EIGENVALUES EVALUATION OF GENERALLY DAMPED ELASTIC DISC BRAKE MODEL LOADED WITH NON-CONSERVATIVE FRICTION FORCE

This paper deals with the evaluation of eigenvalues of a linear damped elastic two-degrees-of-freedom system under a non- onservative loading. As a physical interpretation of a proposed mathematical model, a simplified disk brake model is considered. A spectral analysis is performed to predict an eigenvalues bifurcation, known as the Krein collision, leading to double eigenvalues, one of them having a positive real part causing a vibration instability of the mechanical systems. This defective behaviour of eigenvalues is studied with respect to a magnitude of non-conservative Coulomb friction force, through the variation of the friction coefficient. The influence of a proportional versus general damping on the system stability is further analysed. The generalized non-symmetric eigenvalue problem calculation is employed for spectral analyses, while a modal decomposition is performed to obtain a time-domain response of the system. The analyses are compared with an experiment.

An improved model of contact collision investigation on multi-body systems with revolute clearance joints

An improved contact collision model for the impact analysis of a slider–crank mechanism with revolute clearance joints is presented and discussed in this paper. Clearances in revolute joints are inevitable due to wear, machining tolerance, and local deformations. In view of the energy loss and the stiffness changes during collision, which modifies the coulomb friction force, an improved contact collision force model is established. The influence of clearance size, speed, and friction on the acceleration of slider is analyzed, and the simulation results are compared with the experimental data. The results demonstrate that the parameters of clearance joints have obvious effects on dynamic characteristics, and the improved model can accurately describe the dynamic characteristics of joints with clearance.

Modern explorations of the Brachistochrone-related problem: using the Udwadia–Kalaba approach

Modern explorations regarding the search for a curve subject to a minimization principle and its inverse, namely the search for the minimization object based on a given curve, are made. By using the Udwadia–Kalaba theory, which subsumes the Coulomb friction force as a non-ideal constraint, we obtain the analytic expression of the equation of motion for a particle moving along the Brachistochrone cycloid curve under Coulomb friction. Then we perform the inverse problem for the moving particle. That is, while the Brachistochrone cycloid curve is given, we seek the corresponding minimization object. Both the situations with and without friction are addressed. Finally, we return to the search for a curve subject to a minimization principle to complete the loop. However, this time we presume the minimization object is the total travel time, which was addressed in the classical Brachistochrone problem (hence, frictionless), while recognizing the presence of Coulomb friction. All three analyses come to meet at the special case when there is no friction. Our research reveals profound insights that have not been reported previously. The loop analysis also suggests a new angle for the study of dynamic systems.

Research on Damping and Vibration Characteristic of the Large and Precision NC Rotary Table

According to the current situation of acicular chip and high-frequency chattering of the NC rotary table while gear milling, rigidity and damping performances of the table were analyzed and the damping program of Coulomb friction was bring up. Online test of the gear milling cutting force can be used to establish dynamic model of circumferential vibration of the table with Coulomb friction. Then mechanism of restraining gear cutting chatter by damping of Coulomb friction and acicular chip generating mechanism were exposed. Furthermore, relationship between backlash and rotary rigidity of the table was also analyzed. A kind of floating apparatus with friction damping was designed to optimize circumferential damping of the table by adjusting Coulomb friction force, which reduces the influence of high-frequency chattering on gear milling. As a result, efficiency of gear milling was increased 1.5 times and the noise was reduced from 105dB to 91dB.

Prediction of Dry-Friction Whirl and Whip Between a Rotor and a Stator

This paper addresses recent test results for dry-friction whip and whirl. Authors of these publications suggest that predictions from Black’s 1968 paper (J. Mech. Eng. Sci., 10(1), pp. 1–12) are deficient in predicting their observed transition speeds from whirl to whip and the associated precession frequencies of whirl and whip motion. Predictions from Black’s simple Jeffcott-rotor/point-mass stator are cited. This model is extended here to a multimode rotor and stator model with an arbitrary axial location for rotor-stator rubbing. Predictions obtained from this new model are quite close to experimental observations in terms of the transition from whip to whirl and observed precession frequencies. Paradoxically, nonlinear numerical simulations using Black’s model fail to produce the whirl and whip solutions. The Coulomb friction force in Black’s model has a fixed direction, and Bartha showed in 2000 (“Dry Friction Backward Whirl of Rotors,” Dissertation, THE No. 13817, ETH Zurich) that by making the friction-force direction depend on the relative sliding velocity, nonlinear simulations would produce the predicted whirl solutions. He also showed that Black’s proposed whip solution at the upper precession-frequency transition from whirl to whip was unstable. The multimode extension of Black’s model predicts a complicated range of whirl and whip possibilities; however, nonlinear time-transient simulations (including the sgn function definition for the Coulomb force) only produce the initial whirl precession range, initial whirl-whip transition, and initial whip frequency. Simulation results for these values agree well with predictions. However, none of the predicted higher-frequency whirl results are obtained. Also, the initial whip frequency persists to quite high running speeds and does not (as predicted) transition to higher frequencies. Hence, despite its deficiencies, correct and very useful predictions are obtained from a reasonable extension of Black’s model.

