Normal Vibrations and Friction Under Harmonic Loads: Part I—Hertzian Contacts

1991 ◽  
Vol 113 (1) ◽  
pp. 80-86 ◽  
Author(s):  
D. P. Hess ◽  
A. Soom
Keyword(s):  
Nonlinear Vibrations ◽  
Multiple Scales ◽  
Normal Load ◽  
Perturbation Technique ◽  
Friction Reduction ◽  
Hertzian Contact ◽  
Dynamic Component ◽  
Normal Contact ◽  
Normal Vibrations ◽  
Contact Parameters

Nonlinear vibrations at a Hertzian contact are studied by the perturbation technique known as the method of multiple scales. The vibrations are excited by the dynamic component of an externally applied normal load. Solutions are obtained for both the average and instantaneous contact deflections. As a result of the nonlinear Hertzian stiffness, the average normal contact deflection during oscillations is smaller than the static deflection under the same average load. It is shown that this can result in a reduction of the average area of contact and, by implication, the average friction force in the presence of even small dynamic loads. The parametric dependence of the normal motion on the various contact parameters is investigated. It is shown that the maximum average friction reduction without contact loss is approximately ten percent.

Normal Vibrations and Friction Under Harmonic Loads: Part II—Rough Planar Contacts

1991 ◽  
Vol 113 (1) ◽  
pp. 87-92 ◽  
Author(s):  
D. P. Hess ◽  
A. Soom
Keyword(s):  
Rough Surface ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Normal Load ◽  
Contact Region ◽  
Viscous Damper ◽  
Normal Contact ◽  
Normal Vibrations ◽  
Rigid Mass ◽  
Nonlinear Normal

Nonlinear normal contact vibrations, excited by the application of a dynamic normal load to the contact region formed between rough surfaces, are studied using the method of multiple scales. The planar rough surface is described by the Greenwood and Williamson model. The contact region behaves as a nonlinear spring in parallel with a viscous damper, and supports a rigid mass. It is shown that the average contact separation in the presence of dynamic loading is greater than the static separation under the same average load. In contrast to some previous results, this increase in average separation does not result in a significant change in the average friction force.

On the Frictional Characteristics of Ball Bearings Coated With Solid Lubricants

1999 ◽  
Vol 121 (4) ◽  
pp. 761-767 ◽  
Author(s):  
M. R. Lovell ◽  
M. M. Khonsari
Keyword(s):  
Finite Element ◽  
Friction Force ◽  
Normal Load ◽  
Transversely Isotropic ◽  
Solid Lubricants ◽  
Operating Conditions ◽  
Hertzian Contact ◽  
Ball Bearings ◽  
Normal Contact ◽  
Low Speed

The problem of a ball bearing in normal contact between two transversely isotropic coated substrates is investigated using the finite element method (FEM). A three-dimensional finite element model is developed that accurately determines the steady friction force in low-speed bearing systems containing soft layered solid lubricant films. Extensive numerical results, which are verified using Hertzian contact theory and laboratory experiments, are obtained at 540 operating conditions by varying coating material, coating thickness, normal load, ball material, and ball radius. Friction force results generated from the FEM are normalized by introducing the dimensionless transversely isotropic coating parameter, ξ. A numerical expression for the normalized friction force in coated ball bearings is determined by curvefitting the results of the 540 simulations performed. The relevance of such an expression, as related to the durability of low-speed bearings, is subsequently ascertained and discussed.

Non-linear vibration analysis of a 2-DOF railway vehicle model under random rail excitation

2016 ◽  
Vol 231 (4) ◽  
pp. 591-607 ◽  
Author(s):  
Mohsen Masoomi ◽  
Mohammad Mahdi Jalili
Keyword(s):  
Degrees Of Freedom ◽  
Multiple Scales ◽  
Hertzian Contact ◽  
Railway Vehicle ◽  
Vertical Acceleration ◽  
Normal Contact ◽  
Mean Values ◽  
Rail Track ◽  
Non Linear ◽  
Rail Irregularities

With increase in the axle load and wagon speed, the cost of damage to rail track and wagon components increases significantly. This leads to widespread interests in the investigation of the dynamic interactions of the rail track and the wagon. Rail irregularities are one of the important vibration sources of the rail track structure and train. These irregularities have generally random distribution that are assumed to be stationary random and ergodic processes in space, with Gaussian amplitude probability densities and zero mean values. In this paper, the dynamic response of the railway vehicle due to random irregularity of rail track is analysed. The wagon is modelled as a two degrees-of-freedom non-linear model where includes non-linear spring and linear damper of primary suspension system. The Hertzian contact theory is used to obtain the relationship between normal contact force and the displacement of the mass centre of the wheel. Using the method of multiple scales the analytical approximate response of the railway vehicle due to track irregularities is obtained. The amplitudes of vibrations of the vehicle and the interaction forces between the vehicle and the rail for different line grades and train speeds have been analysed analytically by this model. According to the results, rail irregularities have more effect on the vertical acceleration of the vehicle than the train speed.

