Optimal joint placement and modal disparity in control of flexible structures

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1 Computers and Structures xxx (2007) xxx xxx Optimal joint placement and modal disparity in control of flexible structures Alejandro R. Diaz *, Ranjan Mukherjee Mechanical Engineering Department, Michigan State University, East Lansing, MI 48824, USA Received 5 March 2007; accepted 30 April 2007 Abstract Joints with variable stiffness are used to introduce modal disparity in a frame structure. The joint stiffness can be set at two distinct values and thus the structure can be characterized by two stiffness states. The placement of a small number of these joints on the structure is optimized to achieve a maximum measure of modal disparity. This allows the migration of vibration energy from modes that are not controlled to modes that are, and facilitates the design of simpler and less expensive controllers. An example 3D frame is used for illustration. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Structural control; Modal disparity; Variable stiffness; Switched stiffness 1. Introduction In recent papers, Diaz and Mukherjee [3,4] introduced the concept of modal disparity and discussed how it can be used to gain control over a set of structural modes by directly controlling only a subset of the modes. The methodology relies on a carefully crafted variation in the stiffness of the structure to achieve modal energy redistribution from the modes that are not directly controlled (not-controlled modes) to those that are controlled directly (controlled modes). Stiffness variation was accomplished by adding cables to frame structures and switching the tension in the cables on and off, thus creating two distinct stiffness states, one associated with cables slack and the other with cables taut. After introducing a measure of modal disparity, the number and layout of the cables were determined by solving a topology optimization problem that maximizes modal disparity. In [3] it was shown that energy transfer from not-controlled modes to controlled modes can be enhanced by changing the property of the structure, * Corresponding author. address: diaz@egr.msu.edu (A.R. Diaz). and in particular, through addition of a small number of non-structural masses. The success of a control strategy exploiting modal disparity depends on the ability to generate energy redistribution from not-controlled modes to controlled modes through a change in the stiffness of the system. If a not-controlled mode in one stiffness state is nearly identical to a not-controlled mode in the other stiffness state, then modal energy will drain very slowly from these modes and for most structures this is the more likely scenario. The rate of energy transfer across modes is related to the amount of modal disparity introduced by the stiffness change and this is a property of the structure and of the mechanism introduced to effect the change in stiffness. In this paper we use a more direct mechanism for generating modal disparity: instead of using cables and cable tension, we introduce modal disparity by actively changing the stiffness of a limited number of joints of the structure. We assume that we have a mechanism for varying the stiffness of a joint between two values, affecting only rotational degrees of freedom. Optimization is necessary to determine the more advantageous location of a small number of these special joints. Although we have conceptual designs for the /$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi: /j.compstruc

2 2 A.R. Diaz, R. Mukherjee / Computers and Structures xxx (2007) xxx xxx implementation of variable stiffness joints, this work presents simulation results only. There is prior work on control of large space structures using semi-active joints. For vibration suppression, Gaul and Nitsche [5] designed the joints such that the normal force in the frictional interface can be controlled to improve damping. The normal contact force was varied using a bolt with a piezoelectric washer. A system consisting of two elastic beams connected by a single active joint was considered and the effect of displacement- and velocity-induced friction dynamics was considered in the design of the control law. It was shown that the controller prevents frictional energy, typically stored as potential energy in the bristles, from being returned into the system. Other work done by the same group includes the determination of optimal locations of the semi-active joints and experimental validation of control strategies in a 10-bay truss structure [6]. The idea of stiffness variation has been proposed earlier, for example in [14,2,9] but work is done in the process of switching stiffness in all of these papers. Our work is different in that no work is done in the process of changing the joint stiffness. Control of the structure is not achieved by stiffness variation this is achieved by modal control control is only enhanced by stiffness variation. Stiffness is varied with the objective of changing frequencies and mode shapes in a special way that is