Modelling a spring system with damping force and external forces

Thread starter sss1
Start date Sep 30, 2023
Tags

Damping Spring System

In summary, the modeling of a spring system with damping force and external forces involves analyzing the dynamics of a mass-spring-damper system. This includes the spring's restoring force, which is proportional to its displacement, and the damping force, which opposes motion and is proportional to the velocity. External forces, such as applied loads or friction, are also considered. The system can be described using differential equations that account for these forces, allowing for the prediction of the system's behavior over time, including oscillations and steady-state responses.

Sep 30, 2023

sss1

Homework Statement: I know for a system with no external forces there are conditions for being underdamping, overdamping and critically damped. Is there also such conditions for systems having external forces acting on them also? Specifically, for the example 10y''+9y"+2y'=-2e^(-t/2) with conditions y(0)=0 and y'(0)=0, is the system critically damped?

Relevant Equations: 10y''+9y"+2y'=-2e^(-t/2)

I think its critically damped by looking at the graph of the solution.

Physics news on Phys.org

Sep 30, 2023

kuruman

Science Advisor

Homework Helper

Insights Author

Gold Member

2023 Award

Can you give an example of a system with no external forces that exhibits damping and a system with external forces that also exhibits damping? I do not understand your use of "external force" at least not in the Newtonian sense.

Also, if 10y''+9y"+2y'=-2e^(-t/2), why not 19y''+ 2y'=-2e^(-t/2)? Is there a real difference between y'' and y"?

FAQ: Modelling a spring system with damping force and external forces

What is the general form of the differential equation for a damped spring system with an external force?

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.

How do you determine the type of damping in a spring system?

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.

What is the solution to the differential equation for an underdamped spring system?

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.

How do external forces affect the behavior of a damped spring system?

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.

How can numerical methods be used to solve the differential equation for a damped spring system with external forces?

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.