VIBRATION ANALYSIS OF BEAMS WITH MULTIPLE CONSTRAINED LAYER DAMPING PATCHES

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1 Journal of Sound and Vibration (998) 22(5), VIBRATION ANALYSIS OF BEAMS WITH MULTIPLE CONSTRAINED LAYER DAMPING PATCHES Acoustics and Dynamics Laboratory, Deartment of Mechanical Engineering, The Ohio State University, Columbus, Ohio 432-7, USA (Received 9 June 997, and in final form 3 November 997) A new analytical, energy based aroach is described that redicts the harmonic vibration resonse of a damed beam with multile viscoelastic atches. Each daming atch consists of a metallic constraining layer and an adhesive viscoelastic layer with sectrally-varying material roerties. Since this aroach relates all deformation variables in various layers, only flexural shae functions need to be incororated in the comlex eigenvalue roblem. Consequently flexural, longitudinal and shear deformation eigenvectors can be calculated. In articular, the shear deformation modes of the viscoelastic core rovide useful information regarding the effect of atch daming. The roosed method has been validated by comaring redictions with modal measurements and with those ublished in the literature. Also, an estimation technique is develoed that determines the shear modulus and loss factor roerties of two different viscoelastic materials used in exerimental studies. An uncertainty study is also erformed to establish the error bounds of the estimated material loss factors. Effects of atch boundary conditions, atch cutouts and locations, and mismatched atch combinations are analytically and exerimentally examined. 998 Academic Press Limited. INTRODUCTION Elastic beams with constrained layer viscoelastic material have been analyzed by many investigators, as evident from the studies described in two books on vibration daming by Nashif et al. [] and Sun and Lu [2]. However, much of the rior work has been limited to full coverage, i.e., viscoelastic material added to one or both sides of the beam in a uniform manner. Conversely, only a very few ublications have dealt with artially covered sandwich beams [3 6]. Nokes and Nelson [3] were among the earliest investigators to rovide an analytical solution to the roblem of a artially covered sandwich beam. In their formulation, damed mode shaes are assumed to be the same as the undamed eigenvectors, and the modal loss factor was calculated as the ratio of energy dissiated to the total modal strain energy. A more thorough analytical study was carried out by Lall et al. [4]. In their Rayleigh Ritz aroach, both flexural and longitudinal shae functions were incororated in the eigenvalue roblem for a beam with a single daming atch. In ractice, non-uniform and/or artial daming treatment is necessary because of material, thermal, ackaging, weight or cost constraints. And in some alications multile daming atches at selected locations are more desirable. None of the mathematical models, as available in the literature, aears to be directly alicable to this roblem. Consequently a clear need exists for a more refined analysis which this article attemts to fulfill. Secific objectives are as follows: () develo a new analytical method that considers flexural, longitudinal, rotational and shear deformations in all layers of the sandwich beam, (2) verify the method by comaring results for a single atch with those 22 46X/98/ $25.//sv Academic Press Limited

2 782 reorted in the literature by Lall et al. [4] and Rao [7], (3) estimate the unknown material roerties of viscoelastic material used in the exerimental study, (4) validate the method further by comaring redictions with modal measurements on beams with two mismatched atches, and (5) finally examine critical issues such as the atch boundary conditions, a discontinuity in the material (cutout), and mismatched atch combinations. The method is first described for both thin and thick beams where motion variables for all layers are exressed in terms of the flexural dislacement of the base structure (i.e., beam). Then the formulation is reduced to a thin beam by emloying a Rayleigh Ritz minimization scheme and an eigenvalue roblem of dimension n is obtained where n is the number of admissible functions. This formulation facilitates efficient calculations of various modal deformations in all layers. It should also lead to an imroved understanding of daming system designs. 2. ANALYTICAL FORMULATION 2.. PHYSICAL EXAMPLE The structure of interest is shown in Figure, where N daming atches are attached to the base structure (an elastic beam designated here as layer 3). Each atch of length l is located at x. Layer is a metallic layer while layer 2 is an adhesive caable of dissiating vibratory motions. The viscoelastic nature of the second layer is assumed to be linear and frequency deendent. The comlex-valued Young s modulus (E ) and shear modulus (G ) of the viscoelastic material in atch are reresented by E 2 ()=E 2 ()(+i 2 ()), G 2 ()=G 2 ()(+i 2 ()), (a, b) where i =, 2 is the material loss factor and is the frequency in rad/s. Note that each atch may be different in size and material roerties. The scoe of this article is limited to the harmonic vibration analysis of a sandwich beam, as shown in Figure, with arbitrary boundary conditions. One section of the beam is illustrated in Figure 2 with all relevant variables secified including flexural (w) and longitudinal (u) dislacements as well as rotary () and shear angles (). However, shear deformations in elastic layers (layers and 3) will be ignored in section 3 for the sake of simlification. z l l l + 2 h h 2 h h 2 h + h h 3 x x x + l x Boundary at x = Figure. Beam with constrained layer daming atches. Boundary at x = l

