Estimation of Modal Density of idealized subsystem (Beam) by Theoretical, Experimental and FEM method.

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1 Estimation of Modal Density of idealized subsystem (Beam) by Theoretical, Experimental and FEM method. Vrushali H. Patil Prof. N. S. Hanamapure M. E. Student Professor Tatyasaheb Kore Institute of Engineering & Technology, Warananagar, Kolhapur, Maharashtra ABSTRACT The purpose of this work is to study the response of the idealized subsystem like beam with different material. To study the vibrational response at higher frequency region Statistical Energy Analysis (SEA) method is used. The S.E.A. techniques is used for successful prediction of noise and vibration levels of coupled structural elements and acoustic volumes and it depends to a large extent on an accurate estimate of three parameters: the modal densities, damping loss factors, and coupling loss factors of the subsystems. In this research work, the modal density of idealized subsystem like beam of different materials like mild steel and aluminium obtained by experimental and FEM method and same results are validated by theoretical method. The modal density of one dimensional systems changes with material of the idealized subsystem. Key words: Statistical Energy Analysis (SEA); Modal density; Damping loss factors ; Coupling loss Element Method (FEM). factor; Finite 1.INTRODUCTION The Statistical Energy Analysis (SEA) mostly used to describe the vibro-acoustic behavior of the complex structures. Since the use of deterministic methods, like Finite Element Method (FEM) is limited to the low-frequency range, SEA is proposed for medium and high frequency region to predict the vibrational response of the structure. A structure with many resonances can be modeled in a deterministic model and easily analyzed with statistical nature of SEA method. The model of SEA based on energy balance and power flow among the group of modes in a structure. A complex model can be divided into simple subsystems. [11].The modal densities of these subsystems are then approximated by mathematical expressions for simple structures. This paper shows the comparison of modal densities obtained by theoretical, experimental and FEA method. For this purpose idealized subsystem like beam of different materials like mild steel and aluminium are used to observe the effect of material on the modal densities. 2.BASIC CONCEPTS IN SEA The SEA technique required to divide the complex structure into small subsystems to analyze the energy distribution and energy flow between the coupled subsystems. A subsystem is to be considered as a group of modes having same energy level. The physical coupling between the subsystems is defined properly. Finally the type of external excitation or input power to each of the subsystem is also defined. In addition to that, the power which is dissipated internally in a subsystem is assumed to be proportional to the energy of the subsystem. [11] Energy Equations for SEA methods : 57

2 From above basic concepts, it can be stated that the net energy flow from substructure 1 to substructure 2 varies proportionally to the difference in modal energy, that is the energy per mode in a given frequency band and can be written using fundamental relation of SEA where and are the averaged total energies of the subsystems 1 and 2 respectively. and are the number of modes of the subsystem 1 and 2 in a frequency band respectively (1) and is the centre frequency of the band. The equation (1) can be rewritten as (2) With the reciprocity equation In the equations (2) and (3) and are the coupling loss factors between the subsystems, and and are the modal densities of the subsystem in the band. The energy loss due to the dissipation in each subsystem is directly proportional to the total dynamical energy of the subsystem and can be evaluated as follow E 1 or E 2 (4) Where and are the damping loss factors of the subsystems and and are the time averaged dissipated power in the subsystems. The global SEA equation can be calculated by balancing the time averaged external power input to subsystem 1 with the power dissipated in the subsystem and the net power flow to the subsystem 2 as shown in Fig 1. (3) Fig 1. Power flow between the subsystem Basic SEA model 58

3 The equations for balancing the power of the two subsystems shown in figure 1 can be written as (5) and (6) where the coupling loss factors are as per the equation (3) (7) For N substructures the above equations can be written in the matrix form (8) with (9) where and are the internal damping loss factors of each subsystem. and are the coupling loss factors between subsystems i and j. is the time averaged input power into subsystem i and is the time averaged energy of subsystem i. The solution of equation (8) is possible only if the coupling loss factors fulfil the additional reciprocity equation (9). Therefore an important condition for the SEA is the knowledge of the modal densities of the subsystems. For many simple structures like plate, beams, cylinders the modal densities are evaluated from theoretical formulae s and numerical analysis. However for structures like welded joint, hinged joint and riveted joint an experimental validation with numerical method like FEM is also required. 3. THEORETICAL ANALYSIS The modal density of two dimensional systems like rectangular plates largely dependent on the boundary condition. but for one dimensional system like beam modal density is dependent on material. To estimate the modal density, theoretical formulae are[7]: For a one- dimensional system,(beam) having thickness and length the modal density is given by,[11] (10) where δƒ is average frequency spacing between modal resonances, and it is given by δƒ= (11) where is the radius of gyration and it is given by. To evaluate the modal density, beam with different materials have been taken. The material properties of the beam are as follows. 59

