Questions and Answers of Classical Dynamics Of Particles

Refer 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
Use 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.
Use Simulink to solve Problem 63.In Problem 63
Use 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
Consider 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
Consider 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
Derive the expression for the fluid capacitance of the cylindrical tank shown in Figure.
Derive the expression for the capacitance of the container shown in Figure.
A 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.
The 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
Compare the responses of 2v̇ + v = ġ (t) + g(t) and 2v̇ + v = g(t) if g(t) = 10us(t) and v(0) = 5.
Compare the responses of 5v̇+ v = ġ + g and 5v̇+ v = g if u(0) = 5 and g = 10 for - ¥ ≤ / ≤ ∞.
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
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.
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)?
The 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
Obtain the oscillation frequency and amplitude of the response of the model 3ẍ + 12x = 0 for(a) x(0) = 5 and ẋ (0) = 0 (b) x(0) = 0 and ẋ (0) = 5.
Suppose 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
If 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
If 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
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.
Given the modelẍ (μ + 2) ẋ + (2μ + 5)x = 0Find the values of the parameter \x for which the system isStableNeutrally stableUnstableFor the stable case, for what values of // is the systemUnder
The 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
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
For each of the following models, obtain the free response and the time constant, if any.a. I6ẋ + 14x = 0, x(0) = 6b. 2ẋ + 5x = L5, x(0) = 3c. 3ẋ + 6x = 0, x(0) = -2d. 7ẋ-5x = 0, x(6)=9
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
A 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
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
Compute 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.
A certain system is described by the modelẍ + cẋ +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
A certain system is described by the model9ẍ + cẋ + 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
Derive the fact that the peak time is the same for all characteristic roots having the same imaginary part.
For 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
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
The 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
For the model 2ẋ + 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
Figure shows the response of a system to a step input of magnitude 1000 N. The equation of motion ism + c + kx = f(t)Estimate the values of m, c, and k.
A 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
a. 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 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 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
What 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
Use 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
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.ẍ + 4 ẋ + 8x = 2us(t)
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. 3ẍ +
a. 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
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 ẋ + 5x = 20us,(t), x(0) = 0b. 6 ẋ + 5x = 20u, (t),
a. 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 =
a. 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 =
The following equation has a polynomial input.0. 125ẍ + 0.75ẋ + x = y(t) = -27/800 t3 + 270/800 t2Use Simulink to plot x(t) and y(t) on the same graph. The initial conditions are zero.
The following model has a polynomial input.ẋ1 = -3.x1 + 4x2ẋ2 = -5x1 - x2 + f(t)f(t) = - 5/3 t2 + 25/3 tThe initial conditions are x1(0) = 3 and x2(0) = 7. Use Simulink to plot x1(t). x2(t), and
First 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 =
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
Obtain the total response of the following models.a. 6 ẋ + 5x = 20 us (t), x(0) = 0b. 6 ẋ + 5x = 20us (t), x(0) = 1c. 13 ẋ- 6x = l8us (t), x(0) =-2
A 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
The 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
Use 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),
The model of a certain mass-spring-damper system is10ẍ + cẋ + 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
The model of a certain mass-spring-damper system is13ẍ + 2ẋ + kx = 10 sin ωtDetermine the value of k required so that the maximum response occurs at ω = 4 rad/sec. Obtain the steady-state
Determine the resonant frequencies of the following models. a. T(s) = 7/s (s2 + 6s + 58) b. T(s) = 7/((3s2 + I8s + 174)(2x2 + 8x + 58))
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
(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 Ï‰r, and discuss the effect of the damping c on Mr
For 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
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
Determine the beat period and the vibration period of the model3 ẍ + 75 ẋ = 7 sin 5.2t
Resonance will produce large vibration amplitudes, which can lead to system failure. For a system described by the modelẍ + 64 ẋ = 0.2 sin ωtWhere x is in feet, how long will it take before |x|
The 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
Use 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)
A 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
An 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
An 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
Design a vibrometer having a mass of 0.1 kg, to measure displacements having a frequency near 200 Hz.
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.
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
A 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 Ï‰r, its
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.
Obtain the expressions for the bandwidths of the two circuits shown in Figure.a.b.
