What is Driven oscillator: Definition and 28 Discussions
In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force F proportional to the displacement x:
F
→
=
−
k
x
→
,
{\displaystyle {\vec {F}}=-k{\vec {x}},}
where k is a positive constant.
If F is the only force acting on the system, the system is called a simple harmonic oscillator, and it undergoes simple harmonic motion: sinusoidal oscillations about the equilibrium point, with a constant amplitude and a constant frequency (which does not depend on the amplitude).
If a frictional force (damping) proportional to the velocity is also present, the harmonic oscillator is described as a damped oscillator. Depending on the friction coefficient, the system can:
Oscillate with a frequency lower than in the undamped case, and an amplitude decreasing with time (underdamped oscillator).
Decay to the equilibrium position, without oscillations (overdamped oscillator).The boundary solution between an underdamped oscillator and an overdamped oscillator occurs at a particular value of the friction coefficient and is called critically damped.
If an external time-dependent force is present, the harmonic oscillator is described as a driven oscillator.
Mechanical examples include pendulums (with small angles of displacement), masses connected to springs, and acoustical systems. Other analogous systems include electrical harmonic oscillators such as RLC circuits. The harmonic oscillator model is very important in physics, because any mass subject to a force in stable equilibrium acts as a harmonic oscillator for small vibrations. Harmonic oscillators occur widely in nature and are exploited in many manmade devices, such as clocks and radio circuits. They are the source of virtually all sinusoidal vibrations and waves.
My question also applies to the damped driven oscillator, however for simplicity I will first consider an undamped oscillator.
The equation of motion is $$-kx + F_{0} \cos{\omega t} = m \ddot{x}$$ or in a more convenient form $$\ddot{x} + {\omega_{0}} ^{2}x = \frac{F_{0}}{m} \cos{\omega t}$$The...
I've seen the partition function calculated for the SHO before in a thermodynamics course in order to calculate entropy. Is it possible to calculate it for a driven harmonic oscillator?
So I've derived the equation for the amplitude of a driven oscillator as:
\huge A=\frac{F}{m\sqrt{(\omega_{0}^{2}-\omega_{d}^{2})^{2}+4\gamma^{2}\omega_{d}^{2}}}
Which is what my lecturer has written. Then taking the derivative and setting it to 0 to get the turning point. He makes this leap...
Description of the Problem:
Consider a spring-mass system with spring constant ##k## and mass ##m##. Suppose I apply a force ##F_0 \cos(\omega t)## on the mass, but the frequency ##\omega## is very small, so small that it takes the system, say, a million years to reach a maximum and to go to 0...
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
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
An un-damped harmonic oscillator natural frequency ##\omega_0## is subjected to a driving force, $$F(t)=ame^{-bt}.$$ At time, ##t=0##, ##x=\dot{x}=0##. Find the equation of motion.
Homework Equations
##F=m\ddot{x}##
The Attempt at a Solution
We have...
I started to ponder following problem. I have a driven, damped oscillator where the mass is free to vibrate in y-direction. If I put a wall or a ground near the mass, the mass touches it if the drive amplitude is larger than the distance to the ground. How does this change the normal dynamics. I...
http://depts.washington.edu/uwptms/research.html
In a penning trap where they use sideband detection of the particle, they need to "drive" it. But what does that mean? Is the position of the particle detected anywhere along this process?
Homework Statement
http://imgur.com/a/lv6Uo
Homework Equations
Look below
The Attempt at a Solution
I was unsure where to start. I thought that parseval's theorem may be helpful. I know the Potential energy is equivalent to .5kx^2 and T will be the integral of the force. So i have $$<E> =...
Homework Statement
I am trying to follow a paper, https://arxiv.org/pdf/1410.0710v1.pdf, I want to get the results obtained in equations 5 and 6 but can't quite work out how eq 3 has been diagonalized.
Homework Equations
eq 3
The Attempt at a Solution
As the system is driven i thought I'd...
