of unconstrained layer damping treatments. 13. P. GROOTENHUIS 1972Symposium on Applications of Experimental and Theoretical Structural Dynamics, University of Sourhampron April 1972, Paper 17. Damping mechanisms in structures and some applications of the latest techniques. 14. M. A. K. HAMIK 1973Sound and Vibration 7,20-26. Viscoelastic materials for panel vibration damping. 15. R. LUNDBN 1977 The Swedish Ship Research Foundarion, Report 5612:20. Ship vibration attenuation using unconventional methods.

402

R. LUNDtiN

16. B. AKESSON and H. T~;GNFORS 1978 Chalmers University of Technology, Division of Solid Mechanics, Publication No. 25. PFVIBAT-II: A computer program for plane frame vibration analysis, vols 1 and 2. 17. H. T. MILLER 1971 Lord Manufacturing Company-Division of Lord Corporation, Report PE-002. Damping material application handbook. 18. D. I. G. JONES and W. J. TRAPP 1971 Journalof Sound and Vibration 17,157-185. Influence of additive damping on resonance fatigue of structures. 19. D. I. G. JONES, A. D. NASHIF and M. L. PARIN 1973 Journal of Sound and Vibration 29, 423-434. Parametric study of multiple-layer damping treatments on beams.

APPENDIX:

NOTATION

Note that the “displacements” and “forces” used in this paper are complex amplitudes to be multiplied by exp iwt. In Figure l(a) these complex quantities are symbolically shown as if they were real quantities. b c

damp

E Ff Gb h

i f L, LS, L,

m nb Pl PI t W

x,

Y, 2 a Y Tb bm 11

bu t7 lay 77 rib w

width total cost (or weight) of additive damping modulus of elasticity general response of frame structure, circular frequency general response of frame structure giving maximum response interval function relating cost (or weight) to loss factor height or distance imaginary unit current index area moment of inertia length of beam member, spacer, and gap between spacers mass per unit length of beam member number of beam members displacement component load component time co-ordinate deflection of beam Cartesian co-ordinates constant relating Cd,,&, to volume of damping layer response reduction factor cross-sectional loss factor of beam in bending vibration value of q b resulting in doubled weight of beam value of nb for uniformly distributed damping of the structure loss factor of damping layer column matrix containing loss factors for beam members angular frequency a( )/ax a( )/at

in given frequency