3 DAMPING PATCHES 783 u 2 u γ ψ h h 2 h ψ 2 γ 2 ψ 3 γ 3 δw δx w Undeformed osition u 3 Deformed osition Figure 2. Variables in all layers ENERGY FORMULATION The comlex-valued strain energy (U ) of the system of Figure has contributions from flexural dislacement w (same for each layer), longitudinal dislacements u, u 2 and u 3,and shear deformation, 2 and 3 where suerscrit =,...,N denotes the atch number for layers and 2. Also, refer to Aendix A for the identification of symbols. N U ()= = [ 2 (Dr ) T E (Dr )+ 2 (Dr 2) T E 2 () (Dr 2)] dx + 2 (Dr 3 ) T E 3 (Dr 3 )dx. (2) Note that r, r 2,andr 3 are deformation vectors in which rotations, 2,and 3 are used instead of shear deformations, 2,and 3 : r =w u, r 2 =w 2 u 2, r 3 =w 3 u 3; =,...,N. (3a c) Here D is the differential oerator matrix defined as D =2 /x 2 /x /x. (4) And E, E 2 and E 3 are elasticity matrices that are defined as E =E I G A E A, 2 ()I2 E 2 ()=E G 2 ()A2 E 2 ()A 2, 3 I 3 E 3 =E G 3 A 3 E 3 A 3; =,...,N, (5a c)

4 784 where is the shear correction factor. The real-valued kinetic energy of the system (T) due to flexural, longitudinal and rotary motions is exressed as where N T = [ 2 r T H r + 2 r T 2 H 2 r 2 ]dx + l = T 2 r 3 H 3 r 3 dx, (6) H = A I A, = 2 A2 H 2 2 I2 2 A 2, H 3 = 3 A 3 3 I 3 3 A 3, =,...,N. (7a c) 2.3. RAYLEIGH RITZ METHOD To imlement the Rayleigh Ritz minimization scheme, the flexural dislacement of the beam w is aroximated as w(x, t)=(x)q(t), (8) where q =[q q 2 q k q n ] T is the generalized dislacement vector of the system and =[ 2 k n ] is the flexural shae function vector in which each term k is an admissible function that satisfies the essential boundary conditions of the beam. Recall that energy equations (2) and (6) contain 4N + 3 unknowns: flexural dislacement w, rotation, 2,and 3, and longitudinal dislacements u, u 2, and u 3 for =,...,N. If these unknowns were to be aroximated with n trial functions and to be incororated in the Rayleigh Ritz minimization scheme, the resulting eigenvalue roblem would be of dimension n(4n + 3). An alternative is to assume relationshis between these unknowns; that is, for each flexural admissible function k (x), the corresonding rotational shae functions,k (x), 2,k (x) and 3,k (x) as well as the longitudinal shae functions,k (x), 2,k (x) and 3,k (x) can be calculated by using these relationshis, which will be derived in section 3. With the above assumtion, deformation vectors can be exressed as r = S (x)q(t), r 2 = S 2 (x)q(t), r 3 = S 3 (x)q(t), (9a c) where S, S 2,andS 3 are admissible shae function matrices defined as S =, S 2 = 2 2, S 3 = 3 3; for =,...,N; (a c) ε x = x = Figure 3. Patch boundary conditions at x =. Fixed-end atch, free-end atch where.

5 DAMPING PATCHES 785 and =[,,k,n], 2 =[2, 2,k 2,n], 3 =[ 3, 3,k 3,n ], are the corresonding rotational shae function vectors while =[,,k,n], 2 =[2, 2,k 2,n], 3 =[ 3, 3,k 3,n ] are the corresonding longitudinal shae function vectors. Using equation (9), the strain and kinetic energies can be written as U ()= 2 q T K()q, T = 2 q T Mq, (a, b) where the frequency deendent comlex-valued stiffness (K ) and real-valued mass (M) matrices of the system are N K ()= = [(DS ) T E (DS )+(DS 2) T E 2 () (DS 2)] dx + l (DS 3 ) T E 3 (DS 3 )dx, N M = [S T H S + S T 2 H 2 S 2]dx + l S T 3 H 3 S 3 dx. (2a, b) = The frequency deendent comlex eigenvalue roblem of dimension n can be obtained as Mq +K ()q =. (3) Several aroaches are available in the literature [9, ] for solving eigenvalue roblems of non-roortionally damed systems with frequency deendent arameters, whose eigenvalues and eigenvectors are comlex-valued. Using the method of Rikards et al. [], undamed natural frequencies ( r ) and comosite modal loss factors ( r ) are related to the comlex-valued eigenvalues r of equation (3) in the following manner where r is the modal index: r = Re ( r), r =Im( r)/re ( r); r =,...,n. (4a, b) TABLE System arameters used for examles given in the literature. Refer to Figure for nomenclature Rao [7] Lall et al. [4] Material E and E 3 (Pa) roerties G 2 (Pa) and 3 (kg/m 3 ) (kg/m 3 ) 26 2 Dimensions h 3 5 (mm) h h l 3 Beam boundary Clamed-free Simly suorted conditions on both sides Patch boundary fixed end free end conditions at x =

