A system with a small range of motion has a potential energy of U= $a\dot{θ}^2sin^2(θ)$ and kinetic energy of T=$bθ^4$, where a and b are known positive parameters. Is it possible to find the natural frequency of the system based on this information? I tried using methods like the energy method, but the resulting expression has trigonometric terms, derivatives of position, and other terms that make it difficult to simplify the expression in a form such as $m\ddot{x} + kx = 0$ to be able to find the natural frequencies. The correct answer is $w_n = \sqrt{\frac{b}{a}}$.
I tried to solve the problem using the energy method but am not sure how the resulting expression can be used to find the natural frequency as it is not similar to any general expression.
$d/dt(T+U) = d/dt(a\dot{θ}^2sin^2(θ)+bθ^4)$
$0 = 2a\dot{θ}\ddot{θ}sin^2(θ)+a\dot{θ}^2\ddot{θ}sin(2θ) + 4b\dot{θ}θ^3$
Using the small angle approximation,
$0 = 2a\dot{θ}\ddot{θ}(θ^2)+a\dot{θ}^2\ddot{θ}(2θ)+ 4b\dot{θ}θ^3$
$0 = 2a\ddot{θ}(θ)+a\dot{θ}\ddot{θ}(2)+ 4bθ^2$