d) Substituting the trial solution ï = p cos(wt + ) into the equations of motion, write down the resulting matrix eigenvalue equation. e) Determine the two eigenvalues w² for the matrix equation and the corresponding eigen- vectors.
d) Substituting the trial solution ï = p cos(wt + ) into the equations of motion, write down the resulting matrix eigenvalue equation. e) Determine the two eigenvalues w² for the matrix equation and the corresponding eigen- vectors.
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a b c are fine. please help d and e
Transcribed Image Text:c) Hereafter assume m₁ = m, m² = 2m and k₁ = k2 = k3 = k. Show the equations of
motion in the matrix form is
where M and G are 2 × 2 matrices with
0
(m
1)
M
=
Mi+Gx = 0
0 2m
and G
=
2k
( 24 2 k)
-k 2k
d) Substituting the trial solution x = p cos(wt + p) into the equations of motion, write
down the resulting matrix eigenvalue equation.
e) Determine the two eigenvalues w² for the matrix equation and the corresponding eigen-
vectors.
f) The eigenvectors you found above will not be orthogonal. Explain why.
Transcribed Image Text:A one-dimensional system of two bodies of mass m₁ and m2 and three springs with
Hooke's law constants k₁, k2, and k3 is fixed between two walls as shown in the figure.
The displacements of the two bodies from their equilibrium positions are î₁ and £2, as
shown in the figure.
0000
k₁
X₁
1
m
1
୪୪୪୪
k₂
X2
m2
0000
k₂
a) Write down the total force acting on each of the two bodies. (Note it is not necessary
to determine the equilibrium positions relative to the walls.)
b) Write down the equations of motion for the two bodies.
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