Chapter 11, Symbolic Processing with MATLAB Video Solutions, MATLAB for Engineering Applications | Numerade (2024)

The equation of circle of radius $r$ centered at $x=0, y=0$ is

$$
x^2+y^2=r^2
$$

Use the subs and other MATLAB functions to find the equation of a circle of radius $r$ centered at the point $x=a, y=b$. Rearrange the equation into the form $A x^2+B x+C x y+D y+E y^2=F$, and find the expressions for the coefficients in terms of $a, b$, and $r$.

The equation for a curve called the "lemniscate" in polar coordinates $(r, \theta)$ is

$$
r^2=a^2 \cos (2 \theta)
$$

Use MATLAB to find the equation for the curve in terms of Cartesian coordinates $(x, y)$, where $x=r \cos \theta$ and $y=r \sin \theta$.

The equation for an ellipse centered at the origin of the Cartesian coordinates $(x, y)$ is

$$
\frac{x^2}{a^2}+\frac{y^2}{b^2}=1
$$

where $a$ and $b$ are constants that determine the shape of the ellipse.
$a$. In terms of the parameter $b$, use MATLAB to find the points of intersection of the two ellipses described by

$$
x^2+\frac{y^2}{b^2}=1
$$

and

$$
\frac{x^2}{100}+4 y^2=1
$$

$b$. Evaluate the solution obtained in part (a) for the case: $b=2$.

AG

The equation

$$
r=\frac{p}{1-\epsilon \cos \theta}
$$

describes the polar coordinates of an orbit with the coordinate origin at the sun. If $c=0$, the orbit is circular, if $0<c<1$, the orbit is elliptical. The planets have orbits that are nearly circular; comets have orbits that are highly elongated with $\varepsilon$ nearer to 1 . It is of obvious interest to see whether or not a comet's or asteroid's orbit will intersect that of a planet. For each of the two cases below, use MATLAB to determine whether or not orbits A and B intersect, and if they do, determine the polar coordinates of the intersection point. The units of distance are AU , where 1 AU is the mean distance of the earth from the sun.
a. Orbit $\mathrm{A}: p=1, \varepsilon=0.02$. Orbit $\mathrm{B}: p=0.2, \varepsilon=0.8$.
b. Orbit A: $p=1, e=0.02$. Orbit $\mathrm{B}: p=1.5, \varepsilon=0.6$.

Figure $11.2-2$ in Section 11.2 shows a robot arm having two joints and two links. The angles of rotation of the motors at the joints are $\theta_1$ and $\theta_2$. From trigonometry we can derive the following expressions for the $(x, y)$ coordinates of the hand.

$$
\begin{aligned}
& x=L_1 \cos \theta_1+L_2 \cos \left(\theta_1+\theta_2\right) \\
& y=L_1 \sin \theta_1+L_2 \sin \left(\theta_1+\theta_2\right)
\end{aligned}
$$
Suppose that the link lengths are $L_1=3 \mathrm{ft}$ and $L_2=2 \mathrm{ft}$.
a. Compute the motor angles required to position the hand at $x=3 \mathrm{ft}$, $y=1 \mathrm{ft}$. Identify the elbow-up and elbow-down solutions.
b. Suppose you want to move the hand along a straight, horizontal line at $y=1$, for $2 \leq x \leq 4$. Plot the required motor angles versus $x$. Label the elbow-up and elbow-down solutions.

Use MATLAB to determine all the local minima and local maxima, and all the inflection points where $d y / d x=0$, of the following function.

$$
y=x^4-\frac{28}{3} x^3+9 x^2-5
$$

A certain spherical street lamp emits light in all directions. It is mounted on a pole of height $h$ (see Figure P18). The brightness $B$ at point $P$ on the sidewalk is directly proportional to $\sin \theta$, and inversely proportional to the square of the distance $d$ from the light to the point. Thus

$$
B=\frac{c}{d^2} \sin \theta
$$

where $c$ is a constant. Use MATLAB to determine how high $h$ should be to maximize the brightness at point $P$, which is 30 ft from the base of the pole.

A rocket's mass decreases as it burns fuel. The equation of motion for a rocket in vertical flight can be obtained from Newton's law, and is

$$
m(t) \frac{d v}{d t}=T-m(t) g
$$

where $T$ is the rocket's thrust and its mass as a function of time is given by $m(t)=m_0(1-r / b)$. The rocket's initial mass is $m_0$, the burn time is $b$, and $r$ is the fraction of the total mass accounted for by the fuel.