then exploited in control design. In other related work, Vakakis and co-authors [12,13,7] have investigated an energy pumping phenomenon associated with the addition of a non-linear component to a linear system. In such systems the authors show that energy is transferred from the linear to the non-linear system where it remains trapped. In contrast, our system is linear (but time-varying) and energy is transferred across modes as a result of time-variation of stiffness (resulting in time-dependent modes). This paper is organized as follows. In Section 2 we discuss the model of the structure and the joints and explicitly state our assumptions regarding the model. In Section 3 we pose the optimization problem based on our earlier defined measure of modal disparity [3]. An example that illustrates the method is discussed in Section 3. The paper ends with final remarks and recommendations for further work in Section 4. variable stiffness connection. The stiffness of this connection can be switched between two values, each corresponding to a different state of the structure. The variable stiffness connection affects only rotational degrees of freedom and a weak stiffness connection represents a pinned, truss-like joint while a strong connection represents a rigid, frame-like joint. We assume that at most one bar in each joint may be connected by a variable stiffness connection and that a structure includes no more than m such connections, where m is typically a small number. The modeling of a variable stiffness joint is illustrated in Fig. 1. The figure shows a joint connecting three bars. Bars 1 and 3 are connected rigidly, but bar 2 is connected through a variable stiffness connection that affects the local yy rotation degree of freedom of bar 2. In one stiffness state, this connection is rigid, i.e., h 2 = h 5 = h 8. In the other stiffness state, the rotation of bar 2 along the local yy direction is released and h 2 = h 8 but h 5 is free to take any value. To model this effect, a joint such as the one in Fig. 1 is first modeled as a frictionless hinge with 9 rotational degrees of freedom, all expressed in the local coordinate system of bar 2. The joint stiffness K j is then added to the stiffness matrix of the tree-bar structure. Here we assume that K j is of the form 2 K j ðsþ¼ 6 4 2K 0 0 K 0 0 K kðsþþk 0 0 kðsþ 0 0 K K 0 0 K 0 0 K K 0 0 2K 0 0 K kðsþ 0 0 2kðsÞ 0 0 kðsþ K 0 0 2K 0 0 K K 0 0 K 0 0 2K K 0 0 kðsþ 0 0 kðsþþk K 0 0 K 0 0 2K bar 1 Joint j bar 2 bar 3 (a) Bar 2 has a variable stiffness connection ð1þ 2. Analysis 2.1. Finite element model We assume that our structure is a 3D frame, a collection of bars and joints. Each bar is modeled using standard beam (frame) elements, with six degrees of freedom per node. Typically, several beam elements are used to model one bar. The element stiffness matrix can be found in any standard finite element textbook, e.g. [8]. A joint is simply a location where several bars are connected. In most cases this connection is rigid, but in a small number of joints one of the bars is connected using a bar 1 q 2 q 1 q 5 yy q 4 q 6 xx q 3 Joint zz coordinate system bar 2 q 8 q 9 q 7 bar 3 (b) Degrees of freedom in a variable stiffness joint Fig. 1. A joint with a variable stiffness connection.

3 A.R. Diaz, R. Mukherjee / Computers and Structures xxx (2007) xxx xxx 3 In (1), K models the rigid connection between two bars and is typically a large number, while k(s) is the stiffness of the variable stiffness connection, e.g., kðsþ ¼ K if s ¼ 1 0 if s ¼ 2 Here the variable s is used to represent the state of the switch that turns the variable stiffness joint on (s = 1) or off (s = 2). In a structure where the location of the variable stiffness joints is identified by the index set J, the global stiffness matrix is then formed by the assembly of two matrices: K 0 : the global stiffness matrix of the structure where all joints j 2 J are modeled as frictionless hinges (and degrees of freedom are as shown in Fig. 1(b)) and K J : the assembly of all joint stiffness matrices K j, j 2 J. The index set J identifies which joints, among all joints in the structure, will have variable stiffness connections. The structure stiffness matrix is therefore KðsÞ ¼K 0 þ X K j ðsþ ð2þ j2j A modal analysis of a structure in two stiffness states (s =1 or 2) leads to two eigenvalue problems: Stiffness state s = 1: All variable stiffness joints rigid. K 1 = K(1) and ðk 1 lmþv ¼ 0 ð3þ Stiffness state s = 2: All variable stiffness joints free. K 2 = K(2) and ðk 2 kmþu ¼ 0 ð4þ In (3) and (4), M represents the mass matrix of the structure. Eigenpairs (v (i),l i ) are associated with vibration of the structure with active joints on while (u (i),k i ) are associated with the vibration of the structure with active joints off. In general, the cross products u (i)t Mv (j) are not zero, and this makes it possible for modal energy to migrate between two modes associated with different stiffness states. We take advantage of this fact to divert energy to modes where it can be dissipated by control action. This process is enhanced by the optimal placement