System Dynamics
3rd Edition
ISBN: 9780077509125
Author: Palm
Publisher: MCG
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Chapter 9, Problem 9.35P
A mass-spring-damper system is described by the model mx -I- ci + kx = f(t)
where m = 0.25 slug, c = 2 lb-sec/ft, k = 25 lb/ft, and /(z) (lb) is the externally applied force shown in Figure P9.35. The forcing function can be expanded in a Fourier series as follows:
Find an approximate description of the output x14(/) at steady state, using only those input components that lie within the bandwidth.
Figure P9.3S
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(b) Consider a vertical spring mass-damper-system with mass m = 2 kg, damping coefficient c = 5 N.s/m
and stiffness k = 10 N/m. An external force f(t) = sin3t N is applied to the mass.
(i) Find the static equilibrium position as measured from the un-stretched length of the spring
(ii) Write the equation of motion.
(iii) Find the natural and damped frequencies.
(iv) In steady state, find the time delay in seconds between the mass position and f(t).
(v) Find the steady state maximum acceleration.
Q3.Consider a spring–mass–damper system with m = 80 kg, c = 15 kg/s, and k = 1500 N/m with an impulse force applied to it of 1200 N for 0.01 s.
(C) Calculate the damped natural frequency of the system (D) Calculate the system response to the impulsive force at time t = 1 s
The block diagram given below;
A) Reduce it
B) G1(s)=2/s ; G2(s)=1/4s+2 ; G3(s)=4 ;H(s)=0.5 Given the values of the system; find the time constant, its natural frequency and damping rate and explain what kind of dynamic behavior it exhibits accordingly.
C)Find the poles and zeros of the system according to the values in (B).Is the system stable? Find the unit step response (Inverse Laplace).
Chapter 9 Solutions
System Dynamics
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- 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_forwardA certain mass is driven by base excitation through a spring (Figure P4.13). Its parameter values are m = 100 kg, c = 1000 N * s/m, and k = 10,000 N/m. Determine its peak frequency w_p, it’s peak M_p, and its bandwidth.arrow_forwardGiven an oscillator of mass 2.0kg and spring constant of 180N/m, what is the period without damping? Use numerical methods to model this oscillator with an additional friction force equal to where c is a positive damping constant. Using c=5.0, what is the new period of oscillation. What about for c=10? Assume initial position is 0.2m and initial velocity is zero. Please find the period using the position versus time plot and use the first full cycle of the motion.arrow_forward
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