For a simple pendulum consisting of a spherical mass $m$ anchored to a fixed point $O$ via a mass-less rod of length, perform dimensional analysis on the problem to find the period of oscillation τ (i.e., time scale of the system). Use Newton’s second law to write the governing ordinary differential equation (ODE); a solution to this nonlinear equation should relate θ, or an equivalent variable, to time $t$. Then, for small deviation angles $θ$, use linear approximation to linearize the ODE. Analytically solve this linear ODE to find θ as a function of t. Let $θ = θ_0$ and $\overset{.}{\theta} = 0$ be the initial conditions of the problem. Using the concept of dimensional homogeneity, find the time scale of the problem from the obtained solution. Compare it to the one you found from dimensional analysis.
Since the simple pendulum is a common problem in physics and math, is there any research paper or book where I can study the solution for this problem? Other resources where drag force is also considered would be helpful as well.
For a reference for this problem you can check out Sec 6.7 in Strogatz, nonlinear dynamics and chaos
Eq. 1 and 2 performed the non-dimensionalization of the ODE and found the timescale $\tau$. $$ \frac{d^2\theta}{dt^2}+\frac{g}{L}\sin\theta=0 $$ Let $$ \tau:=\sqrt\frac{L}{g}\\ $$ Then we get can redefine the time variable $\tilde t=t/\tau$ and get the dimensionless equation. $$ \frac{d^2\theta}{d\tilde t^2}+\sin\theta=0 $$ Then we can say $\tau$ is the natural timescale of the system.
Check out that fantastic chapter for more details.