6 786 w.4 u.4.2 (c) u 3.2 (d) γ x/l Figure 4. First three mode shaes for Rao s examle [7]. Flexural modes (w) of the beam, longitudinal modes (u ) of layer, (c) longitudinal modes (u 3) of layer 3, (d) shear modes ( 2) of layer 2. Key: ---, mode ;, mode 2;, mode ADMISSIBLE FUNCTIONS FOR THIN BEAMS In section 2.3, the fundamental relationshis between all 4N + 3 unknowns are assumed in order to obtain an eigenvalue roblem of dimension n. This section exlicitly shows these relationshis by deriving the corresonding shae function of each unknown for a given admissible flexural function. For the sake of simlification, only thin elastic layers ( and 3) are assumed. The following two stes are involved in the variable reduction rocedure. First, the classic sandwich beam theory [7] is emloyed along with the thin elastic layer assumtion to reduce the number of unknowns to N + 2. Second, a secondary TABLE 2 Comarison between Rao s ublished [7] and roosed methods. See Table for arameters Natural frequency (Hz) Modal loss factor Published Proosed Published Proosed Mode (32)* (6 8 3 )* Mode (6869)* (2 7 2 )* Mode (6 497)* (3 8 2 )* * Solution from an aroximate formulation given by Rao [7]

7 DAMPING PATCHES 787 TABLE 3 Comarison between Lall et al. s ublished [4] and roosed methods. See Table for arameters Natural frequency (rad/s) Modal loss factor Published Proosed Published Proosed 2% coverage Mode % coverage Mode % coverage Mode % coverage Mode Mode Mode Mode minimization scheme is used to further reduce the N + 2 unknowns to one flexural shae function vector. 3.. VARIABLE REDUCTION BY USING CLASSIC SANDWICH BEAM THEORY Kerwin s weak core assumtion [8] is alied to longitudinal shae functions and 3 as E A,k /x + E 3 A 3 3,k /x =; =,...,N. (5) Integrating both sides with resect to x, the following exression is obtained:,k = dk e 3,k, (6) where e = E 3 A 3 /(E A) and dk is the constant that relates admissible shae 3,k to the corresonding,k. Next, observing the kinematic relationshi in Figure 2, the longitudinal deformation (2,k) and rotation (2,k) of layer 2 are exressed as 2,k = 2 [( 3,k (h 3 /2) 3,k )+(,k +(h /2),k)], (7) 2,k = (/h 2 )[( 3,k (h 3 /2) 3,k ) (,k +(h /2),k)]. (8) Substituting equation (6) into equations (7, 8), 2,k and 2,k are rewritten as 2,k =(h /4),k (h 3 /4) 3,k + ([ e ]/2) 3,k + (/2)dk, (9) Signal analyzer (HP35665A) Foam Clamed boundary Accelerometer (PCB 39A) Patch Beam Signal generator Electro-magnetic transducer (MAXI-MAG 3A) Figure 5. Schematic of exerimental setu.

8 788 TABLE 4 Proerties of baseline beam and daming atches Stiffness (N/m 2 ) Density (kg/m 3 ) Thickness Patch A Layer E = 8 9 = 772 h /l = Layer 2 G 2 = = 2 h 2/l = Patch B Layer E = 8 9 = 772 h /l = Layer 2 G 2 =3 6 2 = 2 h 2/l = Baseline beam of Layer 3 E 3 = = 735 h 3/l = l = 77 8 mm 2,k = (h /2h 2),k (h 3 /2h 2) 3,k + ([ + e ]/h 2) 3,k (/h 2)d k. (2) For a beam with thin elastic layers whose shear deformations and 3 are ignored, rotations of layer and layer 3 are the same as the sloe of the beam, Therefore, 2,k and 2,k can be rewritten as,k = k /x, 3,k = k /x. (2a, b) 2,k =([h h 3 ]/4) k /x + ([ e ]/2) 3,k + (/2)d k, (22) 2,k = ([h + h 3 ]/2h 2) k /x + ([ + e ]/h 2) 3,k (/h 2)d k. (23) To reduce the 4N + 3 unknowns to N + 2, define transfer matrices V, V 2 and V 3 as S = V, S 2 = V 2, S 3 = V 3, (24a c) where transfer matrices V, V 2,andV 3, as shown below, are derived by using equations (6, 2 23): V =, V x 2 = (h + h 3 ) (e +), 2h2 x h2 h2 (h e h 3 ) e 4 x 2 2 V 3 = x. (25a c) l Figure 6. Benchmark examles. Full free-end atch (benchmark for daming studies based on exerimental measurements), beam without any atch (baseline beam). l

