System Dynamics 3rd Edition solutions

System Dynamics 3
Determine the equivalent resistance \(R_{e}\) of the circuit shown in Figure P6.1, such that \(v_{s}=R_{e} i\). All the resistors are identical and have the resistance R.

System Dynamics 3
Determine the voltage \(v_{1}\) in terms of the supply voltage \(v_{s}\) for the circuit shown in Figure P6.2.

System Dynamics 3
The Wheatstone bridge, like that shown in Figure P6.3, is used for various measurements. For example, a strain gage sensor utilizes the fact that the resistance of wire changes when deformed. If the sensor is one resistance leg of the bridge, then the deformation can be determined from the voltage \(

System Dynamics 3
The power supply of the circuit shown in Figure P6.4 supplies a voltage of 9 V. Compute the current i and the power P that must be supplied.

System Dynamics 3
Obtain the model of the voltage \(v_{1}\), given the current \(i_{s}\), for the circuit shown in Figure P6.5.

System Dynamics 3
(a) Obtain the model of the voltage \(v_{o}\), given the supply voltage \(v_{s}\), for the circuit shown in Figure P6.6. (b) Suppose \(v_{s}(t)=V u_{s}(t)\). Obtain the expressions for the free and forced responses for \(v_{o}(t)\).

System Dynamics 3
(a) Obtain the model of the voltage \(v_{o}\), given the supply voltage \(v_{s}\), for the circuit shown in Figure P6.7. (b) Suppose \(v_{s}(t)=V u_{s}(t)\). Obtain the expressions for the free and forced responses for \(v_{o}(t)\).

System Dynamics 3
(a) Obtain the model of the voltage \(v_{o}\), given the supply voltage \(v_{s}\), for the circuit shown in Figure P6.8. (b) Suppose \(v_{s}(t)=V u_{s}(t)\). Obtain the expressions for the free and forced responses for \(v_{o}(t)\).

System Dynamics 3
(a) The circuit shown in Figure P6.9 is a model of a solenoid, such as that used to engage the gear of a cars starter motor to the engines flywheel. The solenoid is constructed by winding wire around an iron core to make an electromagnet. The resistance R is that of the wire, and the inductance L is

System Dynamics 3
The resistance of a telegraph line is \(R=10 \ \Omega\), and the solenoid inductance is L = 5 H. Assume that when sending a dash, a voltage of 12 V is applied while the key is closed for 0.3 s. Obtain the expression for the current i(t) passing through the solenoid. (See Figure 6.2.15)

System Dynamics 3
Obtain the model of the voltage \(v_{o}\), given the supply voltage \(v_{s}\), for the circuit shown in Figure P6.11.

System Dynamics 3
Obtain the model of the voltage \(v_{o}\), given the supply voltage \(v_{s}\), for the circuit shown in Figure P6.12.

System Dynamics 3
Obtain the model of the current i, given the supply voltage \(v_{s}\), for the circuit shown in Figure P6.13.

System Dynamics 3
Obtain the model of the voltage \(v_{o}\), given the supply current \(i_{s}\), for the circuit shown in Figure P6.14.

System Dynamics 3
Obtain the model of the currents \(i_{1}, i_{2}\), and \(i_{3}\), given the input voltages \(v_{1}\) and \(v_{2}\), for the circuit shown in Figure P6.15.

System Dynamics 3
Obtain the model of the currents \(i_{1}, i_{2}\), and the voltage \(v_{3}\), given the input voltages \(v_{1}\) and \(v_{2}\), for the circuit shown in Figure P6.16.

System Dynamics 3
For the circuit shown in Figure P6.14, determine a suitable set of state variables, and obtain the state equations.

System Dynamics 3
For the circuit shown in Figure P6.15, determine a suitable set of state variables, and obtain the state equations.

System Dynamics 3
For the circuit shown in Figure P6.16, determine a suitable set of state variables, and obtain the state equations.

System Dynamics 3
Use the impedance method to obtain the transfer function \(V_{o}(s) / V_{s}(s)\) for the circuit shown in Figure P6.20.

System Dynamics 3
Use the impedance method to obtain the transfer function \(I(s) / V_{s}(s)\) for the circuit shown in Figure P6.21.

System Dynamics 3
Use the impedance method to obtain the transfer function \(V_{o}(s) / V_{s}(s)\) for the circuit shown in Figure P6.22.

System Dynamics 3
Use the impedance method to obtain the transfer function \(V_{o}(s) / I_{s}(s)\) for the circuit shown in Figure P6.23.

System Dynamics 3
Use the impedance method to obtain the transfer function \(V_{o}(s) / V_{s}(s)\) for the circuit shown in Figure P6.24.

System Dynamics 3
Use the impedance method to obtain the transfer function \(V_{o}(s) / V_{s}(s)\) for the circuit shown in Figure P6.25.

System Dynamics 3
Use the impedance method to obtain the transfer function \(V_{o}(s) / V_{s}(s)\) for the circuit shown in Figure P6.26.

System Dynamics 3
Draw a block diagram of the circuit shown in Figure P6.15. The inputs are \(v_{1}\) and \(v_{2}\). The output is \(i_{2}\).

