Control Systems Engineering
7th Edition
ISBN: 9781118170519
Author: Norman S. Nise
Publisher: WILEY
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Chapter 3, Problem 29P
A single-pole oil cylinder valve contains a spool that regulates hydraulic pressure, which is then applied to a piston that drives a load. The transfer function relating piston displacement, Xp(s) to spool displacement from equilibrium, Xv(s), is given by (Qu, 2010):
where A1=effective area of a the valve’s chamber, Kq=rate of change of the load flow rate with a change in displacement, and
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X
Determine:
k
(i)
(ii)
с
ww
m
E
Figure Q1
Figure Q1 shows a forced spring-mass system with damping, where
mass m = 1 kg, spring constant k = 0.2 N/m, and damping coefficient c
= 0.3 N-s/m.
(a)
This forced spring-mass system with damping can be described
by the following differential equation:
F
d²x(t) c dx(t) k
dt² m dt
+
-+-x(t)==F(t)
m
m
Laplace Transfer Function of this system,
This system's steady state gain, damping ratio and natural
frequency.
1. A spring mass system serving as a shock absorber under a car's suspension, supports the
M
1000 kg mass of the car. For this shock absorber,
k = 1 × 10°N /m and c = 2 × 10° N s/m. The car drives over a corrugated road with force
%3|
F = 2× 10° sin(@t) N . Use your notes to model the second order differential equation suited to this
application. Simplify the equation with the coefficient of x'" as one. Solve x (the general solution) in
terms of w using the complimentary and particular solution method. In determining the coefficients of
your particular solution, it will be required that you assume w – 1z w or 1 – o z -w. Do not
use Matlab as its solution will not be identifiable in the solution entry. Do not determine the value of w.
You must indicate in your solution:
1. The simplified differential equation in terms of the displacement x you will be solving
2. The m equation and complimentary solution xe
3. The choice for the particular solution and the actual particular solution x,…
2. Consider a car suspension, modeled as a mass/spring/damper system (mass m, stiffness k, damping b).
Suppose the height of the chassis is lo at rest, the height of the terrain below the driver varies as h(t),
and the height of the chassis is denoted lo + y(t). (i.e., spring deflection away from rest is y(t) – h(t)).
2
(a) Give the transfer function G(s) = H(s) ·
=
(b) Suppose the ground follows an oscillatory profile h(t) A cos(wx (t)) with magnitude A (in meters)
and frequency w (measured in radians per meter). Suppose the car is traveling at a constant
forward speed v. Using a frequency response analysis strategy, give the amplitude of oscillations
experienced by the driver at steady state as a function of m, k, b, A, w, and v. Hint: You can't
simply consider |G(iw)| to get the amplification in this case.
(c) Suppose the ground varies by A = 5cm, w = 2 rad/m, and you are driving at v = 15 m/s. Using
your answer to part (b), what amplitude of oscillation is felt by the driver when m…
Chapter 3 Solutions
Control Systems Engineering
Ch. 3 - Prob. 1RQCh. 3 - State an advantage of the transfer function...Ch. 3 - Define state variables.Ch. 3 - Define state.Ch. 3 - Define state vector.Ch. 3 - Define state space.Ch. 3 - What is required to represent a system in state...Ch. 3 - 8. An eighth-order system would be represented in...Ch. 3 - If the state equations are a system of first-order...Ch. 3 - Prob. 10RQ
Ch. 3 - What factors influence the choice of state...Ch. 3 - What is a convenient choice of state variables for...Ch. 3 - If an electrical network has three energy-storage...Ch. 3 - Prob. 14RQCh. 3 - Prob. 1PCh. 3 - Represent the electrical network shown in Figure...Ch. 3 - Prob. 3PCh. 3 - Represent the system shown in Figure P3.4 in state...Ch. 3 - Represent the rotational mechanical system shown...Ch. 3 - Represent the system shown in Figure P3.7 in state...Ch. 3 - 8. Show that the system of Figure 3.7 in the text...Ch. 3 - Find the state-space representation in...Ch. 3 - MATLAB ML 10. Repeat Problem 9 using MATLAB....Ch. 3 - For each system shown in Figure P3.9, write the...Ch. 3 - MATLAB ML
12. Repeat Problem 11 using MATLAB....Ch. 3 - 13. Represent the following transfer function in...Ch. 3 - Find the transfer function G(s) = Y(s)/R(s) for...Ch. 3 - MATLAB ML
15. Use MATLAB to find the transfer...Ch. 3 - 17. A missile in flight, as shown in Figure P3.10,...Ch. 3 - Given the dc servomotor and load shown in Figure...Ch. 3 - Prob. 20PCh. 3 - Prob. 23PCh. 3 - Experiments to identify precision grip dynamics...Ch. 3 - State-space representations are, in general, not...Ch. 3 - Figure P3.16 shows a schematic description of the...Ch. 3 - Prob. 28PCh. 3 - A single-pole oil cylinder valve contains a spool...Ch. 3 - Figure P3.17 shows a free-body diagram of an...Ch. 3 - 33. Parabolic trough collector. A transfer...