9 DAMPING PATCHES 789 TABLE 5 Benchmark results for cantilever beams Undamed natural frequency Modal loss factor Theory Exeriment* Theory Exeriment* Mode r f r (Hz) ( rl) 2 ˆ r (%) f r (Hz) ( rl) 2 ˆ r (%) r (%) ˆ r (%) r (%) ˆ r (%) Beam with full free end Patch A Baseline beam (without any atch) * Exerimental database for the full free-end atch is the benchmark case Also, the reduced admissible shae matrices and of equation (24) are defined as =, 3 = d 3 ; =,...,N, (26a, b) where d =[d d k d n ] and each term d k is a constant that relates the admissible shae 3,k to its corresonding,k. Note that this constant (d k ) has been ignored by many rior researchers but it is retained here since it lays an imortant role in determining the longitudinal boundary conditions [] for each atch. For examle, if atch is a fixed-end atch as shown in Figure 3, d must be zero and accordingly it must be eliminated from. As for a free-end atch of Figure 3, d remains undetermined until the secondary minimization scheme is used. The issue of atch boundary conditions will be further examined in section VARIABLE REDUCTIONS BY USING A SECONDARY MINIMIZATION SCHEME Recall that the number of unknowns has been reduced from 4N +3 to N + 2 for the kth admissible function set in matrices and. These N + 2 unknowns are flexural shae functions k, longitudinal shae functions for the base beam 3,k, and the constants d k. Since no exlicit equations are available to relate these unknowns, a secondary minimization scheme is imlemented. First, each admissible function 3,k is aroximated as 3,k = c k, (27) where c k is a coefficient vector to be determined and is the row trial function vector whose terms satisfy essential boundary conditions. The real art of total strain energy of the beam (U k ) exeriencing the deformations of the kth of admissible function set k and k can be exressed as U k ()= 2 K k ()q 2 k, (28)

10 79 where N K k ()= [(DV k) T E (DV k)+(dv 2 k) T E 2 () (DV 2 k)dx = + l (DV 3 k ) T E 3 (DV 3 k )dx (29) is the effective stiffness at any of interest and q k is the corresonding generalized dislacement. Note that in the above analysis the imaginary art of the comlex-valued stiffness is ignored because only kinematic relationshis are of interest. By substituting equations (25 27, 29) into equation (28) and minimizing U k with resect to coefficients of c k and d k, where d k =[dk dk dk N ] T, the set of governing equations can be summarized in matrix form as where AC k = B k, (3) A = Acc A dd cd A dc A, B k k = Bk, Bc d C k = c k d k. (3a c) 5 Comliant Transition Stiff f r (Hz) (c) (d) 3 (e) G 2 (N/m 2 ) 8 Figure 7. Predicted f r G 2 relationshis and measured f r values for estimating G 2 value. Mode, mode 2, (c) mode 3, (d) mode 4, (e) mode 5. Key:, theory;, exeriment;, intersection.

11 DAMPING PATCHES 79.4 η r η 2 8 Figure 8. Predicted r 2 relationshis and measured r values for estimating 2 value. Key:, mode ;, mode 2; ---, mode 3;, mode 4;, mode 5;, measurements. Submatrices of A and sub-vectors of B k are obtained as follows: A cc = l T x E 3 A 3 N x dx + = T x E 3 A 3 e x (e + +) 2 G T 2 A 2 dx, (h2) 2 +)G A cd 2 = A2 (h2) T dx (e 2 (e +)G 2 A 2 (h 2) 2 T dx (e N +)G2 N A2 N (h2 N ) T dx 2 N, +)G A dc 2 = A2 (h2) T dx (e 2 (e +)G 2 A 2 (h 2) η f (Hz) Figure 9. Frequency deendent material loss factor 2 for layer 2 of atch A. Key:, assumed mean; ---, uer limit;, lower limit. 2

12 792 T dx (e N +)G2 N A2 N (h2 N ) T dx T, 2 N A dd = G 2 A 2 l /(h 2) 2 G 2 A 2 l /(h 2) 2 G N 2 A N 2 l N /(h N 2 ) 2, N C B c k = (e +)G2 A2 (h = 2) k x 2 T dx, B d k = C (e +)G 2 A 2 (h 2) 2 k C (e +)G 2 A 2 (h 2) 2 k C N (e N +)G2 N A2 N T N (h2 N ) k, 2 (32a f) where C =(h +2h2 + h 3 )/2 and the satial oerator is defined as f (x)=f (x + l /2) f (x l /2). (33) The coefficients c k and d k of C k can be calculated by C k = A B k, (34) rovided A. As a result, reduced admissible shae matrices and can be determined for a given flexural shae function vector. 4. COMPARISON WITH LITERATURE To validate the roosed formulations, two secific examles found in the existing literature are analyzed first. Table summarizes the system arameters that were used by Rao [7] and Lall et al. [4]. For the resent study, analytical solutions are obtained by using.7 l.65 l 3 η r (%) Mode Figure. Comarisons of measurements and redictions of cantilever sandwich beam with a single atch. Schematic, modal loss factors. Key:, exeriment;, theory; I, variation.