Use the values $T=48,000 \mathrm{~N}, m_0=2200 \mathrm{~kg}, r=0.8, g=9.81 \mathrm{~m} / \mathrm{s}^2$, and $b=40 \mathrm{~s}$.
a. Use MATLAB to compute the rocket's velocity as a function of time, for $t \leq b$.
b. Use MATLAB to compute the rocket's velocity at burnout.

The equation for the voltage $v(t)$ across a capacitor as a function of time is

$$
w(t)=\frac{1}{C}\left[\int_0^t i(O) d t+Q_0\right]
$$

where $i(I)$ is the applied current and $Q_0$ is the initial charge. Suppose that $C=10^{-7} \mathrm{~F}$ and that $Q_0=0$. If the applied current is $i(t)=0.3+0.1 e^{-5 t}$ $\sin (25 \pi t)$, use MATLAB to compute and plot the voltage $v(t)$ for $0 \leq t \leq$ 7 seconds.

The power $P$ dissipated as heat in a resistor $R$ as a function of the current $i(t)$ passing through it is $P=i^2 R$. The energy $E(t)$ lost as a function of time is the time integral of the power. Thus

$$
E(t)=\int_0^t P(t) d t=R \int_0^1 i^2(t) d t
$$
If the current is measured in amperes, the power is in watts and the energy is in joules ( 1 watt $=1$ joule/second). Suppose that a current $i(t)=$ $0.3[1+\sin (3 t)]$ amperes is applied to the resistor.
a. Determine the energy $E(t)$ dissipated as a function of time.
b. Determine the energy dissipated in one minute if $R=2000 \Omega$.

The shape of a cable hanging with no other load other than its own weight is a catcnary curve. A particular bridge cable is described by the catenary $y(x)=15 \cosh ((x-25) / 15)$ for $0 \leq x \leq 55$, where $x$ and $y$ are the horizontal and vertical coordinates measured in feet.

The length $L$ of a curve described by $y(x)$ for $a \leq x \leq b$ can be found from the integral

$$
L=\int_a^b \sqrt{1+\left(\frac{d y}{d x}\right)^2} d x
$$

Determine the length of the cable.

Integrals that cannot be evaluated in closed form sometimes can be evaluated approximately by using a series representation for the integrand. For example, the following integral is used for some probability calculations (see Chapter 7, Section 7.2).

$$
I=\int_0^1 e^{-x^2} d x
$$

a. Obtain the Taylor series for $e^{-x^7}$ about $x=0$, and integrate the first six nonzero terms in the series to find $I$. Use the seventh term to estimate the error.
b. Compare your answer with that obtained with the MATLAB erf( $t$ ) function, defined as

$$
\operatorname{erf}(t)=\frac{2}{\sqrt{\pi}} \int_0^t e^{-x^2} d x
$$

Find the expression for the sum of the geometric series:

$$
\sum_{k=0}^{n-1} r^k
$$

for $r \neq 1$.

The equation for the voltage $y$ across the capacitor of an $R C$ circuit is

$$
R C \frac{d y}{d t}+y=v(t)
$$

where $v(t)$ is the applied voltage. Suppose that $R C=0.3 \mathrm{~s}$ and that the capacitor voltage is initially 5 V . If the applied voltage goes from 0 to 12 V at $t=0$, use MATLAB to determine and plot the voltage $y(t)$ for $0 \leq t \leq 1$ second.

The following equation describes the temperature $T(t)$ of a certain object immersed in a liquid bath of temperature $T_b(t)$.

$$
10 \frac{d T}{d t}+T=T_b
$$

Suppose the object's temperature is initially $T(0)=70^{\circ} \mathrm{F}$ and the bath temperature is $170^{\circ} \mathrm{F}$. Use MATLAB to answer the following questions.
a. Determine $T(t)$.
b. How long will it take for the object's temperature $T$ to reach $168^{\circ} \mathrm{F}$ ?
c. Plot the object's temperature $T(t)$ as a function of time.