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- 0.052 J A block attached to a spring, oscillates on a frictionless horizontal surface with a period of 0.3 s. The time needed by the block to move (for the first time) from position x = -A to x = -A/2 is: %3D 0.3 sec 0.1 sec O 0.2 sec 0.15 sec 0.05 sec A traveling wave on a taut string with a tension force T, is given by the wavearrow_forwardQ1 X Determine: k (i) (ii) m CA Figure Q1 Figure Q1 shows a forced spring-mass system with damping, where mass m = 1 kg, spring constant k = 0.2 N/m, and damping coefficient c = 0.3 N-s/m. F (a) This forced spring-mass system with damping can be described by the following differential equation: d²x(t) c dx(t), k - + − x(t): dt² m dt m + 1 - F(t) == m Laplace Transfer Function of this system, This system's steady state gain, damping ratio and natural frequency. (b) Sketch frequency response of the system from part (a) in the form of Bode plot as accurately as you can.arrow_forwardHarmonic oscillators. One of the simplest yet most important second-order, linear, constant- coefficient differential equations is the equation for a harmonic oscilator. This equation models the motion of a mass attached to a spring. The spring is attached to a vertical wall and the mass is allowed to slide along a horizontal track. We let z denote the displacement of the mass from its natural resting place (with x > 0 if the spring is stretched and x 0 is the damping constant, and k> 0 is the spring constant. Newton's law states that the force acting on the oscillator is equal to mass times acceleration. Therefore the differential equation for the damped harmonic oscillator is mx" + bx' + kr = 0. (1) k Lui Assume the mass m = 1. (a) Transform Equation (1) into a system of first-order equations. (b) For which values of k, b does this system have complex eigenvalues? Repeated eigenvalues? Real and distinct eigenvalues? (c) Find the general solution of this system in each case. (d)…arrow_forward
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- Figure 1 below depicts the popular Spring-Mass-Damper system in which m is the mass, c represented by the dashpot symbol in the figure is called the damping factor, and k is the spring constant. The Spring-Mass-Damper system is with m = 20 kg, c = 20 Ns/m, k = 4000 Px k în с Li m Figure 1: The Spring-Mass-Damper system N/m. Moreover, denote by x(t) the displacement of the spring (from its equilibrium position). The system is acted on by a periodic harmonic force F(t) = Fo sin(wt) where Fo and w are the amplitude and frequency of the harmonic force, respectively. Given that Fo= 100 N and w = 20 rad/s. The Spring-Mass-Damper in Figure 1 is modelled by the following ODE: d²x dx m. +c- + kx = F(t). dt² dt F(t) dx (0) dt (1) Assume that x(0) = 0.01 m and = 0.0. Your duty as an engineer is to analyse this Spring- Mass-Damper system by fulfilling the following requirements: a) Establish the spring displacement trajectory x(t) by solving analytically the ODE equation (1) modelling the…arrow_forward3. Consider the mechanical system shown in Fig. 3. Let V3(t) be the input and the acceleration of the mass be the output. Derive the state equations and the output equation using linear graphs and normal trees. V,(t) В m k2 101 k, Figure 3: A mechanical system with an across-variable sourcearrow_forwardFind the differential equation of the mechanical system in Figure 1(a) To obtain the differential equation of motion of the mass and spring system given in Fig. 1. (a) one may utilize the Newton's law for mass and spring relations defined as shown in Fig. 1. (b) and (c) use f = cv for viscous friction, where v is the velocity of the motion and c is a constant. Z/////// k M F. F, F F F, F, k EF=ma F = k(x, - x,) = kx (b) (c) Figure 1: Mass-spring system (a), Force relations of mass (b) and spring (c)arrow_forward
- 6. The electro-mechanical system shown below consists of an electric motor with input voltage V which drives inertia I in the mechanical system (see torque T). Find the governing differential equations of motion for this electro-mechanical system in terms of the input voltage to the motor and output displacement y. Electrical System puthiy C V V₁ R bac (0) T bac T Motor - Motor Input Voltage - Motor Back EMF = Kbac ( - Motor Angular Velocity - Motor Output Torque = K₂ i Kbacs K₁ - Motor Constants Mechanical System M T Frictionless Supportarrow_forwarda) A vehicle circulates on a road as shown in Figure Q2a. The road profile can be modelled as the input u(t). The vehicle is modelled as a quarter car of mass m, and the suspension has a spring stiffness coefficient k = 2 Nm and a damper of coefficient c = 2 Nm s. Find the position of the mass, y(t) and any time t if the road profile is a unit step. m 一 Road Figure Q2a b) In Figure Q2b, a disk flywheel J of mass m 32 kg and radius r = 0.5 m is driven by an electric motor that when it is working produces an oscillating torque of Tin = 10sin(wt) N- m. The shaft bearings may be modelled as viscous rotary dampers with a damping coefficient of BR = 0.4 N-m-s/rad and stiffness coefficient KR = 2 Nm If the flywheel is at rest at t 0 and the power is suddenly applied to the motor, do the following [Hint:J = mr²/2]: (i) Find the natural frequency of oscillation of the disk expressed in Hz. (ii) Find the damping ratio for this system. (iii) Describe in no more than 3-4 lines the behaviour of the…arrow_forwardGet the equation of motion by drawing the free body diagram of the given systems. a) Get the system's transfer function and find the unit digit answer. Show all decals in detail. m = 1 kg b= 20 Ns/m k = 125 N/m F ww k b) get the transfer function of the system. X(s)/Pg(s) =? Show all decals in detail. resistance R k Pg massless piston area (A) capacitancearrow_forward
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