$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
$$
Generalization: damping, nonlinear spring, and external excitation
$$
mu'' + f(u') + s(u) = F(t),\quad u(0)=I,\ u'(0)=V,\ t\in (0,T]
$$
Input data: \( m \), \( f(u') \), \( s(u) \), \( F(t) \), \( I \), \( V \), and \( T \).
Typical choices of \( f \) and \( s \):
- linear damping \( f(u')=bu \), or
- quadratic damping \( f(u')=bu'|u'| \)
- linear spring \( s(u)=cu \)
- nonlinear spring \( s(u)\sim \sin(u) \) (pendulum)
