Constant damping force on springsystem

[P3X-018]

If we consider a block of mass m attached to a spring, where the system oscillates on a table with friction f, the friction force f on the block would depend on the direction of the velocity, as

$$m\ddot{x} = \begin{cases} -kx+f & \text{if } \dot{x}<0\\ -kx-f & \text{if } \dot{x} > 0 \end{cases}$$

If I just look at one equation at a time and solve them both separatly first, I get equations where the amplitude doesn't drop with time. But that should be the case (energy in that closed system isn't conserved). So how can I solve this system?

 

[OlderDan]

If we consider a block of mass m attached to a spring, where the system oscillates on a table with friction f, the friction force f on the block would depend on the direction of the velocity, as

$$m\ddot{x} = \begin{cases} -kx+f & \text{if } \dot{x}<0\\ -kx-f & \text{if } \dot{x} > 0 \end{cases}$$

If I just look at one equation at a time and solve them both separatly first, I get equations where the amplitude doesn't drop with time. But that should be the case (energy in that closed system isn't conserved). So how can I solve this system?

The direction of $\text{f}$ changes abruptly every time the velocity canges direction. You solve this by solving x(t) for every time interval that has constant $\text{f}$ separately. Solving for each half period separately may appear to give no change in amplitude, as if the mass were hanging on a spring in the presence of gravity, but in fact you are being fooled into thinking that is the case. The change in direction of $\text{f}$ every time the velocity chages direction is going to reduce the amplitude. You can find out the magnitude of this effect from energy considerations, or you can think about what a constant applied force does to the equilibrium position of the oscillator.

 

[kyle8921]

The presence of a constant damping force in a spring system can significantly affect the behavior of the system. In this case, the friction force is dependent on the direction of the velocity, which means that the damping force will change as the block moves back and forth on the spring.

To solve this system, you will need to use a differential equation solver that can handle non-linear systems. You can also use numerical methods, such as Euler's method or Runge-Kutta methods, to approximate the solution.

One important factor to consider is that the damping force will cause the amplitude of the oscillations to decrease over time. This is because the energy in the system is being dissipated by the friction force. Therefore, the amplitude of the oscillations will decrease until the system reaches equilibrium, where the damping force is equal to the restoring force of the spring.

To account for this decrease in amplitude, you will need to include the damping force term in your equations of motion and solve for the time-dependent amplitude. This will give you a more accurate solution that takes into account the effect of the damping force.

In summary, solving a system with a constant damping force on a spring will require a non-linear differential equation solver and the inclusion of the damping force term in the equations of motion. This will allow you to accurately model the behavior of the system and account for the decrease in amplitude over time.