EBK NUMERICAL METHODS FOR ENGINEERS
7th Edition
ISBN: 8220100254147
Author: Chapra
Publisher: MCG
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Chapter 27, Problem 25P
Derive the set of
Note each equation has been divided by the mass. Solve for the eigenvalues and natural frequencies for the following values of mass and spring constants:
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A velocity of a vehicle is required to be controlled and maintained constant even if there are disturbances because of wind, or road surface variations. The forces that are applied on the vehicle are the engine force (u), damping/resistive force (b*v) that opposing the motion, and inertial force (m*a). A simplified model is shown in the free body diagram below.
From the free body diagram, the ordinary differential equation of the vehicle is:
m * dv(t)/ dt + bv(t) = u (t)
Where:
v (m/s) is the velocity of the vehicle,
b [Ns/m] is the damping coefficient,
m [kg] is the vehicle mass,
u [N] is the engine force.
Question:
Assume that the vehicle initially starts from zero velocity and zero acceleration. Then, (Note that the velocity (v) is the output and the force (w) is the input to the system):
1. What is the order of this system?
Harmonic 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)…
The governing equation of motion for a base motion system is given by (assume the
units are Newton)
mä(t)+ci(t)+kx(t) =cY@, cos(@,t) +kY sin(@t)
%3D
Given that m =
180 kg, c = 30 kg/s, Y = 0.02 m,
and
о, — 3.5 rad/s
%3|
1. Use Excel or Matlab to find the largest value of the stiffness, k that makes the
transmissibility ratio less than 0.85
2. Using the value of the stiffness obtained in part (1), determine the transmitted
force to the base motion system using Matlab or Excel.
3. Display and discuss your results.
Chapter 27 Solutions
EBK NUMERICAL METHODS FOR ENGINEERS
Ch. 27 - A steady-state heat balance for a rod can be...Ch. 27 - 27.2 Use the shooting method to solve Prob. 27.1....Ch. 27 - 27.3 Use the finite-difference approach with to...Ch. 27 - 27.4 Use the shooting method to solve
Ch. 27 - Solve Prob. 27.4 with the finite-difference...Ch. 27 - 27.7 Differential equations like the one solved...Ch. 27 - 27.8 Repeat Example 27.4 but for three masses....Ch. 27 - 27.9 Repeat Example 27.6, but for five interior...Ch. 27 - Use minors to expand the determinant of...Ch. 27 - 27.11 Use the power method to determine the...
Ch. 27 - 27.12 Use the power method to determine the...Ch. 27 - Develop a user-friendly computer program to...Ch. 27 - Use the program developed in Prob. 27.13 to solve...Ch. 27 - 27.15 Develop a user-friendly computer program to...Ch. 27 - Use the program developed in Prob. 27.15 to solve...Ch. 27 - 27.17 Develop a user-friendly program to solve...Ch. 27 - Develop a user-friendly program to solve for the...Ch. 27 - 27.19 Use the Excel Solver to directly solve...Ch. 27 - Use MATLAB to integrate the following pair of ODEs...Ch. 27 - The following differential equation can be used to...Ch. 27 - 27.22 Use MATLAB or Mathcad to...Ch. 27 - 27.23 Use finite differences to solve the...Ch. 27 - Solve the nondimensionalized ODE using finite...Ch. 27 - 27.25 Derive the set of differential equations for...Ch. 27 - 27.26 Consider the mass-spring system in Fig....Ch. 27 - 27.27 The following nonlinear, parasitic ODE was...Ch. 27 - A heated rod with a uniform heat source can be...Ch. 27 - 27.29 Repeat Prob. 27.28, but for the following...Ch. 27 - 27.30 Suppose that the position of a falling...Ch. 27 - Repeat Example 27.3, but insulate the left end of...
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