Modeling and Simulation of a Four Bar Mechanism and a Crane using ProM - PDF

[Project-4] [Mechanism Studies] Sasi Bhushan Beera #35763829 Srikanth Avala #35762927 Project4 Four Bar Mechanism Introduction: A four bar linkage or simply a four-bar mechanism is the simplest movable linkage. It consists of four rigid bodies (called bars or links), each attached to two others by single joints or pivots to form a closed loop. If each joint has one rotational degree of freedom (i.e., it is a pivot), then the mechanism is usually planar, and the four-bar is determinate if the positions of any two bodies are known (although there may be two solutions). One body typically does not move (called the ground link, fixed link, or the frame), so the position of only one other body is needed to find all positions. The two links connected to the ground are called grounded links. The remaining one link, not directly connected to the ground link, is called coupler link. In terms of mechanical action, one of the grounded links is selected to be the input link, i.e., the link to which an external force is applied to rotate it. The second grounded link is called the follower link, since its motion is completely determined by the motion of the input link. Four Bar mechanism In the figure shown above the first link (input link) is called Crank, the second link Coupler and the third link is the Follower. Objective: The objective of this project is to simulate the four-bar mechanism using Pro-E and compare the analysis results with the analytical calculations. The dimensions of the Four bar mechanism of interest are shown in the figure below: 0.50 0.50 6.00 Length of the Crank : 6 in Length of the Coupler: 24.7386 in Length of the Follower: 12 in The Crank and the Coupler have to be of negligible mass. So the density is appropriately chosen. The various parameters are tabulated as below: Volume = length * width*thickness + pi * r^2*thickness r: radius of curvature of the ends Link#Length(in)Width(in)Thickness(in) v1DensityVolumeMassCrank610.50.39251.00E-073.39253.3925E-07Coupler24.738610.50.39251.00E-0712.76181.27618E-06Follower1210.50.3925 0.00073246.39250.004681867 The Four bar mechanism built in pro-E is as shown in the figure below: Four Bar in Pro-E Analysis: The Four bar mechanism is simulated in Pro-E and both kinematic and dynamic analysis is done to measure the angle rates and angular acceleration. The Torque and the reaction forces at the Crank-Ground joint are also measured and are shown in the figures below: Initial Configuration: #Angle(rad) Rate(rad/s)Acceleration(rad.s^2)Crankpi/22*pi0Fol ower pi/2TBDTBD The angular rates , accelerations of other joints and torque and reaction forces at the Crank-ground joint are plotted as shown below: W3 vs time W4 vs time W3dot vs time W4dot vs time Fx Fy Moment Analytical Calculations: Notations: cos(th1) : C1 sin(th1) : S1 cos(th2):C2 sin(th2):S2 cos(th4):C4 sin(th4):S4 Closed loop equations: position level l1*C1+l2*C2 = l0+l3*C4 l1*S1+l2*S2 =l3*S4 Differentiating the above set of equations w.r.t time we get equations at velocity level: -l1*S1*w2-l2*S2*w3 = -l3*S4*w4 l1*C1*w2+l2*C2*w3 = l3*C4*w4 Now given w2 we can determine, w3 and w4 at the initial position. Differentiating the above equations w.r.t time we get equations at acceleration level: -l1*S1*α2-l1*C1*(w2^2)-l2*S2* α3-l2*C2*(w3^2) = -l3*S4* α4-l3*C4*(w4^2) l1*C1*α2-l1*S1*(w2^2)+l2*S2* α3-l2*S2*(w3^2) = l3*C4* α4-l3*S4*(w4^2) α3 and α4 can be determined from the above set of equations. Force Calculations: Rocker: F = (I03*w4dot)/(l3*cos(th)) Crank: Rx = -F* cos(th) Ry = -F*sin(th) M = -F*cos(th)*l1 Results: Since pro-E uses relative angles we need to covert them to absolute angles before comparison #Pro-EAnalyticalRelativeAbsoluteAbsolutew3-36000w4180180180α3282.665282.665282.7473α4141.354141.354141.3717 Force Analysis: The hand calculations for the force analysis are submitted in a hand-written format. The results are tabulated as shown below: #Pro-EAnalyticalFx-0.04734290.0462Fy-0.0120.0115Torque0.2840360.2772