What is Damped harmonic motion: Definition and 35 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:






{\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. meher4real

    Damped harmonic motion problem

    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...
  2. J

    What Is the Default Damping Value in PhET's Masses and Springs Simulation?

    PhET: https://phet.colorado.edu/en/simulation/masses-and-springs Any information would be appreciated. Thanks in advance!
  3. Kaushik

    B Forced Oscillations: Pendulum 1 Driving Neighboring Pendulum

    Consider the following setup: In this, let us set the pendulum 1 into motion. The energy gets transferred through the connecting rod and the other pendulum starts oscillating due to the driving force provided by the oscillating pendulum 1. Isn't it? So the neighbouring pendulum starts...
  4. T

    I Solution to a second order differential equation

    I have currently been reading a book called 'Mathematical Methods In Physical Sciences'. Whilest I was looking at the differential section I came across a differential which I have never thought about before, which is of the form...
  5. Incud2

    Find the resistive constant in a critically damped system

    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...
  6. 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...
  7. F

    Is the Total Force in Damped Harmonic Motion Always Opposite to Velocity?

    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}...
  8. G

    Engineering How Does Adjusting Parameter B Affect Servo Motor Friction and System Damping?

    Homework Statement 2. Homework Equations [/B] I'm studying a mathematical behaviour of a servo motor and I need some help to understand it. The output signal is \$\beta (t)\$, representing the angle rotated by the axis at instant t, in relation to the equilibrium position. On the servomotor...
  9. TheBigDig

    Resistance of an oscillating system

    Homework Statement [/B]Homework Equations ##F = -kx = m\ddot{x} ## ## f = \frac{2\pi}{\omega}## ## \omega = \sqrt{\frac{k}{m}} ## ##\ddot{x} + \gamma \dot{x}+\omega_o^2x = 0 ## ##\gamma = \frac{b}{m}## The Attempt at a Solution I'm stuck on part c of this question. Using the above equations I...
  10. C

    Understanding Oscillation with Friction: Problem 130 Solution Clarifications

    Homework Statement So In the following picture is the problem and its solution (Problem 130) Homework Equations KE =1/2 m v2 x:=w^2x The Attempt at a Solution I am confused of the fact that the time from the spring touching t2 is T/4-t where T is the period oscillations while t is the time to...
  11. patrickmoloney

    Finding the Zeros of Damped Harmonic Motion Equations

    Homework Statement Solve the damped harmonic motion system \ddot{x} + 2k\dot{x} + \omega^2 x = 0 with initial conditions \dot{x}=V at x = 0 in the cases (i) \, \omega^2 = 10k^2 (ii) \omega^2 = k^2 (iii) \omega^2 = 5, k = 3 Identify the type of damping, sketch the curve of x versus t>0 in...
  12. V

    I Solve Boom Crane Position with Lagrangian Equation | Wind Loading Included

    Hi, I am trying to model the position of the suspended mass at the end on a boom crane. This is basically a spherical pendulum, however further complicated by the fact that the mass can be hoisted up and down and also the pivot is connected to an arm (boom) which can be rotated up and down and...
  13. RJLiberator

    Mean Input Power & Q value , Damped Harmonic Motion

    Homework Statement Homework EquationsThe Attempt at a Solution I'm working on part a. The numerical value of Q. I have an equation stating that Q = ω_0/ϒ. I don't really know what ϒ is, in other places (http://farside.ph.utexas.edu/teaching/315/Waves/node13.html) it seems like the...
  14. R

    Optimizing Damping Time: Which Application Would Benefit Most?

    Homework Statement "Which of the following applications would have the most benefit from a short damping time?" a. bathroom scale b. child jolly jumper c. suspension on passenger car d. suspension on race car Homework EquationsThe Attempt at a Solution Im assuming that both A and D should be...
  15. nomadreid

    Damped harmonic motion with one end without weight free

    Homework Statement A block on a horizontal surface is attached to two springs whose other ends are fixed to walls. A light string attached to one side of the block initially lies straight across the surface. The other end of the string is free to move. There is significant friction between...
  16. PsychonautQQ

    Energy in damped harmonic motion

    Hey PF, my book either got sloppy in a derivation or I am not connecting two very obvious dots. It gives the energy of the damped harmonic oscillator as E = (1/2)mv^2 + (1/2)kx^2 then takes the derivative with respect to time to get dE/dt. then it gives the differential equation of motion...
  17. L

    Queries on Damped Harmonic Motion

    So we know that SHM can be described as: x(t) = Acos(ωt + ϕ) v(t) = -Aω sin(ωt + ϕ) a(t) = -Aω^2 cos(ωt + ϕ) it can be easily said that the max acceleration (in terms of magnitude) a SHM system can achieve is Aω^2 In Damped Harmonic Motion we know that: x(t) = (A)(e^(-bt/2m))cos(ωt + ϕ) given...
  18. E

    Damped Harmonic Motion - Oscillating Spring

    Homework Statement http://www.mediafire.com/view/?7045cz9au1ci7cd A mountain bike has bad shock absorbers (w0/γ = 10) that oscillate with a period of 0.5 seconds after hitting a bump. If the mass of the bike and rider is 80kg, determine the value of the spring constant k (remembering that...
  19. E

    Understanding Damped Harmonic Motion

    So my professor was discussing the case of a mass suspended from a vertical massless spring in some viscous liquid. He arrives at the equation of motion which was :x: + \frac{b}{m}x. + \frac{k}{m}x = 0 x: is the second derivative of displacement wrt time. similarly x. is the first derivative...
  20. C

    Equation for displacement in damped harmonic motion.

