動吸振器
( ω 1 ω 2 ) = ω m 2 + ( μ + 1 ) ω a 2 ∓ [ ω m 2 + ( μ − 1 ) ω a 2 ] 2 + 4 μ ω a 4 2 {\displaystyle {\dbinom {\omega _{1}}{\omega _{2}}}={\sqrt {\frac {\omega _{m}^{2}+(\mu +1)\omega _{a}^{2}\mp {\sqrt {\left[\omega _{m}^{2}+(\mu -1)\omega _{a}^{2}\right]^{2}+4\mu \omega _{a}^{4}}}}{2}}}}
ここで、
ω a = k a m a , ω m = k m m m , μ = m a m m {\displaystyle \omega _{a}={\sqrt {\frac {k_{a}}{m_{a}}}},\qquad \omega _{m}={\sqrt {\frac {k_{m}}{m_{m}}}},\qquad \mu ={\frac {m_{a}}{m_{m}}}}
である。このとき各質点の振幅倍率は以下のように得られる。
X m x s t = ω m 2 ( ω a 2 − Ω 2 ) ( ω 1 2 − Ω 2 ) ( ω 2 2 − Ω 2 ) {\displaystyle {\frac {X_{m}}{x_{st}}}={\frac {\omega _{m}^{2}(\omega _{a}^{2}-\Omega ^{2})}{(\omega _{1}^{2}-\Omega ^{2})(\omega _{2}^{2}-\Omega ^{2})}}}
α o p t = 1 1 + μ 1 + μ 2 − ζ m ( 4 + μ ) μ 8 ( 1 + μ ) 3 ( 2 + μ ) ( 4 + 3 μ ) + ζ m 2 μ ( 192 + 304 μ + 132 μ 2 + 13 μ 3 ) 8 ( 1 + μ ) 2 ( 4 + 3 μ ) 2 2 ( 2 + μ ) 3 − ζ m 3 b 1 16 μ 3 2 ( 1 + μ ) 5 ( 2 + μ ) 5 ( 4 + 3 μ ) 7 {\displaystyle {\begin{alignedat}{3}\alpha _{opt}=&{\frac {1}{1+\mu }}{\sqrt {1+{\frac {\mu }{2}}}}-\zeta _{m}(4+\mu ){\sqrt {\frac {\mu }{8(1+\mu )^{3}(2+\mu )(4+3\mu )}}}\\&+\zeta _{m}^{2}{\frac {\mu (192+304\mu +132\mu ^{2}+13\mu ^{3})}{8(1+\mu )^{2}(4+3\mu )^{2}{\sqrt {2(2+\mu )^{3}}}}}\\&-\zeta _{m}^{3}{\frac {b_{1}}{16}}{\sqrt {\frac {\mu ^{3}}{2(1+\mu )^{5}(2+\mu )^{5}(4+3\mu )^{7}}}}\\\end{alignedat}}}