Van der Pols equation is a nonlinear model for some

Obtain the transfer function X(s)/F(s) from the block diagram shown in Figure P5.1.

Obtain the transfer function X(s)/F(s) from the block diagram shown in Figure P5.2.

Obtain the transfer function X(s)/F(s) from the block diagram shown in Figure P5.3.

Draw a block diagram for the following equation. The output is X(s); the inputs are F(s) and G(s). \(5 \ddot{x}+3 \dot{x}+7 x=10 f(t)-4 g(t)\)

Draw a block diagram for the following model. The output is X(s); the inputs are F(s) and G(s). Indicate the location of Y (s) on the diagram. \(\dot{x}=y-5 x+g(t) \quad \dot{y}=10 f(t)-30 x\)

Referring to Figure P5.6, derive the expressions for the variables C(s), E(s), and M(s) in terms of R(s) and D(s).

Referring to Figure P5.7, derive the expressions for the variables C(s), E(s), and M(s) in terms of R(s) and D(s).

Use the MATLAB series and feedback functions to obtain the transfer functions X(s)/F(s) and X(s)/G(s) for the block diagram shown in Figure P5.3.

Use the MATLAB series and feedback functions to obtain the transfer functions C(s)/R(s) and C(s)/D(s) for the block diagram shown in Figure P5.6.

Use the MATLAB series and feedback functions to obtain the transfer functions C(s)/R(s) and C(s)/D(s) for the block diagram shown in Figure P5.7

Obtain the state model for the reduced-form model \(5 \ddot{x}+7 \dot{x}+4 x=f(t)\).

Obtain the state model for the reduced-form model \(2 \frac{d^{3} y}{d t^{3}}+5 \frac{d^{2} y}{d t^{2}}+4 \frac{d y}{d t}+7 y=f(t)\)

Obtain the state model for the reduced-form model \(2 \ddot{x}+5 \dot{x}+4 x=4 y(t)\).

Obtain the state model for the transfer-function model \(\frac{Y(s)}{F(s)}=\frac{6}{3 s^{2}+6 s+10}\)

Obtain the state model for the two-mass system whose equations of motion are \(\begin{aligned} m_{1} \ddot{x}_{1}+k_{1}\left(x_{1}-x_{2}\right) & =f(t) \\ m_{2} \ddot{x}_{2}-k_{1}\left(x_{1}-x_{2}\right)+k_{2} x_{2} & =0 \end{aligned}\)

Obtain the state model for the two-mass system whose equations of motion for specific values of the spring and damping constants are \(10 \ddot{x}_{1}+8 \dot{x}_{1}-5 \dot{x}_{2}+40 x_{1}-25 x_{2}=0\) \(5 \ddot{x}_{2}-25 x_{1}+25 x_{2}-5 \dot{x}_{1}+5 \dot{x}_{2}=f(t)\)

Put the following model in standard state-variable form and obtain the expressions for the matrices A, B, C, and D. The output is x. \(2 \ddot{x}+5 \dot{x}+4 x=4 y(t)\)

Given the state-variable model \(\dot{x}_{1}=-5 x_{1}+3 x_{2}+2 u_{1}\) \(\dot{x}_{2}=-4 x_{2}+6 u_{2}\) and the output equations \(y_{1}=x_{1}+3 x_{2}+2 u_{1}\) \(y_{2}=x_{2}\) obtain the expressions for the matrices A, B, C, and D.

Given the following state-variable models, obtain the expressions for the matrices A, B, C, and D for the given inputs and outputs. a. The outputs are \(x_{1}\) and \(x_{2}\); the input is u. \(\dot{x}_{1}=-5 x_{1}+3 x_{2}\) \(\dot{x}_{2}=x_{1}-4 x_{2}+5 u\) b. The output is \(x_{1}\); the inputs are \(u_{1}\) and \(u_{2}\). \(\dot{x}_{1}=-5 x_{1}+3 x_{2}+4 u_{1}\) \(\dot{x}_{2}=x_{1}-4 x_{2}+5 u_{2}\)

Obtain the expressions for the matrices A, B, C, and D for the state-variable model you obtained in Problem 5.16. The outputs are \(x_{1}\) and \(x_{2}\).

The transfer function of a certain system is \(\frac{Y(s)}{F(s)}=\frac{6 s+7}{s+3}\) Use two methods to obtain a state-variable model in standard form. For each model, relate the initial value of the state-variable to the given initial value y(0).

