# Get questions and answers for Classical Dynamics Of Particles

## GET Classical Dynamics Of Particles TEXTBOOK SOLUTIONS

1 Million+ Step-by-step solutions*Q:Refer to Figure 7.10.1. Assume that the resistances obey theRefer to Figure 7.10.1. Assume that the resistances obey the linear relation, so that the mass flow q1 through the left-hand resistance is q1 = (p1 – p)/R1, with a similar linear relation for the right-hand resistance.*

a. Create a Simulink subsystem block for this element.

b. Use the subsystem block to create a Simulink model of the system discussed in Example 7.4.3 and shown in Figure 7.4.3a. Assume that the mass inflow rate qm1 is a step function.

c. Use the Simulink model to obtain plots of h1(t) and h2(t) for the following parameter values: A1 = 2 m2, A2 = 5 m2. R1 = 400 l/(m.s), R2 = 600 l/(m.s), ( = 1000 kg/m3, qmi = 50 kg/s, h1(0) = 1.5 m. and h2(0) = 0.5 m.

Q:Use Simulink to solve Problem 61(b). In Problem 61(b) Use MATLAB toUse Simulink to solve Problem 61(b).

In Problem 61(b)

Use MATLAB to solve the nonlinear equation and plot the water height as a function of time until h(t) is not quite zero.

Q:Use Simulink to solve Problem 63. In Problem 63Use Simulink to solve Problem 63.

In Problem 63

Q:Use Simulink to solve Problem 64. Plot h(t) for bothUse Simulink to solve Problem 64. Plot h(t) for both parts (a) and (b).

In Problem 64

V = 1/3 π (R/H)2 h3

Suppose that the cup’s dimensions are R = 1.5 in. and H = 4 in.

a. If the flow rate from the fountain into the cup is 2 in.3/sec, use MATLAB to determine how long it will 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-2t) in.3/sec, use MATLAB to determine how long will it take to fill the cup to the brim.

Q:Consider the mixing tank treated in Problem 7.6. Generalize theConsider the mixing tank treated in Problem 7.6. Generalize the model to the case where the tank’s volume is V m3. For quality control purposes, we want to adjust the output concentration so by adjusting the input concentration so How much volume should the tank have so that the change in so lags behind the change in si by no more than 20 s?

Q:Consider the liquid-level system shown in Figure 7.3.3. Suppose thatConsider the liquid-level system shown in Figure 7.3.3. Suppose that the height h is controlled by using a relay to switch the How rate qmi, between the values 0 and 50 kg/s. The How rate is switched on when the height is less than 4.5 m 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 l/(m.s),

( = 1000 kg/m3, and h(0) = 1m. Obtain a plot of h(t).

Q:Derive the expression for the fluid capacitance of the cylindricalDerive the expression for the fluid capacitance of the cylindrical tank shown in Figure.

Q:Derive the expression for the capacitance of the container shownDerive the expression for the capacitance of the container shown in Figure.

Q:A rocket sled has the following equation of motion: 6A rocket sled has the following equation of motion: 6 = 2700 – 24v. How long must the rocket fire before the sled travels 2000 m? The sled starts from rest.

Q:The immersed object shown in Figure 8.1.2e is steel andThe immersed object shown in Figure 8.1.2e is steel and has a mass of 100 kg and a specific heat of cp = 500 J/kg. oC. Assume the thermal resistance is R = 0.09°C. s/J. The initial temperature of the object is 20° when it is dropped into a large bath of temperature 80oC. Obtain the expression for the temperature T(t) of the object after it is dropped into the bath.

Q:Compare the responses of 2 + v = (t) +Compare the responses of 2 + v = (t) + g(t) and 2 + v = g(t) if g(t) = 10us(t) and v(0) = 5.

Q:Compare the responses of 5+ v = + gCompare the responses of 5+ v = + g and 5+ v = g if u(0) = 5 and g = 10 for – ( ≤ / ≤ ∞.

Q:Consider the following model: 6 + 3v = (t) +Consider the following model:

6 + 3v = (t) + g(t)

Where v(0) = 0.

a. Obtain the response v(t) if g(t) = us(t).

b. Obtain the response v(t) to the approximate step input g(t) = 1 – e-5t and compare with the results of part (a).

Q:Obtain the response of the model 2 + v =Obtain the response of the model 2 + v = f(t), where f(t) = 5t and v(0) = 0. Identify the transient and steady-state responses.

Q:Obtain the response of the model 9+ 3v = f(t),Obtain the response of the model 9+ 3v = f(t), where f(t) = 7t and v(0) = 0. Is steady-state response parallel to f(t)?

Q:The resistance of a telegraph line is R = 10The resistance of a telegraph line is R = 10 Ω, 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.

Q:Obtain the oscillation frequency and amplitude of the response ofObtain the oscillation frequency and amplitude of the response of the model 3 + 12x = 0 for

(a) x(0) = 5 and (0) = 0

(b) x(0) = 0 and (0) = 5.

Q:Suppose the input f(t) to the following model is aSuppose the input f(t) to the following model is a ramp function: f(t) = at. Assuming that the model is stable, for what values of a, m, c, and k will the steady-state response be parallel to the input? For this case, what is the steady-state difference between the input and the response?

m + c + kx = f(t)

Q:If applicable, compute ζ, τ, ωn and ωd for theIf applicable, compute ζ, τ, ωn and ωd for the following roots, and find the corresponding characteristic polynomial.

a. s = -2 ± 6j

b. s = 1 ± 5j

c. s = -10, -10

d. s = -10

Q:If applicable, compute ζ, τ, ωn and ωd for theIf applicable, compute ζ, τ, ωn and ωd for the dominant root in each of the following sets of characteristic roots.

a. s = -2, -3 ± j

b. s = -3, -2 ± 2j

Q:A certain fourth-order model has the roots s = -2 ±A certain fourth-order model has the roots

s = -2 ± 4j, -10 ± 7j

Identify the dominant roots and use them to estimate the system’s time constant, damping ratio, and oscillation frequency.

Q:Given the model (μ + 2) + (2μ +Given the model

(μ + 2) + (2μ + 5)x = 0

Find the values of the parameter x for which the system is

Stable

Neutrally stable

Unstable

For the stable case, for what values of // is the system

Under damped?

Over damped?

Q:The characteristic equation of the system shown in Figure 8.2.3The characteristic equation of the system shown in Figure 8.2.3 for m = 3 and k = 27 is 3s2 + cs + 27 = 0. Obtain the free response for the following values of damping: c = 0, 9, 18, and 22. Use the initial conditions x (0) = 1 and (0) = 0.

