Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples include viscous drag (a liquid's viscosity can hinder an oscillatory system, causing it to slow down) in mechanical systems, resistance in electronic oscillators, and absorption and scattering of light in optical oscillators. Damping not based on energy loss can be important in other oscillating systems such as those that occur in biological systems and bikes (ex. Suspension (mechanics)). Not to be confused with friction, which is a dissipative force acting on a system. Friction can cause or be a factor of damping.
The damping ratio is a dimensionless measure describing how oscillations in a system decay after a disturbance. Many systems exhibit oscillatory behavior when they are disturbed from their position of static equilibrium. A mass suspended from a spring, for example, might, if pulled and released, bounce up and down. On each bounce, the system tends to return to its equilibrium position, but overshoots it. Sometimes losses (e.g. frictional) damp the system and can cause the oscillations to gradually decay in amplitude towards zero or attenuate. The damping ratio is a measure describing how rapidly the oscillations decay from one bounce to the next.
The damping ratio is a system parameter, denoted by ζ (zeta), that can vary from undamped (ζ = 0), underdamped (ζ < 1) through critically damped (ζ = 1) to overdamped (ζ > 1).
The behaviour of oscillating systems is often of interest in a diverse range of disciplines that include control engineering, chemical engineering, mechanical engineering, structural engineering, and electrical engineering. The physical quantity that is oscillating varies greatly, and could be the swaying of a tall building in the wind, or the speed of an electric motor, but a normalised, or non-dimensionalised approach can be convenient in describing common aspects of behavior.
Question:
I am working with a pair of systems, each of which is a system of damped, driven, coupled harmonic oscillators, and I am trying to figure out what parameters—if any—could result in each system resonating with a different frequency.
I’m wondering if anyone here has any intuitions...
Hi,
so of course Φ0 = 15° and after solving after solving Φ(t=5*T = 5/f) I found γ = 0.012
I need help with b).
If I do 2° = 15° * exp(-0.012t)*cos(2πf*t), I'm not able to find t so I did something else by assuming that the amplitude decreases at a constant rate:
After 5*T = 5*1/f = 18.52 s...
Why are damped oscillation in many books written with equation
\ddot{x}+2\delta \dot{x}+\omega^2 x=0
##\delta## and ##\omega^2## are constants. Why ##2 \delta## many authors write in equation?
Hello,
So about two weeks ago in class we looked at RLC circuits in our E&M course, and short story short... we compared the exchange of energy between the Capacitor and the Inductor (both ideal) to simple harmonic motion. Once the capacitor and inductor are not ideal anymore, we said it's...
I'm trying to find the quality factor of a damped system.
I know 3 points from the graph, ##(t,x): (\frac{\pi}{120},0.5), (\frac{\pi}{80},0), (\frac{\pi}{16},0)##
From this I found that ##T = \frac{\pi}{20}##
##\omega_d = \frac{2\pi}{T} = 40 rad##
Then, from the solution ##x(t) = A_0...
Hi !
Problem :
y = 5 e^-0.25t sin (0.5.t) (m, s). Determine the deviation at a time when the amplitude has
dropped to 1/5 of the original value.
I tried with A=A0 e^-bt=5 e^-0.25t
- Do i need to determine the time here or recreate the deviation equation when A decreased ? I don't understand...
First of all, the problem is not clearly defined as they don't specify if the given mass is the total mass of the car, or just the sprung mass of the car, which is really what's relevant. In any case, with the limited information given, it seems like one is forced to make the assumption that...
Take rightwards as positive.
There are 2 equations of motion, depending on whether ##\frac {dx} {dt} ## is positive or not.
The 2 equations are:
##m\ddot x = -kx \pm \mu mg##
My questions about this system:
Is this SHM?
Possible method to solve for equation of motion:
- Solve the 2nd ODE...
Summary:: What are the Equations of motion for a free damped 2-Dof systrem?
Hello,
I am required to calculate the equations of motion for a 2-dof system as shown in the attached file. The system is undergoing free damped vibrations. I have found the equations of motion for no damping but i...
the differential equation that describes a damped Harmonic oscillator is:
$$\ddot x + 2\gamma \dot x + {\omega}^2x = 0$$ where ##\gamma## and ##\omega## are constants.
we can solve this homogeneous linear differential equation by guessing ##x(t) = Ae^{\alpha t}##
from which we get the condition...
so what I did was e^-(1/10.1)=0.9057
and e^-(1/14.8)=0.93466
Then 0.93466/0.9057 = 1.03198, so the heavier mass dampens 1.03 times more than the lighter mass. If the lighter mass decreases the oscillation to 72.1%, then the heavier mass would be 72.1%*1.03198 = 74.4, but this is wrong. It...
Here are the nonlinear and coupling ordinary differential equations:
I was given values of a, b, and c as well as some initial values for x, y, and z. If ever the equations above are related to the pendulum, I can think of a as the damping factor, b as the forcing amplitude, and c as the...
