Logarithmic decrement of a lightly damped oscillator

In summary, the logarithmic decrement is defined as the natural logarithm of the ratio of successive maximum displacements of a damped oscillator. It is related to the quality factor Q by the equation Q = ωo/(2β). To find the spring constant and damping constant of a damped oscillator, we can use the given values of mass, frequency, and logarithmic decrement. The initial conditions do not affect the solution for the exponential term in the equation of motion, which can be expressed in terms of Q.
  • #1
kraigandrews
108
0

Homework Statement


The logarithmic decrement δ of a lightly damped oscillator is defined to be the natural logarithm of the ratio of successive maximum displacements (in the same direction) of a free damped oscillator. That is, δ = ln(An/An+1) where An is the maximum displacement of the n-th cycle. Derive the simple relationship between δ and Q.
Find the spring constant k and damping constant b of a damped oscillator with mass m, frequency of oscillation f and logarithmic decrement δ.
[Data: m = 4.0 kg; f = 0.9 Hz; δ = 0.029.]
First, the spring constant k...

Also, the damping constant b...


Homework Equations



[itex]\beta[/itex]=b/(2m)
Q=[itex]\omegao/(2\beta)

The Attempt at a Solution



Given the diff eq:

d2x/dt2+2[itex]\beta[/itex](dx/dt)+[itex]\omega[/itex]o2x=0

I can solve this to find x(t), however I feel this is irrelevant because no initial condition or boundary conditions are given, so I am kinda lost here as to where go or to start at for that matter. Any suggestions are greatly appreciated, Thanks
 
Physics news on Phys.org
  • #2
The initial conditions don't matter. Can you write the exponential term of the equation of motion in terms of Q? This term controls the amplitude of the oscillator, so the ratio between successive amplitudes is just the ratio between successive exponential terms.
 

FAQ: Logarithmic decrement of a lightly damped oscillator

1. What is the definition of logarithmic decrement?

The logarithmic decrement of a lightly damped oscillator is the natural logarithm of the ratio of the amplitude of any two successive oscillations.

2. How is the logarithmic decrement calculated?

The logarithmic decrement can be calculated by using the formula ln(An/An+1), where An and An+1 are the amplitudes of two successive oscillations.

3. What does a high value of logarithmic decrement indicate?

A high value of logarithmic decrement indicates a highly damped oscillator, meaning that the amplitude of the oscillations decreases significantly over time.

4. How is the logarithmic decrement related to the damping ratio?

The logarithmic decrement is inversely proportional to the damping ratio. This means that a higher damping ratio will result in a lower value of logarithmic decrement.

5. What is the significance of the logarithmic decrement in real-world applications?

The logarithmic decrement is an important parameter in the study of oscillatory systems and is used to measure the level of damping in these systems. It is particularly useful in engineering applications, such as in the design of shock absorbers and suspension systems.

Similar threads

Replies
4
Views
1K
Replies
2
Views
9K
Replies
8
Views
3K
Replies
1
Views
3K
Replies
4
Views
2K
Replies
2
Views
2K
Replies
1
Views
2K
Back
Top