ELASTODYNAMIC ANALYSIS OF FOUR BAR MECHANISM USING MATLAB AND ANSYS WB

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1 International Journal of Mechanical and Production Engineering Research and Development (IJMPERD ) ISSN Vol., Issue Sep TJPRC Pvt. td., EASTODYNAMIC ANAYSIS OF FOUR BAR MECHANISM USING MATAB AND ANSYS WB 1 MANISH MEHTA & P M GEORGE 1 Research Scholar, SICART, Vallabh Vidyanagar, Gujarat, India & Assistant Prof. of Mech. Engg., G H Patel college of Engg. & Tech., Gujarat, India HOD, Mech. Engg, BVM (Engineering College), Vallabh Vidyanagar, Gujarat, India ABSTRACT It is well-known that the dynamic analysis of mechanisms operating at high speed cannot neglect the effects of link elastic flexibility. In fact this effect may effect the dynamic response of the output link motion, so that the mechanisms may fail to perform their assign tasks effectively. In this paper FEM presented for dynamic analysis of high speed mechanism. Based on this method progrmme was developed in MATAB for determined deformation in coupler link of mechanism. An example problem has been solved. Same example has been taken in ANSYS WB for elastodynamic analysis. Results from MATAB and ANYSYS WB are compared and presented in form of graphs KEYWORDS: MATAB, ANYSYS WB, Dynamic Analysis. INTRODUCTION Generally, all considerations in the force analysis of mechanisms, whether static or dynamic, the links are assumed to be rigid. The complexity of the mathematical analysis of mechanisms with elastic links has been a deferent against giving up the rigidity assumptions. The area of research pertaining to the motion of mechanisms, with consideration of link elasticity and mass distribution has been called kineto- elastodynamics of mechanism. The requirement for machines to run at higher speed brought to the surface many problems, such as balancing and vibrations, which were not serious factor at lower speeds. Therefore the problem they were facing was now to run machines at higher speeds with lower power consumption. New method to understand the elastic behavior of mechanisms was needed. The researcher interest is the development of preferably simple mathematical model which is able to simulate the dynamic behavior of the mechanisms. Since this model make it possible to relate the design and operational parameters of mechanisms to their actual dynamic behavior. FINITE EEMENT MODE In the analysis of the planar four bar elastic mechanisms at a given position, it is assumed that it is a structure composed of discrete members. In this respect, each of the constituent members is regarded as a beam and so the beam theories of bending apply. The effect of shear deformation and rotary inertia are neglected.

2 77 Elastodynamic Analysis of Four Bar Mechanism Using Matlab and Ansys WB EASTIC BEAM EEMENT IN PANE A general beam element, representing a link of a mechanism, is shown in Fig. 1, without the other links. There are two axis of reference; 1. The fixed (OXY) and,. The rotated (Oxy) axis and both have a common origin O. The x axis of the rotated frame is always parallel to the rigid body position of the beam element axis throughout its motion. The beam element is shown by dotted lines is elastically deformed position and its rigid body position by full lines. The elastic deformations of the beam element (link) are completely described by the six generalized nodal displacement coordinates by u 1 to u 6, shown in Fig. 1. These displacements shown in their positive directions with reference to the rigid body position of the beam (link), in Fig. 1 also locate the deformed position P and Q of the end points P and Q.[4] y Y Q Q u 5 u 6 u u 4 P u P u 1 x θ X Figure 1 Rigid and elastic body with coordinate systems By consider the link as a beam element in plane motion. We get the accelerations vectors as follows:...(1) where, the vectors from left to right represent the absolute, rigid body, generalized relative to the rigid body position of link, normal, coriolis and tangential accelerations respectively. In equation (1) the product terms in vectors are considered small compared to corresponding terms in. When it is neglected, equation may be modified as

3 Manish Mehta & P M George 78 Similarly, it may be shown that The unknown displacement field within an element will be interpolated by a linear distribution. This approximation becomes increasingly accurate as more elements are considered in the model. To implement this linear interpolation, linear shape functions will be introduced. MASS AND STIFFNESS MATRICES OF EEMENT The equation of motion of the elastic beam element in Fig. 1 may be described by agrangean s equation. i = 1,,..., 6 Where, have no potential. are the generalized forces acting in the direction of generalized coordinators and Considering the strains associated with the displacement functions and neglecting those due to temperature variations and any strain initially present. The strain energy in matrix form is Then, [ m] = ρ A 1 15 [ k] = EA EA 1EI 6EI 1EI 6EI 4EI 6EI EI EA 1EI 6EI 4EI With the help of agrange s equations, the equation of motion for the beam element may be derived as [4] ASSEMBY OF THE EEMENT In previous section we formulate the mass and stiffness matrices for the element in their local coordinates system. Now system matrices for the whole mechanism are found by define only one global coordinate system for a given mechanism.

