The Mechanics Map is an open textbook for engineering statics and dynamics containing written explanations, video lectures, worked examples, and homework problems. All content is licensed under a creative commons share-alike license, so feel free to use, share, or remix the content. The table of contents below links to all available topics, while the about, instructor resources, and contributing tabs provide information to those looking to learn more about the project in general.
Newtonian Mechanics Video Introduction 1. Statically Equivalent Systems Video Introduction 4. Engineering Structures Video Introduction 5. Friction and Friction Applications Video Introduction 6. Tipping 6. Particle Kinematics Video Introduction 7. Work and Energy in Particles Video Introduction 9. Impulse Momentum Methods Video Introduction Work and Energy Methods Video Introduction Welcome to the Mechanics Map Digital Textbook: The Mechanics Map is an open textbook for engineering statics and dynamics containing written explanations, video lectures, worked examples, and homework problems.
Mechanics Basics: 1. Rigid Body Kinematics: Newton's Second Law for Rigid Bodies: Vector and Matrix Math: A1. Moment Integrals: A2. Video Introduction.Steady friction forces occur in many systems when relative motion takes place between adjacent members. These forces are independent of amplitude and frequency; they always oppose the motion and their magnitude may, to a first approximation, be considered constant.
Dry friction can, of course, just be one of the damping mechanisms present; however, in some systems it is the main source of damping. In these cases the damping can be modelled as in Figure 1. The constant friction force F d always opposethe motion, so that if the body is displaced a distance x 0 to the right and released from rest we have, for motion from right to left only.
The initial conditions were atand at. Substitution into Eqn. At the end of the half cycle right to left, and. That is, there is a reduction in amplitude of per half cycle. From symmetry, for motion from left to right when the friction force acts in the opposite direction to the above, the initial displacement is thereforethat is the reducion in amplitude is per cycle.
This oscillation continues until the amplitude of the motion is so small that the maximum spring force is unable ot overcome the friction force F d. This can happen whenever the amplitude is. The motion is sinusoidal for each half cycle, with successive half cycles centered on points distant and from the origin.
The oscillation ceases with. The zone is called the dead zone.
Identifying Coulomb and Viscous Friction from Free-Vibration Decrements
Now ifso thatfrom which the frequency of oscillation is Hz. That is, the frequency oscillation is not affected by Coulomb friction. Torsional Bar with Coulomb Friction. The free vibration of dynamic systems with viscous damping is characterized by an exponential decay of the oscillation, whereas systems with Coulomb damping possess a linear decay of oscillation. Many real systems have both forms of damping, so that their vibration decay is a combination of exponential and linear functions.
The two damping actions are sometimes amplitude dependent, so that initially the decay is exponential, say, and only towards the end of the oscillation does teh Coulomb effect show. In the anlyses of the cases the Coulomb effect can easily be separated from the total damping to leave the viscous damping alone.
The exponential decay with viscous damping can be checked by plotting the amplitudes on logarithmic-linear axes when the decay should be seen to be linear. If the Coulomb and viscous effects cannot be separated in this way, a mizture of linear and exponential decay functions have to be found by trial and error in order to conform with the experimental data.
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On Coulomb and Viscosity damped single-degree-of-freedom vibrating systems
Motion characteristics are studied for under-damped, critically damped and over-damped systems. Vibration characteristics of an under-damped system are illustrated.
Hysteresis damping and Coulomb damping are also discussed. Key words: viscous damping, logarithmic decrement, critical damping, hysteresis damping, Coulomb damping. It was seen in the preceding chapter that a simple oscillator under idealized conditions without damping will oscillate indefinitely with a constant amplitude at its natural frequency.
In practice, it is not possible to have an oscillator that vibrates indefinitely. In any practical structure frictional or damping forces will always be present in the mechanical energy of the system, whether potential or kinetic energy is transformed to heat energy. In order to account for these forces, we have to make certain assumptions about these forces based on experience. Damping free vibrations. The oscillatory motions considered so far have been for ideal systems, i. In real systems, dissipative forces, such as friction, are present and retard the motion.
Consequently, the mechanical energy of the system diminishes in time, and the motion is said to be damped. One common type of retarding force is proportional to the speed and acts in the direction opposite to the motion. The damping caused by fluid friction is called viscous damping.
The presence of this damping is always modelled by a dashpot, which consists of a piston A moving in a cylinder B as shown in Fig. Equation 3. It has the solution of the form. There are three special cases of damping that can be distinguished with respect to the critical damping coefficient. Over-damped system. There are two constants A and B which can be evaluated using initial conditions. As t increases x decreases.On Coulomb and Viscosity damped single-degree-of-freedom vibrating systems.
N2 - 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. 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. 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. Apparently has friction damping alone an amplitude reducing effect. AB - Attention on friction damping mechanisms could be of interest for vibration reduction, and appears therefore to be desirable.