Q:The characteristic equation of a certain system is 4s2 +The characteristic equation of a certain system is 4s2 + 6ds + 25d2 = 0, where d is a constant,

(a) For what values of d is the system stable?

(b) Is there a value of d for which the free response will consist of decaying oscillations?

Q:For each of the following models, obtain the free responseFor each of the following models, obtain the free response and the time constant, if any.

a. I6 + 14x = 0, x(0) = 6

b. 2 + 5x = L5, x(0) = 3

c. 3 + 6x = 0, x(0) = -2

d. 7-5x = 0, x(6)=9

Q:The characteristic equation of a certain system is s2 +The characteristic equation of a certain system is s2 + 6bs + 5b – 10 = 0, where b is a constant,

(a) For what values of b is the system stable?

(b) Is there a value of b for which the free response will consist of decaying oscillations?

Q:A certain system has two coupled subsystems. One subsystem isA certain system has two coupled subsystems. One subsystem is a rotational system with the equation of motion:

50 dω/dt + 10ω = T(t)

Where T(t) is the torque applied by an electric motor, Figure 8.1.8. The second subsystem is a field-controlled motor. The model of the motor’s field current if in amperes is

0.001dif/dt +5if = v(t)

Where v(t) is the voltage applied to the motor. The motor torque constant is KT = 25 N.m/A. Obtain the damping ratio ζ, time constants, and undamped natural frequency ωn of the combined system.

Q:A certain armature-controlled dc motor has the characteristic equation LaIs2 +A certain armature-controlled dc motor has the characteristic equation

LaIs2 + (RaI + cLa)s + cRa + KbKT = 0

Using the following parameter values:

Kh = KT = 0.1 N- m/A

Ra = 0.6 Ω

I = 6 x 10-5 kg-m2

Ln = 4 x 10-3 H

Obtain the expressions for the damping ratio ζ and un damped natural frequency con in terms of the damping c. Assuming that ζ Q:Compute the maximum percent overshoot, the maximum overshoot, the peakCompute the maximum percent overshoot, the maximum overshoot, the peak time, the 100% rise time, the delay time, and the 2% settling time for the following model. The initial conditions are zero. Time is measured in seconds.

+ 4 + 8x = 2us(t)

Q:A certain system is described by the model + cA certain system is described by the model

+ c +4x = us(t)

Set the value of the damping constant c so that both of the following specifications are satisfied. Give priority to the overshoot specification. If both cannot be satisfied, state the reason. Time is measured in seconds.

Maximum percent overshoot should be as small as possible as and no greater than 20%.

100% rise time should be as small as possible and no greater than 3 s.

Q:A certain system is described by the model 9 + cA certain system is described by the model

9 + c + 4x = us(t)

Set the value of the damping constant c so that both of the following specifications are satisfied. Give priority to the overshoot specification. If both cannot be satisfied, state the reason. Time is measured in seconds.

1. Maximum percent overshoot should be as small as possible as and no greater than 20%.

2. 100% rise time should be as small as possible and no greater than 3 s.

Q:Derive the fact that the peak time is the sameDerive the fact that the peak time is the same for all characteristic roots having the same imaginary part.

Q:For the two systems shown in Figure 8.3.8, the displacementFor the two systems shown in Figure 8.3.8, the displacement y(t) is a given input function. Obtain the response for each system if y(t) = us(t) and m = 3, c = 18, and k = 10, with zero initial conditions.

Q:Suppose that the resistance in the circuit of Figure 8.4.Suppose that the resistance in the circuit of Figure 8.4. l(a) is 3 x l06 Ω. A voltage is applied to the circuit and then is suddenly removed at time t = 0. The measured voltage across the capacitor is given in the following table. Use the data to estimate the value of the capacitance C.

Time t (s)Voltage vc (V)

0……………………………12.0

2……………………………11.2

4……………………………10.5

6……………………………..9.8

8……………………………..9.2

10……………………………8.6

12……………………………8.0

14……………………………7.5

16……………………………7.0

18……………………………6.6

20……………………………6.2

Q:The temperature of liquid cooling in a cup at roomThe temperature of liquid cooling in a cup at room temperature (68oF) was measured at various times. The data are given next.

Time t (sec) Temperature T (°F)

0………………………………..178

500……………………………..150

1000…………………………….124

1500…………………………….110

2000……………………………..97

2500……………………………..90

3000……………………………..82

Develop a model of the liquid temperature as a function of time, and use it to estimate how long it will take the temperature to reach 135°F.

Q:For the model 2 + x = 10 f(t), a. IfFor the model 2 + x = 10 f(t),

a. If x(0) = 0 and f(t) is a unit step, what is the steady-state response xss?. How long does it take before 98% of the difference between x(0) and xss is eliminated?

b. Repeat part (a) with x(0) = 5.

c. Repeat part (a) with x(0) = 0 and f(t) = 20us,(f).

Q:Figure shows the response of a system to a stepFigure shows the response of a system to a step input of magnitude 1000 N. The equation of motion is

m + c + kx = f(t)

Estimate the values of m, c, and k.

Q:A mass-spring-damper system has a mass of 100 kg. ItsA mass-spring-damper system has a mass of 100 kg. Its free response amplitude decays such that the amplitude of the 30th cycle is 20% of the amplitude of the 1st cycle. It takes 60 s to complete 30 cycles. Estimate the damping constant c and the spring constant k.

Q:A. For the following model, find the steady- state responsea. For the following model, find the steady- state response and use the dominant -root approximation to find the dominant response (how long will it take to reach steady-state? does it oscillate?). The initial conditions are zero

d3x/dt3 +22 d2x/dt2 + 131 dx/dt + 110x = us(t)

Q:The following model has a dominant root of s =The following model has a dominant root of s = – 3 ± 5j as long as the parameter μ is no less than 3.

d3y/dt3 + (6 + μ) d2y/dt2 + (34 + 6μ) dy/dt + 34μy = us(t)

Investigate the accuracy of the estimate of the maximum overshoot, the peak time, the 100% rise time, and the 2% settling time based on the dominant root, for the following three cases: (a) μ = 30, (b) μ = 6, and (c) μ = 3. Discuss the effect of μ on these predictions.

