Dynamic Analysis of Elastic Vibration of Connecting Rod of a Slider Crank Mechanism

Transcription

1 Dynamic Analysis of Elastic Vibration of Connecting Rod of a Slider Crank Mechanism K.Venketesh M.Tech student Dept. of Mechanical Engg. NIT, Kurukshetra V.P. Singh Associate Professor Dept. of Mechanical Engg. NIT, Kurukshetra ABSTRACT A partial differential equation for elastic vibration of connecting rod of a slider crank mechanism is developed by using Theorem of motion of centre of mass and viscoelastic model. The partial differential equation is then discretized into two second order ordinary differential equation by assuming suitable two mode approximation. The two equations are then transformed into first order ordinary differential equation with suitable periodic coefficients. Then stability is checked by using Floquet theory. Procedures for analysing stability are developed. Finally a case study is taken involving different material which have different properties. Seeking the current trend composite materials of different proportion is taken. INTRODUCTION Connecting rod is very useful component in slider crank mechanism used to transfer the motion from crank to slider and vice versa. During its operation it is subjected to both sided compression and tension. So the material for connection rod must be tough and have high fatigue strength. So generally forge steel is used. But it is heavy structure, so have high inertia and starting problem. So due to weight considerations so many composites involving aluminium, manganese have been developed. So during transverse vibration its stability analysis is a necessary optimizing technique that must be involved. Zhang jinfu [1] in 2004 developed a partial differential equation of transverse vibration using viscoelastic model which is then discretized into two second order differential equation assuming suitable two mode approximation. In his analysis he is considered crank length to connecting rod length not to be a small parameter. Fallahi et al. [2] in 1995 formulated the equation of elastic vibration for a slider-crank mechanism with crank and connecting rod flexibility using finite element method. Based on the equations, the responses of elastic vibration of the crank and the connecting rod were obtained via numerical simulations. Chen and Chian [3] in 2008 derived the equations of elastic vibration of flexible connecting rod of a slider-crank mechanism by applying Hamilton's principle with all high order terms in the strain energy function being retained. After careful examination of the order of magnitude of each term, the coupled equations were simplified to a single one in terms of the transverse deflection, which turns out to be a Duffing equation under parametric and external excitations simultaneously. Closed-form approximations of the dynamic response of the connecting rod were then derived by using multiple scale method. Jan san chen [4] developed a strain equation by considering the axial load which is time dependent and solved by numerical methods and plotted the response of displacement of one mode approximation with respect to time. DYNAMIC MODELLING AND STABILITY ANALYSIS 86 K.Venketesh, V.P. Singh

2 The crank in this figure rotates at an angular velocity ω. Ox o y o is inertial frame of reference with its origin attached at the centre of the crank. Axy is a moving frame with x-axis passing through the two ends of connecting rod. The position vector r op can be written as Where = angle between x and x0 Component a in y-direction can be expressed as The above figure shows free small element diagram of connecting rod. Applying theorem of motion of centre of mass 87 K.Venketesh, V.P. Singh

3 According to equation of motion with respect to centre of mass Where di= moment of inertia of small element with respect to its mass Substituting the above equation According to mechanics of material bending strain at a distance from median layer of the connecting rod can be expressed as Bending stress of connecting rod can be expressed as E and c are youngs modulus and damping coefficient of material of connecting rod. The bending moment in the connecting rod can be expressed as J=moment of inertia of cross section of connecting rod. The connecting rod is simply supported at both the ends. So two mode approximation can be assumed as l= length of connecting rod 88 K.Venketesh, V.P. Singh

4 q 1 (t) and q 2 (t) are generalized coordinates Multiplying the above equation by sin(πx/l) and sin(2πx/l) respectively and then integrating by parts from x=0 to l, we obtain following ordinary differential equation for connecting rod By introducing following dimensionless parameters The above equation can be nondimensionalized as follows After defining state vector The non dimendionalised equations can be tranformed into ordinary differentilal equation as follows 89 K.Venketesh, V.P. Singh

5 Where Stability characteristics of connecting rod can be analysed as analogous to homogeneous system of above ordinary differential equation. 90 K.Venketesh, V.P. Singh

6 So H(ϴ) can be rewritten as Assuming that the standard fundamental matrix of the system is x=x(θ) that is x=x(θ) is the solution of Where I=identity matrix. PROCEDURE: 1 H(ϴ) from above equation is determined by using different dimensionless numbers. 2 The matrix differential equation is integrated by using Runge-kutta method using MATLAB computational program. 3 The Eigen values of above differentilal equation is determined. 4 According to Floquet theory if all the Eigen values are less than one, elastic vibration of connecting rod is asymptotically stable; if atleast one of eigen value is larger than 1,elastic vibration of connecting rod is unstable ; if some eigen values are less than 1 and the remaining ones are equal to one and if the elementary divisors corresponding to latter are simple, elastic vibration of the connecting rod is stable;if the said divisors are not simple, elastic vibration of the connecting rod is unstable. Cross section of connecting rod 91 K.Venketesh, V.P. Singh

7 Data used in calculation: Mass of connecting rod=50 Gram Crank radius=15 mm Connecting rod length=50 mm Speed of connecting rod=210 rad/sec RESULTS AND ANALYSIS MATERIAL Beryllium alloy Magnesium alloy DENSITY (KG/M^3) YOUNGS MODULUS(Gpa) DAMPING CAPACITY(N- S/M) n Ω b Eigen values Stability ,0.91,0.125,0.125 Asymptotic stable ,0.84,0.625,0.625 unstable Alfasic ,0.994,0.589,0.589 Asymptotic stable Al6061+B4C ,0.63,1.039,1.039 Unstable CONCLUSION The procedure for predicting stability of elastic vibration of connecting rod of a slider crank mechanism are developed upon assuming constant angular velocity. By applying such procedure we can easily determine the connecting rod stability of different materials of different connectring rod proportions. REFERENCES [1] Zhang jinfu, Lui wei, Qin weiyang, He Xinsuo, Stability analysis of elastic vibration of connecting rod of a Slider crank mechanism vol 2 no.1 pp 31-40(2006). [2] Fallaih B, lai s, Venkat c, A finite element formulation of flexible slider crank mechanism using local Coordinate, Asme journal of dynamic system, measurment and control, 117(3) pp: (1994) [3] Chen j chian C.H. on the nonlinear response of a flexible connecting rod, ASME journal of mechanical Design, 125(4) pp (2003) [4] Jan senchen, the role of Lagrangian strain in dynamic response of flexible connecting rod (D.O.I ) [5] S.C. Chu, K.C. Pan, vibration of elastic connecting rod of high speed connecting rod mechanism vol.3(2004) [6] A Prem Kumar, Design and analysis of connecting rod by composite material ISSN [7] Puneetagrawal, Ankitgupta, A comparative study of different material of connecting rod- A review Vol 5, no-1 jan 2015 [8] G.M.SayeedAhmed, Syed Hamza, Design, fabrication and analysis of connecting rod with aluminium Alloys and carbon fibre composite ISSN K.Venketesh, V.P. Singh