The transfer function of a certain system is \(\frac{Y(s)}{F(s)}=\frac{s+2}{s^{2}+4 s+3}\) Use two methods to obtain a state-variable model in standard form. For each model, relate the initial values of the state variables to the given initial values y(0) and \(\dot{y}(0)\).

Use MATLAB to create a state-variable model; obtain the expressions for the matrices A, B, C, and D, and then find the transfer functions of the following models, for the given inputs and outputs. a. The outputs are \(x_{1}\) and \(x_{2}\); the input is u. \(\dot{x}_{1}=-5 x_{1}+3 x_{2}\) \(\dot{x}_{2}=x_{1}-4 x_{2}+5 u\) b. The output is \(x_{1}\); the inputs are \(u_{1}\) and \(u_{2}\). \(\dot{x}_{1}=-5 x_{1}+3 x_{2}+4 u_{1}\) \(\dot{x}_{2}=x_{1}-4 x_{2}+5 u_{2}\)

Use MATLAB to obtain a state model for the following equations; obtain the expressions for the matrices A, B, C, and D. In both cases, the input is f (t); the output is y. a. \(2 \frac{d^{3} y}{d t^{3}}+5 \frac{d^{2} y}{d t^{2}}+4 \frac{d y}{d t}+7 y=f(t)\) b. \(\frac{Y(s)}{F(s)}=\frac{6}{3 s^{2}+6 s+10}\)

Use MATLAB to obtain a state-variable model for the following transfer functions. a. \(\frac{Y(s)}{F(s)}=\frac{6 s+7}{s+3}\) b. \(\frac{Y(s)}{F(s)}=\frac{s+2}{s^{2}+4 s+3}\)

For the following model the output is \(x_{1}\) and the input is f (t). \(\dot{x}_{1}=-5 x_{1}+3 x_{2}\) \(\dot{x}_{2}=x_{1}-4 x_{2}+5 f(t)\) a. Use MATLAB to compute and plot the free response for \(x_{1}(0)=3\), and \(x_{2}(0)=5\). b. Use MATLAB to compute and plot the unit-step response for zero initial conditions. c. Use MATLAB to compute and plot the response for zero initial conditions with the input \(f(t)=3 \sin 10 \pi t\), for \(0 \leq t \leq 2\). d. Use MATLAB to compute and plot the total response using the initial conditions given in Part (a) and the forcing function given in part (c).

Given the state-variable model \(\dot{x}_{1}=-5 x_{1}+3 x_{2}+2 u_{1}\) \(\dot{x}_{2}=-4 x_{2}+6 u_{2}\) and the output equations \(y_{1}=x_{1}+3 x_{2}+2 u_{1}\) \(y_{2}=x_{2}\) Use MATLAB to find the characteristic polynomial and the characteristic roots.

The equations of motion for a two-mass, quarter-car model of a suspension system are \(m_{1} \ddot{x}_{1}=c_{1}\left(\dot{x}_{2}-\dot{x}_{1}\right)+k_{1}\left(x_{2}-x_{1}\right)\) \(m_{2} \ddot{x}_{2}=-c_{1}\left(\dot{x}_{2}-\dot{x}_{1}\right)-k_{1}\left(x_{2}-x_{1}\right)+k_{2}\left(y-x_{2}\right)\) Suppose the coefficient values are: \(m_{1}=240 \mathrm{~kg}, m_{2}=36 \mathrm{~kg}, k_{1}=1.6 \times 10^{4} \mathrm{~N} / \mathrm{m}, k_{2}=1.6 \times 10^{5} \mathrm{~N} / \mathrm{m}, c_{1}=98 \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}\). a. Use MATLAB to create a state model. The input is y(t); the outputs are \(x_{1}\) and \(x_{2}\). b. Use MATLAB to compute and plot the response of \(x_{1}\) and \(x_{2}\) if the input y(t) is a unit impulse and the initial conditions are zero. c. Use MATLAB to find the characteristic polynomial and the characteristic roots. d. Use MATLAB to obtain the transfer functions \(X_{1}(s) / Y(s)\) and \(X_{2}(s) / Y(s)\).

