**Problems**

1. Draw a simulation diagram for the following equation.

**y=5f(t)-7y**

2. Draw a simulation diagram for the following equation.

**5y + 3y + 7y = f(t)**

3. Draw a simulation diagram for the following equation.

**3y+5siny=f(t)**

**Section 9.2**

4. Create a Simulink model to plot the solution of the following equation for 05t5~ .

**10y = 7 sin4t + 5 cos 3t y(O) =4· y(O) = 1**

5. Aprojectile is launched with a velocity of 100 m/s at an angle of 30° 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.

**x=O x(O) = 0**

**y(O) =0**

**x(O) = loocos30°**

**y= -g ~(O) = 100 sin 30**°

Use.the model to plot the projectile’s trajectory y versus x for

05t51Os.

6. The following equation has no analytical solation even though it is linear. x+x=tant x(O) = 0 The approximate solution, which is less accurate for large values of t

**, is x(t) = ~t3 – t2 + 3t – 3 + 3e-‘ **

Create a Simuhnk model to solve this problem and compare its solution with the app~ximate solution over the range 0 5 t 5 1.

7. Construct a Simulink model to plot the solution of the following equation

**forO ~ t ~ 10.**

**x(O) = 2**

where us(t) is a unit-step function (in the Block Parameters window of the .Step block, set the Step time to 0, the Initial value to 0, and the Final value to I).

8. A tank having vertical sides and a bottom area of 100ft2 is used to store water. To fill the tank, water is pumped into the top at the rate given in the following table. Use Simulink to solve for and plot the water height h(t)

for 0 ~ t .:5 10 min.

9. Construct a Simulink model to–plot the -&elution of the following equations

forO ~ t ~ 2.

X2 = 5xI – 7X2 + f(t)

where f(t) = 2t. Use the Ramp block in the Sources library.

10. Construct.a Simnlink model to plot the solution of the following equations

“f9I’O~t~J.

XI = -6xI +4X2+ fl(t)

X2 = 5xI -7X2 + h(t)

where fl (t) is a step function of height 3 starting at t = 0, and h(t) is a step function of height – 3 starting at t = 1. .

·0119

11. Ilse die hIaok to cmde a:BimuliDk 1IIadei to plot the solution of.tiIc:m~:e.gUl_ 0.:5 t 5″6.

3y +y = f(t) y(O)-= 2

12. Construct a Simulink model of the following problem. 5.i + sinx = f(t) x(O) = 0 The forcing function is

{

-5

f(t) = ~(t)

if g(t) s -5

if -5 :::g(t) ::: 5

if g(t) ~ 5

where g(t) = 10 sin 4t.

13. If a mass-spring system has Coulomb friction on the surface rather than viscous friction, its equation of motion is’

my = -ky + f(t) – J.Lmg

my = -ky + f(t) + umg

if Y ~ 0

if Y < 0

where j.L is the coefficient of friction. Develop a Simulink model for the case where m = 1 kg, k = 5 N/m, J.L=0.4, and g = 9.8 rnIs2. Run the simulation for two cases: (a) the applied force f(t) is a step function with a magnitude of ION and (b) the applied force is sinusoidal: f(t) = 10 sin 2.5t. Either the Sign block in the Math Operations library

or the Coulomb and Viscous Friction block in the Discontinuities library can be used, but since there is no viscous friction in this problem, the Sign block is easier to use.

14. A certain.mass, m = 2 kg, moves on a surface inclined at an angle <P = 30° above the horizontal. Its initial velocity is u(O) = 3 rnIs up the incline. An external force of f. = 5 N acts on it parallel to and up the

incline. The coefficient of Coulomb friction is J.L=0.5. Use tire 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. .