13 DAMPING PATCHES admissible functions for the flexural dislacement and 2 trial longitudinal shae functions for each flexural shae function. Rao [7] studied a clamed free beam with a full daming treatment on one side of the beam. The viscoelastic material has a fixed boundary condition at x =. He found the exact solution only for the first mode. Additionally, the first three natural frequencies and modal loss factors were calculated by using an aroximate formulation [7]. It is seen in Table 2 that the results obtained from the resent method are very close to the exact solutions given by Rao. Reasonable agreement is also seen with Rao s aroximations. Lall et al. [4] analyzed a simly suorted beam with a single atch. The following arametric studies were carried out: coverage ratios l /l = 2, 4, 6 and %; atch locations x /l =, 2, 3, and 5 resectively. Comarisons of Table 3 show an excellent match between Lall s and the resent method. Such results are exected since both methods are based on the Rayleigh Ritz aroach. Chief advantage of the roosed formulation, however, is the ease with which mode shaes for all tyes of deformation in any layer can be visualized. Figure 4 shows the first three flexural modes of the beam as well as the corresonding longitudinal modes of layers and 3, and shear mode of layer 2 for Rao s examle [4]. 5. EXPERIMENTAL STUDIES In order to further verify the analytical model as well as to investigate various henomena associated with atch daming, modal tests are carried out on a cantilever beam made of mild steel (Table 4). A eriodic chir as generated within the signal analyzer is fed to a non-contacting magnetic transducer that excites the beam at the free ˆ ω r (%) η ˆ r (%) Mode 4 (c) Mode Figure. Results for a cantilever beam with fixed-end atch. Schematic, normalized eigenvalues, (c) normalized modal loss factors. Key:, exeriment;, theory; --, benchmark (measured value for a free-end atch).

14 794 end, as shown in Figure 5. Structural resonse is measured via a comact accelerometer (of weight g) near the root. Sinusoidal transfer functions are then obtained. No calibration is necessary since only frequency measurements are needed. First five natural frequencies ( f r ) and modal loss factors ( r ) are then extracted using the half-ower bandwidth method []. Two tyes of daming treatment (designated here as Patches A and B) with material roerties and layer thickness as secified in Table 4 are alied in these studies. However, the material roerties of the viscoelastic core are not available. Therefore, a rocess for estimating the material roerties must be develoed before analyzing the damed beam structure. An uncertainty study is also carried out to establish error bounds for estimations. Finally, the rocedure used for obtaining normalized exressions is exlained in this section. 5.. MATERIAL PROPERTY ESTIMATION A material roerty estimation technique is emloyed by combining analytical redictions and measured modal results. Of interest here are the roerties of layer 2: G 2 and 2 since layers and 3 are made of well-known steel. The following rocedure is demonstrated with Patch A as an examle: () Choose one examle and erform an exeriment. The examle case is a cantilever beam with full daming treatment on one side and free atch boundary at x = as shown in Figure 6. Natural frequencies and modal loss factors for the first few modes are then obtained as listed in Table 5. w (c) x/l Figure 2. Flexural dislacement mode shaes. Mode, mode 2, (c) mode 3. Key:, free-end atch;, fixed-end atch;, baseline beam.

15 DAMPING PATCHES 795 (2) Develo f r G 2 relationshis where f r = r /2 is the natural frequency in Hz. With the material loss factor 2 taken as zero, the analytical model is used to redict the variation in f r over a range of G 2 values. A general trend of this relationshi can be seen in Figure 7, where three distinct regions are observed: very comliant, transition and very stiff. These are similar to those reorted in beams with joints [2]. (3) Given measured f r results, find shear modulus G 2 from the grahs. In Figure 7, a horizontal line is drawn at the measured frequency for each mode. A cross mark reresents the intersection of this line and the f r G 2 curve; this yields G 2 at that frequency. Note that in Figure 7, no intersection is found because the measured value is less than the low frequency asymtote of the curve. This is because of the non-ideal claming boundary conditions [3], which esecially affect mode. Therefore, mode is excluded from the shear modulus estimation rocedure. Figure 7(b e) show similar G 2 values over the range of interest. For this articular case, it is safe to assume a sectrally invariant G 2, as listed in Table 4. (4) Develo r 2 relationshis and estimate the material loss factor of layer 2 as a function of frequency. With the assumed G 2 or G 2 ( f ), the analytical model is again used to redict a general relationshi between 2 and the modal loss factors r of the sandwich beam. In Figure 8, such r 2 relationshis are comared with the measured modal loss factors. Again, each cross mark indicates the r value. As a result, a frequency deendent relationshi is obtained for the viscoelastic core of Patch A that is curve fitted to yield: 2,A ( f ) = f f f f 4, γ 2 5 (c) x/l Figure 3. Shear deformation mode shaes of layer 2. Mode, mode 2, (c) mode 3. Key:, free-end atch;, fixed-end atch.

16 796 where f is the frequency in Hz. The same rocedure is erformed on Patch B and the material loss factor is exressed as 2,B ( f ) = f f f f 4. With given material roerties as listed in Table 4, a secific eigenvalue roblem may be constructed for each mode with a articular 2 value at the natural frequency. An iterative rocedure is obviously needed for obtaining eigensolutions [9]. Nonetheless, desite their frequency deendent nature, the material loss factors are assumed to be invariant in the immediate vicinity of an eigenvalue to avoid any iteration []. Exerimental results that will be reorted in the next section validate this assumtion. Also note that the analytical formulation is again used with 2 admissible functions and 2 trial functions for all examle cases UNCERTAINTY OF MATERIAL PROPERTY The determination of 2 is a key to the success of analytical method of the article. However, modal measurements used for the estimation rocedure are affected by many factors including inherent beam daming, microscoic friction at the root and non-erfect bindings between layers. In ractical structures a significant variation in measured r values may be seen. To examine such uncertainties, a 2% tolerance in the daming measurement is assumed. The uer and lower bounds of 2,A ( f ) due to this tolerance are shown in Figure 9. These values are alied to the case of Figure, where a single atch is alied to the cantilever beam from x = 3l to l. A comarison of redicted and measured modal loss factors is shown in Figure. Note that the error bars on redicted r indicate the uncertainty associated with the 2,A ( f ) estimation, while the error.3 l.3 l η ˆ r (%) ˆ ω r (%) (c) Mode Figure 4. Results for a cantilever sandwich beam with a small cutout. Schematic with a free-end atch, normalized eigenvalues, (c) normalized modal loss factors. Key:, exeriment;, theory.

