On Coulomb and Viscosity damped single-degree-of-freedom vibrating systems

J. Jakobsen , Ion Marius Sivebæk

Research output: Chapter in Book/Report/Conference proceedingArticle in proceedingsResearchpeer-review


Attention on friction damping mechanisms could be of interest for vibration reduction, and appears therefore to be desirable. Presentations of textbook analyses on mechanical vibration of a viscosity damped single degree system [mass, spring and eventually damping] are numerous. Often they begin with an assumption of a sin/cos behaviour of mass-amplitude (x) versus time (t) solution to the governing equation [M*acceleration = Sum of forces]. The solutions have all an equal sin/cos form. This may indicate that mass and spring are prime elements of the model and that damping mainly has an amplitude reducing influence.
The amount of analyses of friction damped system is comparatively more limited. The periodic square wave is a frequently occurring type of friction in this type of analyses. This periodic square wave is often named Coulomb friction. It can be resolved in an infinite series of harmonic components with frequencies 1, 3, 5, … times the basic frequency of the square wave and with respective amplitudes: (4/π)∗(1, 1/3, 1/5... )∗Fμ(ωt). Fμ(ωt): the square wave amplitude. The governing equation for the sequence of a free vibration with Coulomb friction damping is nonlinear, but is linear within each ½ period. A complete solution can therefore be made up compounding solutions from ½ periods by inserting end conditions from one ½ period as initial conditions for the following ½ period. – Only spring and Coulomb forces act together.
As a Coulomb force is conceivable as an infinite series of harmonic components the appearance of harmonics could be expected in the behaviour of the amplitude (x) of the mass versus the time (t) in the solution. Some authors may have considered this possibility previously. But the solutions for friction damping can be written as [(x + K1) / K2] cos(ωnt); K1 and K2 are constants, adjustable within each ½ period. ωn is the undamped natural frequency. Apparently has friction damping alone an amplitude reducing effect. Based on these textbook presentations it appears noteworthy that displacement x(t) is described alone with the cos(ωnt) function, that is, a completely pure cosine sequence, without triggering higher harmonics.
One conclusion could be that a free vibration with Coulomb friction damping integrates all harmonics into the constant Coulomb force: Fμnt). This hypothesis of an integrating mechanism may, however, not be valid when the system works, driven by the force Fo(cos(ωt)).
Original languageEnglish
Title of host publicationProceedings of The 17th Nordic Symposium on Tribology
Number of pages9
Publication date2016
Publication statusPublished - 2016
EventThe 17th Nordic Symposium on Tribology - Aulanko, Hämeenlinna, Finland
Duration: 14 Jun 201617 Jun 2016


ConferenceThe 17th Nordic Symposium on Tribology


  • Damped mechanical systems
  • Viscous and Coulomb damping


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