Q:Estimate the maximum overshoot, the peak time, and the riseEstimate the maximum overshoot, the peak time, and the rise time of the unit-step response of the following model if f(t) = 5000us(t) and the initial conditions are zero.

d4y/dt4 + 26 d3y/dt3 + 269 d2y/dt2 + 1524 dx/dt + 4680y = f(t)

Its roots are

s = -3 ±6j, -10 ±2y

Q:What is the form of the unit step response ofWhat is the form of the unit step response of the following model? Find the steady-state response. How long does the response take to reach steady state?

2 d4y/dt4 + 52 d3y/dt3 + 6250 d2y/dt2 + 4108 dx/dt + 1.1202x104y = 5×104 f(t)

Q:Use a software package such as MATLAB to plot theUse a software package such as MATLAB to plot the step response of the following model for three cases: a = 0.2, a = 1, and a = 10. The step input has a magnitude of 2500.

d4y/dt4 +24 d3y/dt3 +225 d2y/dt2 + 900 dx/dt + 2500y = f+a df/dt

Compare the response to that predicted by the maximum overshoot, peak time, 100% rise time, and 2% settling time calculated from the dominant roots.

Q:Use MATLAB to find the maximum percent overshoot, peak time,Use MATLAB to find the maximum percent overshoot, peak time, 2% settling time, and 100% rise time for the following equation. The initial conditions are zero.

+ 4 + 8x = 2us(t)

Q:Use MATLAB to compare the maximum percent overshoot, peak time,Use MATLAB to compare the maximum percent overshoot, peak time, and 100% rise time of the following models where the input fit) is a unit step function. The initial conditions are zero.

a. 3 + 18 + I0x = 10f(t)

b. 3 + 18 + 10x = 10 f(t) + 10f(t)

Q:A. Use MATLAB to find the maximum percent overshoot, peaka. Use MATLAB to find the maximum percent overshoot, peak time, and 100% rise time for the following equation. The initial conditions are zero.

d3x/dt3 + 22 d2x/dt2 + 113 dx/dt +100x = us(t)

b. Use the dominant root pair to compute the maximum percent overshoot, peak time, and 100% rise time, and compare the results with those found in part (a).

Q:Obtain the steady-state response of each of the following models,Obtain the steady-state response of each of the following models, and estimate how long t will take the response to reach steady-state.

a. 6 + 5x = 20us,(t), x(0) = 0

b. 6 + 5x = 20u, (t), x(0) = 1

c. 13-6x = 18us, (t),x(0) = -2

Q:A. Use MATLAB to find the maximum percent overshoot, peaka. Use MATLAB to find the maximum percent overshoot, peak time, and 100% rise time for the following equation. The initial conditions are zero.

d4y/dt4 +26 d3y/dt3 +269 d2y/dt2 + 1524 dy/dt + 4680y = 5000us(t)

b. The characteristic roots are s = – 3 ± 6j -10 ± 2j. Use the dominant root pair to compute the maximum percent overshoot, peak time, and 100% rise time, and compare the results with those found in part (a).

Q:A. Use MATLAB to find the maximum percent overshoot, peaka. Use MATLAB to find the maximum percent overshoot, peak time, and

100% rise time for the following equation. The initial conditions are zero.

d4y/dt4 +14 d3y/dt3 + 127 d2y/dt2 + 426 dx/dt + 962y = 936us(t)

b. Use the dominant root pair to compute the overshoot, peak time, and 100% rise time, and compare the results with those found in part (a).

Q:The following equation has a polynomial input. 0. 125 + 0.75The following equation has a polynomial input.

0. 125 + 0.75 + x = y(t) =-27/800 t3 + 270/800 t2

Use Simulink to plot x(t) and y(t) on the same graph. The initial conditions are zero.

Q:The following model has a polynomial input. 1 = -3.x1 +The following model has a polynomial input.

1 = -3.×1 + 4×2

2 = -5×1 – x2 + f(t)

f(t) = – 5/3 t2 + 25/3 t

The initial conditions are x1(0) = 3 and x2(0) = 7. Use Simulink to plot x1(t). x2(t), and f(t) on the same graph.

Q:First create a Simulink model containing an LTI System blockFirst create a Simulink model containing an LTI System block to plot the unit-step response of the following equation for k = 4. The initial conditions are zero.

5d3x/dt3 +3 d2x/dt2 + 7dx/dt + kxy = us(t)

Then create a script file to run the Simulink model. Use the file to experiment with the value of k to find the largest possible value of k such that the system remains stable.

Q:Figure shows an engine valve driven by an overhead camshaft.Figure shows an engine valve driven by an overhead camshaft. The rocker arm pivots about the fixed point O. For particular values of the parameters shown, the valve displacement x(t) satisfies the following equation.

10-6 + 0.3x = 5Î¸(t)

Where Î¸(t) is determined by the cam shaft speed and the cam profile. A particular profile is

Î¸ (t) = 16 x 103(l04t4 – 200t3 + t2) 0 â¤ t â¤ 0.01

Use Simulink to plot x(t). The initial conditions are zero.

Q:Obtain the total response of the following models. a. 6 +Obtain the total response of the following models.

a. 6 + 5x = 20 us (t), x(0) = 0

b. 6 + 5x = 20us (t), x(0) = 1

c. 13- 6x = l8us (t), x(0) =-2

Q:A certain rotational system has the equation of motion 100 dω/dtA certain rotational system has the equation of motion

100 dω/dt +5ω = T(t)

Where T(t) is the torque applied by an electric motor, as shown in Figure 8.1.8. The model of the motor’s field current if in amperes is

0.002dif/dt + 4if = v(t)

Where v(t) is the voltage applied to the motor. The motor torque constant is KT = 15 N.m/A. Suppose the applied voltage is 12us(t)V. Determine the steady-state speed of the inertia and estimate the time required to reach that speed.

Q:The liquid-level system shown in Figure 8.1.2d has the parameterThe liquid-level system shown in Figure 8.1.2d has the parameter values A = 50 ft2 and R = 60 ft-1 sec-1. If the inflow rate is qv(t) = 10us (t) ft3/sec, and the initial height is 2 ft, how long will it take for the height to reach 15 ft?