A representation of a car’s suspension suitable for modeling the bounce and pitch motions is shown in Figure P5.29, which is a side view of the vehicle’s body showing the front and rear suspensions. Assume that the car’s motion is constrained to a vertical translation x of the mass center and rotation \(\theta\) about a single axis which is perpendicular to the page. The body’s mass is m and its moment of inertia about the mass center is \(I_{G}\). As usual, x and \(\theta\) are the displacements from the equilibrium position corresponding to \(y_{1}=y_{2}=0\). The displacements \(y_{1}(t)\) and \(y_{2}(t)\) can be found knowing the vehicle’s speed and the road surface profile. a. Assume that x and \(\theta\) are small, and derive the equations of motion for the bounce motion x and pitch motion \(\theta\). b. For the values \(k_{1}=1100 \ \mathrm{lb} / \mathrm{ft}, k_{2}=1525 \ \mathrm{lb} / \mathrm{ft}, c_{1}=c_{2}=4 \ \mathrm{lb}-\mathrm{sec} / \mathrm{ft}\), \(L_{1}=4.8 \ \mathrm{ft}, L_{2}=3.6 \ \mathrm{ft}\), m = 50 slugs, and \(I_{G}=1000 \text { slug-ft }{ }^{2}\), use MATLAB to obtain a state-variable model in standard form. c. Use MATLAB to obtain and plot the solution for x(t) and \(\theta(t)\) when \(y_{1}=0\) and \(y_{2}\) is a unit impulse. The initial conditions are zero.

a. Use a MATLAB ode function to solve the following equation for \(0 \leq t \leq 12\). Plot the solution. \(\dot{y}=\cos t \quad y(0)=6\) b. Use the closed-form solution to check the accuracy of the numerical method.

a. Use a MATLAB ode function to solve the following equation for \(0 \leq t \leq 1\). Plot the solution. \(\dot{y}=5 e^{-4 t} \quad y(0)=2\) b. Use the closed-form solution to check the accuracy of the numerical method

a. Use a MATLAB ode function to solve the following equation for \(0 \leq t \leq 1\). Plot the solution. \(\dot{y}+3 y=5 e^{4 t} \quad y(0)=10\) b. Use the closed-form solution to check the accuracy of the numerical method.

a. Use a MATLAB ode function to solve the following nonlinear equation for \(0 \leq t \leq 4\). Plot the solution. \(\dot{y}+\sin y=0 \quad y(0)=0.1\) b. For small angles, sin \(y \approx y\). Use this fact to obtain a linear equation that approximates equation (1). Use the closed-form solution of this linear equation to check the output of your program.

Sometimes it is tedious to obtain a solution of a linear equation, especially if all we need is a plot of the solution. In such cases, a numerical method might be preferred. Use a MATLAB ode function to solve the following equation for \(0 \leq t \leq 7\). Plot the solution. \(\dot{y}+2 y=f(t) \quad y(0)=2\) Where \(f(t)=\left\{\begin{array}{ll} 3 t & \text { for } 0 \leq t \leq 2 \\ 6 & \text { for } 2 \leq t \leq 5 \\ -3(t-5)+6 & \text { for } 5 \leq t \leq 7 \end{array}\right.\)

A certain jet-powered ground vehicle is subjected to a nonlinear drag force. Its equation of motion, in British units, is \(50 \dot{v}=f-\left(20 v+0.05 v^{2}\right)\) Use a numerical method to solve for and plot the vehicle’s speed as a function of time if the jet’s force is constant at 8000 lb and the vehicle starts from rest.

The following model describes a mass supported by a nonlinear, hardening spring. The units are SI. Use \(g=9.81 \mathrm{~m} / \mathrm{s}^{2}\). \(5 \ddot{y}=5 g-\left(900 y+1700 y^{3}\right)\) Suppose that \(\dot{y}(0)=0\). Use a numerical method to solve for and plot the solution for two different initial conditions: (1) y(0) = 0.06 and (2) y(0) = 0.1.

Van der Pol’s equation is a nonlinear model for some oscillatory processes. It is \(\ddot{y}-b\left(1-y^{2}\right) \dot{y}+y=0\) Use a numerical method to solve for and plot the solution for the following cases: 1. \(b=0.1, y(0)=\dot{y}(0)=1,0 \leq t \leq 25\) 2. \(b=0.1, y(0)=\dot{y}(0)=3,0 \leq t \leq 25\) 3. \(b=3, y(0)=\dot{y}(0)=1,0<t<25\)

Van der Pol’s equation is \(\ddot{y}-b\left(1-y^{2}\right) \dot{y}+y=0\) This equation can be difficult to solve for large values of the parameter b. Use b = 1000 and \(0 \leq t \leq 3000\), with the initial conditions y(0) = 2 and \(\dot{y}(0)=0\). Use ode45 to plot the response.