    This is not really a homework problem but rather a question about an equation for displacement in damped harmonic oscillations that I've come across during revision for midterms. In my notes and in various textbooks the equation is given as x=C\mathrm{exp}(-\frac{b}{2m}t)\cdot\mathrm{exp}(\pm...
  21. R

    Damped Harmonic Motion with a Sinusoidal Driving Force

    1. An oscillator with mass 0.5 kg, stiffness 100 N/m, and mechanical resistance 1.4 kg/s is driven by a sinusoidal force of amplitude 2 N. Plot the speed amplitude and the phase angle between the force and speed as a function of the driving frequency and find the frequencies for which the phase...
  22. M

    Damped Harmonic Motion Time Constant?

    Homework Statement A spring with spring constant 17.0 N/m hangs from the ceiling. A 530 g ball is attached to the spring and allowed to come to rest. It is then pulled down 7.00 cm and released. What is the time constant if the ball's amplitude has decreased to 3.50 cm after 41.0...
  23. G

    Mathematica How do I fit damped harmonic motion data (x vs. t) using Mathematica?

    How do I fit the data given below into the standard form (ignore the part saying to use MathCad): http://screensnapr.com/u/apmqkd.png {{{0.002, 0.726}, {0.022, 0.739}, {0.042, 0.75}, {0.062, 0.759}, {0.082, 0.768}, {0.102, 0.776}, {0.121, 0.785}, {0.141, 0.794}, {0.161, 0.802}...
  24. P

    Damped Harmonic Motion Equation

    I am having trouble finding out what the equation for damped harmonic motion is. I have been researching around there there are many small variations on the exponents. I am conducting an experiment which has involved the use of the spring constant from Hooke's Law and have used a hypothesis...
  25. J

    Damped Harmonic Motion on a Spring

    Homework Statement In this problem we will investigate a particular example of damped harmonic motion. A block of mass m rests on a horizontal table and is attached to a spring of force constant k. The coefficient of friction between the block and the table is mu. For this problem we will...
  26. A

    Damped harmonic motion question

    A damped harmonic motion starts from rest at time t=0 with displacement A0 has the equation: x(t) = A0/cos (delta)*e^(-t/tau) *cos (w't + delta) w' is the angular frequency, tau is the time constant and delta is given by: tan (delta) = - (1/w' tau) find the time when the maximum...
  27. S

    Damped Harmonic Motion: Find Ratio & Periods for Decay

    Homework Statement A damped harmonic oscillator has mass m , spring constant k , damping force - cv . (a) Find the ratio of two successive maxima of the oscillations. (b) If the oscillator has Q = 100 , how many periods will it take for the amplitude to decay to 1/ e of it’s initial...
  28. N

    What is the distance traveled in damped harmonic motion?

    http://img13.imageshack.us/img13/9091/53337497.th.jpg Can someone please help me with the problem above? I am unable to start it. Clearly, using the constant acceleration "suvat" equations, 0.5ft^2 is the distance obtainined, however I am unable to proceed. Thanks in advance.
  29. A

    Damped Harmonic Motion: Find Speed at Equilibrium

    The position x(t) of a mass undergoing damped harmonic motion at an angular frequency ω' is described by x(t)=A e^t/τ cos(ώt + delta) where τ is the time constant, A the initial amplitude and delta an arbitrary phase. (a) Find an expression for the speed of the mass as it...
  30. V

    How Do You Calculate the Damping Constant in Damped Harmonic Motion?

    Homework Statement Hi all, A hard boiled egg, with a mass m=51g, moves on the end of a spring, with force constant k=26N/m. It's initial displacement is 0.300m. A damping force F^{}x=-bv^{}x acts on the egg and the amplitude of the motion decreases to 0.106m in a time of 5.45s. Calculate the...
  31. R

    Exploring Damped Harmonic Motion

    Concerning damped harmonic motion (eg. mass on a spring, using cardboard discs as dampers); for the equation (below) of the graph describing the effect of different sized dampers on the time taken for amplitude of oscillations to halve, what would b (y-intercept) and n (gradient) represent...
  32. R

    Exploring b & n in Damped Harmonic Motion

    Homework Statement Concerning damped harmonic motion (mass on a spring using cardboard discs as dampers); for the equation (below) of the graph describing the effect of different sized dampers on the time taken for amplitude of oscillations to halve, what do b (y-intercept) and n (gradient)...
  33. H

    2nd order ordinary differential equation for damped harmonic motion

    Homework Statement the equation of motion for a damped harmonic oscillator is d^2x/dt^2 + 2(gamma)dx/dt +[(omega0)^2]x =0 ... show that x(t) = Ae^(mt) + Be^(pt) where m= -(gamma) + [(gamma)^2 - (omega0)^2 ]^1/2 p =-(gamma) - [(gamma)^2 - (omega0)^2 ]^1/2 If x=x0 and...
  34. B

    Model Damped Harmonic Motion with Y=(e^ax) Sin/Cos bx

    Hey guys, using my knowledge of y=(e^ax) sin bx and y=(e^ax) cos bx, I need to find an example where these functions could be used as a model. I was thinking about damped harmonic motion but had a tough time trying to find an example and how i could relate it to those two graphs, any ideas?
  35. B

    Damped harmonic motion sinusoid equation

    what is the equation? i have something written down in my notes but i really don't get it... x=A(e^-kt)(cos omega t) first of all, how is the amplitude calculated if it decreases over time?? is it averaged? what is e? second of all, to calculate k, you need hooke's law and you need...