ME 205: Homework 6
Instructor: Subhrajit Bhattacharya
Reading assignment: Spectral theorem for symmetric matrices; decoupling systems of linear o.d.e’s with constant
coefficients; Introduction to Fourier series (Sections 8.1, and relevant parts of 8.4 & 11.1–11.4 from Kreyszig).
Solve the following problems on paper and submit: Solve all problems/parts and clearly mark them with
the problem/part number. Only 10 points worth of problems will be selected at random for grading.
Problem 1 :
k5
m3
k3
k4
m1 m
2
k1
k2
Consider the above system, for which you found the equation of motion and its general solution in an earlier
homework. For simplicity, assume that there is no gravity and assume system parameters m1 = m2 = m3 = 5 Kg
and k1 = k2 = k3 = k4 = k5 = 500 N/m.
¨ + B x̄ = 0 into the decoupled form ξ¨¯ + Γξ¯ = 0. Write down the diagonal
Convert the coupled equation of motion x̄
matrix Γ (in numerical terms) and the relationship between the vectors x̄ and ξ¯ (10 points).
Problem 2 :
Consider the following functions:
i. f (t) = sin(3t) − 31 cos( 52 t)
ii. f (t) = 2t − b 2t c
(where, bxc is the floor function, that returns the largest integer less than or equal to x)
For each of the functions, f , above, do the following:
a. Draw the rough sketch of the function.
b. Write down the period of the function.
c. Compute the FourierPseries representationPof the function (i.e., compute the coefficients a0 , a1 , a2 , · · · and
∞
∞
nπ
nπ
b1 , b2 , · · · in f (t) =
n=0 an cos( L t) +
n=1 bn sin( L t) for the appropriately chosen value of L. Hint:
Consult the “Summary of Important Concepts and Formulae” document posted on course-site for the formulae
for computing the coefficients an and bn from the function f .)
d. If yωSf is a specific solution to the equation ÿ + Aẏ + By = sin(ωf t), and yωCf is a specific solution to the
equation ÿ + Aẏ + By = cos(ωf t), write down a specific solution to the equation ÿ + Aẏ + By = f (t) in terms
of yωSf and yωCf (with appropriately chosen values of ωf ).
(3+2+2+3 + 2+2+7+4 points)
1