Underdamped motion (Formula deriving)

[noppawit]

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

 

[tiny-tim]

hi noppawit!

Pe^ix + Qe^-ix

= ((P+Q)/2)cosx + i((P-Q)/2)sinx …

since we know the solution must be real, we can assume P+Q and i(P-Q) are real,

so this is in the form Rsinx + Scosx, which we can rewrite as √(R² + S²)sin(x + tan^-1S/R)

 

[noppawit]

hi noppawit!

Pe^ix + Qe^-ix

= ((P+Q)/2)cosx + i((P-Q)/2)sinx …

since we know the solution must be real, we can assume P+Q and i(P-Q) are real,

so this is in the form Rsinx + Scosx, which we can rewrite as √(R² + S²)sin(x + tan^-1S/R)

Thank you :)