Harmonic oscillation with friction

In summary, the small block is attached to a horizontal spring on a table. The kinetic friction causes two forces on the block, one from the spring and one from the kinetic energy of the block.
  • #1
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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:

[itex]x''(t)=-kx-F_{friction}[/itex]

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.
 
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  • #2
You mean that [itex]F_\mathrm{friction}(t) = \mu x' (t)[/itex]?

How would you solve it without the friction term? I guess you would make an Ansatz for the form of the solution such as [itex]x(t) = e^{\lambda t}[/itex] ?
 
  • #3
CompuChip said:
You mean that [itex]F_\mathrm{friction}(t) = \mu x' (t)[/itex]?

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

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.
 
  • #4
Thank you very much for replying:)
 
  • #5


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!
 

FAQ: Harmonic oscillation with friction

1. What is harmonic oscillation with friction?

Harmonic oscillation with friction is a type of motion in which a system or object moves back and forth repeatedly around a fixed equilibrium position, while also experiencing resistance from frictional forces. It can be described mathematically using the equation x = A cos(ωt + φ), where x is the displacement, A is the amplitude, ω is the angular frequency, and φ is the phase angle.

2. What factors affect the frequency of a harmonic oscillator with friction?

The frequency of a harmonic oscillator with friction is affected by the mass of the object, the spring constant, the damping coefficient, and the initial conditions such as the amplitude and phase angle. The presence of friction also affects the frequency, as it causes the oscillations to gradually decrease in amplitude and frequency over time.

3. How does friction impact the motion of a harmonic oscillator?

Friction acts as a resistive force in harmonic oscillation, reducing the amplitude and frequency of the oscillations over time. This is due to the conversion of mechanical energy into heat energy, causing the system to lose energy and gradually come to rest.

4. Can a harmonic oscillator with friction reach a state of equilibrium?

No, a harmonic oscillator with friction will not reach a state of equilibrium because the presence of friction will always cause the system to lose energy and eventually come to rest. However, the amplitude and frequency of the oscillations will decrease over time and the system will reach a steady state, known as a damped oscillation.

5. How is harmonic oscillation with friction different from simple harmonic motion?

The main difference between harmonic oscillation with friction and simple harmonic motion is the presence of frictional forces. In simple harmonic motion, there is no resistance or friction, so the amplitude and frequency of the oscillations remain constant. In harmonic oscillation with friction, the amplitude and frequency gradually decrease over time due to the presence of friction.

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