The Static and Dynamic Behavior of MEMS Arches Under Electrostatic Actuation

2009 ◽  
Author(s):  
Hassen M. Ouakad ◽  
Mohammad I. Younis ◽  
Fadi M. Alsaleem ◽  
Ronald Miles ◽  
Weili Cui
Keyword(s):  
Multiple Scales ◽  
Perturbation Technique ◽  
Dynamical Behavior ◽  
Primary Resonance ◽  
Reduced Order Model ◽  
Harmonic Load ◽  
Order Model ◽  
Reduced Order ◽  
Electrostatically Actuated ◽  
Model Equations

In this paper, we investigate theoretically and experimentally the static and dynamic behaviors of electrostatically actuated clamped-clamped micromachined arches when excited by a DC load superimposed to an AC harmonic load. A Galerkin based reduced-order model is used to discretize the distributed-parameter model of the considered shallow arch. The natural frequencies of the arch are calculated for various values of DC voltages and initial rises of the arch. The forced vibration response of the arch to a combined DC and AC harmonic load is determined when excited near its fundamental natural frequency. For small DC and AC loads, a perturbation technique (the method of multiple scales) is also used. For large DC and AC, the reduced-order model equations are integrated numerically with time to get the arch dynamic response. The results show various nonlinear scenarios of transitions to snap-through and dynamic pull-in. The effect of rise is shown to have significant effect on the dynamical behavior of the MEMS arch. Experimental work is conducted to test polysilicon curved microbeam when excited by DC and AC loads. Experimental results on primary resonance and dynamic pull-in are shown and compared with the theoretical results.

Motion of masses with asymmetric and symmetric friction. Application to fault sliding

2021 ◽  
Author(s):  
Rui Xiang Wong ◽  
Elena Pasternak ◽  
Arcady Dyskin
Keyword(s):  
Frictional Force ◽  
Normal Load ◽  
Power Spectra ◽  
Inertial Force ◽  
Friction Reduction ◽  
Relative Mass ◽  
Hydraulic Fractures ◽  
Driving Frequency ◽  
Stick Slip ◽  
Friction Forces

<p>This study analyses a situation when a geological fault contains a section of anisotropic gouge with inclined symmetry axes (e.g. inclined layering), Bafekrpour et al. [1]. Such gouge in a constrained environment induces, under compression, asymmetric friction (different friction forces resisting sliding in the opposite directions). The rest of the gouge produces conventional symmetric friction. A mass-spring model of the gouge with asymmetric and symmetric friction sections is proposed consisting of a mass with asymmetric friction connected through a spring to another mass with symmetric friction. These masses are set on a base subjected to vibration. A parametric analysis is performed on this system. Two distinct characteristic regimes were observed: <em>recurrent movement</em> resembling stick-slip motion similar to predicted by [2] and <em>sub-frictional movement</em>. Recurrent movement arises when the inertial force is sufficient to overcome frictional force of a block with symmetric friction. Sub-frictional movement occurs when the inertial force is not sufficient to overcome frictional force of an equivalent system with only symmetric friction. The sub-frictional movement is produced by the force in the connecting spring increased due to the movement of the asymmetric friction block in the direction characterised by low friction. We formulate the criterion at which sub-frictional movement occurs. The occurrence of sub-frictional depends upon the relative mass of the symmetric and asymmetric friction sections, as well as the amplitude and driving frequency of the excitation. Power spectra of the produced vibrations are determined for both regimes. The results can shed light on mechanisms of sliding over pre-existing discontinuities and their effect on seismic event generation and propagation of hydraulic fractures in the presence of discontinuities.</p><p>[1] Bafekrpour,<strong> </strong>E., A.V. Dyskin, E. Pasternak, A. Molotnikov and Y. Estrin (2015), Internally architectured materials with directionally asymmetric friction. <em>Scientific Reports</em>, 5, Article 10732.</p><p>[2] Pasternak, E. A.V. Dyskin and I. Karachevtseva, 2020. Oscillations in sliding with dry friction. Friction reduction by imposing synchronised normal load oscillations. <em>International Journal of Engineering Science</em>, 154, 103313.</p><p><strong>Acknowledgement</strong>. AVD and EP acknowledge support from the Australian Research Council through project DP190103260.</p>

scholarly journals Bifurcation Analysis of a Transversely Cracked Nonlinear Jeffcott Rotor System at Different Resonance Cases

2019 ◽  
Vol 24 (2) ◽  
pp. 284-302 ◽  
Author(s):  
Nasser A. Saeed ◽  
Mostafa Eissa
Keyword(s):  
Multiple Scales ◽  
Perturbation Technique ◽  
Amplitude Spectrum ◽  
Orientation Angle ◽  
Dynamical Behaviour ◽  
Rotor System ◽  
Resonance Case ◽  
Restoring Force ◽  
Jeffcott Rotor ◽  
Harmonic Resonance