Q:Use the following transfer functions to find the steady-state responseUse the following transfer functions to find the steady-state response yss(t) to the given input function f(t).

a. T(s) = Y(s)/F(s) = 25/(14s=18), f(t) = 15sim 1.5t

b. T(s) = Y(s)/F(s) = 15s/(3s+4), f(t) = 5sin2t

c. T(s) = Y(s)/F(s) = (s+50)/(s+150)

d. T(s) = Y(s)/F(s) (33s +100)/(200s +33), f(t) = 8sin 50t

Q:The model of a certain mass-spring-damper system is 10 + cThe model of a certain mass-spring-damper system is

10 + c + 20x = f(t)

How large must the damping constant c be so that the maximum steady-state amplitude of x is no greater than 3, if the input is f(t) = 11 sin cot, for an arbitrary value of ω?

Q:The model of a certain mass-spring-damper system is 13 + 2The model of a certain mass-spring-damper system is

13 + 2 + kx = 10 sin ωt

Determine the value of k required so that the maximum response occurs at ω = 4 rad/sec. Obtain the steady-state response at that frequency.

Q:Determine the resonant frequencies of the following models. a. T(s) =Determine the resonant frequencies of the following models.

a. T(s) = 7/s (s2 + 6s + 58)

b. T(s) = 7/((3s2 + I8s + 174)(2×2 + 8x + 58))

Q:For the circuit shown in Figure, L = 0.1 H,For the circuit shown in Figure, L = 0.1 H, C = 10-6 F, and R = 100 Î©. Obtain the transfer functions I3(s)/V1 (s) and I3(s)/ V2(s). Using asymptotic approximations, sketch the m curves for each transfer function and discuss how the circuit acts on each input voltage (does it act like a low-pass filter, a high-pass filter, or other?).

Q:(a) For the system shown in Figure, m = 1(a) For the system shown in Figure, m = 1 kg and k = 600 N/m. Derive the expression for the peak amplitude ratio Mr and resonant frequency Ïr, and discuss the effect of the damping c on Mr and on Ïr.

(b) Extend the derivation of the expressions for Mr and Ïr to the case where the values of m, c, and k are arbitrary.

Q:For the RLC circuit shown in Figure, C = 10-5For the RLC circuit shown in Figure, C = 10-5 F and L = 5 x 10-3 H. Consider two cases:

(a) R = 10 Î© and (b) R = 1000 Î©. Obtain the transfer function V0(s)/Vs(s) and the log magnitude plot for each case. Discuss how the value of R affects the filtering characteristics of the system.

Q:A model of a fluid clutch is shown in Figure.A model of a fluid clutch is shown in Figure. Using the values I1, = l2 = 0.02 kg.m2, c1 = 0.04 N. m.s/rad, and c2 = 0.02 N.m.s/rad, obtain the transfer function Î©2(s)/T1(s), and derive the expression for the steady-state speed Ï2(t) if the applied torque in N.m is given by

T1 (t) = 4 + 2 sin 1.5t + 0.9 sin 2t

Q:Determine the beat period and the vibration period of theDetermine the beat period and the vibration period of the model

3 + 75 = 7 sin 5.2t

Q:Resonance will produce large vibration amplitudes, which can lead toResonance will produce large vibration amplitudes, which can lead to system failure. For a system described by the model

+ 64 = 0.2 sin ωt

Where x is in feet, how long will it take before |x| exceeds 0.1 ft, if the forcing frequency ω is close to the natural frequency?

Q:The quarter-car weight of a certain vehicle is 625 lbThe quarter-car weight of a certain vehicle is 625 lb and the weight of the associated wheel and axle is 190 lb. The suspension stiffness is 8000 lb/ft and the tire stiffness is 10,000 lb/ft. If the amplitude of variation of the road surface is 0.25 ft with a period of 20 ft, determine the critical (resonant) speeds of this vehicle.

Q:Use asymptotic approximations to sketch the frequency response plots forUse asymptotic approximations to sketch the frequency response plots for the following transfer functions.

a. T(s) = 15/(6s+2)

b. T(s) = 9s/(8s+4)

c. T(s) = 6 (14s+7)/(10s+2)

Q:A certain factory contains a heavy rotating machine that causesA certain factory contains a heavy rotating machine that causes the factory floor to vibrate. We want to operate another piece of equipment nearby and we measure the amplitude of the floor’s motion at that point to be 0.01 m. The mass of the equipment is 1500 kg and its support has a stiffness of k = 2 x 104 N/m and a damping ratio of ζ = 0.04. Calculate the maximum force that will be transmitted to the equipment at resonance.

Q:An electronics module inside an aircraft must be mounted onAn electronics module inside an aircraft must be mounted on an elastic pad to protect it from vibration of the airframe. The largest amplitude vibration produced by the airframe’s motion has a frequency of 40 cycles per second. The module weighs 200 N, and its amplitude of motion is limited to 0.003 m because of space. Neglect damping and calculate the percent of the airframe’s motion transmitted to the module.

Q:An electronics module used to control a large crane mustAn electronics module used to control a large crane must be isolated from the crane’s motion. The module weighs 2 lb.

(a) Design an isolator so that no more than 10% of the crane’s motion amplitude is transmitted to the module. The crane’s vibration frequency is 3000 rpm.

(b) What percentage of the crane’s motion will be transmitted to the module if the crane’s frequency can be anywhere between 2500 and 3500 rpm?

Q:Design a vibrometer having a mass of 0.1 kg, toDesign a vibrometer having a mass of 0.1 kg, to measure displacements having a frequency near 200 Hz.

Q:A motor mounted on a beam vibrates too much when

a. Create a Simulink subsystem block for this element.

b. Use the subsystem block to create a Simulink model of the system discussed in Example 7.4.3 and shown in Figure 7.4.3a. Assume that the mass inflow rate qm1 is a step function.

c. Use the Simulink model to obtain plots of h1(t) and h2(t) for the following parameter values: A1 = 2 m2, A2 = 5 m2. R1 = 400 l/(m.s), R2 = 600 l/(m.s), ( = 1000 kg/m3, qmi = 50 kg/s, h1(0) = 1.5 m. and h2(0) = 0.5 m.

Q:Use Simulink to solve Problem 61(b). In Problem 61(b) Use MATLAB toUse Simulink to solve Problem 61(b).

In Problem 61(b)

Use MATLAB to solve the nonlinear equation and plot the water height as a function of time until h(t) is not quite zero.

Q:Use Simulink to solve Problem 63. In Problem 63Use Simulink to solve Problem 63.

In Problem 63

Q:Use Simulink to solve Problem 64. Plot h(t) for bothUse Simulink to solve Problem 64. Plot h(t) for both parts (a) and (b).