17 DAMPING PATCHES 797 bars on exerimental results indicate the 2% tolerance in measurements. The overla of error bars imlies excellent agreement between measurements and redictions. Only the mean values of 2 are used in subsequent studies; however the robabilistic nature of daming values must be considered when viewing all results NORMALIZATION PROCEDURES Often it is desirable to exress modal results in normalized forms. For examle, the loss factors of a beam with atch daming ( r ) may be normalized with resect to the case where the beam is fully covered with a viscoelastic material ( r,full ): ˆ r = r / r,full (35) in which both values are either redicted or measured. This normalization (given by suerscrit ) can be used to describe the effectiveness of the atch daming concet. Similarly, natural frequencies may be normalized as follows to indicate the mass loading effect ˆ r = r / r,full. (36) However, since different tyes of atches and boundary conditions will be discussed later, it is more aroriate to use a single set of measured results throughout the article as the base for normalization. The resulting normalized natural frequency ˆ r and modal loss factor ˆ r are defined here as ˆ r,b = r / r,b, ˆ r,b = r / r,b (37, 38).3 l.3 l η ˆ r (%) ˆ ω r (%) (c) Mode Figure 5. Results for a cantilever sandwich beam with a large cutout. Schematic with a free-end atch, normalized eigenvalues, (c) normalized modal loss factors. Key:, exeriment;, theory.

18 798 where subscrit b refers to the measured results of a benchmark case: a cantilever beam with full material A daming treatment and free atch end at x = as shown in Figure 6. Yet another exression is used to describe the dimensionless eigenvalue by assuming the damed structure to be an undamed Euler beam. This eigenvalue arameter ( r l) 2 is defined as ( r l) 2 = r l 2 /E 3 / 3. (39) Measured and redicted modal results for the benchmark case as well as the baseline beam (i.e., undamed beam without any daming atch) of Figure 6 are listed in Table 5. Measurements show that fairly high inherent daming is resent in the first mode of the baseline beam. This may be the result of a non-ideal claming condition at x =. Consequently, some caution must be exercised when examining the damed beam results esecially at the first mode. 6. RESULTS AND DISCUSSION 6.. EFFECT OF PATCH BOUNDARY A cantilever beam with a daming atch embedded into the fixed boundary at x = is said to have a fixed-end atch (Figure 3) as oosed to the free-end atch where the material is unconstrained at x =. Practically, this boundary condition is achieved by claming the atch along with the beam at the root. Analytically, a fixed atch end is Large cutout of Figure 5 Small cutout of Figure 4 w (c) x/l Figure 6. Flexural dislacement mode shaes. Mode, mode 2, (c) mode 3. Key:, small cutout;, large cutout; --, baseline beam.

19 DAMPING PATCHES 799 Large cutout of Figure 5 6 Small cutout of Figure 4 γ 2 6 (c) x/l Figure 7. Shear deformation mode shaes of layer 2. Mode, mode 2, (c) mode 3. Key:, small cutout;, large cutout; --, baseline beam. simulated by forcing the column vector d to be zero as described in section 3. Measured and redicted modal characteristics are then normalized and comared with the benchmark case (free-end atch) as shown in Figure. Both measurements and redictions indicate that natural frequencies and loss factors of the beam with fixed-end atch are much higher than those of the beam with a free-end atch, esecially for mode. Nearly coincident flexural mode shaes (Figure 2) of these two cases, as redicted by analytical models, rovide little exlanation for this. However, a closer examination of the shear deformation mode shaes of layer 2, as seen in Figure 3, yields very distinct characteristics between these two cases. The fixed-end atch, acting as an additional constraint, causes the natural frequencies to increase and forces the deformation 2 to be zero at the root. This constraint also results in a higher 2 2 value when.8 ˆ η r Cutout location x /l Figure 8. Effect of cutout locations on normalized loss factors of a sandwich beam. Key: mode 2;, mode 3., mode ;,