of the variable stiffness joints, as discussed in the following optimization problem Switching strategy assumptions The timing for switching the joint stiffness is critical from work energy considerations. A correct timing will also enable the analytical model to closely predict the experimental results and will be useful in control design. Our basic objective is to switch the stiffness to introduce modal disparity without changing the overall energy of the system. To this end, we will switch the stiffness from low stiffness state to high stiffness state, and vice versa, when the deformation of the stiffness element (used to vary joint stiffness) is zero. In terms of Fig. 1, the above discussion implies that switching will take place when h 2 h 5 =0 and h 8 h 5 = 0. A direct consequence of our switching strategy is that the stiffness of all the active joints may not be switched simultaneously and hence the overall system will change stiffness state through a set of intermediate stiffness states. From a modeling standpoint, it needs to be clarified that the switched system will be linear time-varying but piecewise linear time-invariant. Although the stiffness is changed based on the deformation of the system, it is not a continuous function of the deformation, as in a feedback system. Also, the amount of stiffness change does not depend on the deformation at any instant of time it is decided a priori. The deformation only determines the time when the stiffness is changed. Under these circumstances, we propose to use standard modal control in each stiffness state based on their linear time-invariant models and this will guarantee stability of the system in each of the stiffness states. A pertinent question that may arise relates to the frequency with which the stiffness of the system will be switched since fast switching between stable systems can result in instability. However, this will not be the case since modal control will be used to dissipate the energy in the controlled modes after every stiffness switch and this will limit the frequency of switching. 3. Optimization 3.1. Objective function The goal of the optimization is to find the placement of variable stiffness joints that leads to the most efficient dissipation of energy. In Diaz and Mukherjee [3,4] it was argued that the efficiency of this process is related to the matrix C of products c ij = u (i)t Mv (j), as follows. Assume that resources are devoted to control n C1 modes in stiffness state s = 1 and n C2 modes in stiffness state s =2, identified by index sets I C1 and I C2, respectively. Other modes are not controlled, identified using index sets I N1 and I N2. Higher order modes j > n are ignored, either because they will not be excited or because any energy that is associated with these modes will be quickly dissipated through structural damping. The goal is to switch the stiffness of the joints on and off to facilitate the transfer of energy from modes I N1 and I N2 into modes I C1 and I C2. Assume that a disturbance deforms the structure and motion w(t) ensues where wðtþ ¼ Xn i¼1 u i ðtþv ðiþ ¼ Xn i¼1 w i ðtþu ðiþ Parameters u i and w i are the modal amplitudes associated with a modal representation of w using eigenvectors v (i) or u (i), respectively. A measure of the energy in the system is the quantity

4 4 A.R. Diaz, R. Mukherjee / Computers and Structures xxx (2007) xxx xxx K ¼ U T U þ _U T _U ¼ W T W þ _W T _W ð5þ where U ¼fu 1 ;...u n g T and W ¼fw 1 ;...w n g T. Note that U ¼ CW and W ¼ C T U ð6þ Suppose the joints are initially on (s = 1). After modal control and dissipation by structural damping the contributions from all modes except modes i 2 I N1 are removed and the total energy remaining is K 1 ¼ U T R 1T R 1 U þ _U T R 1T R 1 _U ¼ W T C T R 1T R 1 CW þ _W T C T R 1T R 1 C _W ð7þ where R 1 is an n n diagonal matrix with R 1 ii ¼ 1 if i 2 I N1 0 otherwise and (6) was used to obtain the far right expression in terms of W. At time t 1 joints are released and the structure enters its second stiffness state (s = 2). Once again, after modal control and dissipation through structural damping the contributions from all modes except modes i 2 I N2 are removed and K is reduced further to K 2 ¼ W T R 2T C T R 1T R 1 CR 2 W þ _W T R 2T C T R 1T R 1 CR 2 _W where R 2 is now an n n diagonal matrix with R 2 ii ¼ 1 if i 2 I N2 0 otherwise Defining ^C ¼ R 1 CR 2 ð9þ we can write (8) as K 2 ¼ W T ^CT ^CW þ _W T ^CT ^C _W 6 ^C 2 ðw T W þ _W T _WÞ¼ ^C 2 K ð10þ where the 2-norm is used. Therefore after one complete cycle of stiffness switching the energy as measured by K is reduced by a factor controlled by kck. b Thus minimization of kck b leads to an efficient removal of energy from modes that are not controlled (I N1 and I N2 ) by controlling only on a small number of modes (I C1 and I C2 ). To achieve the most advantage per switch, we set a-priori only the number of nodes removed by control action but not which ones. This choice is made part of the optimization. Noting that, if represents the sub-matrix of C with rows i 2 I N1 and columns j 2 I N2, ^C ¼ ð11þ and a suitable