Exploring Damped Harmonic Motion

• romd
In summary, The equation of the graph describing the effect of different sized dampers on the time taken for amplitude of oscillations to halve is ln(T) = n.ln(A) + ln(b).

romd

Concerning damped harmonic motion (eg. mass on a spring, using cardboard discs as dampers); for the equation (below) of the graph describing the effect of different sized dampers on the time taken for amplitude of oscillations to halve, what would b (y-intercept) and n (gradient) represent? (A=area of damper; T=time taken for amplitude to halve)

$$T=b.A^n$$
$$ln(T)=n.ln(A) + ln(b)$$

Thanks

Blech... I vaguely recall this stuff from a control engineering class.

Wikipedia has some useful background:
http://en.wikipedia.org/wiki/Damping

What helps me is to consider limiting cases- the area of the damper going to zero, or infinity, for example. Let's first consider a damper of unit area: A^n is always 1 then. Then b is linearly related to the 'damping time', and is probably connected to the 'damping ratio' of a damped oscillator.

That leaves 'n', which is the effect of varying the area of the damper. Question- how does varying the area change the damping time? is it a linear relationship (n=1)? nonlinear (n>1)? sublinear (n<1)? I don't know the answer, but that's what 'n' represents.

Thanks for the reply!

Now, I have a value for both constants, and both a graph of T against A and one of log(T) against log(A). This is for coursework, and having neglected to find spring constant k or any other potentially useful information, I am having trouble making my interpretation and conclusions 'worthwhile'- other than stating vague implications of the values of b and n. With the data I have would it be possible to find a complete equation for the motion of the spring? Thanks

Well, you may be able to figure out actual values for 'n' and 'b'... What exactly is the purpose of the lab?

I have values but without n and b directly representing anything physical I'm finding it hard to go very in-depth in analysis; for alevel coursework I think more than just a few lines would be needed. The aim was to investigate the effect of damping on SHM of a spring-mass system- I left it vague because at the time didn't know how I would go about it

Oh. That's a different question than what you initially asked, I think. How did you arrive at the equation in your original post? Was it given to you, or did you guess using Excel or something?

It was given to me, as an equation for the graph. Initially I though b and n would represent something physical, but it seems they don't, at least not directly - eg. you said b would be connected to the damping coefficient, but without knowing how I can't write much on it.

Hmmm.. Well, when I get ambiguous comments from reviewers, my strategy is to first re-state the comment as best I can in terms that I do understand, and then provide a response. Sometimes that works, sometimes it doesn't.

So, think about how the damping occurs in your system- demonstrate you understand a damped oscillator. Then, think about what a power-law (T ~ A^n) means physically- that would be impressive to talk about- and then try and relate the two. For example- the area of the disk is related to the mass of the damper, and so is related to 'b' as well: can you re-write the power law in terms of the mass of the damper?

1. What is damped harmonic motion?

Damped harmonic motion is a type of oscillation in which the amplitude of the motion decreases over time due to the presence of a damping force. This force acts to counter the motion and dissipates energy, resulting in the gradual decrease of amplitude.

2. What are the factors that affect the damping of harmonic motion?

The damping of harmonic motion can be affected by several factors, including the type of damping force, the amplitude and frequency of the motion, and the properties of the oscillating object, such as its mass and stiffness.

3. How is damped harmonic motion different from simple harmonic motion?

In simple harmonic motion, the amplitude remains constant while the frequency stays the same. In damped harmonic motion, the amplitude decreases over time, and the frequency may also change due to the damping force.

4. What are some real-life examples of damped harmonic motion?

Some common examples of damped harmonic motion include a swinging pendulum slowed down by air resistance, a car's suspension system reducing the vibrations caused by bumps on the road, and the motion of a mass-spring system with friction.

5. How is damped harmonic motion used in science and engineering?

Damped harmonic motion has various applications in science and engineering, such as in designing shock absorbers for vehicles, studying the behavior of electrical circuits, and analyzing the motion of atoms in a material. It is also used in seismology to study earthquake vibrations and in music to produce different sounds and tones.