15. a. Develop a Simulink model of a thermostatic control system in which the temperature model is

dT

RC- + T = Rq + Ta(t)

dt .

where T is the room air temperature in OF,Ta is the ambient (outside) air temperature in OF,time t is measured in hours, q is the input from the heating system in lb-ft/hr, R is the thermal resistance, and C is the thermal capacitance. The thermostat switches q on at the value qrnax whenever the temperature drops below 69° and switches q to q = 0 whenever the temperature is above 710. The value of qmax indicates the heat output of the heating system. Run the simulation for the case where T(O) = 70° and Ta(t) = 50 + 10 sin(Jrt/12). Use the values R = 5 x 10-5 °F-hrn~ft and

C = 4 x 104 Ib-ftJ°F. Plot the temperatures T and Ta versus t on the same graph, for 0 ~ t ~ 24 hr. Do this for two cases: qmax = 4 x 105 and qmax = 8 x 105 lb-ft/hr, Investigate the effectiveness of each case. b. The integral of q over time is the energy used. Plot f q dt versus t and determine how much energy is used in 24 hr for the case where qmax = 8 X 105.

16. Refer to Problem 15. Use the simulation with q = 8 X 105 to compare the energy consumption and the thermostat cycling frequency for the two temperature bands (69°, 71°) and (68°, 72°).

17. Consider the liquid-level system shown in Figure 9.7-1. The governing equati0’1 based on conservation of mass is (9.7-2). Suppose that the height h is controlled by using a relay to switch the input flow rate between the values 0 and 50 kg/so The flow rate is switched on when the height is less than 4.5 b and is switched off when the height reaches 5.5 m. Create a Simulink model for this application using the values A = 2 m2, R = 400 N· s/(kg· m2), p = 1000 kg/m”, and h(O) = I m. Obtain a plot of h(t).

**Section 9.5 •**

18. Use the Transfer Function block to construct a Simulink model to plot the solution of the following equation for 0 ~ t ~ 4.

2i + 12x + lOx = 5us(t) – 5us(t – 2) x(O) = x(O) = 0

19. Use Transfer Function blocks to construct a Simulink model to plot the solution of the following equations for 0 ~ t ~ 2.

3i + 15x + 18x = f(t) x(O) = x(O) = 0

2y + 16y + 50y = x(t) )'(0) = y(O) = 0

where f(t) = 50us(t)·

20. Use Transfer Function blocks to construct a Simulink model to plot the solution of the following equations for 0 ~ t ~ 2.

3i + 15x + 18x = f(t)

2y + 16y + 50y = x(t)

x(O) = x(O) = 0

)'(0) = )'(0) =0

where f(t) = 50us(t). At the output of the first block there is a dead zone for -1 ~x ~ I. This limits the input to the second block.

21. Use Transfer Function blocks to construct a Simulink model to plot the solution of the following equations for 0 ~ t ~ 2.

3i + 15x + 18x = f(t}

2y + 16y + 50y = x(t)

x(O) =x(O) = 0

)'(0) = y(O) = 0

where f(t) = 50us(t). At the output of the first block there is a saturation that limits .r be Ix I I. This limits the input to the second block

**Section 9.6**

22. Construct a Simulink model to plot the solution of the following equation for 0 T 4. 2i + 12x + lOx2 = 5 sin 0.8t x(O) =x(O) = 0

23. Create a Simulink model to plot the solution of the following equation for 0 t .s 3. x + lOx2 = 2sin4t x(O) = 1

24. Construct a Simulink model of the following problem. lOx + sin x = l(t) x(O) = 0 The forcing function is I(t) = sin2t. The system has the dead-zone nonlinearity shown in Figure 9.5-1.

25. The following model describes a mass supported by a nonlinear, hardening spring. The units are SI. Use g = 9.81 rn/s”. 5y = 5g – (900y + l700i) y(O) = 0.5 )'(0) =0 Create a Simulink model to plot the solution for 0 T 2.

26. Consider the system for lifting a mast shown in Figure P26. The 70-ft long mast weighs 500 lb. The winch applies a force I = 380 lb to the cable. The mast is supported initially at an angle of 30°, and the cable at A is

initially horizontal. The equation of motion of the mast is .. 626,000 25,400(f = -17,500cose + Q sin(l.33 + e)

where Q = y’2020 + 1650cos(l.33 +e) Create and run a Simulink model to solve for and plot e(t) for e(t) ::::

Tf/2 rad.

