An analytical investigation of piecewise linear harmonic oscillators
Sponsored Links
This presentation is the property of its rightful owner.
1 / 5

An Analytical Investigation of Piecewise-linear Harmonic Oscillators PowerPoint PPT Presentation

  • Uploaded on
  • Presentation posted in: General

An Analytical Investigation of Piecewise-linear Harmonic Oscillators. Brendan Jones Christian Fadul. Introduction.

Download Presentation

An Analytical Investigation of Piecewise-linear Harmonic Oscillators

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

An Analytical Investigation of Piecewise-linear Harmonic Oscillators

Brendan Jones

Christian Fadul


The equation of motion for a linear oscillator does not have a single analytical solution. As the components of the system in motion make intermittent contact with each other, an interesting piecewise linear form is examined in the motion of the system. The system under consideration shows a single degree of freedom that is periodically forced. Due to the complexity of expressing the position as a function of time, the piecewise function was solved by using a numerical integrator.

Hand Calculations

First we drew the FBD’s

LMB: mx'' = -c(x') – h(x) + F cos(ωt)

Mass normalized motion equation: x'' + ξ(x') + H(x) = A cos(ωt)

Simplification for integrator:

x1 = x

x2 = x'

x'1 = x2

x'2 = -ξ(x2) - H(x1) + A cos(ωt)

Experimental Results

x’’ = (1/2)*(-(1/2)*ξ+(1/2)*sqrt(ξ^2-4*H))^2*exp((-(1/2)*ξ+(1/2)*sqrt(ξ^2-4*H))*x)*(-r*ξ^2*ω^2-sqrt(ξ^2-4*H)*ξ*r*ω^2+2*r*H*ω^2-2*r*H^2+2*A*H)*(-sqrt(ξ^2-4*H)*ξ+ξ^2-4*H)/((2*ξ^2*ω^2-8*H*ω^2+ξ^4-6*H*ξ^2+8*H^2-ξ^3*sqrt(ξ^2-4*H)+4*sqrt(ξ^2-4*H)*ξ*H)*H)+(1/2)*(-(1/2)*ξ-(1/2)*sqrt(ξ^2-4*H))^2*exp((-(1/2)*ξ-(1/2)*sqrt(ξ^2-4*H))*x)*(-r*ξ^2*ω^2+sqrt(ξ^2-4*H)*ξ*r*ω^2+2*A*H+2*r*H*ω^2-2*r*H^2)*(sqrt(ξ^2-4*H)*ξ+ξ^2-4*H)/((sqrt(ξ^2-4*H)*ξ+ξ^2-2*H+2*ω^2)*H*(ξ^2-4*H))+(-(H-ω^2)*cos(ω*x)*ω^2-sin(ω*x)*ω^3*ξ)*A/(ω^4+(ξ^2-2*H)*ω^2+H^2).

As you can see, the analytical form of the acceleration expression is very complex.


With the knowledge obtained throughout the course it is possible to derive the equations of motion for this system. Using specific conditions for the system, different situations can be observed and examined. Through the examination of each of the special conditions of the system, the entire process is automatically checked for its accuracy. In this specific case, the original derivation of the equation of motion agrees with the solution set that was obtained in the experimental calculations. Also, we can cross check with the work done by Kisliakov and Popov [1] and Arnold [2] to verify that the solution obtained matches the solution set of the previous research.

  • Login