Department of Mechanical Engineering Manufacturing Engineering. Abstract Attention on friction damping mechanisms could be of interest for vibration reduction, and appears therefore to be desirable. Keywords Damped mechanical systems Viscous and Coulomb damping. JakobsenJ. Proceedings of The 17th Nordic Symposium on Tribology.Manuscript received August ; revised May Associate Editor: R. Cheng, G.
Identifying Coulomb and Viscous Friction from Free-Vibration Decrements
September 20, October ; 4 : — In this paper, a mass-spring-friction oscillator subjected to two harmonic disturbing forces with different frequencies is studied for the first time. The friction in the system has combined Coulomb dry friction and viscous damping. Two kinds of steady-state vibrations of the system—non-stop and one-stop motions—are considered. The existence conditions for each steady-state motion are provided. Using analytical analysis, the steady-state responses are derived for the two-frequency oscillating system undergoing both the non-stop and one-stop motions.
The focus of the paper is to study the influence of the Coulomb dry friction in combination with the two frequency excitations on the dynamic behavior of the system.
From the numerical simulations, it is found that near the resonance, the dynamic response due to the two-frequency excitation demonstrates characteristics significantly different from those due to a single frequency excitation. Furthermore, the one-stop motion demonstrates peculiar characteristics, different from those in the non-stop motion. Sign In or Create an Account. Sign In. Advanced Search.
Article Navigation. Close mobile search navigation Article navigation. VolumeIssue 4. Previous Article Next Article. Technical Papers. This Site. Google Scholar. Jean W. Zu, Associate Professor Jean W.
Zu, Associate Professor. Author and Article Information. Gong Cheng, Research Fellow. Oct4 : 8 pages. Published Online: September 20, Article history Received:. Views Icon Views. Issue Section:. You do not currently have access to this content.This study focuses on an algorithm for the simultaneous identification of Coulomb and viscous damping effects from free-vibration decrements in a damped linear single degree-of-freedom DOF mass-spring system.
Analysis shows that both damping effects can indeed be separated. Numerical study of a combined-damping system demonstrates a perfect match between the simulation parameters and the estimated values. Experimental study includes two types of real systems. The method is applied to an experimental industrial bearing. Experimental results are compared with numerical simulations to illustrate the reliability of this method.
An analysis provides conservative bounds on error estimates. An example of the effect of quantization error on the estimations is included. This is a preview of subscription content, log in to check access. Rent this article via DeepDyve. Helmholtz, H. Translation by A.
Ellis of Die Lehre von dem Tonempfindungen, fourth edition, ; first edition published in I, reprinted by Dover, New York, Lorenz, H. Jacobsen, L. Watari, A. Feeny, B. Dahl, P. Canudas de Wit C. Harnoy, A. Liang, J. Polycarpou, A. Ruina, A. Bazant ed. Dupont, P. Shaw, S. Download references. Reprints and Permissions. Nonlinear Dynamics 16, — Download citation. Issue Date : August Search SpringerLink Search.
Abstract This study focuses on an algorithm for the simultaneous identification of Coulomb and viscous damping effects from free-vibration decrements in a damped linear single degree-of-freedom DOF mass-spring system.
Immediate online access to all issues from Subscription will auto renew annually. Taxes to be calculated in checkout. References 1.Friction can also provide vibration damping. In this case, however, the damping is not proportional to the magnitude of velocity.
It only depends on the direction of velocity. The above equation does not include velocity. We know that kinetic friction acts to oppose motion, however, so a more complete expression would be:. Where sgn is the "sign" function, a function that captures the sign direction of velocity.
The above equation then indicates that the direction of friction is always opposite the direction of velocity, but the magnitude of velocity does not make a difference in the magnitude of friction. If we plot the response, we can see that there are several differences from a system with viscous damping.
Some differences when compared to viscous damping include: The system oscillates at the natural frequency of the system, not a damped natural frequency. The bounding curves are linear, not exponential.
The system does not return to zero. If we consider our simple linear mass-spring system, the magnitude of F f does not change with velocity, unlike with viscous damping. If we plotted both types of damping for the same system, we would get the following:.
Note that the viscous damping has more reduction in amplitude earlier despite relatively light dampingbut continues oscillating past the point when the friction-damped system has stopped specific relative values are dependant on the values of damping constant and coefficients of friction. Also note that since the viscous damping is relatively light, the difference in period between the two plots is quite small in this example.
The system is initially perturbed by a distance of 0. The system is initially perturbed by a distance of 1. Determine the displacement distance where the system will come to rest. Response of the system in friction damping. Comparison to Viscous Underdamped System If we consider our simple linear mass-spring system, the magnitude of F f does not change with velocity, unlike with viscous damping.
If we plotted both types of damping for the same system, we would get the following: Response of the system in friction damping and in viscous damping, for the same initial conditions x 0v 0spring constants and masses.
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