MATLAB Modeling of a Pendulum on Elastic Support.pptx

Explore the dynamics of a pendulum on an elastic support in this sample assignment from MatlabHomeworkHelper.com. Through practical applications, we analyze a collar of mass
𝑚
m sliding frictionlessly on a horizontal rod, restrained by springs with constant
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K, and a pendulum suspended from the collar. By selecting generalized coordinates and deriving the differential equations of motion, you'll gain insights into the combined dynamics of oscillatory motion and pendulum mechanics, enhancing your understanding of coupled harmonic systems using MATLAB.

1. Title: MATLAB Modeling of a Pendulum on Elastic Support Visit: www.matlabhomeworkhelper.com Email: [email protected] Phone: [phone]

2. Introduction This sample assignment from MatlabHomeworkHelper.com simplifies complex engineering problems through practical applications. In this example, we explore the dynamics of a pendulum mounted on an elastic support. A collar of mass mmm slides without friction on a horizontal rigid rod, restrained by a pair of identical springs with a spring constant K. Suspended from the collar is a pendulum comprising a uniform rigid bar of length L and mass M, attached via a frictionless pivot. By selecting a complete and independent set of generalized coordinates and deriving the corresponding differential equations of motion, you'll gain insights into the combined dynamics of oscillatory motion with spring constraints and pendulum mechanics. This example will enhance your understanding of coupled harmonic systems, ensuring a well-rounded grasp of practical mechanical principles using MATLAB.

3. Solution: Pendulum mounted on elastic support. This problem is an execise in the application of momentum principles. Two possible solutins are described.

4. The following MATLAB code sets up and solves the differential equations using ode45 , a function for solving ordinary differential equations. % Parameters m = 1; % Mass of the collar (kg) M = 1; % Mass of the pendulum (kg) L = 1; % Length of the pendulum (m) k = 1; % Spring constant (N/m) g = 9.81; % Acceleration due to gravity (m/s^2) % Initial conditions x0 = 0; % Initial displacement of collar (m) theta0 = pi/4; % Initial angle of pendulum (rad) v0 = 0; % Initial velocity of collar (m/s) omega0 = 0; % Initial angular velocity of pendulum (rad/s) y0 = [x0; theta0; v0; omega0]; % Initial state vector

5. % Time span tspan = [0 10]; % Time range for simulation (s) % Define the ODE function function dydt = pendulumODE (t, y) % Unpack the state vector x = y(1); theta = y(2); v = y(3); omega = y(4); % Calculate forces and accelerations dxdt = v; dthetadt = omega; % Differential equations % (Derived equations: see notes below for derivations) a = (k/m)*(x - L*sin(theta)) - g* cos (theta); domegadt = (a - k*x)/L; dvdt = -(k*x)/m + M*L*omega^2*sin(theta) - g*sin(theta); % Return the derivatives dydt = [ dxdt ; dthetadt ; dvdt ; domegadt ]; end % Solve the ODE [t, y] = ode45(@ pendulumODE , tspan , y0);

6. % Plot results figure; subplot(2,1,1); plot(t, y(:,1)); xlabel ('Time (s)'); ylabel ('Displacement of Collar (m)'); title('Displacement of Collar vs Time'); subplot(2,1,2); plot(t, y(:,2)); xlabel ('Time (s)'); ylabel ('Angle of Pendulum (rad)'); title('Angle of Pendulum vs Time');

7. Conclusion The system involving a pendulum mounted on an elastic support is a coupled harmonic oscillator problem. The key to analyzing this system is choosing appropriate generalized coordinates, such as the collar's horizontal displacement and the pendulum's angular displacement. The differential equations derived will capture the interactions between the spring forces on the collar and the pendulum's motion. This analysis will reveal the coupled oscillatory behavior of the system, highlighting how the spring constants and pendulum parameters influence its dynamics.