Dynamics cheat sheet

x 1
x 2
k 1 x 1
k 2 x 2 x 1
m
k 2 x 2 x 1
m 2
1
f 1 t
f 2 t
F m 1 x 1
k 1 x 1 k 2 x 2 x 1 f 1 t m 1 x 1
F m 2 x 2
k 2 x 2 x 1 f 2 t m 2 x 2
Hence, from the above the equations of motion are
or
In Matrix form
⎡ ⎤ ⎧
⎣ m 1 0 ⎨
⎦
0 m
⎩
2
m 1 x ′′
1 + k 1 x 1 − k 2 (x 2 − x 1 ) = f 1 (t)
m 2 x ′′
2 + k 2 (x 2 − x 1 ) = f 2 (t)
m 1 x ′′
1 + x 1 (k 1 + k 2 ) − k 2 x 2 = f 1 (t)
m 2 x ′′
2 + k 2 x 2 − k 2 x 1 = f 2 (t)
x ′′
1
x ′′
2
⎫ ⎡
⎤ ⎧ ⎫ ⎧ ⎫
⎬
⎭ + ⎣ k 1 + k 2 −k 2
⎨x ⎦ 1 ⎬ ⎨
−k 2 k
⎩
2 x
⎭ = f 1 (t) ⎬
⎩
2 f 2 (t)
⎭
The above two EOM are coupled in stiffness, but not mass coupled. Using short notations, the above is
written as
[M]{x ′′ } + [K]{x} = {f}
Modal analysis now starts with the goal to decouple the EOM and obtain the fundamental shape functions that
the system can vibrate in. To make these derivations more general, the mass matrix and the stiffness matrix are
written in general notations as follows
⎡ ⎤ ⎧
⎣ m 11 m 12
⎨
⎦
m 21 m
⎩
22
x ′′
1
x ′′
2
⎫ ⎡ ⎤ ⎧ ⎫ ⎧ ⎫
⎬
⎭ + ⎣ k 11 k 12
⎨x ⎦ 1 ⎬ ⎨
k
⎩
22 x
⎭ = f 1 (t) ⎬
⎩
2 f 2 (t)
⎭
k 21
The mass matrix [M] and the stiffness matrix [K] must always come out to be symmetric. If they are not
symmetric, then a mistake was made in obtaining them. As a general rule, the mass matrix [M] is PSD (positive
definite matrix) and the [K] matrix is positive semi-definite matrix. The reason the [M] is PSD is that x T [M]{x}
represents the kinetic energy of the system, which is typically positive and not zero. But reading some other
references 1 it is possible that [M] can be positive semi-definite. It depends on the application being modeled.
1.2 Step 2, solving the eigenvalue problem, finding the natural frequencies
The first step in modal analysis is to solve the eigenvalue problem det ( [K] − ω 2 [M] ) = 0 in order to determine
the natural frequencies of the system. This equations leads to a polynomial in ω and the roots of this polynomial
1 http://en.wikipedia.org/wiki/Fundamental_equation_of_constrained_motion
4