# Constant damping force on springsystem

• P3X-018
In summary, the block's oscillation on a table with friction depends on the direction of its velocity, causing the friction force to change abruptly. To solve the system, you must separately solve for each half period with a constant friction force, which will result in a decrease in amplitude due to the change in direction of the force. Energy considerations or considering the effect of a constant applied force on the equilibrium position can help determine the magnitude of this decrease.
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?

P3X-018 said:
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.

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.

## 1. What is a constant damping force on a springsystem?

A constant damping force on a springsystem is a force that is applied to a spring in order to counteract its natural tendency to oscillate or vibrate. This force is typically caused by friction or air resistance and helps to dampen the motion of the spring, resulting in a more stable and controlled movement.

## 2. How does a constant damping force affect the behavior of a springsystem?

A constant damping force can significantly reduce the amplitude of the spring's oscillations and also decrease the time it takes for the spring to come to rest. This results in a more controlled and predictable motion of the spring.

## 3. What are the benefits of using a constant damping force on a springsystem?

There are several benefits of using a constant damping force on a springsystem. It can help to reduce unwanted vibrations and ensure a smoother and more stable motion. It can also prevent damage to the spring and other components of the system by reducing the stress and strain caused by excessive oscillations.

## 4. How is a constant damping force calculated and applied to a springsystem?

The amount of constant damping force required for a specific springsystem can be calculated using mathematical equations that take into account factors such as the mass of the object attached to the spring, the spring constant, and the desired level of damping. This force can then be applied using various methods such as adding a damper or adjusting the position of the spring in the system.

## 5. Are there any limitations or drawbacks to using a constant damping force on a springsystem?

While a constant damping force can greatly improve the behavior and stability of a springsystem, there are some limitations and drawbacks to consider. The force applied may not be constant over time due to factors such as temperature changes or wear and tear on the damping mechanism. Additionally, using too much damping force can restrict the motion of the spring too much, resulting in decreased performance or even failure of the system.

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