Damped and undamped oscillations (integration?)

In summary, the conversation discusses a problem about oscillations with a force of -c⋅x acting on a masspoint m. The Newton's equation of motion is set up as m \cdot \ddot{x} + c \cdot x = 0 and solved using an exponential approach x(t) ∼ e^{λt}. The initial conditions are also taken into account. The conversation then moves on to adding a friction force to the system and solving for the path x(t) in various cases of underdamping, critical damping, and overdamping. The conversation concludes with a question about how to get from the solution in exponential form to the solution in the form x(t) = A \cdot e^{-ϒt}
  • #1
JulienB
408
12

Homework Statement



Hi everybody! I'm doing a problem about oscillations, and I must admit that a few things are still unclear to me about that subject. Can someone maybe help me?

a) A onedimensional masspoint m is oscillating under the influence of the force [tex]F(x) = -c⋅x (c > 0).[/tex] What is the Newton's equation of motion for the system and integrate it with an exponential approach [tex]x(t) ∼ e^{λt}.[/tex] Find the integration constant for the initial conditions [tex]x(t = 0) = x_0[/tex] and [tex]\dot{x}(t = 0) = \dot{x_0}.[/tex]
b) A friction force [tex]-ϒ \dot{x} (ϒ > 0)[/tex] is added to the system of a), that means a total force [tex]F(x,\dot{x}) = -cx - ϒ\dot{x}[/tex] is acting upon the masspoint m. Similarly to question a), find the path [tex]x(t)[/tex] and distinguish it between the cases:
[tex] 0 < \frac{ϒ}{m} < 2ω_0 : \mbox{underdamping}[/tex]
[tex] \frac{ϒ}{m} = 2ω_0 : \mbox{critical damping}[/tex]
[tex] 0 < 2ω_0 < \frac{ϒ}{m} : \mbox{overdamping}[/tex]

The Attempt at a Solution



a) So first I set up my equation of motion:

[tex] m \cdot \ddot{x} = -c \cdot x ⇔ m \cdot \ddot{x} + c \cdot x = 0 ⇔ \ddot{x} + \frac{c}{m} x = 0 \\
\mbox{Let } ω_0 = \sqrt{\frac{c}{m}} ⇒ \boldsymbol{\ddot{x} + ω_0^2 x = 0} [/tex]
Then I rewrite the equation in the exponential form and solve for λ:
[tex]x(t) = α \cdot e^{λt} ⇒ α \cdot λ^2 \cdot e^{λt} + α \cdot ω_0^2 \cdot e^{λt} = 0 ⇔ λ^2 + ω_0^2 = 0 \\
⇒ \boldsymbol{λ = ± i \cdot ω_0} ⇒ x(t) = α \cdot e^{i ω_0 t} + α^{*} \cdot e^{-i ω_0 t}
[/tex]
Then do I have to integrate the equation of motion by substituting x(t) by that expression? Why would we want to do that anyway? To get the velocity? It's the first time I'm ever asked to integrate an equation of motion, and there is no explanation whatsoever with the problem about why should one do that. Anyway I gave it a go:

[tex]\int -α \cdot ω_0^2 \cdot e^{i ω_0 t} + α \cdot ω_0^2 \cdot e^{-i ω_0 t} dt = -α \cdot ω_0^2 \int e^{i ω_0 t} dt+ α \cdot ω_0^2 \int e^{-i ω_0 t} dt \\
= i \cdot α \cdot ω_0 (e^{i ω_0 t} + \frac{1}{e^{-i ω_0 t}}) + C[/tex]

Ehem...Not sure what is that, if it's correct and what to do from there! :wideeyed: I do not know what α is, and all those imaginary units kind of scare me! Some help/hint here would be very appreciated. :)

Thanks a lot in advance for your answers!Julien.
 
Physics news on Phys.org
  • #2
Hi Julien,
JulienB said:
Then do I have to integrate the equation of motion
You've done that by solving ! You assumed a solution of the given form and found it satisfies the differential equation if ##\lambda = i\omega## and also if ##\lambda = -i\omega##. Since the equation is linear, ##x(t) = A e^{i\omega t} + B e^{-i\omega t}## also satisfies the equation of motion

[edit] I think you already imposed "x(t) is real" to get a restriction the integration constatns.
You restricted yourself with ##a## and ##a^*## if you mean ##a^*## is the complex conjucate of ##a##.
 
  • Like
Likes JulienB
  • #3
@BvU Aaah then no integration needed! That question really confused me :wideeyed:

Okay so then if I resume from there:

[tex]x(t) = Ae^{iω_0t} + Be^{-iω_0t} ⇒ \dot{x}(t) = iω_0Ae^{iω_0t} - iω_0Be^{-iω_0t}[/tex]

The initial conditions can then be expressed in terms of A and B:

[tex]x(t = 0) = x_0 = A + B \\
\dot{x}(t = 0) = \dot{x}_0 = iω_0A - iω_0B[/tex]

When I solve for A and B I get:

[tex]A = x_0 - B ⇒ \dot{x_0} = x_0iω_0 - 2Biω_0 ⇒ B = \frac{x_0}{2} - \frac{\dot{x_0}}{2iω_0} ⇒ A = \frac{x_0}{2} + \frac{\dot{x_0}}{2iω_0}[/tex]

Does that make sense? I won't pretend I perfectly understand what I'm doing, so I also don't really know if I have to go further or if that was the question. :DD

Thanks a lot for your answer!Julien.
 
