Control Systems Engineering
7th Edition
ISBN: 9781118170519
Author: Norman S. Nise
Publisher: WILEY
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Textbook Question
Chapter 2, Problem 52P
A system’s output, c, is related to the system’s input, r, by the straight-line relationship,
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Q4: (M4)
LE
Velocity: V [km/hr]
Wavelength: L [m]
M
k
y(t)
Model
Height: H [m]
The figure above shows a model of a person riding a unicycle that contains a spring
under its seat. The spring constant is k = 10,600 N/m. Assume that damping is minimal,
the wheel of the unicycle has no mass and is not a spring, the unicyle always stays
perfectly upright, and the person is represented by a rigid mass M = kg.
a) When the unicycle is being ridden at speed V = 10 km/hr over the sinusoidal bumpy
terrain shown above, with bump spacing L=0.6 m and bump height H 0.05 m, what
will be the steady-state peak-to-peak amplitude of the motion y(t) [m] of the person
riding the unicycle?
b) Recalculate the steady-state peak-to-peak amplitude of the motion for 2.5, 5, and 20
km/hr. Will the rider have difficulty reaching speeds above 5 km/hr?
If a system, * + 10x + 21x = 4f(t) is converted into a state-space model, what would
be the state (A), input (B), and output (C) matrices?
[where input = f (t) and output = x(t)]
%3D
A) A = |
[-21
l, B = | and C = [0 1]
В
-10
1
B) A = 21
ol, B =
and C = [1 0]
%3D
C) A =
= Al and C = [1 0]
В
%3D
-21
D) None of the above
Consider the state space representation of the following system
[6]-[2][]+8-0
X2 (1)
y()=[11][26]
y(t) = [1
X2
u(t)
Find x(t) and y(t) of this systems by u(t)= unit step function.
=[7]
x(0) = Xo =
Chapter 2 Solutions
Control Systems Engineering
Ch. 2 - Prob. 1RQCh. 2 - Prob. 2RQCh. 2 - Prob. 3RQCh. 2 - Define the transfer function.Ch. 2 - Prob. 5RQCh. 2 - What do we call the mechanical equations written...Ch. 2 - If we understand the form the mechanical equations...Ch. 2 - Why do transfer functions for mechanical networks...Ch. 2 - What function do gears perform?Ch. 2 - What are the component parts of the mechanical...
Ch. 2 - The motor’s transfer function relates armature...Ch. 2 - Summarize the steps taken to linearize a nonlinear...Ch. 2 - Prob. 1PCh. 2 - Prob. 2PCh. 2 - Prob. 3PCh. 2 - Prob. 4PCh. 2 - Prob. 5PCh. 2 - Prob. 6PCh. 2 - Prob. 7PCh. 2 - A system is described by the following...Ch. 2 - For each of the following transfer functions,...Ch. 2 - Write the differential equation for the system...Ch. 2 - Write the differential equation that is...Ch. 2 - Prob. 12PCh. 2 - Use MATLAB to generate the MATLAB ML transfer...Ch. 2 - Repeat Problem 13 for the MATLAB following...Ch. 2 - Use MATLAB to generate the partial fraction...Ch. 2 - Use MATLAB and the Symbolic Math Symbolic Math...Ch. 2 - Prob. 17PCh. 2 - Prob. 18PCh. 2 - Prob. 19PCh. 2 - Repeat Problem 19 using nodal equations. [Section:...Ch. 2 - Prob. 22PCh. 2 - Prob. 23PCh. 2 - Prob. 24PCh. 2 - Prob. 25PCh. 2 - Prob. 26PCh. 2 - Prob. 27PCh. 2 - Prob. 28PCh. 2 - Prob. 29PCh. 2 - Write, but do not solve, the equations of motion...Ch. 2 - For the unexcited (no external force applied)...Ch. 2 - For each of the rotational mechanical systems...Ch. 2 - For the rotational mechanical system shown in...Ch. 2 - Find the transfer function, 1sTs , for the system...Ch. 2 - For the rotational mechanical system with gears...Ch. 2 - For the rotational system shown in Figure P2.21,...Ch. 2 - Prob. 37PCh. 2 - Find the transfer function, Gs=4s/Ts , for the...Ch. 2 - For the rotational system shown in Figure P2.24,...Ch. 2 - Prob. 40PCh. 2 - Given the rotational system shown in Figure P226,...Ch. 2 - In the system shown in Figure P2.27, the inertia,...Ch. 2 - Prob. 43PCh. 2 - Given the combined translational and rotational...Ch. 2 - Prob. 45PCh. 2 - The motor whose torque-speed characteristics are...Ch. 2 - A dc motor develops 55 N-m of torque at a speed of...Ch. 2 - 48. In this chapter, we derived the transfer...Ch. 2 - Prob. 49PCh. 2 - Find the series and parallel analogs for the...Ch. 2 - Find the series and parallel