The equation of motion for a pendulum whose base is accelerating horizontally with an acceleration a(t) is \(L \ddot{\theta}+g \sin \theta=a(t) \cos \theta\) Suppose that \(g=9.81 \mathrm{~m} / \mathrm{s}^{2}\), L = 1 m, and \(\dot{\theta}(0)=0\). Solve for and plot \(\theta(t)\) for \(0 \leq t \leq 10 \mathrm{~s}\) for the following three cases. a. The acceleration is constant: \(a=5 \mathrm{~m} / \mathrm{s}^{2}\), and \(\theta(0)=0.5 \ \mathrm{rad}\). b. The acceleration is constant: \(a=5 \mathrm{~m} / \mathrm{s}^{2}\), and \(\theta(0)=3 \ \mathrm{rad}\). c. The acceleration is linear with time: \(a=0.5 t \mathrm{~m} / \mathrm{s}^{2}\), and \(\theta(0)=3 \ \mathrm{rad}\).

Suppose the spring in Figure P5.40 is nonlinear and is described by the cubic force-displacement relation. The equation of motion is \(m \ddot{x}=c(\dot{y}-\dot{x})+k_{1}(y-x)+k_{2}(y-x)^{3}\) where m = 100, c = 600, \(k_{1}=8000\), and \(k_{2}=24000\). Approximate the unit-step input y(t) with \(y(t)=1-e^{-t / \tau}\), where \(\tau\) is chosen to be small compared to the period and time constant of the model when the cubic term is neglected. Use MATLAB to plot the forced response x(t).

Create a Simulink model to plot the solution of the following equation for \(0 \leq t \leq 6\). \(10 \ddot{y}=7 \sin 4 t+5 \cos 3 t \quad y(0)=4 \quad \dot{y}(0)=1\)

A projectile is launched with a velocity of 100 m/s at an angle of \(30^{\circ}\) above the horizontal. Create a Simulink model to solve the projectile’s equations of motion, where x and y are the horizontal and vertical displacements of the projectile. \(\ddot{x}=0 \quad x(0)=0 \quad \dot{x}(0)=100 \cos 30^{\circ}\) \(\ddot{y}=-g \quad y(0)=0 \quad \dot{y}(0)=100 \sin 30^{\circ}\) Use the model to plot the projectile’s trajectory y versus x for \(0 \leq t \leq 10 \mathrm{~s}\).

In Chapter 2 we obtained an approximate solution of the following problem, which has no analytical solution even though it is linear. \(\dot{x}+x=\tan t \quad x(0)=0\) The approximate solution, which is less accurate for large values of t, is \(x(t)=\frac{1}{3} t^{3}-t^{2}+3 t-3+3 e^{-t}\) Create a Simulink model to solve this problem and compare its solution with the approximate solution over the range \(0 \leq t \leq 1\).

Construct a Simulink model to plot the solution of the following equation for \(0 \leq t \leq 10\). \(15 \dot{x}+5 x=4 u_{s}(t)-4 u_{s}(t-2) \quad x(0)=2\)

Use Simulink to solve Problem 5.18 for zero initial conditions, \(u_{1}\) a unit-step input, and \(u_{2}=0\).

Use Simulink to solve Problem 5.18 for the initial conditions \(x_{1}(0)=4\), \(x_{2}(0)=3\), and \(u_{1}=u_{2}=0\).

Use Simulink to solve Problem 5.19a for zero initial conditions and u = 3 sin 2t.

Use Simulink to solve Problem 5.26c.

Use the Transfer Function block to construct a Simulink model to plot the solution of the following equation for \(0 \leq t \leq 4\). \(2 \ddot{x}+12 \dot{x}+10 x^{2}=5 u_{s}(t)-5 u_{s}(t-2) \quad x(0)=\dot{x}(0)=0\)

Construct a Simulink model to plot the solution of the following equation for \(0 \leq t \leq 4\). \(2 \ddot{x}+12 \dot{x}+10 x^{2}=5 \sin 0.8 t \quad x(0)=\dot{x}(0)=0\)

Use the Saturation block to create a Simulink model to plot the solution of the following equation for \(0 \leq t \leq 6\). \(3 \dot{y}+y=f(t) \quad y(0)=2\) where \(f(t)=\left\{\begin{array}{ll} 8 & \text { if } 10 \sin 3 t>8 \\ -8 & \text { if } 10 \sin 3 t<-8 \\ 10 \sin 3 t & \text { otherwise } \end{array}\right.\)

Construct a Simulink model of the following problem. \(5 \dot{x}+\sin x=f(t) \quad x(0)=0\) The forcing function is \(f(t)=\left\{\begin{array}{ll} -5 & \text { if } g(t) \leq-5 \\ g(t) & \text { if }-5 \leq g(t) \leq 5 \\ 5 & \text { if } g(t) \geq 5 \end{array}\right.\) where g(t) = 10 sin 4t.

