# Theoretical model for damped harmonic oscillation.

• Daniel Sellers
In summary, the conversation discusses the topic of determining the damping constant for a model of a vertically oscillating mass on a spring. The individual is struggling to predict the damping constant and is seeking advice on how to approach the problem. Suggestions are made to model the drag force as linear or quadratic with respect to velocity, and to obtain the parameters experimentally. The individual has also sought help from teachers and plans to move forward with a strategy suggested by them. It is mentioned that a theoretical model does not necessarily have to predict the constants and can simply focus on predicting the shape of the relationship.
Daniel Sellers

## Homework Statement

Hello all,

I have a question regarding the damping constant for a model of a vertically oscillating mass on a spring. I have read through one or two similar questions on this site but I think I can manage to be a little more specific about what I'm asking.

I am in a physics lab course and have collected data from a motion detector for a mass on a spring by itself, and two sets of data for which wide, flat objects were affixed to the mass (with some metal disks removed from the mass to offset the additional mass from the 'damper').

I am asked to create a theoretical model for the equation of motion and compare it to our collected data. The issue I'm having is when I try to predict the damping constant, b. I am hoping to find an expression for drag force which is linear with respect to velocity; the constant be needs to be such that multiplying it by v gives units of force.

The model of drag force I have learned previously is dependent on (v^2)/2 and so throws off my model. I have looked into Stoke's equation since our system never reaches very large velocity, but treatments of Stoke's seem to specify that it applies only small spherical objects moving slowly enough that the airflow relative to the object is completely laminar.

So my question is this; is there a way predict b (even with somewhat low accuracy) from the area and possibly an average velocity which gives be in units of kg/s? or alternatively, is there a way to take the fairly simple drag force equation [(CA{rho}v^2)/2] and make it dependent on v instead of v^2?

Or any other ideas for how I might make a reasonable guess at b for the sake of creating a theoretical model to compare my data to. I am not versed in fluid dynamics beyond the basics like Bernouli's and the equation of continuity for laminar flow, but I will make every attempt to learn the necessary math and physics if someone could point me in the right direction.

2. Homework Equations ; unknown (or stated above)

3. Attempt at the solution;

Algebraic manipulation and seemingly endless forum searches.

A theoretical model does not necessarily have to predict the constants. It might be enough to predict the shape of the relationship (linear or quadratic) and obtain the parameters experimentally.
Why do you think it should be linear with velocity? Have you calculated the Reynolds number?

BvU
@haruspex: I believe the constant b should be linear with velocity because the model for damped harmonic oscillation is a solution to the equation of motion for a harmonic oscillator, where the force responsible for damping is described as F = -bv. If the constant b then changes with velocity it introduces a whole new set of problems lab report (which I'm not sure I have time to learn and write about before this is due.)

I have spoken to teachers at my university since posting this question who have suggested I model the drag force as if one of the velocity terms is constant, based on the approximate average of the velocity of the system. I am going to move forward with this strategy, but if anyone would like to discuss it here, please feel free!

I have been given all the help I need, I believe. Thanks for your reply.

Daniel Sellers said:
the constant b should be linear with velocity because the model for damped harmonic oscillation is a solution to the equation of motion for a harmonic oscillator, where the force responsible for damping is described as F = -bv.
Yes, that is the classical model referred to as a damped harmonic oscillator, but I see no reason why the physical set up you describe should be dominated by viscous forces. You wrote that you are asked to create a theoretical model for the actual experiment. Quadratic seems more likely.
And, yes, quadratic is far harder to deal with.

Edit: more thoughts...
A theoretical model can just be a differential equation. It does not have to be solvable analytically. If you have the time, I suggest setting up a quadratic model in software (just a spreadsheet would do) and see which can be tuned to give the better fit to the data.

Last edited:

## What is a theoretical model for damped harmonic oscillation?

A theoretical model for damped harmonic oscillation is a mathematical representation of a system that undergoes oscillatory motion with a gradually decreasing amplitude due to the presence of a damping force. It takes into account the mass, spring constant, damping coefficient, and initial conditions of the system.

## How is the damping coefficient related to the rate of decay in a damped harmonic oscillator?

The damping coefficient is directly proportional to the rate of decay in a damped harmonic oscillator. This means that as the damping coefficient increases, the rate of decay also increases, resulting in a faster decrease in amplitude of the oscillations.

## Can a damped harmonic oscillator exhibit overshoot?

Yes, a damped harmonic oscillator can exhibit overshoot, which is when the amplitude of the oscillations momentarily exceeds the equilibrium position before returning to it. This can happen in cases of heavy or critical damping where the system does not have enough energy to overcome the damping force.

## What is the difference between underdamping, critical damping, and overdamping in a damped harmonic oscillator?

Underdamping occurs when the damping force is less than the critical damping force, resulting in oscillations with a decreasing amplitude. Critical damping occurs when the damping force is equal to the critical damping force, resulting in oscillations that decay to the equilibrium position without any overshoot. Overdamping occurs when the damping force is greater than the critical damping force, resulting in oscillations that quickly decay to the equilibrium position without any overshoot.

## How does a change in the mass or spring constant affect the oscillations in a damped harmonic oscillator?

A change in the mass or spring constant affects the frequency of the oscillations in a damped harmonic oscillator. A decrease in mass or increase in spring constant will result in a higher frequency, while an increase in mass or decrease in spring constant will result in a lower frequency. However, the rate of decay and type of damping will remain the same.

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