Harmonic oscillation with friction

[nos]

Hello,

I want to include kinetic friction into the harmonic oscillator.
A small blocks is attached to a horiontal spring on a table.
Because there is kinetic friction there are two forces on the blok that we need to describe the oscillation.

First, the force that the spring exerts and second, the kinetic friction.
The kinetic friction is always opposite to the velocity but it is not proportional to velocity.

The differential equation is:

$x''(t)=-kx-F_{friction}$

How can I rewrite Fore of friction to solve this equation? force of friction is not just a constant since it depends on the direction of the velocity.

 

[CompuChip]

You mean that $F_\mathrm{friction}(t) = \mu x' (t)$?

How would you solve it without the friction term? I guess you would make an Ansatz for the form of the solution such as $x(t) = e^{\lambda t}$ ?

 

[AlephZero]

You mean that $F_\mathrm{friction}(t) = \mu x' (t)$?

How would you solve it without the friction term? I guess you would make an Ansatz for the form of the solution such as $x(t) = e^{\lambda t}$ ?

Most engineers (at least in the UK and US) would call that "viscous damping", not "friction".

For the Coulomb model of friction, F in the OP's equation is constant, and its sign depends on the sign of the velocity.

You can easily solve the two separate cases where F is positive or negative. The solution is the same as if the mass and spring was vertical, and F was the weight of the mass.

For the complete solution, you start with one of the two solutions (depending on the initial conditiosn) until the velocity = 0, then you switch to the other solution, and so on. You can't easily get a single equation that gives the complete solution in one "formula" for x.

The graph of displacement against time will look like a sequence of half-oscillations of simple harmonic motion, with amplitudes that decrease in a linear progression (not exponentially). The mass will stop moving after a finite number of half-osciillations, at some position where the static friction force can balance the tension in the spring.

 

[nos]

Thank you very much for replying:)

 

[PJC01]

Hello,

Thank you for your inquiry. Including kinetic friction into the harmonic oscillator is an interesting idea. In order to solve the equation, you can rewrite the force of friction as a function of velocity. This can be done by using the coefficient of kinetic friction, μ, which is a constant that depends on the materials in contact and the surface area. The force of friction can then be written as μmg, where m is the mass of the block and g is the acceleration due to gravity. This force acts in the opposite direction of the velocity, so it can be written as -μmgv. This can then be substituted into the differential equation you provided, resulting in:

x''(t)=-kx-μmgv

Solving this equation will give you the position of the block as a function of time, taking into account the effects of both the spring and the kinetic friction. I hope this helps. Good luck with your research!