analogs for the...Ch. 2 - A system’s output, c, is related to the system’s...Ch. 2 - Prob. 53PCh. 2 - Consider the differential equation...Ch. 2 - 55. Many systems are piecewise linear. That is,...Ch. 2 - For the translational mechanical system with a...Ch. 2 - 57. Enzymes are large proteins that biological...Ch. 2 - Prob. 58PCh. 2 - Figure P2.36 shows a crane hoisting a load....Ch. 2 - 60. In 1978, Malthus developed a model for human...Ch. 2 - 61. In order to design an underwater vehicle that...Ch. 2 - 62. The Gompertz growth model is commonly used to...Ch. 2 - A muscle hanging from a beam is shown in Figure...Ch. 2 - A three-phase ac/dc converter supplies dc to a...Ch. 2 - Prob. 65P
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- 28. Find the transfer function, G(s) = X1(s)/F(s), for the translational mechanical system shown in Figure P2.13. [Section: 2.5] 2 N-s/m X3(1) 2 N-s/m (1)'x- [4 kg 2 N-s/m 6 N/m 6 N/m 4 kg 0000 4 kg "Frictionless FIGURE P2.13 USE MATRIX METHODarrow_forwardThe state transmission matrix of the system whose state-space [3²₁] = [0²2 J]+[]u a. b. C. O 0 cosh at c. Ø(t) = [ a sinh at/a cosh a. ¢(t) = [sinhat cosh at a. Ø(t) = [a cosh at sinh at b. Ø(t) = [a [a cosh at a sinh at sinhat cosh at] sinhat/a] cosh at [/a] sinh at/a] a cosh at sinh at att cosh atarrow_forwardO 1::09 O [Template] Ho... -> Homework For the system shown in figure below, Find the range of K for stable system. R K(s + 2) C s(s +5)(s² + 2s + 5) IIarrow_forward
- 3. In this problem, you are going to analyze the dynamics of a rotational mechanical system shown in Figure below (this is also covered in Lecture Notes #3 of M. Mert Ankarali [1]). In this system input the external torque t(t), and output is the angular velocity of the load wL(t). JR WR OR K JL OL WL T DL DR The state-space representation of this system is provided in the Lecture Notes #3 [1]. Find the transfer function of the dynamical system. Find another (minimal) state-space representation for the system.arrow_forward32. For the rotational mechanical system with gears shown in Figure P2.18, find the transfer function, G(s) = 03(s)/T(s). The gears have inertia and bear- ing friction as shown. [Section: 2.7] T(t) to |N1 小D N2 N3 2, D2 Jz, D3 03(1) N4 J4. D4 J5. D5 FIGURE P2.18 sairarrow_forward01(1) 18 N-m-s/rad T(t) 02(1) 1 N-m-s/rad kg-m? 3 kg-m2 3 N-m/rad 9 N-m/rad (a) Figure P2.16a OJohn Wiley & Sons, Inc. All rights reserved. The transfer function where the output Theta, is s? +5s +1 0,(s) а. Н(s)3D T(s) s' +30s +80s² +s+15 4 s' +5s +1 0,(s) b. H(s)= T(s) 25s* + 30s' +80s? + s +15 4 5s? +9s+9 0,(s) c. H(s)= T(s) 15s* +30s +95s² + s +27 5s +9s +9 0,(s) T(s) 15s +32s' +95s² +99s+27 d. H(s)= 4 Select one: O a.c O b. a O .b O d.d 立arrow_forward
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- 26. For the system shown in Figure P4.8, a step torque is applied at 01 (t). Find a. The transfer function, G(s) = 02(s)/T(s). b. The percent overshoot, settling time, and peak time for 02(t). [Section: 4.6] T(t) 01(1) 02(1) ff 1.07 kg-m2 1.53 N-m-s/rad 1.92 N-m/rad FIGURE P4.8arrow_forwardConsider the following state space system 1 B = 1 C =[1 0] D=[0] -5 -6 1- Check the controllability of the system. 2- Check the observability of the systemarrow_forwardk₁ B₁ Fs(t) ww k2 12 m B2 Figure 4: A translational system 2. Consider a translational system shown in Fig. 4. Answer the following questions. (a) Draw a linear graph and write down all the elemental equations. (Don't draw the normal tree yet.) (b) From the elemental equation you write down, identify three variables that can potentially serve as state variables and explain why. (c) Are these potential state variables independent of each other? If not, use either conti- nuity or compatibility equation to prove it. How would you choose your state variables? (d) Draw a normal tree to see if there is any dependent energy storage element. What are the state variables according to your normal tree? Are they consistent with the explanation in the Part (c)?arrow_forward
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