4 79 Elastodynamic Analysis of Four Bar Mechanism Using Matlab and Ansys WB In four bar mechanism of Fig., in which for the analysis purpose each link is to be consider as an element and joints of all links are to be consider as node. For the finite element analysis prepare the stiffness and mass matrix for all elements and these elements mass and stiffness matrixes are systematically superposed to develop the stiffness and mass matrix of the mechanism [4]. Afterwards solve the coupled differential equation of motion with help of modal analysis. Solutions of equation give a displacement of each element at particular at that instant. With the help of displacement calculate the strain and stress at desire point on links.[4] If the system oriented coordinates {U} are used as the generalized coordinates to describe the structural deformation of the linkage from its rigid body position, as in Fig.. In first, second and the third terms on the left hand side of agrange s equation is reduce to respectively. In matrix form, the equations of motion are Figure. Four bar mechanism with displacement and three elements In the absence of damping forces in the mechanism and of external forces on the follower, the equation of motion may be written in matrix form as below [4] STRESS CACUATIONS The axial strains at the neutral axis is The axial stress is

5 Manish Mehta & P M George 8 SOUTION OF EQUATION OF MOTION The system is excited by a periodic forcing function. The method is use for solving nine couple differential equations are modal analysis. Application of modal analysis for linear system only requires symmetry of the stiffness and mass matrix. In coupling equation first step is to determine the natural frequencies and mode shapes. When the principle coordinates are used as dependent variables, the governing differential equations are uncoupled. Each equation can be solved by any appropriate method with the help of developed programme in MATAB. DESIGN PROBEM Specifications of four bar linkage for analysis are as follows: Parameters Fixed link (1) Crank () Coupler () Follower (4) ength (mm) C/S Area (mm ) Area moment of Inertia (mm 4 ) Modulus of Elasticity, E = 7.1 x 1 4 MPa Density = 77 kg/m Crank speed =. rad/sec For the finite-element analysis of a mechanism total number of elements equal to three and total number of degrees of freedom equal to nine. RESUTS AND CONCUSIONS Figure Result from MATAB

6 81 Elastodynamic Analysis of Four Bar Mechanism Using Matlab and Ansys WB Figure 4 Results from ANSYS WB This paper avoids the selection of the element type and model shape function of the displacement and prepared the equation of motion for uniform cross section links. By assuming the straight beam of link, mechanism stiffness and mass matrix are prepared. Uncoupled differential equations are solved by using modal analysis method. Programme for generating the mass matrix, stiffness matrix and rigid body angular velocity and acceleration was developed in MATAB. Programming was also done for solving the uncoupled differential equations. After solving the equations strain in coupler are calculate at different position of crank and plot the graph of it by programming in MATAB. For more accurate results take two or three elements per link. This design problem was also solved in ANSYS WB. Model has prepared in Pro E and for analysis exported to ANSYS for analysis. Result from ANSYS WB and MATAB were presented in form graphs in this paper REFERENCES [1] A G Erdman and G N Sandor, Kineto elastodynamics- A review of the state of the art and trends. Mech. & Mach. Theory, Vol. 7 pp 19-, 197

7 Manish Mehta & P M George 8 [] V Masurekar and K N Gupta, Theoretical and experimental kineto elastodynamic analysis of high speed linkage. Mech. & Mach. Theory, Vol. 4 pp 5-4, 1989 [] D A Turcic and A Midha, Generalized equations of motion for the dynamic analysis of elastic mechanism systems. Journal of dynamic systems, measurement, and control, Vol. 16 pp 4-48, 1984 [4] Manish Mehta and Dr Anurag Verma, Theoretical kineto elastodynamic analysis of mechanism by finite element method using MATAB. proceeding of 4th International Conference on Advances in Mechanical Engineering (ICAME) during rd - 5 th September 1 organized by Mechanical Engineering Department, SVNIT, Surat. [5] G N Sander and A G Erdman, Advanced Mechanism Design: Analysis and Synthesis Vol. [6] T R Chandrupatla and A D Belegundu, Introduction to Finite Elements in Engineering [7] J S Rao and K Gupta, Introductory course on Theory and Practice of Mechanical Vibrations