EBK SYSTEM DYNAMICS
3rd Edition
ISBN: 8220100254963
Author: Palm
Publisher: MCG
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Chapter 5, Problem 5.15P
Obtain the state model for the two-mass system whose equations of motion are
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3) For the mechanical system shown below find a state variable
representation of the equations of motion
→→→→ X(t)
M
K
K
Fmmmm
⇒ F(t)
1. Consider the simplified model we discussed in class about a car on a bumpy road; see Fig. 1.
The car has a mass m supported by stiffness k and damping c. The road gives a displacement
excitation R(t) to the car. The transfer function from R(t) to the car displacement y(t) is
Hy(s) =
=
Y(s)
R(s)
=
cs+k
ms² + cs+k
(1)
Let the acceleration of the car be a(t) = ÿ(t). Determine the frequency response function
Ga(w) from R(t) to a(t). Plot |Ga(w) as a function w. Hint: Analyze |Ga(w)| for w > wn.
m
y(t)
Suspension
k₁
y(t)
Head
m
k
Air
Bearing
x(t)
k2
R(t)
Disk Surface
Figure 1: A simple model of
a car on a bumpy
road
Figure 2: Suspension in computer
hard disk drives
1. Consider the simplified model we discussed in class about a car on a bumpy road. The car has
a mass m supported by stiffness k and damping c. The road gives a displacement excitation
R(t) to the car. The transfer function from R(t) to the car displacement y(t) is
Hy(s)
-
Y(s)
R(s)
cs + k
ms² + cs+k
(1)
(a) Determine the driving point impedance ZR(s), which is the ratio of the velocity R(t) to
the force acting on the wheel from the road.
(b) Engineer X is testing the car in the Vehicle Dynamics Lab of a start-up company
GF.com. Because of insufficient cash flow, Engineer X only has the equipment to mea-
sure ZR(s), and identify the zeros and poles of ZR(s). But eigineer X wants to find the
poles of Hy(s). To do so, should Engineer X use the zeros or poles of ZR(s) instead?
Explain why?
m
y(t)
R(t)
Figure 1: A simple model of a car on
a bumpy road
Chapter 5 Solutions
EBK SYSTEM DYNAMICS
Ch. 5 - Prob. 5.1PCh. 5 - Obtain the transfer function Xs/Fs from the block...Ch. 5 - Obtain the transfer function Xs/Fs from the block...Ch. 5 - Draw a block diagram for the following equation....Ch. 5 - Draw a block diagram for the following model. The...Ch. 5 - Referring to Figure P5.6, derive the expressions...Ch. 5 - Prob. 5.7PCh. 5 - Prob. 5.10PCh. 5 - Obtain the state model for the reduced-form model...Ch. 5 - Obtain the state model for the reduced-form model...
Ch. 5 - Prob. 5.13PCh. 5 - Obtain the state model for the transfer-function...Ch. 5 - Obtain the state model for the two-mass system...Ch. 5 - Prob. 5.16PCh. 5 - Put the following model in standard state-variable...Ch. 5 - Given the state-variable model...Ch. 5 - Given the following state-variable models, obtain...Ch. 5 - Prob. 5.20PCh. 5 - The transfer function of a certain system is...Ch. 5 - Prob. 5.22PCh. 5 - Prob. 5.23PCh. 5 - Use MATLAB to obtain a state model for the...Ch. 5 - Use MATLAB to obtain a state-variable model for...
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- Question 3. Consider a mass-spring translational mechanical system in series. The spring has a nonlinear characteristics such that the relationship between the spring force and the spring displacement can be described mathematically f.(t) = 3x. Assume that the applied force is f(t) = 6+ 8f (t), where 8f(t) is a small force about the 6 Newton constant value. Assuming the output to be the displacement of the mass, obtain the state space representation of the system about the equilibrium displacement.arrow_forwardmechanical vibrations 3m i+4cx+ 2kx = 4cj+3ky For the system given above, obtain the state-s pace representation,arrow_forwardThe state z(t) of a dynamical system is solution of equation ż(t) + aż(t) + 25z(t) = 12, with a = 7.3. Calculate the Peak time of the response.arrow_forward
- 4. Consider a mass of 500 g placed at the end of a spring with stiffness constant 100 N/m and hanging at rest. The mass is displaced downward by 1.0 cm and released from rest. When the motion sets in, a time-varying force F(t) = cos³ 2t is applied to the spring-mass system. Solve the nonhomogeneous 2nd order differential equation representing the motion. That is, solve for x(t). [Note: Take the downward direction as the negative x-axis.]arrow_forwardObtain the state model for the two-mass system whose equations of motion for specific values of the spring and damping constants are 15x₁ + 7x₁ − 4x₂ + 30x₁ – 15x2 = 0 6x2 - 15x₁ + 15x₂ − 4x₁ + 4x₂ = f(t)arrow_forward2. Express the equation of motion of the damped 2DOF system depicted below, where p, q, and r are the horizontal coordinates and u₁, u2, and u3 are the forcing functions. Represent your equation of motion in form of the state space representation with the general form below. k₁ Ρ ալ √ x = Ax + Bu y = Cx + Du r q ԱՅ Աշ КА k3 k2 ww ww M3 M2 C4 C3 C₁ M1 ми C2arrow_forward
- Correct and complete solution please don't copyarrow_forwardConsider the following mechanical system: →y k +f m (?)ap 9+ + ky(t) = f (t) dt d'y(t) dy(t) m dt? Obtain the state space model of the system with input f(t) and output y(t). Calculate the system matrices for m = 1, k = 1 and b = 2. 3.arrow_forwardThe following figure shows a machine of mass m mounted on a vibration isolator. The machine (starting at rest at t=0) is subjected to a sinusoidal excitation force p(t) = P sin wt. Solve for the transfer function X(s)/P(s). Then state in words the process you would then follow to determine the force transmissibility, TR. p(t) = P sin wi m b₂arrow_forward
- The figure that is attached illustrates a system in which a force F is applied to a mass m2 that is connected to another mass m1 via a spring and a damper, and mass m1 is connected to a wall via a damper. The equations of motion that govern the time evolution of the mass displacements, y1(t)and y2(t), are given below. A) Define 4 state variables of the system (xi, i = 1,…,4) as phase variables, and define the control input u. Convert the two equations of motion above into four 1st-order ODEs that are functions of the state variables and the control input.(b) Write the four 1st -order ODEs from part (a) in state-space form,x = Ax+Bu.arrow_forwardConsider a spring mass system with following parameter. m = 1, c = 4, k = 5, force = e-3* cos 2x Find the transient motion of the system assuming that it is initially at rest state. Show the details of your work.arrow_forward1. The equations of motion of this system are ÿ + 3y + 4y - 32 - 4Z = 0 Ż +52 +6Z-5ý - 6y = f(t) * = A + Bū y = Cx+Dū Put these equations into state variable form and express the model as a matrix vector equation if output of the system is y. Energy storage element m1 m2 k₁ k₂ State variable *1=ý x₂ = Ż x3 = y x4 =Z k₁ my D k₂ C₂H m₂arrow_forward
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