This work focuses on the dynamical behaviour and bifurcations of a vertically supported Jeffcott rotor system having a transverse crack and nonlinear stiffness characteristics at the primary, sub-harmonic, and super-harmonic resonance cases. The nonlinear restoring force due to the bearing-clearance, the crack breathing, the disc eccentricity, and the orientation angle between the crack and imbalance direction are considered in the system model. The equations governing the system motion are derived and solved analytically by applying the Multiple Scales Perturbation Technique (MSPT). The slow-flow modulating equations are obtained and the spinning speed response curve is plotted. The whirling orbit and amplitude spectrum are constructed in the three considered resonance cases. The acquired results provide a better understanding of the main reasons of the super- and sub-harmonic resonance excitations. In additions, we concluded that the suitable resonance case that can be used for early detections of the cracks in the rotating shafts is the sub-harmonic resonance case. Finally, the obtained results are confirmed numerically and compared with the work published in the literature

An Experimental and Theoretical Investigation of Nonlinear Vibrations of Piezoelectrically-Driven Microcantilevers

2006 ◽  
Author(s):  
S. Nima Mahmoodi ◽  
Nader Jalili
Keyword(s):  
Natural Frequency ◽  
Piezoelectric Layer ◽  
Nonlinear Vibrations ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Galerkin Approximation ◽  
Closed Form Solution ◽  
Cubic Form ◽  
Dynamic Representation ◽  
Microcantilever Beam

The nonlinear vibrations of a piezoelectrically-driven microcantilever beam are experimentally and theoretically investigated. A part of the microcantilever beam surface is covered by a piezoelectric layer, which acts as an actuator. Practically, the first resonance of the beam is of interest, and hence, the microcantilever beam is modeled to obtain the natural frequency theoretically. The bending vibrations of the beam are studied considering the inextensibility condition and the coupling between electrical and mechanical properties in piezoelectric materials. The nonlinear term appears in the form of quadratic due to presence of piezoelectric layer, and cubic form due to geometry of the beam (mainly due to the beam's inextensibility). Galerkin approximation is utilized to discretize the equations of motion. The obtained equation is simulated to find the natural frequency of the system. In addition, method of multiple scales is applied to the equations of motion to arrive at the closed-form solution for natural frequency of the system. The experimental results verify the theoretical findings very closely. It is, therefore, concluded that the nonlinear approach could provide better dynamic representation of the microcantilever than previous linear models.

Interaction Model for “Shelled Particles” and Small-Strain Modulus of Granular Materials

2018 ◽  
Vol 85 (10) ◽  
Author(s):  
Shun-hua Zhou ◽  
Peijun Guo ◽  
Dieter F. E. Stolle
Keyword(s):  
Elastic Modulus ◽  
Contact Stiffness ◽  
Scaling Law ◽  
Interaction Model ◽  
Contact Problems ◽  
Hertzian Contact ◽  
Spherical Particles ◽  
Thin Coating ◽  
Normal Contact ◽  
Laboratory Test Results

The elastic modulus of a granular assembly composed of spherical particles in Hertzian contact usually has a scaling law with the mean effective pressure p as K∼G∼p1/3. Laboratory test results, however, reveal that the value of the exponent is generally around 1/2 for most sands and gravels, but it is much higher for reclaimed asphalt concrete composed of particles coated by a thin layer of asphalt binder and even approaching unity for aggregates consisting of crushed stone. By assuming that a particle is coated with a thin soft deteriorated layer, an energy-based simple approach is proposed for thin-coating contact problems. Based on the features of the surface layer, the normal contact stiffness between two spheres varies with the contact force following kn∼Fnm and m∈[1/3,  1], with m=1/3 for Hertzian contact, m=1/2 soft thin-coating contact, m=2/3 for incompressible soft thin-coating, and m=1 for spheres with rough surfaces. Correspondingly, the elastic modulus of a random granular packing is proportional to pm with m∈[1/3,  1].

Analysis and Research on Fractal Model of Normal Contact Stiffness of Joint Interfaces

2011 ◽  
Vol 328-330 ◽  
pp. 336-345
Author(s):  
Guo Sheng Lan ◽  
Xue Liang Zhang ◽  
Hong Qin Ding ◽  
Shu Hua Wen ◽  
Zhong Yang Zhang
Keyword(s):  
Numerical Simulation ◽  
Fractal Dimension ◽  
Normal Load ◽  
Contact Stiffness ◽  
Fractal Model ◽  
Normal Contact ◽  
Fractal Models ◽  
Normal Contact Stiffness

Through the analysis and research on three fractal models of normal contact stiffness of joint interfaces, the differences between them can be found. Furthermore, numerical simulation was carried out to obtain the complicated nonlinear relations between normal contact stiffness and the normal load. The results show that the normal contact stiffness increases with the normal load, decreases with G but complicatedly varies with D. According to different fractal dimension, we can chose an appropriate one among the three fractal models of normal contact stiffness of joint interfaces when describing normal contact stiffness of joint interfaces.

Sign in / Sign up

Export Citation Format

Share Document