MODAL ANALYSIS OF BEAM THROUGH ANALYTICALLY AND FEM

MODAL ANALYSIS OF BEAM THROUGH ANALYTICALLY AND FEM Ankit Gautam 1, Jai Kumar Sharma 2, Pooja Gupta 3 1,3 Department of civil Engineering, MAIIT, Kota (India) 2 Department of Mechanical Engineering, Rajasthan Technical University, Kota (India) ABSTRACT Modal analysis has become a major technique to determine dynamic characteristics of engineering structures and its components. It is a process by which the natural frequencies, mode shapes and damping factor of structures can be determined with a relative ease. In the present work for beam the natural frequencies and mode shapes are obtained through analytical and FEM. The natural frequencies and mode shape are determined throughfiniteelement modeling (FEM) using ANSYS 14.5 and analytical. A good correlation between the FEM and theoretical results is observed. Keyword: Analytical, numerical, beam, finite element modeling, modal analysis I INTRODUCTION The finite element modeling study for engineering component always provided a major contribution to our efforts to control and understand the many vibration phenomena s encountered in practice [1]. The modal analysis process has two types of method to analysis the structures. First is theoretical modal analysis and second is experimental modal analysis. In the theoretical modal analysis method (fig.1) the spatial properties of the modal (mass, stiffness and damping) are given and using them modal and response modal are obtained. Fig: 1 Theoretical route to vibration analysis 373 P a g e

In theoretical modal analysis one cannot forecast accurate boundary conditions, actual rigidity and damping for complex engineering structures and component. So the calculated results often have certain error with actual result [1, 2].The Numerical modal analysis method using the Finite element modeling software ANSYS enables engineers to get a better understanding of dynamic properties of structures [3]. The experimental modal analysis path from response modal (fig 2) and not so often ending with spatial modal. Experimental modal analysis used to derive the modal of a linear time-invariant vibratory system. Modal analysis using vibrometer is non- destructive testing, based on vibration response of the structures. For excitation impact hammer is widely used in modal analysis [4]. It is well known that for structures falling under resonant conditionssmall force can result in large deformation, and possibly, damage can be induced in the structure. The interaction between the inertial and elastic properties of the materials causes resonant vibration in the structures [5]. Modal is frequently used to find mode of vibration of machine component in the structure. Fig. 2 Experimental route to vibration analysis In this paper the modal parameters i.e. natural frequencies and mode shape for the beam are determined usingfinite element modeling software ANSYS 14.5. The result of thus obtained natural frequencies is then compared with theoretically calculated values. II THEORTICAL MODAL ANALYSIS OF BEAM 2.1 Cantilever beam: fixed - free Consider an Euler-Bernoulli uniform cantilever beam undergoing transverse vibration condition as shown in Fig.3. x X=0 L X=L Fig.3 Cantilever Beam For free vibrations the equation of motion of beam can be given as [6] (1) 374 P a g e

0 (2) where (3) Free vibration solution can be found using the method of separation of variable. Assuming solution as (4) From equation (2) and (4) (5) Where can be shown as a constant. The equation (5) can be written as two equations (6) Where (7) (8) From the equation (7) the natural frequency of beam can be written as (9) The solution of equation (8) is (10) Where A and B are constant that can be determined from the initial boundary conditions. Assuming the solution of equation (6) as (11) Using (6) and (11) one can obtain the general solution 375 P a g e

(12) Where, the constants can be determined from the boundary conditions. For a cantilever beam the transverse deflection and its slope must be zero at the fixed end and at free end the bending moment and shear force must be zero. Thus the boundary conditions become (13) (14) (15) (16) Substituting the equation from (13) to (16) in equation (12) to obtain (17) Equation (17) is the frequency equation. This transcendental equation can be solved to obtain the value of for the cantilever beam the values are given in Table 1 [5]. and Table1 The value of Mode Mode 1 1.875104 Mode 2 4.694091 Mode 3 7.854757 2.2 Free-Free beam The beam is free at both end as shown in fig. 4. At a free end, the bending moment and shear force are zero. Hence, the boundary condition of the beam can be stated as 376 P a g e

X=0 X=L Figure: 4 Free-Free beam =0 (18) (19) =0(20) (21) ] Equation(18)and(19) require that In equation(18), (20)and(21) lead to + and in equation (25) and (26) the determine formed by their coefficient is set equal to zero. (27) Equation (27) is the frequency equation. This transcendental equation can be solved to obtain the value of for the free-free beam the values are given in Table 2[5]. Table:2 The Value of and Modal Modal 1 0 Modal 2 4.730041 Modal 3 7.853205 2.3 Fixed-Fixed Beam At a fixed end, the transverse of the displacement are zero hence, the boundary condition are given by W (0)=0 (28) 377 P a g e

