Hello! I need this equation solved and modeled in MATLAB (all questions are in the file). The basic premise is using two ODE functions for the down and upswing motion I believe. Thank you!
ME 2101 Project – Spring 21 A mechanical system is designed with two particle masse ??1 and ??2. Particle ??1 is released from rest at the top of a concave quarter-circular path of radius ?? = 1 m with center at O and friction coefficient ?? = 0.2. The particle then collides with particle ??2 = 1 kg that is attached to the end of a pendulum with massless rod of length ?? = 0.5 m. The pendulum hinge has viscous friction coefficient ?? = 0.2 Nm/(rad/s) such that it applies the moment of −???? 2 where ??2 is the pendulum angle measured CCW from its equilibrium position (i.e. swing-up angle). (a) Initially an experiment is performed to determine the coefficient of restitution. The mass selected for the experiment was ??1 = 2 kg and it was observed that the pendulum swung up 80o before stopping and returning. What is the coefficient of restitution ??? (b) In the next phase, the designer wants to know how to choose ??1 if (s)he wants to precisely swing up the pendulum to a desired maximum angle. To do this, (s)he runs a series of simulations with ??1 values at 0.5, 1, 1.5, 2, 3, 4, 5, 6, 7, 8, 9, 10 kg and records the maximum ??2 angles in an Excel file. What does the trend look like? Submit the report as a single pdf file with the following included: 1) A title page with “ME2101 – Project” at the top, followed by your name, course number, and date on the subsequent lines. 2) The equation derivation steps, free-body diagrams, and the final equations circled. 3) Plot of the time history swing-up angle ??2(??) in degrees for part (a) with the value of ?? written on the plot in MATLAB. 4) A plot of maximum swing-up angles ??2 in degrees vs. ??1 for part (b) with the trend explanation or equation (this part is easier if you use Excel). 5) A printout of your mfile. O R g ? ? 1 ? ? 2 ? ? ? ? ? ? 2 ? ? 3/29/21 1:29 PM C:\Users\hashrafi\Documents\Acad...\MHW3.m 1 of 1 %%%%%%%%%%%% ME 2101 - MHW3- S21 %%%%%%%%%%%% % variable order z = [x; xdot]; zdot = [xdot, xddot] clear; clc; global mA mB k b mA = 1; mB = 2; k = 300; b = 30; vA0 = 6; v0 = mA*vA0/(mA+mB); z0 = [0; v0]; t = 0:.01:2; [t,z] = ode45(@MHW3_der,t,z0); % initial conditions for combined mass A&B % integration duration % MATLAB integration function ode45 figure(1) plot(t,z(:,1),'b-','LineWidth',1.5) grid xlabel('time (s)') ylabel('x (m)') %% derivative function % variable order z = [x; xdot]; zdot = [xdot, xddot] function zdot = MHW3_der(~,z) global mA mB k b x = z(1); xdot = z(2); xddot = -k*x/(mA+mB) - b*xdot/(mA+mB); zdot = [xdot; xddot]; end 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 time (s) -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 x (m ) X 1.4 Y -9.0567e-05 hashrafi Typewritten Text Reach equilibrium point: x=0 at about t=1.4 s ME 2101 –MATLAB Assignment 4 The disk with its center hinged at pin O, fixed to ground, rotates clockwise at constant angular speed ??0. The rod AB has a length of 3?? and is hinged at both ends, one at pin A fixed to the disk and the other to pin B fixed to the collard. Pin A is located at distance ?? from O. Part 1 – Derive the position, velocity and acceleration of collar B in terms of ?? and ??0. Do this part by hand and neatly write your derivations and equations. Part 2 – Write a MATLAB program and plot the position, velocity and acceleration of collar B vs. ??, assuming ??(0) = ??/4, ??0 = 2?? rad/s, and ?? = 0.5 m. Hint: You must define a time array in MATLAB with sufficiently small step size starting at 0 and ending such that the disk rotates two revolutions. Ultimately all variables are functions of time. Note also that this homework does not required integration; i.e., no need to call ode45. ?? ?? 3?? 1.9?? 4/19/21 1:15 PM C:\Users\hashrafi\Documents\Acad...\MHW4.m 1 of 1 %%%%%%%%%%%% ME 2101 - MATLAB Homework 4 %%%%%%%%%%%% clear; clc; r = 0.5; l = 3*r; d = 1.9*r; w0 = 2*pi; phi0 = pi/4; t = 0:0.01:2; phi = phi0 - w0*t; theta = acos((d-r*cos(phi))/l); yB = r*sin(phi) - l*sin(theta); thetadot = r*w0*sin(phi)./(l*sin(theta)); vyB = -r*cos(phi)*w0-l*cos(theta).*thetadot; thetaddot = (l*cos(theta).*thetadot.^2-r*w0^2*cos(phi))./(l*sin(theta)); ayB = -r*w0^2*sin(phi)+l*sin(theta).*thetadot.^2-l*cos(theta).*thetaddot; figure(1) subplot(311) plot(phi,yB) grid xlabel('\phi (rad)') ylabel('y_B (m)') subplot(312) plot(phi,vyB) grid xlabel('\phi (rad)') ylabel('v_{yB} (m/s)') subplot(313) plot(phi,ayB) grid xlabel('\phi (rad)') ylabel('a_{yB} (m/s^2)') -12 -10 -8 -6 -4 -2 0 2 (rad) -2 -1 0 y B ( m ) -12 -10 -8 -6 -4 -2 0 2 (rad) -2 0 2 4 6 v y B ( m /s ) -12 -10 -8 -6 -4 -2 0 2 (rad) -100 -50 0 a y B ( m /s 2 )