System Dynamics 3
Draw a block diagram of the circuit shown in Figure P6.16. The inputs are \(v_{1}\) and \(v_{2}\). The output is \(v_{3}\).

System Dynamics 3
Obtain the transfer function \(V_{o}(s) / V_{i}(s)\) for the op-amp system shown in Figure P6.29.

System Dynamics 3
Obtain the transfer function \(V_{o}(s) / V_{i}(s)\) for the op-amp system shown in Figure P6.30.

System Dynamics 3
Obtain the transfer function \(V_{o}(s) / V_{i}(s)\) for the op-amp system shown in Figure P6.31.

System Dynamics 3
Obtain the transfer function \(V_{o}(s) / V_{i}(s)\) for the op-amp system shown in Figure P6.32.

System Dynamics 3
Obtain the transfer function \(V_{o}(s) / V_{i}(s)\) for the op-amp system shown in Figure P6.33.

System Dynamics 3
(a) Obtain the transfer function \(\Theta(s) / V_{i}(s)\) for the DArsonval meter. (b) Use the final value theorem to obtain the expression for the steady-state value of the angle \(\theta\) if the applied voltage \(v_{i}\) is a step function.

System Dynamics 3
(a) Obtain the transfer function \(\Omega(s) / T_{L}(s)\) for the field-controlled motor of Example 6.5.2. (b) Modify the field-controlled motor model in Example 6.5.2 so that the output is the angular displacement \(\theta\), rather than the speed \(\omega\), where \(\omega=\dot{\theta}\). Obta

System Dynamics 3
Modify the motor model given in Example 6.5.2 to account for a gear pair between the motor shaft and the load. The ratio of motor speed to load speed \(\omega_{L}\) is N. The motor inertia is \(I_{m}\) and the motor damping is \(c_{m}\). The load inertia is \(I_{L}\) and the load damping is \(c_{L}\)

System Dynamics 3
The derivation of the field-controlled motor model in Section 6.5 neglected the elasticity of the motor-load shaft. Figure P6.37 shows a model that includes this elasticity, denoted by its equivalent torsional spring constant \(k_{T}\). The motor inertia is \(I_{1}\), and the load inertia is \(I_{2}\

System Dynamics 3
Figure P6.38 is the circuit diagram of a speed-control system in which the dc motor voltage \(v_{a}\) is supplied by a generator driven by an engine. This system has been used on locomotives whose diesel engine operates most efficiently at one speed. The efficiency of the electric motor is not as sen

System Dynamics 3
The parameter values for a certain armature-controlled motor are \(\begin{array}{c} K_{T}=K_{b}=0.2 \mathrm{~N} \cdot \mathrm{m} / \mathrm{A} \\ c=5 \times 10^{-4} \mathrm{~N} \cdot \mathrm{m} \cdot \mathrm{s} / \mathrm{rad} \quad R_{a}=0.8 \ \Omega \end{array}\) The manufacturers data states

System Dynamics 3
The parameter values for a certain armature-controlled motor are \(\begin{array}{c} K_{T}=K_{b}=0.05 \mathrm{~N} \cdot \mathrm{m} / \mathrm{A} \\ R_{a}=0.8 \ \Omega \\ L_{a}=3 \times 10^{-3} \mathrm{H} \quad I=8 \times 10^{-5} \mathrm{~kg} \cdot \mathrm{m}^{2} \end{array}\) where I includes t

System Dynamics 3
The parameter values for a certain armature-controlled motor are \(\begin{array}{c} K_{T}=K_{b}=0.2 \mathrm{~N} \cdot \mathrm{m} / \mathrm{A} \\ c=5 \times 10^{-4} \mathrm{~N} \cdot \mathrm{m} \cdot \mathrm{s} / \mathrm{rad} \quad R_{a}=0.8 \ \Omega \\ L_{a}=4 \times 10^{-3} \mathrm{H} \quad I=5

System Dynamics 3
The following measurements were performed on a permanent magnet motor when the applied voltage was \(v_{a}=20 \mathrm{~V}\). The measured stall current was 25 A. The no-load speed was 2400 rpm and the no-load current was 0.6 A. Estimate the values of \(K_{b}, K_{T}, R_{a}\), and c.

System Dynamics 3
A single link of a robot arm is shown in Figure P6.43. The arm mass is m and its center of mass is located a distance L from the joint, which is driven by a motor torque \(T_{m}\) through spur gears. Suppose that the equivalent inertia felt at the motor shaft is \(0.215 \mathrm{~kg} \cdot \mathrm{m}^

System Dynamics 3
A conveyor drive system to produce translation of the load is shown in Figure P6.44. Suppose that the equivalent inertia felt at the motor shaft is \(0.05 \mathrm{~kg} \cdot \mathrm{m}^{2}\), and that the effect of Coulomb friction in the system produces an opposing torque of \(3.6 \mathrm{~N} \cdot

System Dynamics 3
Consider the accelerometer model in Section 6.7. Its transfer function can be expressed as \(\frac{Y(s)}{Z(s)}=-\frac{s^{2}}{s^{2}+(c / m) s+k / m}\) Suppose that the input displacement is z(t) = 10 sin 120t mm. Consider two cases, in SI units: (a) k/m = 100 and c/m = 18 and (b) \(k

