Vibration Absorber Design via Frequency Response Function Measurements

Transcription

1 Proceedings of the IMAC-XXVIII February 1 4, 2010, Jacksonville, Florida USA 2010 Society for Experimental Mechanics Inc. Vibration Absorber Design via Frequency Response Function Measurements N. Nematipoor, M.R. Ashory, E. Jamshidi Department of Mechanical Engineering, Semnan University, P.O. Box: , Semnan, Iran ABSTRACT Vibration absorbers are usually designed using the Finite Element (FE) model of structures. However, the FE models of vibrating systems are not always available due to the complexity of structure. Moreover, the FE models of structures are not accurate due to the joint problem or numerical errors. Modal testing is an experimental approach to build the mathematical model of structures. It is generally believed that the modal models are more accurate than FE models, because the test structure is modeled by direct measurement from the structure. In this paper, a method is proposed to design the vibration absorbers using the measured Frequency Response Functions (FRFs) of the primary structure. A translational absorber is designed using the method proposed in this paper to absorb the vibration amplitude of a cantilever beam. The experimental results show that the designed absorber is effective and suppresses the amplitude of vibration considerably. 1. Introduction In recent years, control of the vibration amplitude of flexible structures using tuned vibration absorbers has attracted much attention andhas been studied by many authors. Jacquot[1] proposed a method eliminates vibration of harmonically excited Euler-Bernoulli beam. The method gives the vibration absorber parameters based on a single mode of beam. So it was limited in application.özgüven and Candir[2]extended the previous method to suppress any two resonances. They performed a min-max optimization for the response in any desired mode using assumed-modes approach.manikanahally and Crocker [3] give the optimized stiffness and damping parameters for a certain chosen mass to suppress significant modes in which the absorbers are tuned to operate.keltie and Cheng [4]used point masses to absorb the vibration amplitude of any location of structure. Their method find the optimized location of certain point mass to reduce the vibration level at the desired location of the structure. Ozer and Royston[5]proposed a method gives the vibration absorber parameters mounted to a damped multi-degrees-of-freedom structure based on Sherman-Morrison matrix inversion formula. The method is capable to minimize the overall vibration amplitude of a multi-degrees-of-freedom system.cha and Pierre[6] proposed a method to impose a single node using normal modes of a supported linear structure by mounting a chain of absorbers.cha[7]extended his work by using a set of sprung masses and rotational absorbers to enforce one or more fixed nodes for any supported linear structure subjected to harmonic excitations. In this paper, a method is proposed to suppress the vibration amplitude of an arbitrary location on a linear structure subjected to harmonic excitation. The method is beneficial because there is no need to have theoretical or FE model and it is not related to a certain geometry.in order to validate the proposed method a numerical and experimental case studies were performed. In both cases a tuned sprung mass absorber was considered as dynamic vibration absorber and for chosen absorber stiffness, its mass value was calculated.

2 2. Theory Changes in a Frequency Response Function (FRF) of a system due to mounting an absorber, using direct substructuring technique SMURF, can be estimated as follows: Fig. 1(a) indicates a structure which has been rigidly connected to the second system of an absorber. The connection is at point j and in x direction. The force excites the structure at point i and the response is measured at point l. Fig. 1(b) indicates the free body diagram of the system. Figure1.Modification of the structure by attaching an absorber The system is presumed to be linear, andthe equations governing the coupled system are: = + (1) = + (2) = (3) Where isthe receptance for DOFs i and j,and is the receptance of the attached absorber. and the constraining equations are: = (4) + =0 (5) Elimination of the reaction forces and displacements at the connection point results in: =( ) = ( ). (6) ( in which ) is the modified receptance function between points l and i when the second system absorber is ( connected to the original system at point j. From equation (6) one can immediately relate ) and the original receptances, as: ( α ) = α Whereα li, α ji, α lj andα jj are the original receptances, namely for the system without the absorber. Multiplying both sides by ω 2, and the numerator and denominator of the fraction on the right hand side of equation (9) by ω 2,the relation between accelerances is: (7) ( ) = (8)