Prediction of Dry-Friction Whirl and Whip Between a Rotor and a Stator

This paper addresses recent test results for dry-friction whip and whirl. Authors of these publications suggest that predictions from Black’s 1968 paper are deficient in predicting their observed transition speeds from whirl to whip and the associated precession frequencies of whirl and whip motion. Predictions from Black’s simple Jeffcott-rotor/point-mass stator are cited. This model is extended here to a multi-mode rotor and stator model with an arbitrary axial location for rotor-stator rubbing. Predictions obtained from this new model are quite close to experimental observations in terms of the transition from whip to whirl and observed precession frequencies. Paradoxically, nonlinear numerical simulations using Black’s model fail to produce the whirl and whip solutions. The Coulomb friction force in Black’s model has a fixed direction, and Bartha showed in 2000 that by making the friction-force direction depend on the relative sliding velocity, nonlinear simulations would produce the predicted whirl solutions. He also showed that Black’s proposed whip solution at the upper precession-frequency transition from whirl to whip was unstable. Results presented here show that Black’s whirl solutions are unstable for all whirl precession frequencies, not just the whirl-whip transition frequency. The multi-mode extension of Black’s model predicts a complicated range of whirl and whip possibilities; however, nonlinear time-transient simulations (including the sgn function definition for the Coulomb force) only produce the initial whirl precession range, the initial whirl-whip transition, and the initial whip frequency. Simulation results for these values agree well with predictions. However, none of the predicted higher-frequency whirl results are obtained. Also, the initial whip frequency persists to quite high running speeds and does not (as predicted) transition to higher frequencies. Hence, despite its deficiencies, correct and very useful predictions are obtained from a reasonable extension of Black’s model.

Flow-Coupled Vibrations of Rotor and Seal

We study the flow-coupled vibrations of an unbalanced rotor and a floating sealing ring. The rotor is high speed; its operating regimes are above second critical velocity. The ring is subjected to Coulomb friction force, which prevents its motion about the casing. At rotor rotation, the intensive vibrations of rotor and floating ring take place and the hydrodynamic forces in clearance between them are dominated. Non-impact and impact regimes are revealed and the latter are very dangerous for safety. In the stationary case, the analytical solutions for impact regimes are obtained; these show that synchronous oscillations of rotor and floating ring may occur. The non-stationary oscillations during startup and shutdown are also considered. Domains for the trouble-free non-impact regimes by various rotor velocities are determined. Rotor trajectories with impacts are submitted.

Dynamics of a Coulomb Damped Helical Spring: A Finite Element Approach

This work presents the results of a mathematical modeling to study the dynamic behavior of a helical spring under a periodic excitation induced by a rotating cam. The spring is sleeved over a mandrel; thereby it is further subjected to a Coulomb damping force as it oscillates. Helical springs expand radially when they are compressed. The effect of this radial expansion is included in the mathematical model. Standard wave equation that includes variable Coulomb damping was used to examine the vibratory behavior of the spring. Numerical solution to the no-friction, constant-friction, and varying-friction forces were obtained from the wave equation, using Explicit Finite Difference method. Finite Element was used to model the radial expansion of the spring to determine the variations of the Coulomb friction force. The spring response to the prescribed cam excitation, under the variable Coulomb friction force, was found not to be significantly different from that of a previously assumed constant friction force, for the cases that were studied in this work. In case of postulating a variable damping force the residual vibrations of spring loops are slightly higher than of the constant damping force.

Forced Vibration of a Base-Excited Single-Degree-of-Freedom System With Coulomb Friction

Using the “stopping region of motion” concept, a brief analytical technique is worked out for the behavior of the linear forced vibratory system under the influence of a Coulomb friction force. The following points are clarified by the above technique: 1. The behavior of the system is completely determined by the three non-dimensional parameters of nondimensional friction force, frequency ratio and damping ratio. 2. The vibratory system undergoes a periodic vibration with stopping periods when the mass cannot move. These stopping periods increase at lower exciting frequencies, owing to Coulomb friction. 3. The relation between the kind of motion occurring in the system and the above three parameters can be obtained theoretically and verified experimentally.