Proof for peak time for an underdamped second-order system

Question

I've been attempting to prove the taught result that:

$$T_p=\frac{\pi}{\omega_n\sqrt{1-\zeta^2}}\textrm{.}$$

From the time-domain form of an underdamped response, we have

$$c(t)=K_1e^{-\sigma_dt}\cos(\omega_dt-\phi)\textrm{.}$$

In an attempt to find maxima, the first derivative has been found as

$$\frac{dc(t)}{dt}=K_1\left(-\sigma_de^{-\sigma_dt}\cos(\omega_dt-\phi)-\omega_de^{-\sigma_dt}\sin(\omega_dt-\phi)\right)\textrm{.}$$

From here, assuming \$K_1\textrm{, }e^{-\sigma_dt}\ne0\$, the maxima are found at

$$\omega_d\sin(\omega_dt-\phi)+\sigma_d\cos(\omega_dt-\phi)=0\textrm{.}$$

Re-arranging for \$t\$, we find

$$t=\frac{-\tan^{-1}\left(\frac{\sigma_d}{\omega_d}\right)+\phi +n\pi}{\omega_d}\textrm{.}$$

The denominator clearly aligns with the taught result, as the damped frequency is related as such to the natural frequency. However, I do not see how the numerator aligns to \$\pi\$. Where does the \$\tan\$ term go? I'm aware that we're hunting for the first peak, so it makes sense for \$n=0/1\$? Any help would be appreciated.

Answer 1

I was assuming that the \$\phi\$ phase constant was arbitrary, but it is the result of a contraction from: $$\cos\omega_n\sqrt{1-\zeta^2}t+\frac{\zeta}{\sqrt{1-\zeta^2}}\sin\omega_n\sqrt{1-\zeta^2}t\textrm{.}$$ This can be represented as $$\frac{1}{\sqrt{1-\zeta^2}}\cos\left(\omega_n\sqrt{1-\zeta^2}-\phi\right),$$ with \$\phi=\tan^{-1}{\frac{\zeta}{\sqrt{1-\zeta^2}}}\$.

It is also fairly trivial to find that $$\frac{\sigma_d}{\omega_d}=\frac{\zeta\omega_n}{\omega_n\sqrt{1-\zeta^2}}=\frac{\zeta}{\sqrt{1-\zeta^2}},$$ so the \$\tan^{-1}\$ term and the \$\phi\$ term cancel in the original equation, leaving $$\frac{n\pi}{\omega_n\sqrt{1-\zeta^2}}.$$ For \$n=0\$, this is the lowest point before the response begins, so for \$n=1\$, this gives the first peak value, as expected. Additional thanks to other answers which gave guidance.

Answer 2

The inverse tangent is cancelled by the phase shift angle. To see it you should present EVERY term as expressions of the natural frequency and the damping factor. The basic 2nd order low-pass system doesn't have other parameters.