objective function is f ðxþ ¼ min I N1 ;I N2 ð8þ ð12þ Design variables x control the position on the structure of the variable stiffness switches, specifically, the index set J in (2). This is discussed next Design variables Minimization of the objective function f in (13) is achieved by selecting an optimal location of a small number of variable stiffness joints. As discussed in Section 2 and illustrated in Fig. 1, we assume that at most one bar in each joint may be connected by a variable stiffness connection and that a structure includes no more than m such connections. In a structure with M possible joint locations, the complete layout can be determined from the vector x ¼fx 1 ;...; x M g T of non-negative integers where x j controls the nature of the connection of bars in joint j. Specifically, b if bar b of joint j has a variable stiffness connection x j ¼ 0 if all bars in joint j are rigidly connected Each variable x j takes values in [0,1,...,n j ], where n j is the number of bars connected at joint j Optimization problem statement Find x ¼fx 1 ;...; x M g T 2 Z M that Minimizes f ðxþ ¼ min Subject to XM j¼1 I N1 ;I N2 vðx j Þ 6 m 0 6 x j 6 n j ð13þ where m is the maximum number of allowed variable stiffness joints in the structure and vðx j Þ¼ 1 if x j > 0 0 if x j ¼ 0 The objective function depends on x because c ij = u (i)t Mv (j) and eigenvectors u (i) depend on x via (2) and (4) since the index set J identifying the location of the variable stiffness joints depends on x. Note that since in stiffness state s =1 all joints are rigid, eigenvectors v (j) are independent of x Solution scheme As stated, problem (13) cannot be solved using standard, gradient based optimization techniques. A re-casting of the problem may lead to a formulation amenable to optimization by a gradient based approach, e.g., using an approach similar to the SIMP formulation used in topology optimization. However, while such re-formulation of the problem is conceivable, it is unlikely that a formulation can be found that avoids the difficulties associated with the existence of multiple local optima. In the end, a standard, gradient based implementation will demand significant user intervention, or the use of computationally intensive (and less standard ) methods for non-convex optimization. For problems of moderate size (e.g., around 1000 elements) it is possible to solve the optimization problem (13) on a desktop computer in reasonable time using a simple-minded genetic algorithm, trading elegance for expedi-

5 A.R. Diaz, R. Mukherjee / Computers and Structures xxx (2007) xxx xxx 5 ency. This was our approach. The design variables are easily coded using an integer chromosome representation. The only item that remains is the evaluation of the individual fitness function. Evaluation of the objective function f In the evaluation of f in (13) for a fixed x (fixed variable stiffness joint locations) one needs to know which modes should be controlled for maximum effect. In the problem we assume that data for the problem are, n, the highest mode relevant in the problem (e.g., energy in higher modes if dissipated by structural damping), n C1 and n C2 respectively, the number of modes controlled in each stiffness state and n C max the highest mode controlled. The actual modes controlled in each state are defined by the index sets I C1 and I C2 (recall that I N1 and I N2 in (13) are index sets of modes that are relevant but not controlled, i.e., I N1 =(I C1 ) 0 and I N2 =(I C2 ) 0 where () 0 denotes complement in {1,2,...,n}). Form the data it is clear that there are a total n of C max n C max ways to select I n C1 n C1 and I C2. To illustrate, if the largest mode controlled is the eight mode C2 (n Cmax = 8) and a total of four modes are controlled (n C1 = n C2 = 2), there are 784 possible combinations of modes to control. Evaluation of f requires the solution of that many small eigenvalue problems, since f ðxþ ¼ min C IN1 ;I I N1 ;I N2 ¼ min max 1 k N2 I N1 ;I N2 k where 1 k is the kth eigenvalue of the (typically small) n C1 n C2 matrix A ¼ C T I N1 ;I N2 The computational burden of forming all possible matrices A and computing their eigenvalues is typically not significant, when compared with the solution of the eigenvalue problem (4). Evaluation of the genetic algorithm merit function To complete the implementation of a genetic algorithm solution one must define a merit function that reflects both the objective function f and the resource constraint X M j¼1 vðx j Þ 6 m which limits the maximum allowable number of variable stiffness joints. One way to do this is simply to penalize solutions with more than m variable stiffness joints while allowing some infeasible solutions in the population. We opt for the merit function! f GA ðxþ ¼f ðxþþp max XM j¼1 vðx j Þ m; 0 where p is a penalty parameter, a small scalar. Since 0 < f(x) 6 1, finding a suitable p is not difficult. Typically we use p = 0.1, adding roughly a 10% penalty to f for each additional variable stiffness joint. 