20 8 ˆ η r Cutout size l /l Figure 9. Effect of cutout size on normalized loss factors of a sandwich beam. Key:, mode ;, mode 2;, mode 3. integrated over the atch length, which imlies more energy dissiation. This discreancy in shear deformation mode shaes is very noticeable for mode, but not as significant for modes 2 and 3 as shown in Figure EFFECT OF CUTOUTS IN DAMPING TREATMENT A cutout, no matter how small it is, essentially creates a beam with two distinct daming atches. Resulting modal characteristics are investigated by using two examle cases. Figure 4 shows a cantilever sandwich beam with a small cutout (with 3% of beam length) located at 3l from the root (x = ). Figure 5 shows a similar beam excet that the cutout is 3% of the beam length but still located at 3l. Measured and redicted eigenvalues and modal loss factors are normalized and lotted in Figures 4 and 5. It is observed that the small cutout case yields more daming than the large cutout one, as one would intuitively exect. A higher flexural amlitude is found near the cutout location of the large cutout case, esecially for the third mode (Figure 6). Figure 7 shows shear deformation mode shaes in the core material. Note that a higher 2 2 value, when integrated over the atch length, indicates increased energy dissiation. Parametric studies are carried out analytically in order to further investigate the effect of cutout size and cutout location. Figure 8 shows normalized loss factors of the first three modes for a sandwich beam with a 3% cutout at various locations x /l. It is seen that the loss factor value is very sensitive to the cutout location, esecially for lower modes. Modal loss factors as affected by cutout size with a given cutout location ( 3l from the root) are A.4 l.2 l B.22 l A.4 l B.22 l.7 l Figure 2. Cantilever beam with mismatched atches. Case A, Case B, (c) Case C. Note that Case C is a combination of Cases A and B..2 l.7 l (c)

21 DAMPING PATCHES 8 lotted in Figure 9. Again, a monotonic decrease in r is observed as the cutout size is increased EFFECT OF TWO MISMATCHED PATCHES A cantilever beam with two mismatched atches is examined by using three examle cases, as shown in Figure 2, to see whether the daming atches introduce daming in an additive manner. A beam with a single Patch A of length 4l located at 2l from the root is designated as Case A, and a beam with a single Patch B of 22l at 7l from the root as Case B. Then both atches A and B are alied simultaneously; this is designated as Case C. Measured and redicted eigenvalues and loss factors are normalized and listed in Table 6. Flexural dislacement mode shaes are shown in Figure 2. A simle additive effect in modal daming can be exressed as r,c = r,a + r r,b, (4) where subscrits A, B and C are the case designations defined earlier. According to the analytical model, equation (4) may not work since the resulting mode shaes are not the same because of the mass loading effect. Therefore, a modified exression is introduced to describe the additive effect as r,c = r r,a + r r,b, (4) where r and r are the weighting factors for mode r. Note that r and r are obtained analytically and samle values are listed in Table 7. It is seen that r and r. This indicates that Patch B rovides more daming in Case C than it does in Case B. This may be exlained by looking at the shear deformation mode shaes of layer 2 in Figure 22, where 2 of Case C has higher absolute values in the Patch B region than that in the Patch B region of Case B. TABLE 6 Results for a cantilever beam with mismatched atches. Refer to Figure 2 Eigenvalue ( rl) 2 Normalized loss factor ˆ r Mode Theory Exeriment Theory (%) Exeriment (%) Beam with Patch A (Case A) Beam with Patch B (Case B) (c) Beam with Patches A and B (Case C)

22 82 Patch A Patch B w (c) x/l Figure 2. Flexural dislacement mode shaes. Mode, mode 2, (c) mode 3. Key:, beam with Patch A;, beam with Patch B;, beam with Patches A and B. Additionally, the issue of inherent daming in the exerimental study needs to be resolved. Measured modal loss factors are considered to have contributions from inherent daming of baseline beam and alied atch daming, i.e., r,ai = r,i + r,a, r,bi = r,i + r,b, r,ci = r,i + r,c, (42a c) where the subscrit I indicates the inherent daming. The values of inherent daming r,i are found from the modal measurements on the baseline beam. Modal daming of Case C is then estimated with weighting factors similar to equation (39) as r,ci = r,i + r r,a + r r,b = r,i + r ( r,ai r,i )+ r ( r,bi r,i ). (43) Note that one may also develo a simle additive estimation rocedure where r = r =. Modal loss factors that secifically exclude inherent daming for Cases A and B are TABLE 7 Weighting factors r and r for equation (4) as derived from analytical models Mode r r r

23 25 DAMPING PATCHES γ (c) x/l Figure 22. Shear deformation mode shaes of layer 2. Mode, mode 2, (c) mode 3. Key:, beam with Patch A;, beam with Patch B;, beam with Patches A and B. normalized and listed in Table 8. The values of ˆ r,ci are first estimated by using equation (4) with r and r taken from the analytical results (Table 7) and then with r = r =. Both sets of estimations (weighted and simle additive) are then comared with measured results of Case C. Table 8 shows that both estimation methods comare well with measurements. Similar studies can be carried out on other atch atterns. TABLE 8 Comarison between estimated and measured ˆ r,ci for the examle of Figure 2 Mode r ˆ r,i (%) ˆ r,a = ˆ r,ai ˆ r,i (%) ˆ r,b = ˆ r,bi ˆ r,i (%) Estimation of ˆ r,a and ˆ r,b Mode r ˆ r,ci = ˆ r,i + ˆ r,a + ˆ r,b (%) ˆ r,ci = ˆ r,i + ˆ r,a + ˆ r,b (%) Measured ˆ r,ci (%) Comarison of estimated and measured ˆ r,ci

