FINITE ELEMENT ANALYSIS OF BEAM

Transcription

1 YMCA University of Science & Technology, Faridabad, Haryana, Oct 9-, FINITE ANALYSIS OF Hasan Zakir Jafri, I.A. Khan, S.M. Muzakkir, Research Scholar, Faculty of Engineering & Technology, Jamia Millia Islamia -, Faculty of Engineering & Technology, Jamia Millia Islamia Maulana Mohammad Ali Jauhar Marg, Jamia Nagar, New Delhi-5 Abstract With the increasing demand for newer products, subsequent design changes and introduction of newer models, such as in case of automobile industry, there is a need of quick and reliable method of product design and analysis. The finite element Analysis (FEA) has gained a lot of popularity in the present scenario as they can be used for quick design and analysis. But the FE predictions are often called into question as they are, at times, in conflict with test results because of inaccuracies in FE models which can arise due to use of incorrect modeling of boundary conditions, incorrect modeling of joints, and difficulties in modeling of damping etc. This paper presents the work that shows the need of correct selection of finite elements and nodes in order to get a correct prediction of dynamic behavior of the body or machine part under consideration which saves time and provides fairly good results. A Cantilever beam is modeled using Finite Element method using different nodes and elements and then the results are compared with analytical solution which shows that the significant error may arise in FE models. Thus there is a need of choosing the right elements and nodes while performing analysis, in order to predict the correct dynamic behavior as the analytical results may not be available or may be too comple to obtain. Keywords: FEM, ANSYS.. Introduction Most of the beam theories were introduced as early as 9, and the problem of the transversely vibrating beam was formulated in terms of the partial differential equation of motion, an eternal forcing function, boundary conditions and initial conditions. However, work on this subject was done in a patchwork fashion by showing parts of the solution at a time, and there is no paper that presents the complete solution. The most complete study was done by Traill-Nash and Collar []. An eact formulation of the beam problem was first investigated in terms of general elasticity equations by Pochhammer (876) and Chree (889) []. They derived the equations that describe a vibrating solid cylinder. Since it is not practical to solve the full problem because it yields more information than usually needed in applications, therefore, the approimate solutions for transverse displacement are sufficient. The Euler-Bernoulli model includes the strain energy due to the bending and the kinetic energy due to the lateral displacement. Many advances on the elastic curves were made by Euler []. The Euler-Bernoulli beam theory is the most commonly used because it is simple and provides reasonable engineering approimations for many problems. However, the Euler-Bernoulli model tends to slightly overestimate the natural frequencies. This problem is eacerbated for the natural frequencies of the higher modes. Also, the prediction is better for slender beams than non-slender beams. Timoshenko (9, 9) [, 5] proposed a beam theory which adds the effect of shear as well as the effect of rotation to the Euler-Bernoulli beam. The Timoshenko model is a major improvement for non-slender beams and for high-frequency responses where shear or rotary effects are not negligible. Following Timoshenko, several authors have obtained the frequency equations and the mode shapes for various boundary conditions. Some are Kruszewski (99) [6], Traill-Nash and Collar (95) [], Dolph (95) [7], and Huang (96) [8]. Assumptions made by all models are as follows.. One dimension (aial direction) is considerably larger than the other two.. The material is linear elastic.. The Poisson effect is neglected.. The cross-sectional area is symmetric so that the neutral and centroidal aes coincide. 5. Planes perpendicular to the neutral ais remain perpendicular after deformation. 6. The angle of rotation is small so that the small angle assumption can be used. The common approach currently used in the structural analysis is the finite element method. The elements used in the finite element method are usually void of dynamics. The consequence is that hundreds and thousands of elements are needed to represent large fleible structures in order to acquire analytical accuracy. To avoid the large dimensionality the current practice is to reduce the order of the model for structural system identification and control synthesis. This approimation, however, can lead to system instability due to the dynamics which are ignored. In contrast, distributed parameter modeling seems to offer a viable alternative to the finite element approach for modeling large fleible structures. The essential difference between the distributed parameter approach and the finite element approach is that instead of the approimate shape functions by interpolating polynomials the solutions to the PDE's are used to describe the structural elements []. Finite Element Analysis 9