The following equation describes the motion of a mass connected to a spring, with viscous friction on the surface.

$$
m \overline{\mathrm{y}}+c \dot{\mathrm{y}}+k \mathrm{y}=f(t)
$$

where $f(t)$ is an applied force. The position and velocity of the mass at $t=0$ are denoted by $x_0$ and $v_0$. Use MATLAB to answer the following questions.
a. What is the free response in terms of $x_0$ and $v_0$ if $m=3, c=18, k=102$ ?
$b$. What is the free response in terms of $x_0$ and $v_0$ if $m=3, c=39$, and $k=120$ ?

The equation for the voltage $y$ across the capacitor of an RC circuit is

$$
R C \frac{d y}{d t}+y=v(t)
$$

where $v(t)$ is the applied voltage. Suppose that $R C=0.2 \mathrm{~s}$ and that the capacitor voltage is initially 2 s . If the applied voltage is $v(t)=10\left[2-e^{-t}\right.$ $\sin (5 \pi t)]$. use MATLAB to compute and plot the voltage $y(t)$ for $0 \leq$ $t \leq 5 \mathrm{~s}$.

The following equation describes a certain dilution process, where $y(t)$ is the concentration of salt in a tank of fresh water to which salt brine is being added.

$$
\frac{d y}{d t}+\frac{2}{10+2 t} y=10
$$

Suppose that $y(0)=0$. Use MATLAB to compute and plot $y(t)$ for $0 \leq t \leq 20$.

The following equation describes the motion of a certain mass connected to a spring, with viscous friction on the surface.

$$
3 y+18 y+102 y=f(t)
$$

where $f(t)$ is an applied force. Suppose that $f(t)=0$ for $t<0$ and $f(t)=10$ for $t \geq 0$.
a. Use MATLAB to compute and plot $y(t)$ when $y(0)=y(0)=0$.
b. Use MATLAB to compute and plot $y(t)$ when $y(0)=0$ and $y(0)=10$.

The following equation describes the motion of a certain mass connected to a spring, with viscous friction on the surface.

$$
3 y+39 y+120 y=f(t)
$$

where $f(t)$ is an applied force. Suppose that $f(t)=0$ for $t<0$ and $f(t)=10$ for $t \geq 0$.
a. Use MATLAB to compute and plot $y(t)$ when $y(0)=y(0)=0$.
$b$. Use MATLAB to compute and plot $y(t)$ when $y(0)=0$ and $y(0)=10$.

The equations for an armature-controlled dc motor are the following. The motor's current is $i$ and its rotational velocity is $\omega$,

$$
\begin{aligned}
& L \frac{d i}{d t}=-R i-K_e \omega+v(t) \\
& I \frac{d \omega}{d t}=K_T i-c \omega
\end{aligned}
$$
where $L, R$, and $I$ are the motor's inductance, resistance, and inertia, $K_T$ and $K_e$ are the torque constant and back emf constant, $c$ is a viscous damping constant, and $v(t)$ is the applied voltage.

Use the values $R=0.8 \Omega, L=0.003 \mathrm{H}, K_T=0.05 \mathrm{~N} \cdot \mathrm{m} / \mathrm{A}, K_{\mathrm{e}}=0.05$ $\mathrm{V} \cdot \mathrm{s} / \mathrm{rad}, c=0$, and $I=8 \times 10^{-5} \mathrm{~N} \cdot \mathrm{m} \cdot \mathrm{s}^2$.

Suppose the applied voltage is 20 V . Use MATLAB to compute and plot the motor's speed and current versus time. Choose a final time large enough to show the motor's speed becoming constant.

The RLC circuit described in Problem 23 and shown in Figure P23 has the following differential equation model.

$$
L C v_o+R C \dot{v}_o+v_o=R C v_1(t)
$$

Use the Laplace transform method to solve for the unit-step response of $v_o(t)$ for zero initial conditions, where $C=10 \times 10^{-5} \mathrm{~F}$ and $L=5 \times$ $10^{-3} \mathrm{H}$. For the first case (a broadband filter), $R=1000 \Omega$. For the second case (a narrowband filter), $R=10 \Omega$. Compare the step responses of the two cases.

