Broadband Vibration Response Reduction Using FEA and Optimization Techniques

Transcription

1 Broadband Vibration Response Reduction Using FEA and Optimization Techniques P.C. Jain Visiting Scholar at Penn State University, University Park, PA A.D. Belegundu Professor of Mechanical Engineering, Penn State University, 330 Leonhard Bldg., University Park, PA Abstract This paper discusses vibration response reduction using optimization. Specifically, we consider structures that are vibrating due to broadband excitation. The concept of a single absorber to reduce vibration at a fixed frequency is, of course, well known [1]. Several researchers, including the author, have discussed the use of several absorbers to target multiple resonance peaks [2-7]. However, achieving this using commercial FEA software is advantageous for engineers in industry, particularly because the full analysis and graphics capability of the software can be harnessed for engineering problems, and because a variety of design concepts can be simulated. Here, we discuss how one may formulate and optimize using ANSYS 7.0 software [8]. Results for plate vibration problems are presented. Introduction We consider a vibrating structure whose FE model has already been created. The structure is subjected to a harmonic excitation F e iωt. As a result, critical point (s) vibrate with displacement response equal to x e iωt. Based on this, metrics relating to either total kinetic energy, or mean acceleration, PSD s etc can be computed. Here, we focus on kinetic energy (KE). The design approach is to attach tuned vibration absorbers at given locations, and optimally determine the parameters of the absorbers to minimize KE. The only constraint imposed is that total added mass be less than, say, 30% of the mass of the original (baseline) structure. The parameters of each absorber are its point stiffness and mass, or k a and m a. Thus, if there are 10 absorbers, then we have a total of 20 design variables in the optimization problem. The objective function is defined as the sum of peaks of KE response spectrum. This definition has found to be effective for lightly damped structures. To obtain this sum of peaks, ANSYS is first used to solve the generalised eigenvalue problem for all resonances ω i in the range of interest, [ω L, ω U ]. Then, harmonic analysis based on mode superposition is used to analyze for the response at every 1Hz interval, with clustering at resonances. From this response spectrum, information at only the resonances, are extracted. Total KE for the structure is then defined as KE structure = e KE(e) where, for each element, adopting a lumped mass matrix, KE(e) = ½ (m e h i / 4 ) * V i V i with V = element velocity vector = (i * ω j * u i ), u i = element displacement vector, and h denotes Hermitian (complex conjugate).

2 Procedure A scripting language (APDL) ANSYS Parametric Design Language is used to carry out optimization. A finite element model is generated within the PREP7 module of ANSYS 7.0. The plate is modeled using the SHELL63 element. This element has six degrees of freedom at each node corresponding to three translations and three rotations. Absorber springs are modeled using short massless BEAM4 elements. Mass is modelled using MASS21 element. After applying boundary conditions and loads, the modal analysis (ANTYPE,MODAL) followed by harmonic analysis (ANTYPE,HARMIC) using mode superposition (HROPT,MSUP) are carried out. Displacement response over the desired frequency range [ Hz] is obtained. This displacement is used to obtain kinetic energy as discussed above, by coding the relevant equations within ANSYS using APDL. The total kinetic energy has been minimized using optimizer available within ANSYS 7.0. ANSYS Optimizer The Sub problem approximation method [9] available within ANSYS 7.0 is used for optimization. The sub problem approximation method uses only the values of the dependent variables, and not their gradients. For this method, the program establishes the relationship between the objective function and the design variables by curve fitting. Curve fitting is done by calculating the objective function for several sets of design variables and performing a least squares fit between the data points. The resulting curve or surface is an approximation. Each optimization loop generates a new data point, and the objective function approximation gets updated. This approximation is minimized during analysis. Design Example An aluminum plate of dimensions 290 mm x 290 mm and thickness of mm is considered. The plate is clamped on all four edges. A sinusoidal force of 0.03 N acts at the centre of the plate, in the frequency range Hz. One percent of critical damping is assumed. The finite element model is shown in Figure 1 ( FE Model of Clamped Plate ). A total of 30 absorbers have been distributed over the plate as shown in Figure 2 ( Location of (30) Spring-Mass Absorbers on Plate ). Initially the stiffness and mass for each absorber are 3000 N/m & 3.4 g, respectively. Both the plate without absorbers and with absorbers have been analyzed. Lower and upper limits on design variables are given below: Design variables: 30 spring stiffness ( range n/m ) 30 masses ( range grams ) Total number = 60

3 Figure 1. FE Model of Clamped Plate Figure 2. Location of (30) Spring-Mass Absorbers on Plate

4 Analysis Harmonic analysis and optimization runs have been performed as described in procedure section. Analysis Results & Discussion The optimization has resulted in optimum values of stiffness and mass for each of the thirty absorbers. Total mass of all the absorbers at optimum is 70 gram. Mass of the plate without absorbers is 231 g. Kinetic energy vs frequency plots before and after optimization is shown in the Figure 3 (Kinetic Energy Before and After Optimization). The kinetic energy (objective function) has been reduced by 74 %. The displacement response vs frequency for three cases viz. plate without absorber, plate with absorber at initial design, and plate with absorber at optimum are shown in Figure 4 (Displacement Response). The displacement response at resonances has been reduced by 72 %. Figure 4. Displacement Response Figure 5 ( Individual Tuned Absorber Frequencies ) shows the individual tuned absorber frequencies at optimum. That is, a plot of k a /m a for each absorber is shown. This illustrates the fact that the optimizer distributes the absorbers over the frequency band to target multiple response peaks.

5 Figure 5. Individual Tuned Absorber Frequencies Conclusion It is concluded that use of multiple tuned absorbers, is an effective way of passive vibration reduction over a broad frequency band. Graphical and analysis power of the commercial FEA code like ANSYS is attractive, compared to in-house codes used in the literature. Further, a methodology based on use of an ANSYS can be utilized for vast variety of absorber concepts to be tried out prior to building a prototype. For instance, the absorbers can be distributed plates, shells, etc that are attached to the base structure. Size, shape and topology of the attachments can be varied for optimum response. References (1) J. Ormondroyd and J.P. Den Hartog, The theory of dynamic vibration absorber, Transaction of the ASME, Journal of Applied Mechanics 50 (7), 1927 (2) E. W Constans, Ashok D. Belegundu, Gary H. Koopmann, Optimally designed shell enclosures with tuned absorbers for minimizing sound power, Optimization & Engineering, 1, 67-86, 2000 (3) E.W. Constans, G. H. Koopmann and A. D. Belegundu, The use of modal tailoring to minimize the radiated sound power of vibrating shells : theory & experiment, Journal of Sound and Vibration, 217 (2), , 1998 (4) R.L. St. Pierre Jr. and G.H. Koopman, A design method for minimizing the sound power radiated from plates by adding optimally sized, discrete masses, ASME Journal of Mechanical Design, 117, , 1995

6 (5) M. D. Grissom, Quiet Product Design Using Optimized Broadband Vibration Absorbers:, Ph.D. Thesis, Penn State University, (6) M. J. Brennan, Characteristics of a Wideband Neutralizer, Noise Control Engineering Journal, 45 (5), , 1997 (7) G. Maidanik, Power Dissipated in a Spring Mass attached to a Master structure, JASA, 98(6) , 1995 (8) ANSYS Inc., ANSYS package version 7.0, Canonsburgh PA, USA (9) ANSYS, Inc. Theory Reference