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System Dynamics

The partial fraction expansion reslut by MATLAB for a function F(s) is provided below. Find its inverse Laplace transform.

Derive the equations of motion for the two-mass system shown, and then express the equations of motion in the matrix form.

For the spring-loaded pendulum shown in the figure in the textbook, assume the mass of the rod of the pendulum is not negligible and it has the same mass, m, as the point mass at the end of the rod. Use energy method to obtain the equation of motion and find the natural frequency of the system for small angle of oscillation theta.

The mechanical system shown in the figure is initially at rest. The displacement x of mass m is measured from the rest position. At t=0, mass m is set into motion by an impulsive force whose strength is unity. Using MATLAB, plot the response curve x(t) versus t when m=10 kg, b=20 Ns/m, and k=50N/m. Plot the response for 6 seconds using "impulse" command.

Consider the mechanical system shown in the figure. Plot the response curve x(t) versus t with MATLAB when the mass m is pulled slightly downward, generating the initial conditions x(0)=0.05m and xdot(0)=1 m/s, and released at t=0. The displacement x is measured from the equilibrium position before m is pulled downward. Assume that m=1 kg, b1=4 Ns/m, k1=6N/m, and k2=10N/m. (a) Plot the exact solution of response for 5 s using the "residue" command. (b) Plot the response for 5 s using the "impulse" command. (c) Plot the response for 5 s using the "step" command.

Consider the system defined by the following matrix. This system involves two inputs and two outputs. Four transfer functions are involved: Y1(s)/U1(s), Y2(s)/U1(s), Y1(s)/U2(s), and Y2(s)/U2(s). (When considering input u1, we assume that input u2 is zero, and vice versa.) Obtain the transfer matrix (consisting of the preceding four transfer functions of the system. (a) Obtain the transfer matrix G(s) by hand. (b) Verify your answers by using ss2tf in MATLAB. (c) Explicityly write out the four individual transfer functions.

For the torsianl system shown, obtain the transfer function G(s).

The mechanical system shown is a simple model of a rack-pinion mechanism. The rack rolls without slipping horizontally on the pulley, triggering the pulley's rotational motion. Derive the equation of motion for the system using the rack's displacement x as the variable, and then find the natural frequency of the system.

Derive the equations of motion for the two-mass system shown, and then express the equations of motion in the matrix form.

For the spring-loaded pendulum shown in the figure in the textbook, assume the mass of the rod of the pendulum is not negligible and it has the same mass, m, as the point mass at the end of the rod. Use energy method to obtain the equation of motion and find the natural frequency of the system for small angle of oscillation theta.

The mechanical system shown in the figure is initially at rest. The displacement x of mass m is measured from the rest position. At t=0, mass m is set into motion by an impulsive force whose strength is unity. Using MATLAB, plot the response curve x(t) versus t when m=10 kg, b=20 Ns/m, and k=50N/m. Plot the response for 6 seconds using "impulse" command.

Consider the mechanical system shown in the figure. Plot the response curve x(t) versus t with MATLAB when the mass m is pulled slightly downward, generating the initial conditions x(0)=0.05m and xdot(0)=1 m/s, and released at t=0. The displacement x is measured from the equilibrium position before m is pulled downward. Assume that m=1 kg, b1=4 Ns/m, k1=6N/m, and k2=10N/m. (a) Plot the exact solution of response for 5 s using the "residue" command. (b) Plot the response for 5 s using the "impulse" command. (c) Plot the response for 5 s using the "step" command.

Consider the system defined by the following matrix. This system involves two inputs and two outputs. Four transfer functions are involved: Y1(s)/U1(s), Y2(s)/U1(s), Y1(s)/U2(s), and Y2(s)/U2(s). (When considering input u1, we assume that input u2 is zero, and vice versa.) Obtain the transfer matrix (consisting of the preceding four transfer functions of the system. (a) Obtain the transfer matrix G(s) by hand. (b) Verify your answers by using ss2tf in MATLAB. (c) Explicityly write out the four individual transfer functions.

For the torsianl system shown, obtain the transfer function G(s).

The mechanical system shown is a simple model of a rack-pinion mechanism. The rack rolls without slipping horizontally on the pulley, triggering the pulley's rotational motion. Derive the equation of motion for the system using the rack's displacement x as the variable, and then find the natural frequency of the system.

Find the inverse Laplace transform for the given function F(s)=2e^-s/(s+3)^2-e^-3s/(s^2+4)

For the given function, F(s)=2(s+2)/(s(s+1)(s+3)), (a) find f(0) by the initial value theorem, (b) find f(infinity) by the final value theorem.

For the first order mechanical system shown in the figure in the textbook, obtain the time constant of the system by using the following system parameters k1=10 N/m, k2=5 N/m, b1=2 Ns/m. The input and output displacements, xi and xo, are measured from the equilibrium positions.

Consider the liquid-level system shown in the figure. At steady state, the inflow rate and outflow rate are both Q and the heads of tanks 1, 2, and 3 are Hbar1, Hbar2, and Hbar3 respectively, where Hbar1=Hbar2. At t=0, the inflow rate is changed from Qbar to Qbar+qi. Assuming that h1, h2, and h3 are small changes, obtain the transfer function Qo(s)/Qi(s). (a) Set up the dynamic equations for the three-tank system, but don't derive the transfer function. (b) Obtain the state space representation of the system. (Define the matrices A, B, C, and D.) Use the following state variables: x1=h1, x2=h2, x3=h3, with input qi and output qo.

