Consider a system of two toy railway cars (i.e., frictionless masses) connected to each other by two springs, one of which is attached to the wall, as shown in the figure. Let x and y be the displacement of the first and second masses from their equilibrium positions. Suppose the masses are m1 = 2 kg and m2 = 1 kg, and the spring constants are ki = 64 N/m and k2 = 32 N/m. a. Set up a system of second-order differential equations that models this situation. This situation is the same model as discussed in class except that there is no third spring, i.e., kz = 0. b. Compute the eigenvalues of the coefficient matrix A and find a corresponding eigenvector for each one. You must get all six entries correct to receive credit.
Consider a system of two toy railway cars (i.e., frictionless masses) connected to each other by two springs, one of which is attached to the wall, as shown in the figure. Let x and y be the displacement of the first and second masses from their equilibrium positions. Suppose the masses are m1 = 2 kg and m2 = 1 kg, and the spring constants are ki = 64 N/m and k2 = 32 N/m. a. Set up a system of second-order differential equations that models this situation. This situation is the same model as discussed in class except that there is no third spring, i.e., kz = 0. b. Compute the eigenvalues of the coefficient matrix A and find a corresponding eigenvector for each one. You must get all six entries correct to receive credit.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Consider a system of two toy railway cars (i.e., frictionless masses) connected to each
other by two springs, one of which is attached to the wall, as shown in the figure. Let x
and y be the displacement of the first and second masses from their equilibrium
positions. Suppose the masses are m1 = 2 kg and m2 = 1 kg, and the spring
constants are k1 = 64 N/m and k2 = 32 Ñ/m.
a. Set up a system of second-order differential equations that models this situation. This situation is
the same model as discussed in class except that there is no third spring, i.e., k3 = 0.
x"
b. Compute the eigenvalues of the coefficient matrix A and find a corresponding eigenvector for each
one. You must get all six entries correct to receive credit.
The eigenvalue
O has an eigenvector
The eigenvalue
has an eigenvector
c. Calculate the natural frequencies w of the system and enter them as a comma separated list.
d. Find the general solution to this system of differential equations. Use a1, a2, b1 , b2 to denote
arbitrary constants, and enter them as a1, a2, b1, b2.
x(t) =
y(t) =
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