3 As the aim of mounting, of the absorber is vanishing the vibration at node l, ( ) is assumed to be zero consequently: 0= Therefore, A mm can be given as: = So the vibration absorber impedance is calculated from equation (10). Equation (10) is a general equation for modifying a system by the connection of an absorber. For a simple sprung mass absorber (Figure 2), the dynamic stiffness matrix is: [ ] = (9) (10) (11) Figure2.Sprung mass absorber The receptance matrix is obtained by inversing the dynamic stiffness matrix, as: [ ] = [ ] = Therefore, α mm can be given as: = And A mm can be given by: = = SubstitutingA mm from equation (14) into equation (10) we have: = (12) (13) (14) (15) For a certain j andk, m can be obtained from equation (15). 3. Numerical case study A theoretical cantilever steel beam was considered for the numerical case study. Thebeam had dimensions mm and the density and modulus of elasticity were assumed to be respectively 7870 kg /m 3 and 206 GN/m 2. The beam was discretized to 30 elements of 2 nodded 4 Degrees Of Freedom (DOFs) beam elements. The beam was excited by a harmonic forcepoint i(figure 3) with the frequency ofω=1180 rad/s (187.8 Hz). The absorber mounted at point j.in order to suppress the vibration amplitude at point l.

4 Figure3.Numerical case study of the beam Substituting α ij,α jj, α li and α lj in equation (15) and considering the absorber stiffness to be N/m, the absorber mass(m) was obtained. Figure (4) shows the FRF (α li ) before and after mounting the absorber. It can be seen that the absorber is capable tovanish the vibration amplitude in the exaction frequency ω=1180 rad/s (187.8 Hz) X/F (db) Frequency (Hz) Figure4.Computed α li with (dotted line) and without (solid line) absorber. 4. Experimental Case Study A steel beam withthe dimensions of mm was tested in the free-free boundary conditions as shown in Figure (5).The beam was excited by a shaker type 4808at point I, j and the responses were measured at two points l and j by accelerometer type DJBA120V.

5 Figure5. Schematic of the test configuration The beam was excited at the frequency of 79 Hz at point i. The aim was to suppress the vibration amplitude at point l by attaching a sprung mass absorber at point j. Two testswere carried out on the beam. At first the beam was excited at point l to obtainα li and α ij. Then the beam was excited at point j to obtain α lj and α jj. Substituting α ij, α jj, α li and α lj in equation (15) andconsidering the absorber stiffness k j to be N/m, the absorber mass (m) was calculated to be 42gr.To assess the method,α li is plotted before and after mounting the vibration absorber in Figure(6).It can be seen from Figure (6) that the amplitude vibration decreases considerably around the excitation frequency 79 Hz. Particularly the vibration amplitude at the excitation frequency 79 Hz decreases approximately 85%.

6 30 Without absorber With absorber X/F (db) Frequency (Hz) Figure6.α li obtained from experiment with (dotted line) and without (solid line) absorber. 5. Conclusions In this paper, a new method was proposedto design a vibration absorber using the measured FRFs of the primary structure.the advantage of this method is that the vibration absorber can be designed by using the experimental results of modal testing and there is no need to have any theoretical or finite element model. Also, the method is not restricted to any certain geometry or boundary conditions. It can be used for any linear structure with any complexity. In addition, the only required data of the structure to design the vibration absorber is four measured FRFs from the modal testing. One numerical and one experimental case studywere carried out showing the effectiveness of method in suppressing the vibrationamplitude at the desired point. The vibration amplitude decreased by 100% in the numerical case study. However, vibration amplitude decrement in the experimental case study wasabout 85%. 6. References [1]JacquotR.G., Optimal dynamic vibration absorbers for general beam systems, Journal of Sound and Vibration 60 (4), , 1978 [2] zg venh.n., CandirB., Suppressing the first and second resonances of beams by dynamic vibration absorbers, Journal of Soundand Vibration 111 (3), ,.1986 [3] ManikanahallyD.N., CrockerM.J., Vibration absorbers for hysteretically damped mass-loaded beams, Journal of Vibration andacoustics 113, ,.1991 [4]KeltieR.F., ChengC.C., Vibration reduction of a mass-loaded beam, Journal of Sound and Vibration 187 (2), ,.1995

7 [5] OzerM.B., RoystonT.J., Application of Sherman Morrison matrix inversion formula to damped vibration absorbers attached tomulti-degree of freedom systems, Journal of Sound and Vibration 283 (3 5), ,.2004 [6] ChaP.D., PierreC., Imposing nodes to the normal modes of a linear elastic structure, Journal of Sound and Vibration 219 (4), ,.1998 [7] ChaP.D., ZhouX., Imposing points of zero displacements and zero slopes along any linear structure during harmonic excitations, Journal of Sound and Vibration 297(1-2),55-71,2006