Solve z^6=i. That is finding all six sixth roots of the complex number i.

Linearize the following function z=2xy^2-x^3y about the point (xo,yo)=(1,1).

Obtain the Laplace transform of the function defined by f(t)=0 for t less than 0 and f(t)=cos(2wt)cos(3wt) for t greater than 0.

Find the Laplace transform of f(t) defined in the figure.

For the given function, F(s)=2(s+2)/(s(s+1)(s+3)), (a) find f(0) by the initial value theorem, (b) find f(infinity) by the final value theorem.

For the first order mechanical system shown in the figure in the textbook, obtain the time constant of the system by using the following system parameters k1=10 N/m, k2=5 N/m, b1=2 Ns/m. The input and output displacements, xi and xo, are measured from the equilibrium positions.

Consider the liquid-level system shown in the figure. At steady state, the inflow rate and outflow rate are both Q and the heads of tanks 1, 2, and 3 are Hbar1, Hbar2, and Hbar3 respectively, where Hbar1=Hbar2. At t=0, the inflow rate is changed from Qbar to Qbar+qi. Assuming that h1, h2, and h3 are small changes, obtain the transfer function Qo(s)/Qi(s). (a) Set up the dynamic equations for the three-tank system, but don't derive the transfer function. (b) Obtain the state space representation of the system. (Define the matrices A, B, C, and D.) Use the following state variables: x1=h1, x2=h2, x3=h3, with input qi and output qo.

Solve z^6=i. That is finding all six sixth roots of the complex number i.

Linearize the following function z=2xy^2-x^3y about the point (xo,yo)=(1,1).

Obtain the Laplace transform of the function defined by f(t)=0 for t less than 0 and f(t)=cos(2wt)cos(3wt) for t greater than 0.

Find the Laplace transform of f(t) defined in the figure.

For the given complex function F(z)=(z-7-10i)/(z(z+1-2i)), determine its magnitude and angle at z=2-2i. Use MATLAB to check the results.

Find the inverse Laplace transform for the given function F(s)=(s^2+3s-3)/(s+4)^3 .

For the given second order ordinary differential equation, xdoubledot+4xdot+40x=f(t), x(0)=0.1, xdot(0)=-0.1, (a) Obtain the solution x(t) by hand using Laplace transform, (b) Graph the solution x(t) only for the first 8 seconds using MATLAB.

Derive the equations of motion for the torsion system shown, where T is the external torque applied on the J1 shaft. Then express the equations of motion in matrix form.

Consider the mechanical system shown in the figure. The system is initially at rest. The displacements x1 and x2 are measured from their respective equilibrium positions before the input u is applied. Assume that b1=1Ns/m, b2=10Ns/m, k1=4N/m, and k2=20N/m. Obtain the displacement x2(t) when u is a step force input of 2 N. Derive the transfer function and then plot the response for a ramp input u(t)=2t N. a) Plot the response for 5s using lsim command. b) Plot the reponse for 5s using impulse command. c) Plot the response for 5s using step command.

Obtain a state-space representation of the mechanical system shown in the figure, where u1 and u2 are the inputs and y1 and y2 are the outputs. The displacements y1 and y2 are measured from their respective equilibrium positions.

For the given function F(s), use residue in MATLAB to obtain the partial fraction expansion and then find its inverse Laplace transform f(t) by hand based on the output.

Compute z for the following cases and express the answers in both rectangular form and polar (or Euler) form. Note: Show your detailed step-by-step work.

Find the inverse Laplace transform for the given function F(s)=(s^2+3s-3)/(s+4)^3 .

For the given second order ordinary differential equation, xdoubledot+4xdot+40x=f(t), x(0)=0.1, xdot(0)=-0.1, (a) Obtain the solution x(t) by hand using Laplace transform, (b) Graph the solution x(t) only for the first 8 seconds using MATLAB.

Derive the equations of motion for the torsion system shown, where T is the external torque applied on the J1 shaft. Then express the equations of motion in matrix form.

Consider the mechanical system shown in the figure. The system is initially at rest. The displacements x1 and x2 are measured from their respective equilibrium positions before the input u is applied. Assume that b1=1Ns/m, b2=10Ns/m, k1=4N/m, and k2=20N/m. Obtain the displacement x2(t) when u is a step force input of 2 N. Derive the transfer function and then plot the response for a ramp input u(t)=2t N. a) Plot the response for 5s using lsim command. b) Plot the reponse for 5s using impulse command. c) Plot the response for 5s using step command.

Obtain a state-space representation of the mechanical system shown in the figure, where u1 and u2 are the inputs and y1 and y2 are the outputs. The displacements y1 and y2 are measured from their respective equilibrium positions.

For the given function F(s), use residue in MATLAB to obtain the partial fraction expansion and then find its inverse Laplace transform f(t) by hand based on the output.

Compute z for the following cases and express the answers in both rectangular form and polar (or Euler) form. Note: Show your detailed step-by-step work.

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