System Dynamics 3
An electromagnetic microphone has a construction similar to that of the speaker shown in Figure 6.7.2, except that there is no applied voltage and the sound waves are incoming rather than outgoing. They exert a force \(f_{s}\) on the diaphragm whose mass is m, damping is c, and stiffness is k. Develo

System Dynamics 3
Consider the speaker model developed in Example 6.7.1. The model, whose transfer function is given by equation (3) in that example, is third order and therefore we cannot obtain a useful expression for the characteristic roots. Sometimes inductance L and damping c are small enough to be ignored. If L

System Dynamics 3
The parameter values for a certain armature-controlled motor are \(\begin{aligned} K_{T} & =K_{b}=0.2 \mathrm{~N} \cdot \mathrm{m} / \mathrm{A} \\ c & =5 \times 10^{-4} \mathrm{~N} \cdot \mathrm{m} \cdot \mathrm{s} / \mathrm{rad} \quad R_{a}=0.8 \ \Omega \\ L_{a} & =4 \times 10^{-3} \ \mathrm{H}

System Dynamics 3
Consider the motor whose parameters are given in Problem 6.48. Use MATLAB to obtain a plot of the response of \(i_{a}(t)\) and \(\omega(t)\) if the applied voltage is the modified step \(v_{a}(t)=10\left(1-e^{-100 t}\right) \mathrm{V}\). Determine the peak value of \(i_{a}(t)\).

System Dynamics 3
Consider the circuit shown in Figure P6.50. The parameter values are \(R=10^{3} \ \Omega, C=2 \times 10^{-6} \mathrm{~F}\), and \(L=2 \times 10^{-3} \ \mathrm{H}\). The voltage \(v_{1}\) is a step input of magnitude 5 V, and the voltage \(v_{2}\) is sinusoidal with frequency of 60 Hz and an amplitude

System Dynamics 3
The parameter values for a certain armature-controlled motor are \(\begin{aligned} K_{T} & =K_{b}=0.2 \mathrm{~N} \cdot \mathrm{m} / \mathrm{A} \\ c & =3 \times 10^{-4} \mathrm{~N} \cdot \mathrm{m} \cdot \mathrm{s} / \mathrm{rad} \quad R_{a}=0.8 \ \Omega \\ L_{a} & =4 \times 10^{-3} \ \mathrm{H}

System Dynamics 3
The parameter values for a certain armature-controlled motor are \(\begin{aligned} K_{T} & =K_{b}=0.05 \mathrm{~N} \cdot \mathrm{m} / \mathrm{A} \\ c & =0 \quad R_{a}=0.8 \Omega \\ L_{a} & =3 \times 10^{-3} \mathrm{H} \quad I=8 \times 10^{-5} \mathrm{~kg} \cdot \mathrm{m}^{2} \end{aligned}\)

System Dynamics 3
A single link of a robot arm is shown in Figure P6.43. The arm mass is m and its center of mass is located a distance L from the joint, which is driven by a motor torque \(T_{m}\) through spur gears. Suppose that the equivalent inertia felt at the motor shaft is \(0.215 \mathrm{~kg} \cdot \mathrm{m}^

System Dynamics 3
Consider the circuit shown in Figure P6.50. The parameter values are \(R=10^{4} \ \Omega, C=2 \times 10^{-6} \mathrm{~F}\), and \(L=2 \times 10^{-3} \ \mathrm{H}\). The voltage \(v_{1}\) is a single pulse of magnitude 5 V and duration 0.05 s, and the voltage \(v_{2}\) is sinusoidal with frequency of

System Dynamics 3
Consider the circuit shown in Figure P6.55. The parameter values are \(R=2 \times 10^{4} \ \Omega\) and \(C=3 \times 10^{-6} \mathrm{~F}\). The voltage \(v_{s}\) is \(v_{s}(t)=12 u_{s}(t)+3 \sin 120 \pi t \mathrm{~V}\). The initial conditions are zero. Use Simulink to obtain a plot of the responses \

System Dynamics 3
The parameter values for a certain armature-controlled motor are \(\begin{aligned} K_{T} & =K_{b}=0.2 \mathrm{~N} \cdot \mathrm{m} / \mathrm{A} \\ c & =5 \times 10^{-4} \mathrm{~N} \cdot \mathrm{m} \cdot \mathrm{s} / \mathrm{rad} \quad R_{a}=0.8 \ \Omega \\ L_{a} & =4 \times 10^{-3} \ \mathrm{H}

System Dynamics 3
The parameter values for a certain armature-controlled motor are \(\begin{aligned} K_{T} & =K_{b}=0.05 \mathrm{~N} \cdot \mathrm{m} / \mathrm{A} \\ c & =0 \quad R_{a}=0.8 \Omega \\ L_{a} & =3 \times 10^{-3} \mathrm{H} \quad I=8 \times 10^{-5} \mathrm{~kg} \cdot \mathrm{m}^{2} \end{aligned}\)