# Resonance - working out damping constant and damping factor

• Anabelle37
In summary, the conversation is about a lab report on resonance and finding the damping constant and damping factor. The person has graphed amplitude number (n) vs ln(A) and is trying to use the equation 5 to obtain an equation for t in terms of wo, gamma, and the observed amplitude number n. They are struggling with understanding the negative slope of the graph and how it relates to gamma. They also mention confusion about amplitude number (n) and its relationship to time.
Anabelle37

## Homework Statement

I'm doing a lab report on resonance and I'm trying to find the damping constant and damping factor. I measured as many amplitudes(A) for successive oscillations as possible and graphed amplitude number (n) vs. ln(A)

Part 1:Using equation 5, Plot a graph of amplitude number n vs ln(A) (the amplitude in
any units of convenience). The starting amplitude should be marked as n = 0.

Part 2:Now, by assuming that phi = 0, derive an equation for t in terms of wo, gamma and the observed amplitude number n. The starting amplitude corresponds to n = 0, so the next amplitude would be n =1 etc. Once you obtain the equation for t, rewrite equation (5), so that an equation relating n and y is obtained. By taking natural log of both sides, an equation of a straight line can now be obtained. Now do some algebra to obtain expression for gamma interms of the slope of this straight line. Measure the slope of your graph and hence get the value of gamma.

## Homework Equations

equation 5: y=(Ce-gamma*t).(cos(w't - phi))

where w' = sqrt(wo2 - gamma2)

## The Attempt at a Solution

So I've graphed amplitude number n vs ln(A) and it gives negative gradients. i used this value as my gamma but because its negative value it gives me a value of < 1 for my damping constant, d and d should be >1

is the slope of the graph not meant to be gamma??
i cannot figure out how to do part 2 of the hint (rearranging formulas to get necessary expressions)

$$A=ce^{-\gamma t}$$
and
$$lnA=lnc -\gamma t$$
$$lnA/dt=-\gamma$$
You get negative slope from graph
$$-|a|$$
It is equal to
$$-|a|=-\gamma$$
and
$$|a|=\gamma$$

Ok thanks heaps.
Just confused about one thing...my graph was amplitude number (n) vs. ln(A) not time versus ln(A). does gamma still equal the positive value of the slope from that graph?

Please explain what is amplitude number (n).
n depends on time.
May be you measure amplitude A when cos(w't)=1 and phase is w't=2 pi n.
You could also measure when phase is w't= pi n - when one amplitude is
positive and other negative.

I would recommend double checking your calculations and graphing to ensure accuracy. It is possible that there may have been a mistake in your measurements or calculations. Additionally, make sure you are using the correct units and equations for your specific experiment.

For part 2 of the hint, you can start by setting phi = 0 and rearranging equation 5 to solve for t. Then, plug this value for t back into equation 5 and rearrange it to solve for gamma. This should give you an equation relating n and gamma. By taking the natural log of both sides, you should be able to get an equation in the form of y = mx + b, where m is the slope and b is the y-intercept. The slope of this line should be equal to -gamma, so by measuring the slope of your graph, you can solve for gamma. Again, make sure to double check your calculations and units to ensure accuracy.

## 1. What is resonance?

Resonance is a phenomenon that occurs when a vibrating system is subjected to an external force that matches its natural frequency. This causes the system to vibrate with a larger amplitude, potentially leading to destructive effects.

## 2. What is the damping constant?

The damping constant, also known as the damping coefficient, is a measure of how quickly a vibrating system loses energy over time. It is typically represented by the Greek letter "ζ" (zeta).

## 3. How is the damping constant calculated?

The damping constant can be calculated by dividing the damping ratio by the natural frequency of the system. The damping ratio is the ratio of the actual damping (resistance to motion) to the critical damping (minimum amount of damping required to prevent oscillation).

## 4. What is the damping factor?

The damping factor is a dimensionless measure of how quickly a vibrating system's amplitude decays over time. It is related to the damping constant and can be calculated by taking the square root of the damping constant.

## 5. How does damping affect resonance?

Damping plays a critical role in controlling resonance. A higher damping constant or factor will result in a smaller amplitude of vibration and a quicker dissipation of energy, reducing the effects of resonance. In contrast, a lower damping constant or factor can lead to destructive resonance and potential damage to the system.

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