Modeling and simulation of engineering systems

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An engineer wants to develop mathematical models for one engineering system. A simplified mechanical model of a vibration isolation system is shown in Figure. Damper b1 connects mass m to the fixed overhead horizontal surface. The vibration mounts that support the mass on the moving base are modeled by lumped stiffness k and viscous friction b2. The displacement of the base z in (t) is the input to the system and z is the vertical displacement of mass m as measured from the static equilibrium position
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1. Derive the mathematical model of this mechanical system
2. Use MATLAB’s command to solve the differential equation
3. Use MATLAB’s command to plot the evolution of vibration of that mechanical engineering system for t in the interval [0,50]. Take k = 500N/m, b1 b2 = 0. 3N/m/s and m= 200kg
4. To evaluate the effect of the damping parameter b1, Use MATLAB’s command to plot in the same graph the evolution of vibration when b1 = 0 N/m/s and b1 = 1.5 N/m/s
5. Analyze your result

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Desmond 2025-01-30T13:10:43+05:00 1 Answer 30 views Beginner
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MathJax Example

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  1. Mechanical Engineering Expert
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    2025-01-30T13:35:14+05:00
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    clc;
    clear;
    close all;

    % Given parameters
    k = 500; % N/m
    b2 = 0.3; % N/m/s
    m = 200; % kg

    % Time range
    tspan = [0 50];

    % Initial conditions: Assume initial displacement and velocity
    z0 = [0.1; 0]; % Initial conditions [z(0), z_dot(0)]

    % Input conditions (assume zero input for simplicity)
    z_in = 0;
    dz_in = 0;

    % Define function to solve ODE for different b1 values
    b1_values = [0, 1.5];

    figure;
    hold on;

    for i = 1:length(b1_values)
    b1 = b1_values(i);
    b = b1 + b2;

    % Define the system of equations
    odeFunc = @(t, z) [z(2); (b2 * dz_in + k * z_in – b * z(2) – k * z(1)) / m];

    % Solve ODE using ode45
    [t, Z] = ode45(odeFunc, tspan, z0);

    % Plot results
    plot(t, Z(:,1), ‘DisplayName’, sprintf(‘b1 = %.1f N/m/s’, b1));
    end

    % Graph settings
    xlabel(‘Time (s)’);
    ylabel(‘Displacement (m)’);
    title(‘Evolution of Vibration for Different Damping Parameters’);
    legend;
    grid on;
    hold off;

    Best answer

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