Why is my analysis of critically damped motion wrong?
x'' + y*x' + wo²x = 0
Choosing a complex number z as z = A*e^i(wt+a) and putting on the equation calling x as the real part of Z
w = ( i*y +- (4wo²-y²)^(1/2) )/2 (bhaskara)
2wo = y (critical)
w = iy/2
z = A*e^i(ity/2 + a)
z = A*e^(-yt/2...
First of all, i tried to find w, the angular frequency, by calculating the oscillations from ta to tc, there is ~ 20 oscillations coursed.
so,
w = 2*pi*20/(tc-ta)
ta = 0, tc = 0 + 5.2 ms
And tried to find the factor gama y by A(t) = A*cos(Φ + wt)*e^(-yt/2)
A(0) = 2.75u = A*cos(Φ)
1u = A*cos(Φ...
Hello folks,
So the solution of the equation of motion for damped oscillation is as stated above. If we were to take an specific example such as:
$$\frac{d^2x}{dt^2}+4\frac{dx}{dt}+5x=0$$
then the worked solution to the second order homogeneous is...
Hello everyone.
I'm currently trying to solve the damped harmonic oscillator with a pseudospectral method using a Rational Chebyshev basis
$$
\frac{d^2x}{dt^2}+3\frac{dx}{dt}+x=0, \\
x(t)=\sum_{n=0}^N TL_n(t), \\
x(0)=3, \\
\frac{dx}{dt}=0.
$$
I'm using for reference the book "Chebyshev and...
So in my textbook on oscillations, it says that angular frequency can be defined for a damped oscillator. The formula is given by:
Angular Frequency = 2π/(2T), where T is the time between adjacent zero x-axis crossings.
In this case, the angular frequency has meaning for a given time period...
Hi,
So the main question is: How to deal with power loss in E-M waves numerically when we are given power loss in dB's?
The context is that we are dealing with the damped wave equation: \nabla ^ 2 \vec E = \mu \sigma \frac{\partial \vec E}{\partial t} + \mu \epsilon \frac{\partial ^ 2 \vec...
c = Critically Damped factor
c = 2√(km)
c = 2 × √(150 × .58) = 18.65
Friction force = -cv
Velocity v = disp/time = .05/3.5
Friction force = - 18.65 * .05/3.5 = -.27 N
I am not sure if above is correct. Please check and let me know how to do it.
I found the steady state solution as
F_0(mw_0^2 - w^2m)Coswt/(mwy)^2 + (mw_0^2 -w^2m)^2
+ F_0mwySinwt/(mwy)^2 + (mw_0^2 -w^2m)^2
But I'm not sure how to sketch the amplitude and phase? Do I need any extra equations?
Hi,
for ease of reference this posting is segmented into :
1. Background
2. Focus
3. Question
1. Background:
Regarding (one, linear, second-order, homogeneous, ordinary, differential) equation describing the force in a non-driven, damped oscillation:
F = m.a = -k.x - b.v
F =...
This is another application of using Taylor recurrences (open access) to solve ODEs to arbitrarily high order (e.g. 10th order in the example invocation). It illustrates use of trigonometric recurrences, rather than the product recurrences in my earlier Lorenz ODE posts.
Enjoy!
#!/usr/bin/env...
The quarter car system is represented by the above picture and I currently have all of the equations of motion and constants for each spring, mass, damper, distance, and moment of inertia. How can I find the frequency response with this information and knowing both tires hit a pothole of height...
A pendulum with no friction/resistance/damping (i.e. in a vacuum) will swing indefinitely.
Does a pendulum with damping effects ever truly stop oscillating? That is, does the graph tend to infinity or actually reach a value of 0, i.e. the equilibrium position?
Thanks for your time.
Homework Statement
A critically damped simple harmonic oscillator starts from an amplitude of 5.0 cm and comes to rest at equilibrium 3.5 s later. The SHO is made of a 0.58 kg mass hanging from a spring with spring constant 150 N/m. Assuming the friction force is in the vertical direction, how...
Homework Statement
[/B]
Let us assume that neutral atoms or molecules can be modeled as harmonic oscillators in some cases. Then, the equation of the displacement between nucleus and electron cloud can be written as
$$\mu\left(\frac{d^x}{dt^2}+\gamma\frac{dx}{dt}+\omega_0^2x\right)=qE.$$
where...
Homework Statement
A block is acted on by a spring with spring constant k and a weak friction force of constant magnitude f . The block is pulled distance x0 from equilibrium and released. It oscillates many times and eventually comes to rest.
Show that the decrease of amplitude is the same...
Hello,
in every book and on every website (e.g. here http://farside.ph.utexas.edu/teaching/315/Waves/node13.html) i found for driven harmonic osciallation the same solution for phase angle:θ=atan(ωb/(k−mω^2)) where ω is driven freq., m is mass, k is spring constant. I agree with it =it follows...