Create a Simulink model to plot the solution of the following equation for \(0 \leq t \leq 3\). \(\dot{x}+10 x^{2}=2 \sin 4 t \quad x(0)=1\)

Construct a Simulink model of the following problem. \(10 \dot{x}+\sin x=f(t) \quad x(0)=0\) The forcing function is f (t) = sin 2t. The system has the dead-zone nonlinearity shown in Figure 5.6.6.

The following model describes a mass supported by a nonlinear, hardening spring. The units are SI. Use \(g=9.81 \mathrm{~m} / \mathrm{s}^{2}\). \(5 \ddot{y}=5 g-\left(900 y+1700 y^{3}\right) \quad y(0)=0.5 \quad \dot{y}(0)=0\). Create a Simulink model to plot the solution for \(0 \leq t \leq 2\).

Consider the system for lifting a mast, discussed in Chapter 3 and shown again in Figure P5.56. The 70-ft-long mast weighs 500 lb. The winch applies a force f = 380 lb to the cable. The mast is supported initially at an angle of \(\theta=30^{\circ}\), and the cable at A is initially horizontal. The equation of motion of the mast is \(25,400 \ddot{\theta}=-17,500 \cos \theta+\frac{626,000}{Q} \sin (1.33+\theta)\) where \(Q=\sqrt{2020+1650 \cos (1.33+\theta)}\) Create and run a Simulink model to solve for and plot \(\theta(t)\) for \(\theta(t) \leq \pi / 2 \ \mathrm{rad}\).

A certain mass, m = 2 kg, moves on a surface inclined at an angle \(\phi=30^{\circ}\) above the horizontal. Its initial velocity is v(0) = 3 m/s up the incline. An external force of \(f_{1}=5 \mathrm{~N}\) acts on it parallel to and up the incline. The coefficient of dynamic friction is \(\mu_{d}=0.5\). Use the Coulomb Friction block or the Sign block and create a Simulink model to solve for the velocity of the mass until the mass comes to rest. Use the model to determine the time at which the mass comes to rest.

If a mass-spring system has Coulomb friction on the horizontal surface rather than viscous friction, its equation of motion is \(m \ddot{y}=-k y+f(t)-\mu m g \quad \text { if } \dot{y} \geq 0\) \(m \ddot{y}=-k y+f(t)+\mu m g \quad \text { if } \dot{y}<0\) where \(\mu\) is the coefficient of friction. Develop a Simulink model for the case where m = 1 kg, k = 5 N/m, \(\mu=0.4\), and \(g=9.8 \mathrm{~m} / \mathrm{s}^{2}\). Run the simulation for two cases: (a) the applied force f (t) is a step function with a magnitude of 10 N, and (b) the applied force is sinusoidal: f (t) = 10 sin 2.5t.

Redo the Simulink suspension model developed in subsection 5.6.2, using the spring relation and input function shown in Figure P5.59, and the following damper relation. \(f_{d}(v)=\left\{\begin{array}{cl} -500|v|^{1.2} & v \leq 0 \\ 50 v^{1.2} & v>0 \end{array}\right.\) Use the simulation to plot the response. Evaluate the overshoot and undershoot.

Consider the system shown in Figure P5.60. The equations of motion are \(\begin{aligned} m_{1} \ddot{x}_{1}+\left(c_{1}+c_{2}\right) \dot{x}_{1}+\left(k_{1}+k_{2}\right) x_{1}-c_{2} \dot{x}_{2}-k_{2} x_{2} & =0 \\ m_{2} \ddot{x}_{2}+c_{2} \dot{x}_{2}+k_{2} x_{2}-c_{2} \dot{x}_{1}-k_{2} x_{1} & =f(t) \end{aligned}\) Suppose that \(m_{1}=m_{2}=1, c_{1}=3, c_{2}=1, k_{1}=1\), and \(k_{2}=4\). a. Develop a Simulink simulation of this system. In doing this, consider whether to use a state-variable representation or a transfer function representation of the model. b. Use the Simulink program to plot the response \(x_{1}(t)\) for the following input. The initial conditions are zero. \(f(t)=\left\{\begin{array}{ll} t & 0 \leq t \leq 1 \\ 2-t & 1<t<2 \\ 0 & t \geq 2 \end{array}\right.\)