4. Example This example illustrates the approach using the structure shown in Fig. 2. The structure is m and built from aluminum 6061 tubing with outside diameter 22.2 mm and inside diameter 20.4 mm. The structure is supported at four points, as shown. Each bar in the structure is discretized using four beam elements. The complete structure is modeled using 540 elements and 449 nodes. It is assumed that the largest contributing mode is mode 12 and that the highest mode controlled is mode 8. We solve the problem first by allowing six variable stiffness connections (m = 6 in (13)) which can be placed on any joint except for the four fixed corners. Only four modes are controlled, two in each state (n C1 = n C2 = 2). A simple genetic algorithm [1] was used. Using a multipopulation approach with 4 sub-populations of 50 individuals each, after 500 generations the algorithm produces the solution shown in Fig. 3. For this design the objective function f(x) = and the best modes to control are I C1 = {3,4} and I C2 = {1,2}. Fig. 4 illustrates how energy added to not-controlled modes is transferred to controlled modes where it is dissipated. Initially, energy is added to not-controlled modes and energy is normalized so that K = 1 at t = 0. Subsequent reductions in K through 20 stiffness switches are shown in Fig. 4. The figure shows a significant reduction in the energy remaining in non-controlled modes (modes 5 through 12). Clearly, this reduction in energy content in all twelve relevant modes is achieved even though only four modes are directly controlled. This is only possible because fixed Fig. 2. Example geometry and boundary conditions. variable stiffness joint Fig. 3. A solution with six variable stiffness joints.

6 6 A.R. Diaz, R. Mukherjee / Computers and Structures xxx (2007) xxx xxx control 1+1 modes control 2+2 modes control 3+3 modes Λ Λ Number of stiffness switches Fig. 4. Reduction in energy after 20 stiffness switches using 6 variable stiffness joints. Λ joints 4 joints 6 joints 8 joints Number of stiffness switches Fig. 5. Effect of the maximum number of variable stiffness joints. 0 of the significant modal disparity that resulted from careful placement of the variable stiffness joints. Fig. 5 shows the effect of changing the maximum allowable number of variable stiffness joints. The figure shows energy dissipation associated with solutions to problem (13) obtained using 2, 4, 6, and 8 variable stiffness joints. The objective function for these results are f(x) = , f(x) = , f(x) = , and f(x) = , respectively for m = 2, 4, 6, and 8. These results show that in this structure, the number of joints does not seem to have a very significant impact of the rate of energy dissipation, provided that at least 4 variables stiffness joints are used. Finally, Fig. 6 shows the effect of changing the number of controlled modes. The figure shows solutions to (13) obtained using a total of 2, 4, and 6 controlled modes (one half of the modes in stiffness state 1, the other in stiffness state 2). The objective function for these results are f(x) = , f(x) = and f(x) = , respectively. As expected, the impact of increasing the number of controlled modes is significant. Note that even though it is possible to reduce the total energy in all modes by controlling only two modes, the rate of decay is very slow. The result suggests that when designing the complete system, a good design will be a compromise solution that balances the total cost of (expensive) controllers and (more economical) variable stiffness joints to achieve a desired control performance at a minimum cost. 5. Final remarks Number of stiffness switches Fig. 6. Effect of the number of controlled modes. This work suggests that variable stiffness joints can be used effectively to introduce modal disparity in flexible structures and, as a result, to simplify and reduce the cost of implementing control strategies for vibration suppression. The optimization problem arising from the present problem is somewhat complex in that the design variables are discrete and the topography of the objective function is complicated by the presence of many local optima. A simple genetic algorithm is sufficient to address these difficulties in problems of moderate size, but problems of larger dimensions may require a different approach if computational resources are scarce. Fortunately, current and foreseeable trends point to increasing access to inexpensive computing power, diminishing the importance of this issue. Nevertheless, it may be interesting in future work to focus on the development of a re-formulation of the problem that makes it more amenable to solution by a more rigorous and reliable approach. An interesting possibility could be to cast the problem as an integer programming, 0 1 problem and apply a strategy along the lines of recent work by Stolpe [10] and Stolpe and Rasmussen [11]. The structure of the present problem prevents application of these methods in their present form but it is conceivable that with modifications (to both the present problem and to the solution methods themselves) a more robust and rigorous formulation may be found. This remains a very interesting research topic for future work.

