A mass m on a spring with constant k satisfies the differential equation (see Section 3.7) mu′′+ku=0,mu″+ku=0, where u(t) is the displacement at time t of the mass from its equilibrium position. a.Let x1 = u, x2 = u′, and show that the resulting system is x′=(01−km0)x.x′=01−km0x. b.Find the eigenvalues of the matrix for the system in part a c.Sketch several trajectories of the system. Choose one of your trajectories, and sketch the corresponding graphs of x1 versus t and x2 versus t. Sketch both graphs on one set of axes. d.What is the relation between the eigenvalues of the coefficient matrix and the natural frequency of the spring‑mass system?
A mass m on a spring with constant k satisfies the differential equation (see Section 3.7) mu′′+ku=0,mu″+ku=0, where u(t) is the displacement at time t of the mass from its equilibrium position. a.Let x1 = u, x2 = u′, and show that the resulting system is x′=(01−km0)x.x′=01−km0x. b.Find the eigenvalues of the matrix for the system in part a c.Sketch several trajectories of the system. Choose one of your trajectories, and sketch the corresponding graphs of x1 versus t and x2 versus t. Sketch both graphs on one set of axes. d.What is the relation between the eigenvalues of the coefficient matrix and the natural frequency of the spring‑mass system?
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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A mass m on a spring with constant k satisfies the differential equation (see Section 3.7)
mu′′+ku=0,mu″+ku=0,
where u(t) is the displacement at time t of the mass from its equilibrium position.
a.Let x1 = u, x2 = u′, and show that the resulting system is
x′=(01−km0)x.x′=01−km0x.
b.Find the eigenvalues of the matrix for the system in part a
c.Sketch several trajectories of the system. Choose one of your trajectories, and sketch the corresponding graphs of x1 versus t and x2 versus t. Sketch both graphs on one set of axes.
d.What is the relation between the eigenvalues of the coefficient matrix and the natural frequency of the spring‑mass system?
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