Homework Statement
Force F = const is applied to H.O. initially at rest with mass m, freq w0, damping T. Find x(t). Find work as function of time.
Homework Equations
mx'' + Tx' + kx = F for F= Constant
The Attempt at a Solution
First obtain complimentary solution for free H.O. which I get...
Homework Statement
Consider two masses m connected to each other and two walls by three springs with spring constant k. The left mass is subject to a driving force ## F_d\cos(2 \omega t) ## and the right to ## 2F_d\cos(2 \omega t) ##
Homework Equations
Writing out the coupled equations:
$$...
Homework Statement
A damped harmonic oscillator is driven by an external force of the form $$F_{ext}=F_0sin(\omega t)$$
Show that the steady state solution is given by $$x(t)=A(\omega)sin(\omega t-\phi)$$
where $$ A(\omega)=\frac{F_0/m}{[(\omega_0^2-\omega^2)^2+4\gamma^2\omega^2]^{1/2}} $$
and...
Hello
Need some help simplifying this. In relation to a driven oscillator I am looking for wmax for which the amplitude is maximum. I understand the theory and I am just missing something with the maths and I know its probably something so stooooopid I am missing >< I'm taking the derivative...
Homework Statement If both k of the spring and m are doubled while the damping constant b and driving force magnitude F0 are kept unchanged, what happens to the curve, which shows average power P(ω)?
Does the curve:
a) The curve becomes narrower (smaller ω) at the same frequency;
b) The curve...
Homework Statement
Consider a damped oscillator, with natural frequency ω_naut and damping constant both fixed, that is driven by a force F(t)=F_naut*cos(ωt).
a) Find the rate P(t) at which F(t) does work and show that the average (P)avg over any number of complete cycles is mβω2A2. b)...
I found via this forum the hint to use the inverse squared equation to differentiate to find the resonance frequency from the amplitude equation (equilibrium not transient solution). Thank you! (AlephZero?)
When substituting the resulting frequency for the resonance into the amplitude...
Homework Statement
Given a simple pendulum with a mass on the end and a massless string. The support point for the pendulum is moved laterally with an amplitude D at the resonant frequency. The damping is from the air and is considered viscous i.e. not turbulent. The difference between the...
Homework Statement
The oscillator is driven by a force F(t) = mAcos(wt). Plot the amplitude D of oscillations, in units of the maximum (resonant) amplitude D(max), as a function of w in units of w_0. (In other words, plot D/D(max) versus w/w_0.) Find Q.
\beta=(1/6)w_0
Homework...
The driven oscillator mechanism explains the swing of a child that drives it up higher.
I am using the same mechanism with a double pendulum hammer.
The hammer will increase its amplitude, and if so we have to stop it at its high point preventing it from flipping over.
If we use a spring, it...
Homework Statement
http://fatcat.ftj.agh.edu.pl/~i7zebrow/rysunek.jpg
tring constant is k,
object mass is M_{1}
Damping friction is b
and we wiggle the top end of spring in the above diagram with amount Asin(wt)
(Where A is a amplitude and w is a frequency).
Homework Equations
Spring...
Homework Statement
"The equation mx'' + kx = F0 * Sin (wt) governs the motion of an undamped harmonic oscillator driven by a sinusoidal force of angular frequency w. Show that the steady-state solution is
x = F0 * Sin (wt) /(m * (w0^2 - w^2))
Homework Equations
x(t) = xta(t) +...
Homework Statement
A driven oscillator with mass m, spring constant k, and damping coefficient b is is driven by a
force F_{o}cos(\omega t). The resulting steady-state oscillations are described by x(t) = Re{\underline{A}e^{i\omega t}} where:
\underline{A} = \frac{F_{0}/m}{(\omega_{o}^{2} -...
Hi everyone. I have a project where I need to find a situation this is, or is similar to, a damped oscillator. That is, the Differential Equation (DE) for the system must follow:
x'' + ax' + bx = 0
And, further, it must have some situation corresponding to being 'driven' or 'forced', that...