W(L)=0 (30) Equation (28) and (29) Equation (30) and (31) Equation (33)and(34) denote a system of two homogeneous algebraic equation determine of the coefficients of in equation (33) and(34) to zero. Frequency equation as Equation (35) is the frequency equation. This transcendental equation can be solved to obtain the value of for the fixed-fixed beam the values are given in Table 3 [5]. Table:3 The Value of and Modal Modal 1 4.730041 Modal 2 7.853205 Modal 3 10.995608 The material and geometric parameter used for theoretical and FEM analysis of beam are tabulated in Table 4. Table 4: Material and geometric parameter Material Parameter Geometric Parameter E = 20.5 L = 2m B=0.3m V=0.33 H=0.1m IV NUMERICAL MODAL ANALYSIS OF BEAM The three-dimensional finite element model of beam is constructed in ANSYS 14.5 and then computational modal analysis is performed to generate natural frequencies and mode shapes [10]. The geometric and material parameter is taken from the Table 4 [8]. Solid 185 element are adopted for beam analysis. Relevant boundary conditions are applied at the end of beam [9]. The FEM results are compared to theoretical results. 378 P a g e

V RESULTS AND DISCUSSION The results of the theoretical natural frequencies and FEM obtained natural frequencies of mild steel, beam are calculated using the material properties and dimensions of the beam given in Table 4. The theoretical natural frequencies are calculated using the equation (9) and the finite element natural frequencies are determined using ANSYS 14.5. Table 5 depicts the theoretical, numerical and natural frequencies of the beam. To validate FEM result theoretical modal analysis carried out. Table: 5 Theoretical and numerically natural frequency of beam End conditions Mode Analytically Natural FEMNatural frequency (Hz) frequency (Hz) Free - Free 1 0 0 2 130.90 130.59 3 356.3339 356.00 Fixed - Free 1 20.54 20.818 2 129.24 129.25 3 357.56 357.05 Fixed - Fixed 1 132.04 132.04 2 357.3022 357.80 3 687.72 687.19 The numerical mode shape and corresponding natural frequency obtained using ANSYS are shown in fig.5-7. (1) (2) (3) Figure 5: Mode shape and corresponding frequency for free - free boundary condition 379 P a g e

(1) (2) (3) Figure 6: Mode shape and corresponding frequency for fixed - free boundary condition (1) (2) (3) Figure 7: Mode shape and corresponding frequency for fixed - fixed boundary condition VI CONCLUSION In this paper the theoretical and numerical modal analysis of beam are performed. The numerical results are obtained using ANSYS 14.5. The numerical and theoretical results are found to have extremely good correlation. From the analysis if the beam is in fixed-fixed boundary condition is vibrate naturally more than other end supports. The overall analysis show that the results are all well within reasonable error. REFERENCES [1] D. J. Ewins Modal testing : theory and practice (Research study press Ldt, England1984) [2] Jimin He and Zhi-Fang Fu Modal Analysis (Butterworth-Heinemann, Replika Press Pvt Ltd, Dehli 2001) 380 P a g e

[3] LingmiZhang An Overview of Operational Modal Analysis: Major Developmentand Issues (Nanjing University of Aeronautics & Astronautics, China RuneBrinckerPalleAnderse) [4] D.Ravi Prasad and D.R. Seshu, A Study on Dynamic Characteristics of Structural Materials using Modal Anaylsis, Asian Journal of Civil Engineering(Building and Housing), 9(2), 2008,141-152. [5] WalunjPrashant S., V.N.Chouguleb, Anirban C. Mitrac,Investigation on modal parameters of rectangular cantilever beamusing Experimental modal analysis, Materials Today: Proceedings,2 2015,2121 2130. [6] S. S RaoMechanica vibration (Pearson Education, Inc., publishing as Prentice Hall, 2011) [7] Sandeep Kumar Parashar, Utz von Wagner, Peter Hagedorn, Finite element modeling of nonlinear vibration behaviour of piezo-integrated structures, Computers and Structures, 119, 2013, 37 47. [8] Vipin Kumar, Kapil Kumar Singh, ShwetanshuGaurav, Analysis of Natural Frequencies for Cantilever Beam with I- and T- Section Using Ansys, International Research Journal of Engineering and Technology (IRJET), 02, 2015 [9] S. Pedrammehr, H. Farrokhi, A. KhaniSheykh Rajab, S. Pakzad, M. Mahboubkhah, M. M. Ettefagh and M.H. Sadeghi, Modal Analysis of the Milling Machine Structure through FEM and Experimental Test, Advanced Materials Research, 383-390,2012,6717-672. [10] Jai Kumar Sharma and Sandeep Kumar Parashar, Experimental investigation using vibrometer and modal analysis of cantilever beam.ijirse,02,,2016, 128-136. 381 P a g e