- #1
noppawit
- 27
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Homework Statement
I'm trying to derive the equation of motion of underdamped motion.
Homework Equations
From F = m\ddot{x}
-kx - c\dot{x} = m\ddot{x}
m\ddot{x} + c\dot{x} + kx = 0
x_{h}(t) = ae^{\lambda t}
\dot{x}(t) = \lambda ae^{\lambda t}
\ddot{x}(t) = \lambda^2 ae^{\lambda t}
Therefore, for underdamped: \lambda = -\zeta\omega_{n} \pm i\sqrt{1-\zeta^{2}}
x_{h}(t) = a_{1}e^{(-\zeta\omega_{n} + i\sqrt{1-\zeta^{2}}) t} + a_{2}e^{(-\zeta\omega_{n} - i\sqrt{1-\zeta^{2}}) t}
x_{h}(t) = e^{-\zeta\omega_{n} t}(a_{1}e^{i\sqrt{1-\zeta^{2}}) t} + a_{2}e^{- i\sqrt{1-\zeta^{2}}) t}) <---- From here (1)
x_{h}(t) = e^{-\zeta\omega_{n} t}(A sin(\omega_{n}t + \phi)) <---- To here (2)
From (1) to (2), how can it become like that? I tried from Euler Equation that e^{i\theta} = cos(\theta) + i sin(\theta), but I still cannot derive from (1) to (2)
P.S.\omega_{d} is damped natural frequency.
Noppawit