The differential equation model for a certain speed control system for a vehicle is

$$
\ddot{v}+\left(1+K_p\right) \dot{v}+K_l v=K_p \dot{v}_d+K_l v_d
$$

where the actual speed is $v$, the desired speed is $v_d(t)$, and $K_p$ and $K_{/}$are constants called the "control gains." Use the Laplace transform method to find the unit-step response [that is, $v_d(t)$ is a unit-step function]. Use zero initial conditions. Compare the response for three cases:
a. $K_p=9, K_I=50$
b. $K_p=9, K_I=25$
c. $K_p=54, K_l=250$

The differential equation model for a certain position control system for a metal cutting tool is

$$
\begin{aligned}
\frac{d^3 x}{d t^3} & +\left(6+K_D\right) \frac{d^2 x}{d t^2}+\left(11+K_p\right) \frac{d x}{d t}+\left(6+K_b\right) x \\
& =K_D \frac{d^2 x_d}{d t^2}+K_p \frac{d x_d}{d t}+K_l x_d
\end{aligned}
$$

where the actual tool position is $x$, the desired position is $x_d(t)$, and $K_p$, $K_b$ and $K_D$ are constants called the "control gains." Use the Laplace

The differential equation model for the motor torque $m(t)$ required for a certain speed control system is

$$
4 \ddot{m}+4 K \ddot{m}+K^2 m=K^2 \dot{v}_d
$$

where the desired speed is $v_d(t)$, and $K$ is a constant called the "control gain."
a. Use the Laplace transform method to find the unit-step response (that is, $v_d(t)$ is a unit-step function). Use zero initial conditions.
$b$. Use symbolic manipulation in MATLAB to find the value of the gain $K$ that minimizes the peak torque that must be supplied by the motor. In addition, compute the value of the peak torque.

Find the characteristic polynomial and roots of the following matrix.

$$
\mathbf{A}=\left[\begin{array}{rr}
-6 & 2 \\
3 k & -7
\end{array}\right]
$$

Use the matrix inverse and the left-division method to solve the following set.

$$
\begin{aligned}
4 c x+5 y & =43 \\
3 x-4 y & =-22
\end{aligned}
$$

AG

The currents $i_1, i_2$, and $i_3$ in the circuit shown in Figure P50 are described by the following equation set if all the resistances are equal to $R$.

$$
\left[\begin{array}{rrr}
2 R & -R & 0 \\
-R & 3 R & -R \\
0 & R & -2 R
\end{array}\right]\left[\begin{array}{l}
i_1 \\
i_2 \\
i_3
\end{array}\right]=\left[\begin{array}{l}
v_1 \\
0 \\
v_2
\end{array}\right]
$$

where $v_1$ and $v_2$ are applied voltages. The other two currents can be found from $i_4=i_1-i_2$ and $i_5=i_2-i_3$.
a. Use both the matrix inverse method and the left-division method to solve for the currents in terms of the resistance $R$ and the voltages $v_1$ and $v_2$.
b. Find the numerical values for the currents if $R=1000 \Omega, v_1=100 \mathrm{~V}$, and $v_2=25 \mathrm{~V}$

The equations for the armature-controlled de motor shown in Figure P51 are the following. The motor's current is $i$ and its rotational velocity is $\omega$.

$$
\begin{aligned}
& L \frac{d i}{d t}=-R i-K_e \omega+v(t) \\
& I \frac{d \omega}{d t}=K_T i-c \omega
\end{aligned}
$$

where $L_{,} R$, and $/$ are the motor's inductance, resistance, and inertia, $K_T$ and $K_e$ are the torque constant and back emf constant, $c$ is a viscous damping constant, and $v(t)$ is the applied voltage.
a. Find the characteristic polynomial and the characteristic roots.
b. Use the values $R=0.8 \Omega, L=0.003 \mathrm{H}, K_T=0.05 \mathrm{~N} \cdot \mathrm{m} / \mathrm{A}, K_e=$ $0.05 \mathrm{~V} \cdot \mathrm{s} / \mathrm{rad}$, and $I=8 \times 10^{-5} \mathrm{~N} \cdot \mathrm{m} \cdot \mathrm{s}^2$. The damping constant $c$ is often difficult to determine with accuracy. For these values find the expressions for the two characteristic roots in terms of $c$.
c. Using the parameter values in part (b), determine the roots for the following values of $c$ (in newton meter second): $c=0, c=0.01$, $c=0.1$, and $c=0.2$. For each case, use the roots to estimate how long the motor's speed will take to become constant, and discuss whether or not the speed will oscillate before it becomes constant.