In Problem 64

V = 1/3 π (R/H)2 h3

Suppose that the cup’s dimensions are R = 1.5 in. and H = 4 in.

a. If the flow rate from the fountain into the cup is 2 in.3/sec, use MATLAB to determine how long it will 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-2t) in.3/sec, use MATLAB to determine how long will it take to fill the cup to the brim.

Q:Consider the mixing tank treated in Problem 7.6. Generalize theConsider the mixing tank treated in Problem 7.6. Generalize the model to the case where the tank’s volume is V m3. For quality control purposes, we want to adjust the output concentration so by adjusting the input concentration so How much volume should the tank have so that the change in so lags behind the change in si by no more than 20 s?

Q:Consider the liquid-level system shown in Figure 7.3.3. Suppose thatConsider the liquid-level system shown in Figure 7.3.3. Suppose that the height h is controlled by using a relay to switch the How rate qmi, between the values 0 and 50 kg/s. The How rate is switched on when the height is less than 4.5 m 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 l/(m.s),

( = 1000 kg/m3, and h(0) = 1m. Obtain a plot of h(t).

Q:Derive the expression for the fluid capacitance of the cylindricalDerive the expression for the fluid capacitance of the cylindrical tank shown in Figure.

Q:Derive the expression for the capacitance of the container shownDerive the expression for the capacitance of the container shown in Figure.

Q:A rocket sled has the following equation of motion: 6A rocket sled has the following equation of motion: 6 = 2700 – 24v. How long must the rocket fire before the sled travels 2000 m? The sled starts from rest.

Q:The immersed object shown in Figure 8.1.2e is steel andThe immersed object shown in Figure 8.1.2e is steel and has a mass of 100 kg and a specific heat of cp = 500 J/kg. oC. Assume the thermal resistance is R = 0.09°C. s/J. The initial temperature of the object is 20° when it is dropped into a large bath of temperature 80oC. Obtain the expression for the temperature T(t) of the object after it is dropped into the bath.

Q:Compare the responses of 2 + v = (t) +Compare the responses of 2 + v = (t) + g(t) and 2 + v = g(t) if g(t) = 10us(t) and v(0) = 5.

Q:Compare the responses of 5+ v = + gCompare the responses of 5+ v = + g and 5+ v = g if u(0) = 5 and g = 10 for – ( ≤ / ≤ ∞.

Q:Consider the following model: 6 + 3v = (t) +Consider the following model:

6 + 3v = (t) + g(t)

Where v(0) = 0.

a. Obtain the response v(t) if g(t) = us(t).

b. Obtain the response v(t) to the approximate step input g(t) = 1 – e-5t and compare with the results of part (a).

Q:Obtain the response of the model 2 + v =Obtain the response of the model 2 + v = f(t), where f(t) = 5t and v(0) = 0. Identify the transient and steady-state responses.

Q:Obtain the response of the model 9+ 3v = f(t),Obtain the response of the model 9+ 3v = f(t), where f(t) = 7t and v(0) = 0. Is steady-state response parallel to f(t)?

Q:The resistance of a telegraph line is R = 10The resistance of a telegraph line is R = 10 Ω, 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.

Q:Obtain the oscillation frequency and amplitude of the response ofObtain the oscillation frequency and amplitude of the response of the model 3 + 12x = 0 for

(a) x(0) = 5 and (0) = 0

(b) x(0) = 0 and (0) = 5.

Q:Suppose the input f(t) to the following model is aSuppose the input f(t) to the following model is a ramp function: f(t) = at. Assuming that the model is stable, for what values of a, m, c, and k will the steady-state response be parallel to the input? For this case, what is the steady-state difference between the input and the response?

m + c + kx = f(t)

Q:If applicable, compute ζ, τ, ωn and ωd for theIf applicable, compute ζ, τ, ωn and ωd for the following roots, and find the corresponding characteristic polynomial.

a. s = -2 ± 6j

b. s = 1 ± 5j

c. s = -10, -10

d. s = -10

Q:If applicable, compute ζ, τ, ωn and ωd for theIf applicable, compute ζ, τ, ωn and ωd for the dominant root in each of the following sets of characteristic roots.

a. s = -2, -3 ± j

b. s = -3, -2 ± 2j

Q:A certain fourth-order model has the roots s = -2 ±A certain fourth-order model has the roots

s = -2 ± 4j, -10 ± 7j

Identify the dominant roots and use them to estimate the system’s time constant, damping ratio, and oscillation frequency.

Q:Given the model (μ + 2) + (2μ +Given the model

(μ + 2) + (2μ + 5)x = 0

Find the values of the parameter x for which the system is

Stable

Neutrally stable

Unstable

For the stable case, for what values of // is the system

Under damped?

Over damped?

Q:The characteristic equation of the system shown in Figure 8.2.3The characteristic equation of the system shown in Figure 8.2.3 for m = 3 and k = 27 is 3s2 + cs + 27 = 0. Obtain the free response for the following values of damping: c = 0, 9, 18, and 22. Use the initial conditions x (0) = 1 and (0) = 0.

Q:The characteristic equation of a certain system is 4s2 +The characteristic equation of a certain system is 4s2 + 6ds + 25d2 = 0, where d is a constant,

(a) For what values of d is the system stable?

(b) Is there a value of d for which the free response will consist of decaying oscillations?

Q:For each of the following models, obtain the free responseFor each of the following models, obtain the free response and the time constant, if any.

a. I6 + 14x = 0, x(0) = 6

b. 2 + 5x = L5, x(0) = 3

c. 3 + 6x = 0, x(0) = -2

d. 7-5x = 0, x(6)=9

Q:The characteristic equation of a certain system is s2 +The characteristic equation of a certain system is s2 + 6bs + 5b – 10 = 0, where b is a constant,

(a) For what values of b is the system stable?

(b) Is there a value of b for which the free response will consist of decaying oscillations?

Q:A certain system has two coupled subsystems. One subsystem isA certain system has two coupled subsystems. One subsystem is a rotational system with the equation of motion:

50 dω/dt + 10ω = T(t)

Where T(t) is the torque applied by an electric motor, Figure 8.1.8. The second subsystem is a field-controlled motor. The model of the motor’s field current if in amperes is

0.001dif/dt +5if = v(t)

Where v(t) is the voltage applied to the motor. The motor torque constant is KT = 25 N.m/A. Obtain the damping ratio ζ, time constants, and undamped natural frequency ωn of the combined system.

