The general form of the differential equation for a damped spring system with an external force is: \( m \frac{d^2x}{dt^2} + c \frac{dx}{dt} + kx = F(t) \), where \( m \) is the mass, \( c \) is the damping coefficient, \( k \) is the spring constant, \( x \) is the displacement, and \( F(t) \) is the external force as a function of time.
The type of damping is determined by the damping coefficient \( c \) relative to the critical damping coefficient \( c_{crit} = 2\sqrt{mk} \). If \( c < c_{crit} \), the system is underdamped. If \( c = c_{crit} \), the system is critically damped. If \( c > c_{crit} \), the system is overdamped.
For an underdamped system (\( c < c_{crit} \)), the solution to the differential equation is: \( x(t) = e^{-\frac{c}{2m}t} \left( A \cos(\omega_d t) + B \sin(\omega_d t) \right) \), where \( \omega_d = \sqrt{\frac{k}{m} - \left(\frac{c}{2m}\right)^2} \) is the damped natural frequency, and \( A \) and \( B \) are constants determined by initial conditions.
External forces can drive the system and influence its motion. If the external force \( F(t) \) is periodic, such as \( F(t) = F_0 \cos(\omega t) \), the system can exhibit forced oscillations. The response of the system will depend on the frequency of the external force relative to the natural frequency of the system, potentially leading to resonance if the frequencies match.
Numerical methods, such as the Euler method, Runge-Kutta methods, or finite difference methods, can be used to solve the differential equation. These methods discretize the time variable and iteratively compute the system's state at each time step, providing an approximate solution to the differential equation when an analytical solution is difficult or impossible to obtain.