For the circuit shown in Figure, L = 0.1 H and C = 10-6 F. Investigate the effect of the resistance R on the bandwidth, resonant frequency, and resonant peak over the range 100 ‰¤ R v 1000 Î©.
Figure 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
For 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.
The voltage shown in Figure is produced by applying a sinusoidal voltage to a full wave rectifier. The Fourier series approximation to this function isSuppose this voltage is applied to a series RC
The voltage shown in Figure is called a square wave. The Fourier series approximation to this function isSuppose this voltage is applied to a series RC circuit whose transfer function isVo(s)/Vs(s) =
The displacement shown in Figure (a) is produced by the cam shown in part (b) of the figure. The Fourier series approximation to this function isFor the values m = 1 kg, c = 98 N.s/m, and k = 4900
Given the model 0.5 + 5y = f(t) With the following Fourier series representation of the input f(t) = sin 4t + 4 sin 8t + 0.04 sin 12t + 0.06 sin 16t + ... Find the steady-state response yss(t) by
A mass-spring-damper system is described by the modelm + c + kx = f (t)Where m = 0.25 slug, c = 2 lb-sec/ft, k = 25 lb/ft, and f(t) (lb) is the externally applied force shown in Figure. The forcing
An input vs(t) = 20 sin cot V was applied to a certain electrical system for various values of the frequency ω, and the amplitude |vo| of the steady-state output was recorded. The data are shown in
An input f(t) = 15 sin cot N was applied to a certain mechanical system for various values of the frequency ω, and the amplitude ω of the steady-state output was recorded. The data are shown in the
The following data were taken by driving a machine on its support with a rotating unbalance force at various frequencies. The machine's mass is 50 kg, but the stiffness and damping in the support are
For the system shown in Figure, m1 = m2, k1 = k2, and k1/m1 = 64 N/(m.kg). Obtain the transfer function X1(s)/Y(s) and its Bode plots. Identify the resonant frequencies and bandwidth.
A single-room building has four identical exterior walls, 5 m wide by 3 m high, with a perfectly insulated roof and floor. The thermal resistance of the walls is R = 4.5 x 10-3 K/W.m2. Taking the
For the system shown in Figure, I1 = I2, cT1 = cT2, cT1/I1 = 0.1 rad-1.s-1, and kT/11 = 1 s-2. Obtain the transfer function ÆŸ1(s)/((s) and its Bode plots. Identify the resonant
9.41 A certain mass is driven by base excitation through a spring (see Figur). Its parameter values are m = 50 kg, c = 200 N.s/m, and k = 5000 N/m.Determine its resonant frequency Ï‰r, its
The transfer functions for an armature-controlled dc motor with the speed as the output areΩ(s)/V(s) = KT/((Is+c)(Ls + R) + KbKT)Ω(s)/Td(s) = (Ls + R)/((Is+ c)(Ls + R) + KbKT)A certain motor has
The transfer function of the speaker model derived in Chapter 6 is, for c = 0, X(s)/V(s) = Kf /(mLs3 +mRs2 + (kL + KfKb)s + kR) Where x is the diaphragm's displacement and v is the applied voltage. A
The following is a two-mass model of a vehicle suspension.m11 + c11 + k1x1 - c12 - k1x2 = 0m22 + c12 + (k1 + k2)x2 - c11 - k1x1 = k2yMass m1 is one-fourth the mass of the car body, and m2 is
For the rotational system shown in Figure I = 2 kg.m2 and c = 4 N.m.s/rad. Obtain the transfer function Î©(s)/T(s), and derive the expression for the steady-state speed cos5 (t) if the applied
For the system shown in Figure the bottom area is A = 4Ï€ ft2 and the linear resistance is R = 1500 sec-1ft-1. Suppose the volume inflow rate isqvi = 0.2 + 0.1 sin0.002t ft3/secObtain the
Use 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) = 10/((10s + l)(4s+ 1), f(t) = 10sin 0.2t b. T(s) = Y(s)/F(s) =
Use 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) = 8/(s(s2 + 10s+ 100), f(t) = 6sin 9t b. T(s) = Y(s)/F(s) = 10/s2(s
The model of a certain mass-spring-damper system is10ẍ + cẋ + 20s = f(t)Determine its resonant frequency ωr and its peak magnitude Mr if (a) ζ = 0.1 and (b) ζ = 0.3.

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