Q:A certain armature-controlled dc motor has the characteristic equation LaIs2 +A certain armature-controlled dc motor has the characteristic equation

LaIs2 + (RaI + cLa)s + cRa + KbKT = 0

Using the following parameter values:

Kh = KT = 0.1 N- m/A

Ra = 0.6 Ω

I = 6 x 10-5 kg-m2

Ln = 4 x 10-3 H

Obtain the expressions for the damping ratio ζ and un damped natural frequency con in terms of the damping c. Assuming that ζ Q:Compute the maximum percent overshoot, the maximum overshoot, the peakCompute the maximum percent overshoot, the maximum overshoot, the peak time, the 100% rise time, the delay time, and the 2% settling time for the following model. The initial conditions are zero. Time is measured in seconds.

+ 4 + 8x = 2us(t)

Q:A certain system is described by the model + cA certain system is described by the model

+ c +4x = us(t)

Set the value of the damping constant c so that both of the following specifications are satisfied. Give priority to the overshoot specification. If both cannot be satisfied, state the reason. Time is measured in seconds.

Maximum percent overshoot should be as small as possible as and no greater than 20%.

100% rise time should be as small as possible and no greater than 3 s.

Q:A certain system is described by the model 9 + cA certain system is described by the model

9 + c + 4x = us(t)

Set the value of the damping constant c so that both of the following specifications are satisfied. Give priority to the overshoot specification. If both cannot be satisfied, state the reason. Time is measured in seconds.

1. Maximum percent overshoot should be as small as possible as and no greater than 20%.

2. 100% rise time should be as small as possible and no greater than 3 s.

Q:Derive the fact that the peak time is the sameDerive the fact that the peak time is the same for all characteristic roots having the same imaginary part.

Q:For the two systems shown in Figure 8.3.8, the displacementFor the two systems shown in Figure 8.3.8, the displacement y(t) is a given input function. Obtain the response for each system if y(t) = us(t) and m = 3, c = 18, and k = 10, with zero initial conditions.

Q:Suppose that the resistance in the circuit of Figure 8.4.Suppose that the resistance in the circuit of Figure 8.4. l(a) is 3 x l06 Ω. A voltage is applied to the circuit and then is suddenly removed at time t = 0. The measured voltage across the capacitor is given in the following table. Use the data to estimate the value of the capacitance C.

Time t (s)Voltage vc (V)

0……………………………12.0

2……………………………11.2

4……………………………10.5

6……………………………..9.8

8……………………………..9.2

10……………………………8.6

12……………………………8.0

14……………………………7.5

16……………………………7.0

18……………………………6.6

20……………………………6.2

Q:The temperature of liquid cooling in a cup at roomThe temperature of liquid cooling in a cup at room temperature (68oF) was measured at various times. The data are given next.

Time t (sec) Temperature T (°F)

0………………………………..178

500……………………………..150

1000…………………………….124

1500…………………………….110

2000……………………………..97

2500……………………………..90

3000……………………………..82

Develop a model of the liquid temperature as a function of time, and use it to estimate how long it will take the temperature to reach 135°F.

Q:For the model 2 + x = 10 f(t), a. IfFor the model 2 + x = 10 f(t),

a. If x(0) = 0 and f(t) is a unit step, what is the steady-state response xss?. How long does it take before 98% of the difference between x(0) and xss is eliminated?

b. Repeat part (a) with x(0) = 5.

c. Repeat part (a) with x(0) = 0 and f(t) = 20us,(f).

Q:Figure shows the response of a system to a stepFigure shows the response of a system to a step input of magnitude 1000 N. The equation of motion is

m + c + kx = f(t)

Estimate the values of m, c, and k.

Q:A mass-spring-damper system has a mass of 100 kg. ItsA mass-spring-damper system has a mass of 100 kg. Its free response amplitude decays such that the amplitude of the 30th cycle is 20% of the amplitude of the 1st cycle. It takes 60 s to complete 30 cycles. Estimate the damping constant c and the spring constant k.

Q:A. For the following model, find the steady- state responsea. For the following model, find the steady- state response and use the dominant -root approximation to find the dominant response (how long will it take to reach steady-state? does it oscillate?). The initial conditions are zero

d3x/dt3 +22 d2x/dt2 + 131 dx/dt + 110x = us(t)

Q:The following model has a dominant root of s =The following model has a dominant root of s = – 3 ± 5j as long as the parameter μ is no less than 3.

d3y/dt3 + (6 + μ) d2y/dt2 + (34 + 6μ) dy/dt + 34μy = us(t)

Investigate the accuracy of the estimate of the maximum overshoot, the peak time, the 100% rise time, and the 2% settling time based on the dominant root, for the following three cases: (a) μ = 30, (b) μ = 6, and (c) μ = 3. Discuss the effect of μ on these predictions.

Q:Estimate the maximum overshoot, the peak time, and the riseEstimate the maximum overshoot, the peak time, and the rise time of the unit-step response of the following model if f(t) = 5000us(t) and the initial conditions are zero.

d4y/dt4 + 26 d3y/dt3 + 269 d2y/dt2 + 1524 dx/dt + 4680y = f(t)

Its roots are

s = -3 ±6j, -10 ±2y

Q:What is the form of the unit step response ofWhat is the form of the unit step response of the following model? Find the steady-state response. How long does the response take to reach steady state?

2 d4y/dt4 + 52 d3y/dt3 + 6250 d2y/dt2 + 4108 dx/dt + 1.1202x104y = 5×104 f(t)

Q:Use a software package such as MATLAB to plot theUse a software package such as MATLAB to plot the step response of the following model for three cases: a = 0.2, a = 1, and a = 10. The step input has a magnitude of 2500.

d4y/dt4 +24 d3y/dt3 +225 d2y/dt2 + 900 dx/dt + 2500y = f+a df/dt

Compare the response to that predicted by the maximum overshoot, peak time, 100% rise time, and 2% settling time calculated from the dominant roots.

Q:Use MATLAB to find the maximum percent overshoot, peak time,Use MATLAB to find the maximum percent overshoot, peak time, 2% settling time, and 100% rise time for the following equation. The initial conditions are zero.

+ 4 + 8x = 2us(t)

Q:Use MATLAB to compare the maximum percent overshoot, peak time,Use MATLAB to compare the maximum percent overshoot, peak time, and 100% rise time of the following models where the input fit) is a unit step function. The initial conditions are zero.

a. 3 + 18 + I0x = 10f(t)

b. 3 + 18 + 10x = 10 f(t) + 10f(t)

Q:A. Use MATLAB to find the maximum percent overshoot, peaka. Use MATLAB to find the maximum percent overshoot, peak time, and 100% rise time for the following equation. The initial conditions are zero.

d3x/dt3 + 22 d2x/dt2 + 113 dx/dt +100x = us(t)

b. Use the dominant root pair to compute the maximum percent overshoot, peak time, and 100% rise time, and compare the results with those found in part (a).