4 Sr. No. Material Breadth Mm Table 1.Material properties of the beam Width mm Length mm Density(ρ) Kg/m 3 Poisson s Ratio (ν) Lonitudinal velocity ( ) m/sec 1 Mild steel Aluminum EXPERIMENTAL ANALYSIS To estimate the modal density of the beam experimentally using cantilever condition, the experimental set up is made as shown in figure 1. One edge of the beam is clamped with the fixture so that cantilever condition is achived. The accelerometer is mounted on the face of the beam to sense the excitations. The hammer is used to excite the beam. The connections of hammer and accelerometer given to the FFT analyser through the cables. After exciting the beam an accelerometer mounted on the beam senses the vibrations and transmits the signals to the FFT. Then FFT process the signals and gives diferent spectrums as an output. By observing frequency spectrum the number of resonant modes within specific frequency band, the experimental modal density is calculated. Figure 1:Experimental set up for cantilever beam 5. FINITE ELEMENT ANALYSIS To estimate the modal density by FEM method ANSYS software is used. For this geometric model of the beam is prepared and then with the help of pre-processor meshing of the model is done. To find out the resonant frequencies occurring in the particular frequency band modal analysis of the beam is done. 60

5 Figure 2.Modeling and meshing of the beam Figure 3. Cantilever type boundary conditions for beam 6.RESULTS AND DISCUSSION 61

6 Figure 4. Comparison of experimental results two beams for cantilever condition. Figure 5. Comparison of FEM results of two beams for cantilever condition. 1. From graphs in figure4. and 5. it is clear that modal density of aluminum beam is slightly higher than the M.S.beam one of the reason is that the difference in material properties of the beam. 62

7 Figure 6. Comparison of modal density of a cantilever mild steel beam From figure 6. it is observed that, 1.As the frequency increases the value of modal density decreases within lower frequency range, but it converges to a constant value in higher frequency region, because in the higher frequency range due to high modal overlap the number of resonant modes occurring are less for comparatively higher frequency band. 2. Numerical (FEM) value of modal density is higher than the experimental values and theoretical values. Figure 7. Comparison of modal density of a cantilever Aluminium beam 63

8 From figure7. it is observed that, As the frequency increases the value modal density decreases within lower frequency range, but it converges to a constant value in higher frequency region, because in the higher frequency range due to high modal overlap the number of resonant modes occurring are less for comparatively higher frequency band. 7. CONCLUSION Due to structural damping phenomenon the number of resonant modes obtained in case of experimental method are less so that the modal density obtained by experimental method is less as compared to theoretical and FEM method. These results are helpful for guiding the selection of the proper material in automotive and ship manufacturing industries. REFERENCES [1] P. Ramachandran, S. Narayanan, Evaluation of modal density, radiation efficiency and acoustic response of longitudinally stiffened cylindrical shell, Journal of Sound and Vibration 304 (2007) [2] V. Cotoni, R.S. Langley, P.J. Shorter, A statistical energy analysis subsystem formulation using finite element and periodic structure theory Journal of Sound and Vibration 318 (2008) [3] S. Finnveden, Evaluation of modal density and group velocity by a finite element method Journal of Sound and Vibration 273 (2004) [4] K. Renji, Experimental modal densities of honeycomb Sandwich panels at high frequencies Journal of Sound and vibration 237(1) (2000) [5] Kranthi Kumar Vatti, Damping Estimation Of Plates For Statistical Energy Analysis Thesis submitted for M.S. at University of Kansas, March, [6] H. Jeong, B. Ahn and C. Shin, Experimental Statistical Energy Analysis Applied To A Rolling Piston- Type Rotary Compressor, International Compressor Engineering Conference. (2002) Paper [7] G. Xie, D.J. Thomson, C.J.C. Jones, Mode count and modal density of structural systems: relationship with boundary conditions, Journal of Sound and Vibration 274 (2004) [8] Benjamin Elie, François Gautier, Bertrand David, Estimation of mechanical properties of panels based on modal density and mean mobility measurements Elsevier Publication (2013) [9] A.M. Fareed, G. Schimdit, F. Wahl Experimental identification of Modal Density Parameters of Light Weight Structure Institute of Mechanics, University of Magdeburg. (2001) [10] B.L. Clarkson, M.F. Ranky, Modal Density of Honeycomb Plates, Journal of Sound and Vibration (1983) 91 (1), [11] R.H. Lyon, R.G. Dejong, Theory and Application of Statistical Energy Analysis, Second edition, Butterworth-Heinemann, London,