Dynamics cheat sheet

7 Some common formulas
These definitions are used throughout the derivations below.
ξ = c c r
=
c
2 √ km = c
2ω n m
u st = F static deflection
√
k
k
ω n =
m
ω D = ω n
√
1 − ξ 2 note: not defined for ξ > 1 since becomes complex
r = ϖ ω n
T d = 2π damped period of oscillation
ω
{ d
−1
τ = , −1 }
time constants where λ i are roots of characteristic equation
λ 1 λ2
1
β =
magnification factor
√(1 − r 2 ) 2 + (2rξ) 2
β max when r = √ 1 − 2ξ 2
1
β max =
2ξ √ 1 − ξ 2
y n
= e ζωn2π
ω D
y
( n+1
)
yn
ln = ζ2π
y n+1
( )
1
M ln yn
= ζ2π
y n+M
small damping
⇒
√
ζ2π
1−ζ 2
e
⇒ e ζ2π
This table shows many cycles it takes for the peak to decay by half its original value as a function of the
damping ζ. For example, we see that when ζ = 2.7% then it takes 4 cycles for the peak (i.e. displacement) to
reduce to half its value.
data = Table[{i, (1/i Log[2]/(2*Pi)*100)}, {i, 1, 20}];
TableForm[N@data,
TableHeadings -> {None, {Column[{"number of cycles",
"needed for peak", "to decay by half"}], "\[Zeta] (%)"}}]
31