7 A.R. Diaz, R. Mukherjee / Computers and Structures xxx (2007) xxx xxx 7 The correct timing for changing the stiffness of each active joint will be determined using sensor measurements. These sensors are needed in addition to active elements that will have to be used for changing the stiffness of the joints. The idea proposed in this paper will therefore require additional hardware, although control system hardware may be reduced since fewer actuators and sensors will be required for control of a few select modes. The expectation here is that the new control methodology will have certain advantages over existing and traditional control strategies. For example, it is expected that the new control methodology will funnel energy from the non-controlled modes, such as the higher-order modes, to the controlled modes, which are typically lower-order modes, and thus sidestep the problem of spillover. However, to substantiate such advantages of our control methodology, we will have to conduct further investigations and experimentation and this will be part of future research. Finally, the question always arises as to which structures are more likely to be good candidates for modification resulting in modal disparity. So far we have been unable to identify specific criteria that distinguish structures that can be significantly enhanced by the addition of special joints or other mechanisms. This remains an open question of both theoretical and practical significance. References [1] Chipperfield AJ, Fleming PJ, Pohlheim H, Fonseca C. The Genetic Algorithm Toolbox for MATLAB, 2. UK: Department of Automatic Control and Systems Engineering of The University of Sheffield; [2] Corr LR, Clark WW. A novel semi-active multi-modal vibration control law for a piezoceramic actuator. J Vib Acoust-Trans ASME 2003;125(2): [3] Diaz AR, Mukherjee R. Modal disparity enhancement through optimal insertion of non-structural masses. Struct Multidiscip Optim 2006;1(1):1 12. [4] Diaz AR, Mukherjee R. A topology optimization problem in control of structures using modal disparity. J Mech Des 2006;128(3): [5] Gaul L, Nitsche R. Vibration control by interface dissipation in semiactive joints. Z Angew Math Mech 2000;80:S45 8. [6] Gaul L, Albrecht H, Wirnitzer J. Semi-active friction damping of large space truss structures. Shock Vib 2004;11(3 4): [7] Kerschen G, Lee YS, Vakakis AF, McFarland DM, Bergman LA. Irreversible passive energy transfer in coupled oscillators with essential nonlinearity. SIAM J Appl Mater 2006;66(2): [8] Logan D. A first course in the finite element method. 3rd edition. Pacific Grove, CA: Brooks/Cole; [9] Ramaratnam A, Jalili N. A switched stiffness approach for structural vibration control: theory and real-time implementation. J Sound Vib 2006;291(1 2): [10] Stolpe M. Global optimization of minimum weight truss topology problems with stress, displacement, and local buckling constraints using branch-and-bound. Int J Num Meth Engrg 2004;61(8): [11] Stople M, Rasmussen ML. Global optimization of large topology design problems using branch and bound methods. In: Mota Soares C, Rodrigues H, Ambrosio J, editors. Proceedings of the III European conference on computational mechanics solids, structures and coupled problems in engineering. Lisbon, Portugal; [12] Vakakis AF, Gendelman O. Energy pumping in nonlinear mechanical oscillators: Part II Resonance capture. J Appl Mech-Trans ASME 2001;68(1):42 8. [13] Vakakis AF, Manevitch LI, Gendelman O, Bergman L. Dynamics of linear discrete systems connected to local, essentially non-linear attachments. J Sound Vib 2003;264(3): [14] Walsh PL, Lamancusa JS. A variable stiffness vibration absorber for minimization of transient vibrations. J Sound Vib 1992;158(2):