2 YMCA University of Science & Technology, Faridabad, Haryana, Oct 9-, (FEA) on other hand can be used for solving such problems by using available computer programs such as described in [5-6].. Methodology The Finite Element method is used to solve physical problems in engineering analysis and design. The idealization of physical problem to mathematical model requires certain assumptions that together lead to differential equation governing the mathematical model. The FEA solves this mathematical model. Here the mass matri formulation is carried out. A two-element cantilever beam is used in order to develop the consistent mass matri. The si degrees of freedom lumped mass is used for constructing the lumped mass matri The global mass matri is built up as an assemblage of element mass matrices. A method analogous to static condensation, Guyan reduction, is used to reduce the size of the two-element cantilever problem. The model is then solved for its eigenvalues using Guyan reduction. Element Stiffness Matri The element stiffness matri is developed by using basic strength of materials techniques to analyse the forces required to displace each degree of freedom a unit value in the positive direction: Beam Node Definitions The two-beam elements are made identical, with the same E, I and length; the global stiffness matri can then be rewritten as: k g 6 8 For solving the dynamics of the cantilever beam, a mass matri is developed to complete the equations of motion. For a beam finite element, there are a number of different mass matri formulations as discussed in [- 5] out of which Consistent mass distributed mass is used. 9

3 YMCA University of Science & Technology, Faridabad, Haryana, Oct 9-, Consistent mass The consistent mass matri for a beam element is a filled matri. The filled matri can be combined with other consistent mass matrices of other elements of the structure, in the same manner as the element stiffness matrices are combined, to yield the final global mass matri. The element consistent mass matri for a prismatic beam is, with mass per unit length m and length l. m e ml Assuming the two elements have the same properties and lengths, the global mass matri becomes: 56m m 5m m m m m m 5m m m 5m m mg m m 8m m m 5m m 56m m m m m m Taking into account the two constrained degrees of freedom at the built in end, we can eliminate the first two rows and columns: m g m 5m m 8m m m 5m m 56m m m m m m Having the mass and stiffness matrices allows us to solve the eigenvalue problem for the homogeneous equations of motion: mg ž + kgz [ ] It is better to reduce the problem down to size. Therefore, the Guyan reduction [5] will be used to reduce the size of the problem. Two element cantilever eigen value closed form solution using guyan reduction Repeating the rearranged global stiffness matri from the static run, 8 k g EI 6 6 Equation of motion is: mss + kss [] 9

4 YMCA University of Science & Technology, Faridabad, Haryana, Oct 9-, 9 + l EI l 6EI l 6EI l 9EI 75 7ml 7 ml 7 ml 75 58ml Pre multiplying the equation of motion by identity matri + 567ml 8988EI 567ml 5988EI 567ml 96EI 567ml 8EI Rewriting without the identity matri: + 567ml 8988EI 567ml 5988EI 567ml 96EI 567ml 8EI 567ml 8988EI 567ml 5988EI 567ml 96EI 567ml 8EI Using a symbolic algebra program to solve for the eigenvalues, ml 78) EIm( π f ± eq. The above equation is used to solve the natural frequency of cantilever beam which is listed in table []. Analytical modeling of cantilever beam The analytical solution for resonant frequencies will be obtained from the eact analysis to confirm the validity of FE Model solution. The analytical solution of resonant frequencies of continuous systems is given in references [-5]. The dimensions of the cantilever beam are as follows: Thickness of the cantilever beam.5 m. Width. m Length.9 m The equation for solving the natural frequency of a cantilever beam is given by:

5 YMCA University of Science & Technology, Faridabad, Haryana, Oct 9-, λi EI (i,,,...) Eq. ππ M fi Using the following properties of the material: Young s modulus, E 6 GPa Poisson s ratio ν.. Density ρ7.8 kg/m The values of constants are: λ.875 λ.69 λ 7.85 For this set of data the analytical solutions are obtained and listed in table. Fig. First three mode shapes obtained by ANSYS. Results The results obtained by solving the equation and and the solution obtained from ANSYS software using different Elements are shown in the table below, the following results are obtained. Table : Natural Frequency of different FE Models MOD NO. EXACT SOLN