27. The equation describing the water height h in a spherical tank with a drain at the bottom is Suppose the tank’s radius is r = 3 m and that the circular drain hole of area A has a radius of 2 ern. Assume that Cd = 0.5, and that the initial water height is h(O) = 5 m. Use g = 9.81 rn/s2. Use Simulink to solve the ‘nonlinear equation and plot the water height as a function of time until h(l) = O.

28. A cone-shaped paper drinking cup (like the kind used at water fountains) has a radius R and a height H. If the water height in the cup is h, the water volume is given by

1 (R)2 V = -71: – h3

3 H.

Suppose that the cup’s dimensions are R = 1.5 in. and H = 4 in. Q. If the flow rate from the fountain into the cup is 2 in. 3/sec, use Simulink to determine how long will it take to fill the cup to the brim. b. If the flow rate from the fountain into the cup is given by 2(1 – e-21) in.3/sec, use Simulink to determine how long will it take

to fill the cup to the brim.

**Section 9.7**

29. Refer to Figure 9.7-2. Assume that the resistances obey the linear relation, so that the mass flow ql through the left-hand resistance is ql = (PI – p)/ RI; with a similar linear relation for the right-hand resistance.

Q. Create a Simulink subsystem block for this element. b. Usb the subsystem block to create a Simulink model of the system shown in Figure 9.7-5. Assume that the mass inflow rate is a step function. c. Use the Simulink model to obtain plots of h I(t) and h2(1) for the following parameter values: A I= 2 rrr’, A2 = 5 m2, R I=

400 N· s/(kg· rrr’), R2 = 600 N . s/(kg . m2), p = 1000 kg/rrr’, qmi = 50 kg/s, hi (0) = 1.5 m, and h2(0) = 0.5 m.

30. Q. Use the subsystem block developed in Section 9.7 to construct a Simulink model of the system shown in Figure P30. The mass inflow rate is a step function.

b. Use the Simulink model to obtain plots of h1Jt) and h2(1) for the following parameter values: AI = 3 ft2, A2 = 5 ft2, RI = 30 lb-sec/islug-fr’), R2 =40 lb-sec/tslug-ft”), p = 1.94 slug/fr’, qmi = 0.5 slug/see, hi (0) = 2 ft, and h2(0) = 5 ft.

31. Consider Figure 9.7-7 for the case where there are three RC loops with the values RI = R3 = 104 n, R2 = 5 X 104 n, CI = C3 = 10-6 F, and C2 = 4 X 10-6 F.

Q. Develop a subsystem block for one RC loop.

b. Use the subsystem block to construct a Simulink model of the entire system of three loops. Plot V3(t) over 0 ~ t 3; for VI(t) = 12sin lOt V.

32. Consider Figure 9.7-8 for the case where there are three masses. Use the values m, = m3 = 10 kg, m2 = 30 kg, k, = k4 = 104 N/m, and

ki = ks = 2 x 104 N/m.

Q. Develop a subsystem block for one mass.

b. Use the subsystem block to construct a Simulink model of the entire system of three masses. Plot the displacements of the masses over . 0 t .2 s for if the initial displacement of m, is 0.1 m.

**Section 9.8**

33. Refer to Figure P30. Suppose there is a dead time of 10 see between the outflow of the top tank and the lower tank. Use the subsystem block developed in Section 9.7 to create a Simulink model of this system. Using

the parameters given in problem 30, plot the height h, and h2 versus time.

**Section 9.9**

34. Redo the Simulink suspension model developed in Section 9.9, using the spring relation and input function shown in Figure P34, and the following damper relation.

Use the simulation to plot the response. Evaluate the overshoot and undershoot.

• Consider the system shown in Figure P35. The equations of motion are

mixi + (CI +C2)XI + (kl +k2)xl – C2X2 – k2X2 = 0

m2x2 +C2X2 +k2X2 – C2XI – k2xI = f(l)

Suppose that ml = m2 = 1. CI = 3. C2 = 1. k, = 1. and k2 = 4.

Q. Develop a Simulink model 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 model to plot the response XI (I) for the following input. The initial conditions are zero.