Q:Obtain the steady-state response of each of the following models,Obtain the steady-state response of each of the following models, and estimate how long t will take the response to reach steady-state.

a. 6 + 5x = 20us,(t), x(0) = 0

b. 6 + 5x = 20u, (t), x(0) = 1

c. 13-6x = 18us, (t),x(0) = -2

Q:A. Use MATLAB to find the maximum percent overshoot, peaka. Use MATLAB to find the maximum percent overshoot, peak time, and 100% rise time for the following equation. The initial conditions are zero.

d4y/dt4 +26 d3y/dt3 +269 d2y/dt2 + 1524 dy/dt + 4680y = 5000us(t)

b. The characteristic roots are s = – 3 ± 6j -10 ± 2j. Use the dominant root pair to compute the maximum percent overshoot, peak time, and 100% rise time, and compare the results with those found in part (a).

Q:A. Use MATLAB to find the maximum percent overshoot, peaka. Use MATLAB to find the maximum percent overshoot, peak time, and

100% rise time for the following equation. The initial conditions are zero.

d4y/dt4 +14 d3y/dt3 + 127 d2y/dt2 + 426 dx/dt + 962y = 936us(t)

b. Use the dominant root pair to compute the overshoot, peak time, and 100% rise time, and compare the results with those found in part (a).

Q:The following equation has a polynomial input. 0. 125 + 0.75The following equation has a polynomial input.

0. 125 + 0.75 + x = y(t) =-27/800 t3 + 270/800 t2

Use Simulink to plot x(t) and y(t) on the same graph. The initial conditions are zero.

Q:The following model has a polynomial input. 1 = -3.x1 +The following model has a polynomial input.

1 = -3.×1 + 4×2

2 = -5×1 – x2 + f(t)

f(t) = – 5/3 t2 + 25/3 t

The initial conditions are x1(0) = 3 and x2(0) = 7. Use Simulink to plot x1(t). x2(t), and f(t) on the same graph.

Q:First create a Simulink model containing an LTI System blockFirst create a Simulink model containing an LTI System block to plot the unit-step response of the following equation for k = 4. The initial conditions are zero.

5d3x/dt3 +3 d2x/dt2 + 7dx/dt + kxy = us(t)

Then create a script file to run the Simulink model. Use the file to experiment with the value of k to find the largest possible value of k such that the system remains stable.

Q:Figure shows an engine valve driven by an overhead camshaft.Figure shows an engine valve driven by an overhead camshaft. The rocker arm pivots about the fixed point O. For particular values of the parameters shown, the valve displacement x(t) satisfies the following equation.

10-6 + 0.3x = 5Î¸(t)

Where Î¸(t) is determined by the cam shaft speed and the cam profile. A particular profile is

Î¸ (t) = 16 x 103(l04t4 – 200t3 + t2) 0 â¤ t â¤ 0.01

Use Simulink to plot x(t). The initial conditions are zero.

Q:Obtain the total response of the following models. a. 6 +Obtain the total response of the following models.

a. 6 + 5x = 20 us (t), x(0) = 0

b. 6 + 5x = 20us (t), x(0) = 1

c. 13- 6x = l8us (t), x(0) =-2

Q:A certain rotational system has the equation of motion 100 dω/dtA certain rotational system has the equation of motion

100 dω/dt +5ω = T(t)

Where T(t) is the torque applied by an electric motor, as shown in Figure 8.1.8. The model of the motor’s field current if in amperes is

0.002dif/dt + 4if = v(t)

Where v(t) is the voltage applied to the motor. The motor torque constant is KT = 15 N.m/A. Suppose the applied voltage is 12us(t)V. Determine the steady-state speed of the inertia and estimate the time required to reach that speed.

Q:The liquid-level system shown in Figure 8.1.2d has the parameterThe liquid-level system shown in Figure 8.1.2d has the parameter values A = 50 ft2 and R = 60 ft-1 sec-1. If the inflow rate is qv(t) = 10us (t) ft3/sec, and the initial height is 2 ft, how long will it take for the height to reach 15 ft?

Q:Use the following transfer functions to find the steady-state responseUse the following transfer functions to find the steady-state response yss(t) to the given input function f(t).

a. T(s) = Y(s)/F(s) = 25/(14s=18), f(t) = 15sim 1.5t

b. T(s) = Y(s)/F(s) = 15s/(3s+4), f(t) = 5sin2t

c. T(s) = Y(s)/F(s) = (s+50)/(s+150)

d. T(s) = Y(s)/F(s) (33s +100)/(200s +33), f(t) = 8sin 50t

Q:The model of a certain mass-spring-damper system is 10 + cThe model of a certain mass-spring-damper system is

10 + c + 20x = f(t)

How large must the damping constant c be so that the maximum steady-state amplitude of x is no greater than 3, if the input is f(t) = 11 sin cot, for an arbitrary value of ω?

Q:The model of a certain mass-spring-damper system is 13 + 2The model of a certain mass-spring-damper system is

13 + 2 + kx = 10 sin ωt

Determine the value of k required so that the maximum response occurs at ω = 4 rad/sec. Obtain the steady-state response at that frequency.

Q:Determine the resonant frequencies of the following models. a. T(s) =Determine the resonant frequencies of the following models.

a. T(s) = 7/s (s2 + 6s + 58)

b. T(s) = 7/((3s2 + I8s + 174)(2×2 + 8x + 58))

Q:For the circuit shown in Figure, L = 0.1 H,For the circuit shown in Figure, L = 0.1 H, C = 10-6 F, and R = 100 Î©. Obtain the transfer functions I3(s)/V1 (s) and I3(s)/ V2(s). Using asymptotic approximations, sketch the m curves for each transfer function and discuss how the circuit acts on each input voltage (does it act like a low-pass filter, a high-pass filter, or other?).

Q:(a) For the system shown in Figure, m = 1(a) For the system shown in Figure, m = 1 kg and k = 600 N/m. Derive the expression for the peak amplitude ratio Mr and resonant frequency Ïr, and discuss the effect of the damping c on Mr and on Ïr.

(b) Extend the derivation of the expressions for Mr and Ïr to the case where the values of m, c, and k are arbitrary.