6 YMCA University of Science & Technology, Faridabad, Haryana, Oct 9-, The analytical model was solved to obtain the Natural Frequency from the analytical epression. It gives the following first three Natural Frequency of 5.5, and 899 Hz. The first three Natural Frequencies for a one element FE model are determined as 5.65, 55.9 and 57Hz, indicating huge differences between the second and third Natural Frequency of analytical model and one element FE model. This is due to the gross approimation in the FE model. The percentage error comes out to be.776 %, 6.796% and.86 % in first, second and third Natural Frequency respectively. The first three Natural Frequencies for a two element FE model are determined as 5.7, 5.5 and 88 Hz and are fairly close to the analytical model. The percentage error comes out to be -.58%,.85 and 6.9% in first, second and third Natural Frequencies respectively. The first three Natural Frequencies for a ten element FE model using ANSYS are determined as 5., 98.8 and 89.8 Hz and are much closer to the analytical model. The percentage error comes out to be -.57%, -.65 and.9679% in first, second and third Natural Frequencies respectively. This clearly establishes that as the number of elements is increased the percentage error is reduced further. Therefore a FE model with large number of elements is desirable to obtain accurate results. The first three Natural Frequencies for a ten element FE model using ANSYS are determined as 5., 98.6 and Hz and are much closer to the analytical model. The percentage error comes out to be -.57%, -.579% and.9977% in first, second and third Natural Frequency respectively. As seen from the above, the error for the first natural frequency is fairly constant and is around -.5 % but for the second natural frequency the error is quite high for one element FE model but is quite less for the higher number of nodes. Same trend can be seen for the third natural frequency of higher number of nodes. Table : Percentage error in different FE models MOD NO Conclusion The eact solution obtained from the differential equations gives us the basis to correlate the results obtained using the FE solution. As the table. shows that one element FE solution gives very crude results as epected because the error involved in analysis is more and it further increases as the modes are increased. As the number of nodes are increased the results tends to improve further but it shows a wavy pattern because of rounding off errors involved in adding different beam elements. The following conclusions may be drawn from the results thus obtained.. A huge amount of errors may be involved in the FE analysis if element selection is not proper thus needs a considerable eperience for making the choice of selecting the elements.. The time involved in analysis, which adds to the cost of analysis and processing time also depends upon the choice of the elements.. For complicated or huge structures as used in mechanical and civil engineering the analytical solution may not be available or may be too comple to obtain. Therefore it may be concluded that for comple structures or real structures the choice of elements in FEA has a great impact on results and should be carefully chosen. 95

7 YMCA University of Science & Technology, Faridabad, Haryana, Oct 9-, References. R.W.Traill-nash and A.R.Collar, 95, Quarterly Journal of Mechanics and Applied Mathematics 6, 86-. The effects of shear fleibility and rotatory inertia on the bending vibrations of beams. A.E.H.Love 97 Treatise on the Mathematical theory of Elasticity. New York: Dover Publications, Inc.. S.P.Timoshenko, 95, History of Strength of Materials. New York: Dover Publications, Inc.. S.P.Timoshenko, 9, Philosophical Magazine, 7. On the correction for shear of the differential equation for transverse vibrations of bars of uniform cross-section 5. S.P.Timoshenko, 9, Philosophical Magazine, 5. On the transverse vibrations of bars of uniform cross-section 6. E.T.Kruszewski, 99, National Advisory Committee for Aeronautics, 99. Effects of transverse shear and rotary inertia on the natural frequencies of a uniform beam 7. C.L.Dolph, 95, Quarterly of Applied Mathematics, On the Timoshenko theory of transverse beam vibrations 8. T.C.Huang, 96, Journal of Applied Mechanics, The effect of rotatory inertia and of shear deformation on the frequency and normal mode equations of uniform beams with simple end conditions 9. M.Levinson, 98, Journal of Sound and vibration 77, -. Further results of a new beam theory. M.Levinson, 98, Journal of Sound and vibration 7, A new rectangular beam theory. J.Y. Shen, Jen K. Huang and Lawrence W. Taylor, Jr., "Timoshenko Beam Modeling for Parameter Estimation of NASA Mini-Mast Truss", the Journal of Vibration and Acoustics, Transaction of the ASME, Vol. 5, Jan. 99, pp Thomson, W.T. Vibration Theory and Applications George Allen and Unwin Ltd., 97.. Den Hartong, J.P., Mechanical Vibrations McGraw-Hill Book Co. th ed., New York Rao, J.S and Gupta, K. Introductory Course on Theory and Practice of Mechanical Vibrations Wiley Eastern Limited (India) Stolarski, T., Nakasone, Y., Yoshimoto, S. Engineering Analysis with ANSYS Software Elsevier Butterworth-Heinemann, Oford, Hatch, R.Michael, Vibration Simulation Using MATLAB and ANSYS CRC Press, Florida 96