Modeling, simulation and analysis of engineering system

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Modeling, simulation and analysis of engineering system

Figure shows a 1-DOF mass-damper idealized form of engineering mechanical system. Displacement x is measured from an equilibrium position where the damper is at the “neutral” position. External force fa(t) =Acos3t is applied directly to mass m.

1. Derive the mathematical model of the mechanical system with position x as the dynamic variable.

2. Derive the mathematical model with velocity v(t)= x(t) as the dynamic variable.

3. Use MATLAB’s command to solve the differential equation of velocity.

4. Use MATLAB’s command to plot the evolution of velocity with time t in the interval [0, 20]. b = 0. 3N/m/s and m= 200kg, A=2.

5. Use MATLAB’s command to plot the evolution of velocity with time t

6. Use MATLAB’s command to plot in the same graph the evolution of position x when External force =0 and

when A= 10N

7. Analyze your result

Figure shows a 1-DOF mass-damper idealized form of engineering mechanical system. Displacement x is measured from an equilibrium position where the damper is at the “neutral” position. External force fa(t) =Acos3t is applied directly to mass m.

1. Derive the mathematical model of the mechanical system with position x as the dynamic variable.

2. Derive the mathematical model with velocity v(t)= x(t) as the dynamic variable.

3. Use MATLAB’s command to solve the differential equation of velocity.

4. Use MATLAB’s command to plot the evolution of velocity with time t in the interval [0, 20]. b = 0. 3N/m/s and m= 200kg, A=2.

5. Use MATLAB’s command to plot the evolution of velocity with time t

6. Use MATLAB’s command to plot in the same graph the evolution of position x when External force =0 and

when A= 10N

7. Analyze your result

Group 4: Turbine generator

Figure shows a wind turbine generator used for transforming mechanical energy into electrical energy. For this problem, let us assume that the turbine inertia J1 and generator inertia J2 are both rigidly connected to their respective gears in the train

the mathematical model of this rotational mechanical system (wind turbine generator system)

(J_{1} + 1/(N ^ 2) * J_{2}) tilde theta 1 +(b 1 + 1 N^ 2 b 2 ) hat theta 1 =T aero – 1 N T gen

We can write the system model in a more compact form by defining the equivalent or “composite” inertia and friction coeficient as

J c1 = J_{1} + 1/(N ^ 2) * J_{2} b c1 = b_{1} + 1/(N ^ 2) * b_{2}

1. Derive mathematical model in terms of angular velocity of the turbine Where omega_{1} = theta_{1} is angular velocity of the turbine and hat omega_{1} = overline theta 1

2. Use MATLAB’s command to solve the differential equation of angular velocity of the turbine

3. Use MATLAB’s command to plot the evolution of angular velocity of the turbine of that mechanical engineering system for t in the interval [0,20). Take J C2 = 50sl , b epsilon3 =0. si and N = 1 T gan =10.SI and T aaro =25 S1

4. Use MATLAB’s command to plot in the same graph the evolution of of angular velocity of the turbine when T aerv = 0Sl and T arrea = 25Sl

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Desmond 2025-01-30T21:26:18+05:00 1 Answer 16 views Beginner
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  1. Mechanical Engineering Expert
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    2025-01-30T23:04:05+05:00
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    % Parameters
    Jc1 = 50; % Equivalent inertia
    bc1 = 0.5; % Equivalent friction coefficient
    N = 1; % Gear ratio
    Tgen = 10; % Generator torque
    Taero1 = 25; % Aerodynamic torque case 1
    Taero2 = 0; % Aerodynamic torque case 2

    % Define the differential equations
    domega_dt1 = @(t, omega) (Taero1 – (1/N)*Tgen – bc1*omega) / Jc1;
    domega_dt2 = @(t, omega) (Taero2 – (1/N)*Tgen – bc1*omega) / Jc1;

    % Initial condition
    omega0 = 0; % Initial angular velocity

    % Time span
    tspan = [0 20];

    % Solve the differential equations
    [t1, omega1] = ode45(domega_dt1, tspan, omega0);
    [t2, omega2] = ode45(domega_dt2, tspan, omega0);

    % Plot the evolution of angular velocity
    figure;
    plot(t1, omega1, ‘b’, t2, omega2, ‘r’);
    xlabel(‘Time (s)’);
    ylabel(‘Angular Velocity (rad/s)’);
    title(‘Evolution of Angular Velocity of the Turbine’);
    legend(‘T_{acro} = 25’, ‘T_{acro} = 0’);
    grid on;

    % Analyze the results
    fprintf(‘Analysis:n’);
    fprintf(‘1. When T_{acro} = 25, the turbine accelerates to a higher angular velocity due to the greater driving torque.n’);
    fprintf(‘2. When T_{acro} = 0, the turbine”s angular velocity decreases over time due to friction and the generator torque, eventually stabilizing at a lower velocity or stopping.n’);

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