Q:For the RLC circuit shown in Figure, C = 10-5For the RLC circuit shown in Figure, C = 10-5 F and L = 5 x 10-3 H. Consider two cases:

(a) R = 10 Î© and (b) R = 1000 Î©. Obtain the transfer function V0(s)/Vs(s) and the log magnitude plot for each case. Discuss how the value of R affects the filtering characteristics of the system.

Q:A model of a fluid clutch is shown in Figure.A model of a fluid clutch is shown in Figure. Using the values I1, = l2 = 0.02 kg.m2, c1 = 0.04 N. m.s/rad, and c2 = 0.02 N.m.s/rad, obtain the transfer function Î©2(s)/T1(s), and derive the expression for the steady-state speed Ï2(t) if the applied torque in N.m is given by

T1 (t) = 4 + 2 sin 1.5t + 0.9 sin 2t

Q:Determine the beat period and the vibration period of theDetermine the beat period and the vibration period of the model

3 + 75 = 7 sin 5.2t

Q:Resonance will produce large vibration amplitudes, which can lead toResonance will produce large vibration amplitudes, which can lead to system failure. For a system described by the model

+ 64 = 0.2 sin ωt

Where x is in feet, how long will it take before |x| exceeds 0.1 ft, if the forcing frequency ω is close to the natural frequency?

Q:The quarter-car weight of a certain vehicle is 625 lbThe quarter-car weight of a certain vehicle is 625 lb and the weight of the associated wheel and axle is 190 lb. The suspension stiffness is 8000 lb/ft and the tire stiffness is 10,000 lb/ft. If the amplitude of variation of the road surface is 0.25 ft with a period of 20 ft, determine the critical (resonant) speeds of this vehicle.

Q:Use asymptotic approximations to sketch the frequency response plots forUse asymptotic approximations to sketch the frequency response plots for the following transfer functions.

a. T(s) = 15/(6s+2)

b. T(s) = 9s/(8s+4)

c. T(s) = 6 (14s+7)/(10s+2)

Q:A certain factory contains a heavy rotating machine that causesA certain factory contains a heavy rotating machine that causes the factory floor to vibrate. We want to operate another piece of equipment nearby and we measure the amplitude of the floor’s motion at that point to be 0.01 m. The mass of the equipment is 1500 kg and its support has a stiffness of k = 2 x 104 N/m and a damping ratio of ζ = 0.04. Calculate the maximum force that will be transmitted to the equipment at resonance.

Q:An electronics module inside an aircraft must be mounted onAn electronics module inside an aircraft must be mounted on an elastic pad to protect it from vibration of the airframe. The largest amplitude vibration produced by the airframe’s motion has a frequency of 40 cycles per second. The module weighs 200 N, and its amplitude of motion is limited to 0.003 m because of space. Neglect damping and calculate the percent of the airframe’s motion transmitted to the module.

Q:An electronics module used to control a large crane mustAn electronics module used to control a large crane must be isolated from the crane’s motion. The module weighs 2 lb.

(a) Design an isolator so that no more than 10% of the crane’s motion amplitude is transmitted to the module. The crane’s vibration frequency is 3000 rpm.

(b) What percentage of the crane’s motion will be transmitted to the module if the crane’s frequency can be anywhere between 2500 and 3500 rpm?

Q:Design a vibrometer having a mass of 0.1 kg, toDesign a vibrometer having a mass of 0.1 kg, to measure displacements having a frequency near 200 Hz.

Q:A motor mounted on a beam vibrates too much when

A motor mounted on a beam vibrates too much when it runs at a speed of 6000 rpm. At that speed the measured force produced on the beam is 60 lb. Design a vibration absorber to attach to the beam. Because of space limitations, the absorber’s mass cannot have an amplitude of motion greater than 0.08 in.

Q:The supporting table of a radial saw weighs 160 lb.The supporting table of a radial saw weighs 160 lb. When the saw operates at 200 rpm, it transmits a force of 4 lb to the table. Design a vibration absorber to be attached underneath the table. The absorber’s mass cannot vibrate with amplitude greater than 1 in.Q:A certain mass is driven by base excitation through aA certain mass is driven by base excitation through a spring (see Figure). Its parameter values are m = 200 kg, c = 2000 N.s/m, and k = 2 x 104 N/m. Determine its resonant frequency Ïr, its resonance peak Mr, and its bandwidth.

Q:A certain series RLC circuit has the following transfer function. T(s)A certain series RLC circuit has the following transfer function.

T(s) = I(s)/V(s) = Cs/(LCs2 + RCs + 1)

Suppose that L = 300 H, R = 104 Ω, and C = 10-6 F. Find the bandwidth of this system.

Q:Obtain the expressions for the bandwidths of the two circuitsObtain the expressions for the bandwidths of the two circuits shown in Figure.

a.

b.

Q:Figure is a representation of the effects of the tideFigure is a representation of the effects of the tide on a small body of water connected to the ocean by a channel. Assume that the ocean height h, varies sinusoidally with a period of 12 hr with an amplitude of 3 ft about a mean height of 10 ft. If the observed amplitude of variation of h is 2 ft, determine the time constant of the system and the time lag between a peak in hi and a peak in .

Q:For the circuit shown in Figure 9.1.7a, can values beFor the circuit shown in Figure 9.1.7a, can values be found for Rl, R, and C to make a low-pass filter? Prove your answer mathematically.

Q:The voltage shown in Figure is produced by applying aThe voltage shown in Figure is produced by applying a sinusoidal voltage to a full wave rectifier. The Fourier series approximation to this function is

Suppose this voltage is applied to a series RC circuit whose transfer function is

Vo(s)/Vs(s) = 1/(RCs+1)

where R = 600Î© and C = 10-6 F. Keeping only those terms in the Fourier series whose frequencies lie within the circuit’s bandwidth, obtain the expression for the steady-state voltage vo(t).

Suppose this voltage is applied to a series RC circuit whose transfer function is

Vo(s)/Vs(s) = 1/(RCs + 1)

where R = 103Î© and C = 10-6 F. Keeping only those terms in the Fourier series whose frequencies lie within the circuit’s bandwidth, obtain the expression for the steady-state voltage vo(t).

For the values m = 1 kg, c = 98 N.s/m, and k = 4900 N/m, keeping only those terms in the Fourier series whose frequencies lie within the system’s bandwidth, obtain the expression for the steady-state displacement x(t).

a.

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