24 84 7. CONCLUSION This aer resents a refined method for analyzing the harmonic resonse of beams with multile constrained-layer viscoelastic atches. Initially the method is develoed for thick beams, but subsequently it is restricted to thin beams. The classic sandwich beam theory and a secondary minimization scheme are emloyed to derive kinematic relationshis between flexural dislacement and other deformations in all layers. This aroach requires the inclusion of only flexural shae functions in the comlex eigenvalue roblem. Nonetheless, eigenvectors can be related to flexural, longitudinal and shear mode shaes, some of which can not be exerimentally measured. Most imortant of all, the knowledge of shear deformation modes in the viscoelastic core rovides an imroved understanding of the effect of atch daming. The roosed model can be alied to either fully or artially covered sandwich beams. It is successfully validated by comaring results with the examles described by Rao [7] and Lall et al. [4]. Several daming configurations are then exerimentally and analytically studied. Excellent agreement between theory and exeriment is seen for all examles. Some imortant atch daming issues have been clarified esecially through an examination of modal deformations. In order to identify the unknown roerties of the viscoelastic material used in this article, an estimation rocedure has been roosed. The frequency-deendent material loss factor and stiffness are estimated by combining analytical arametric studies with modal measurements from beam tests. An uncertainty study has also been carried out to establish the error bounds of these estimations. Future work will extend this formulation to thick beams and lates. Imortant design issues, including the otimization of atch atterns for imroved daming erformance, also need to be investigated. ACKNOWLEDGMENT This research has been suorted by the Army Research Office (URI Grant DAAL 3-92-G-2; ; Project monitor: Dr. T. L. Doligalski). REFERENCES. A. D. NASHIF, D. I. G. JONES and J. P. HENDERSON 985 Vibration Daming. New York: John Wiley. 2. C. T. SUN and Y. P. LU 995 Vibration Daming of Structural Elements. New Jersey: Prentice Hall. 3. D. S. NOKES and F. C. NELSON 968 Shock and Vibration Bulletin 38, art 3, 5. Constrained layer daming with artial coverage. 4. A. K. LALL, N. T. ASNANI and B. C. NAKRA 988 Journal of Sound and Vibration 23, Daming analysis of artially covered sandwich beams. 5. H. DEWA, Y. OKADA and B. NAGAI 99 JSME International Journal Series III 34, Daming characteristics of flexural vibration for artially covered beams with constrained viscoelastic layers. 6. C. LEVY and Q. CHEN 994 Journal of Sound and Vibration 77, Vibration analysis of a artially covered, double sandwich-tye, cantilever beam. 7. D. K. RAO 978 Journal of Mechanical Engineering Science 2, Frequency and loss factors of sandwich beams under various boundary conditions. 8. E. M. KERWIN. JR. 959 Journal of the Acoustical Society of America 3, Daming of flexural waves by a constrained viscoelastic layer. 9. R. M. LIN and M. K. LIM 996 Journal of the Acoustical Society of America, Comlex eigensensitivity-based characterization of structures with viscoelastic daming.. R. RIKARDS, A. CHATE and E. BARKANOV 993 Comuters and Structures 47, 5 5. Finite element analysis of daming the vibration of laminated comosites.

25 ˆ DAMPING PATCHES 85. P. TROMPETTE, D. BOILLOT and M.-A. RAVANEL 978 Journal of Sound and Vibration 6, The effect of boundary conditions on the vibration of a viscoelastically damed cantilever beam. 2. R. SINGH, M. L. LIAW, J. E. FARSTAD and S.-W. KUNG995 The American Society of Mechanical Engineers. Paer No. 95-WA/NCA-28. Determination of joint stiffness through vibration analysis of beam assemblies. 3. H. VINAYAK and R. SINGH 996 Journal of Sound and Vibration 92, Eigensolutions of annular-like elastic disks with intentionally removed or added material. APPENDIX A: LIST OF SYMBOLS A cross sectional area A governing equation matrix a, b, c coefficient vectors a, b, c coefficients B governing equation vector C coefficient vector C thickness arameter (h +2h 2 + h 3)/2 d satial matrix d satial constant E elasticity matrix E Young s modulus e elasticity ratio E 3 A 3 /E A f frequency (Hz) G shear modulus H inertia matrix h thickness I area moment of inertia i K stiffness matrix l length M mass matrix N total number of atches n total number of shae functions q generalized dislacement vector q generalized dislacement r deformation vector S admissible shae function matrix T kinetic energy u in-lane or longitudinal dislacement U otential or strain energy V transfer matrix w flexural dislacement x, z satial coordinates shear deformation loss factor shear correction factor eigenvalue rotational shae function vector rotational shae function OPERATORS D Im Re SUPERSCRIPTS T SUBSCRIPTS mass density longitudinal shae function vector longitudinal shae function reduced admissible shae matrix flexural shae function vector flexural shae function trial function vector for longitudinal dislacement rotation vector rotation frequency (rad/s) differential oerator matrix imaginary art real art a satial oerator differential oerator atch number transose comlex valued normalized quantity A, B tye of daming atch b measurement of the benchmark case I inherent daming i layer number k admissible function number r modal index layer (elastic constraining layer) 2 layer 2 (viscoelastic constrained layer) 3 layer 3 (base structure: beam)