  • #4
You've answered a) in full, but perhaps it becomes more eye-pleasing if you rewrite a little bit using ##e^{i\phi} = \cos\phi + i\sin\phi##: you get something like ##x_0\cos(\omega t) + {\dot x_0\over \omega} (\sin\omega t)## (Do check me here !)

Now part b) will reveal why this approach as in part a) is so useful.

The harmonic oscillator is a very relevant item for a lot of Physics with a capital P, from solid state phics, quantuum mechanics all the way up to the very front of field theory. But also in acoustics, mechanics, etc, etc. You name it. reason: the F(x) = - kx is the simplest desciption (first order approach) of any equilibrium.

So by all means carry on and ask if stuck.​
 
  • Like
Likes JulienB
  • #5
@BvU Thanks a lot for your help! I will try to solve b) on my own and will post my progress again, probably tomorrow.Julien.
 
  • #6
@BvU (Btw your expression with cos and sin was correct :wink:)
 
  • #7
I started b) and got that:

[tex]
m \cdot \ddot{x} = -ϒ \cdot \dot{x} - c \cdot x ⇔ \ddot{x} + \frac{ϒ}{m} \dot{x} + \frac{c}{m} x = 0 \\
\mbox{Let } 2δ = \frac{ϒ}{m} ⇒ \ddot{x} + 2δ\dot{x} + ω_0^2x = 0 \\
x(t) = α \cdot e^{λt} ⇒ λ^2\cdot α \cdot e^{λt} + 2δ \cdot λ \cdot e^{λt} + ω_0^2 \cdot α \cdot e^{λt} = 0 \\
⇔ λ^2 + 2δλ + ω_0^2 = 0 ⇒ λ = -δ ± \sqrt{δ^2 - ω_0^2}
[/tex]

Okay that's something. :) Now I try to go further for the case "underdamped":

[tex] 0 < 2δ < 2ω_0 : \\
\mbox{Let } ω^2 = ω_0^2 - δ^2 ⇒ λ = -ϒ ± \sqrt{-ω^2} = -ϒ ± iω \\
⇒ x(t) = A \cdot e^{-δt + iωt} + B \cdot e^{-δt - iωt} = e^{-δt}(A \cdot e^{iωt} + B \cdot e^{-iωt})
[/tex]

Then I am afraid I encounter something I do not understand again. Although that looks a lot like in a), in my book it says that there is only one solution A and they write x(t) like that:

[tex]
x(t) = A \cdot e^{-ϒt} cos(ωt + φ)
[/tex]

Did I do something wrong? How can I go from the last expression to that one?Thanks a lot in advance for your help.Julien.
 

1. What is the difference between damped and undamped oscillations?

Damped oscillations refer to a type of motion where the amplitude of the oscillation decreases over time due to the presence of a dissipative force, such as friction. Undamped oscillations, on the other hand, do not experience any decrease in amplitude and continue to oscillate with a constant frequency.

2. What causes damped oscillations?

Damped oscillations are caused by the presence of a dissipative force, such as friction or air resistance, which acts against the motion of the oscillating object. This force causes energy to be lost from the system, leading to a decrease in amplitude over time.

3. How do you calculate the damping ratio for a damped oscillation?

The damping ratio, represented by the Greek letter "zeta" (ζ), is calculated by dividing the actual damping coefficient by the critical damping coefficient. The critical damping coefficient is equal to two times the square root of the mass of the object multiplied by the spring constant. The actual damping coefficient can be determined experimentally by measuring the decrease in amplitude over time.

4. Can an undamped oscillation be achieved in a real-world system?

In theory, an undamped oscillation can be achieved in a system by eliminating all sources of dissipative forces. However, in reality, it is impossible to completely eliminate all sources of friction and other dissipative forces. Therefore, truly undamped oscillations do not occur in real-world systems.

5. What role does integration play in studying damped and undamped oscillations?

Integration is used in the mathematical analysis of damped and undamped oscillations to model the behavior of the system over time. By using differential equations and integrating them, we can determine the position, velocity, and acceleration of the oscillating object at any given time. This helps us understand the behavior of the system and make predictions about its future motion.

Similar threads

  • Introductory Physics Homework Help
Replies
2
Views
449
  • Introductory Physics Homework Help
Replies
10
Views
912
  • Introductory Physics Homework Help
Replies
3
Views
741
  • Introductory Physics Homework Help
Replies
6
Views
805
  • Introductory Physics Homework Help
Replies
8
Views
561
  • Introductory Physics Homework Help
Replies
17
Views
372
  • Introductory Physics Homework Help
Replies
4
Views
540
  • Introductory Physics Homework Help
Replies
24
Views
256
  • Introductory Physics Homework Help
Replies
13
Views
1K
  • Introductory Physics Homework Help
Replies
2
Views
799
Back
Top