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.

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  1. R

    Driven oscillator amplitude steady state X(t) = ##Asin(\omega t + \delta)##

    I found ## \frac{\gamma}{2} = 7##, ##\gamma = 14## ##\omega_0^2 = \omega_d^2 + \frac{\gamma^2}{4} = 25## ##\omega_0 = \omega = 25##, thus ##\delta = \frac{\pi}{2}## ##A = \frac{\frac{F_0}{m}}{\sqrt((\omega_0^2 - \omega^2)+ \gamma^2\omega^2)} = 0.04## Thus, ##X(t) = 0.04sin(25t + \frac{\pi}{3} -...
  2. E

    Help with the phase of the solution for a driven oscillator

    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...
  3. L

    A Partition function for a driven oscillator?

    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?
  4. SuchBants

    Finding the max frequency of a driven 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...
  5. Samama Fahim

    Frequency of Undamped Driven Oscillator near Zero

    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...
  6. K

    I Phase angle of a damped driven harmonic oscillation

    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...
  7. Phantoful

    Damped harmonic oscillator for a mass hanging from a spring

    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...
  8. B

    Damped driven oscillator

    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...
  9. V

    Exponentially driven harmonic oscillator

    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...
  10. N

    A Driven Harmonic Oscillator where Mass Hits Ground

    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...
  11. J

    I What does it mean to "drive a particle" in a penning trap?

    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?
  12. B

    Average energy of a damped driven oscillator

    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> =...
  13. J

    Diagonalize a coupled damped driven oscillator

    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...
  14. N

    (Hard) work done by damped, driven oscillator as function of time

    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...
  15. P

    Particular Solution of A Coupled and Driven Oscillator

    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: $$...
  16. T

    Steady State Solution of Forced, Damped Harmonic Oscillator

    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...
  17. N

    I Driven forced oscillation wmax for Amax stuck on maths ><

    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...
  18. B

    Short Change Resonance of a Damped, driven oscillator

    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...
  19. D

    MHB Fourier series damped driven oscillator ODE

    $$ -\sum_{n = 0}^{\infty}n^2\omega^2C_ne^{in\omega t} + 2\beta\sum_{n = 0}^{\infty}in\omega C_ne^{in\omega t} + \omega_0^2\sum_{n = 0}^{\infty}C_ne^{in\omega t} = \sum_{n = 0}^{\infty}f_ne^{in\omega t} $$ How can I justify removing the summations and solving for $C_n$? $$...
  20. C

    The rate at which a damped, driven oscillator does work

    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)...
  21. B

    SH driven oscillator amplitude at resonance equation

    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...
  22. S

    How Does Damping Affect the Resonant Amplitude of a Driven Pendulum?

    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...
  23. W

    Driven oscillator with damping

    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...
  24. M

    Can a Driven Oscillator Mechanism Enhance a Double Pendulum Hammer's Swing?

    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...
  25. D

    What is the Differential Equation for a Forced Driven Oscillator with Damping?

    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...
  26. E

    How Does the Driven Oscillator ODE Describe Long-Term Motion?

    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) +...
  27. S

    What Is the Relationship Between Damping and Resonance in Driven Oscillators?

    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} -...
  28. qspeechc

    Analogy to Damped , Driven Oscillator.

    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...
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