Homework Statement
This problem is taken from Problem 2.3, Introduction to Vibration and Waves, by H.J. Pain and P. Rankin:
A critically mechanical system consisting of a pan hanging from a spring with a damping. What is the value of damping force r if a mass extends the spring by 10cm without...
Hello,
I have a question regarding Damped Harmonic Motion and I was wondering if anyone out there could help me out? Under normal conditions, gravity will not have an affect on a damped spring oscillator that goes up and down. Gravity will just change the offset, and the normal force equation...
Hi there.
I have a question about the damped pendulum. I am working on an exercise where I have already numerically approximated the solution for a simple pendulum without dampening. Now, the excercise says that I can simply change the code of this simple situation to describe a pendulum with...
Homework Statement
The question I am working on is number 3 in the attached file. There are two initial conditions given: at time = 0, x(t) = D and x'(t) = v 'in the direction towards the equilibrium position'. Does that last statement mean that when I substitute the second IC in, I should...
Homework Statement
Show that the ratio of two successive maxima in the displacement of a damped harmonic oscillator is constant.(Note: The maxima do not occur at the points of contact of the displacement curve with the curve Aeˆ(-yt) where y is supposed to be gamma.
2. Homework Equations The...
Homework Statement
Homework Equations
Complex number solutions
z= z0eαt
Energy equations and Q (Quality Factor)
The Attempt at a Solution
For this question, I followed my book's "general solution" for dampened harmonic motions, where z= z0eαt, and then you can solve for α and eventually...
Homework Statement
I have a project in university that's about creating a simplified model of a washing machine in the program ADAMs View. Here is a picture of how it's constructed: https://imgur.com/a/zZzS5
So basically to oversimplify the problem I've understood that the rotating mass will...
Homework Statement
Reading chapter 4 of Morin's "Introduction to classical mechanics" I came across to the explanation of the damped harmonic motion.
The mass m is subject to a drag force proportional to its velocity, ##F_f = -bv ##.
He says that the total force of the mass is ##F= -b \dot{x}...
Homework Statement
I'm reading the textbook section covering damped series RLC circuits (provided below). I'm wondering why the author stipulates "When R is small..."
Homework Equations
Given above.
The Attempt at a Solution
Given above.
Any gentle and courteous comments are welcome!
Hi there I am really new into programiing thing, and I am trying to make program of this problem usin FORTRAN
I want the output to be Time,poisiton,velocity and contact force
and i already know
= amplitude
U = velocity of the car
d = rail distance
and so far this is my program...
Hello All,
I have come across a problem, which has troubled me for some time now. What needs to be done is the following:
A mass on a rod 0.6m (mass less) has a mass of 1 kgr attached at the end of it. The rod needs to be rotated 60 degrees, within t=120 sec (see image). What I would like to...
Hello,
I am attempting to solve the 1 d heat equation using separation of variables.
1d heat equation:
##\frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial x^2}##
I used the standard separation of variables to get a solution. Without including boundary conditions right now...
Homework Statement
[/B]
Question 3.9
Homework Equations
equation for dampened ocillation[/B]
The Attempt at a Solution
In case this might appear confusing, I derived(with respect to t) the equation for dampened oscillation given above and tried to solve for when it equaled zero expecting...
In driven SHM, we ignore an entire section of the solution to the differential equation claiming that it disappears once the system reaches a steady state. Can someone elaborate on this?
Homework Statement
Homework EquationsThe Attempt at a Solution
I tried differentiating both sides of 3 and re-arranging it such that it started to look like equation 2, however i got stuck with 2 first order terms z' and couldn't find a way to manipulate it into a function z.
I then tried...
I need to design a multi-layer cantilever beam with alternate visco-elastic and elastic layers. The goal is a beam with 18" unsupported and 4" glued to a firm support. In operation, this firm support will be vibrating in the audio range and the beam wants to damp out those vibrations and not...
If you consider b^2/m > 4*k, you can get the solution by using classic method (b = damping constant, m = mass and k = spring constant) otherwise you have to use complex numbers. How have the references books proved the solution for this differential equation?
Homework Statement
The car circulates on a section of road whose profile can be approximated by a sinusoidal curve with the wavelength of 5.0 m. The mass of the car is 600.0 kg, and each wheel is equipped with a constant spring
k = 5000 Nm-1 and a damper with constant b = 450 Nm-1s.
Calculate...
Homework Statement
Homework EquationsThe Attempt at a Solution
After the release the block will move towards right and friction will be towards the left.
##M\ddot x = f - kx##
Solving for ##x##,
##x = A\cos (\omega t) + B\sin(\omega t) + f/k##
Initial conditions are ##x(0) = x_0, \dot...