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.

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.

Perfect Mechanical Sealing in Rough Elastic Contacts: Is it Possible?

2012 ◽  
Author(s):  
Ilya I. Kudish ◽  
Donald K. Cohen ◽  
Brenda Vyletel
Keyword(s):  
Elastic Modulus ◽  
Rough Surface ◽  
Applied Load ◽  
Contact Region ◽  
Effective Elastic Modulus ◽  
Necessary Condition ◽  
Surface Irregularity ◽  
Normal Contact ◽  
Multiply Connected ◽  
Sharp Edges

Generally, it is assumed that under any applied force there will always be some gap between the surfaces in a contact of rough elastic surfaces resulting in a discontinuous (i.e. multiply connected) contact. The presence of gaps along the line contact relates to the ability to form an adequate mechanical seal across an interface. This paper will demonstrate that for a twice continuously differentiable rough surface with sufficiently small asperity amplitude and/or sufficiently large applied load and/or sufficiently low material elastic modulus singly connected contacts exist. Solution of a contact problem for a rough elastic half-plane and a perfectly smooth rigid indenter with sharp edges is considered. First, considered a problem with artificially created surface irregularity and it is shown that for such a surface the contact region is always multiply connected. An exact solution of the problem for an indenter with sharp edges resulting in a singly connected contact region is considered and it is conveniently expressed in the form of a series in Chebyshev polynomials. A sufficient (not necessary) condition for a contact of an indenter with sharp edges and a rough elastic surface to be singly connected is derived. The singly connected contact condition depends on the surface micro-topography, material effective elastic modulus, and applied load. It is determined that in most cases a normal contact of a twice continuously differentiable rough surface with sufficiently small asperity amplitude and/or sufficiently large applied load is singly connected.

Nonlinear Normal Modes of a Continuous System With Quadratic Nonlinearities

1995 ◽  
Vol 117 (2) ◽  
pp. 199-205 ◽  
Author(s):  
A. H. Nayfeh ◽  
S. A. Nayfeh
Keyword(s):  
Boundary Conditions ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Normal Modes ◽  
Continuous System ◽  
Nonlinear Normal Modes ◽  
Continuous Systems ◽  
Quadratic Nonlinearities ◽  
Nonlinear Normal ◽  
Quadratic And Cubic Nonlinearities

We use two approaches to determine the nonlinear modes and natural frequencies of a simply supported Euler-Bernoulli beam resting on an elastic foundation with distributed quadratic and cubic nonlinearities. In the first approach, we use the method of multiple scales to treat the governing partial-differential equation and boundary conditions directly. In the second approach, we use a Galerkin procedure to discretize the system and then determine the normal modes from the discretized equations by using the method of multiple scales and the invariant manifold approach. Whereas one- and two-mode discretizations produce erroneous results for continuous systems with quadratic and cubic nonlinearities, all methods, in the present case, produce the same results because the discretization is carried out by using a complete set of basis functions that satisfy the boundary conditions.

Some Observations of Mode Localizations in Flexibly Coupled Two Rotors

2005 ◽  
Author(s):  
Yohta Kunitoh ◽  
Hiroshi Yabuno ◽  
Tsuyoshi Inoue ◽  
Yukio Ishida
Keyword(s):  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Rotor System ◽  
The Other ◽  
Whirling Motion ◽  
Averaged Equations ◽  
Weakly Coupled ◽  
Nonlinear Normal

Mode localizations in a weakly coupled two-span rotor system are theoretically and experimentally discussed. One rotor has a slight unbalance and the other one is well-assembled. First, the equations governing the whirling motions of the coupled rotors are expressed due to nonlinearity in each span and the weakness of the coupling. The averaged equations are obtained by the method of multiple scales and it is shown that the nonlinear normal modes are bifurcated from the linear normal modes. It results from this bifurcation that the number of nonlinear normal modes exceeds the equivalent degree of freedom of the two-span rotor system, i.e., 2-degree under the assumption that the trajectory of the whirling motion is circle. Also, it is theoretically clarified that whirling motion caused by the unbalance in the rotor is localized in the rotor with unbalance or in one without unbalance depending on the rotational speed. Furthermore, these mode localizations are experimentally confirmed.

Numerical Analysis of Normal Contact Stiffness of Rough Surfaces

2016 ◽  
Vol 846 ◽  
pp. 300-305
Author(s):  
Chong Pu Zhai ◽  
Yi Xiang Gan ◽  
Dorian Hanaor
Keyword(s):  
Rough Surface ◽  
Rough Surfaces ◽  
Normal Load ◽  
Contact Stiffness ◽  
Surface Characterisation ◽  
Small Surface ◽  
Normal Contact ◽  
Contact Behaviour ◽  
Normal Contact Stiffness ◽  
Stress Dependent

A numerical model was proposed to investigate the contact behaviour of a solid with a rough surface squeezed against a rigid flat plane. We considered simulated hierarchical surface structures as well as scanned surface data obtained by the profilometry of isotropically roughened specimens. The simulated and treated surfaces were characterised using statistical and fractal parameters. The evolution of contact stiffness under increasing normal compression was analysed through the total truncated area at varying heights, in order to relate contact mechanics to different surface parameters employed for surface characterisation. For a relatively small surface interference, the predicted stress-dependent normal contact stiffness of both scanned and simulated surfaces is in good agreement with experimental observation from nanoindentation tests, revealing a power-law function of the normal load, with the exponent of this relationship closely depending on the fractal dimension of rough surfaces. The numerical results show that the amplitude of a fractal rough surface mainly contributes to the magnitude of the contact stiffness at a given normal load.

Modeling Tangential Contact of Rough Surfaces With Elastic- and Plastic-Deformed Asperities

2017 ◽  
Vol 139 (5) ◽  
Author(s):  
Dong Wang ◽  
Chao Xu ◽  
Qiang Wan
Keyword(s):  
Rough Surface ◽  
Normal Load ◽  
Experimental Results ◽  
Partial Slip ◽  
Tangential Load ◽  
Stick Slip ◽  
Normal Contact ◽  
Tangential Contact ◽  
Bilinear Relation ◽  
Proposed Model

A new tangential contact model between a rough surface and a smooth rigid flat is proposed in this paper. The model considers the contribution of both elastically deformed asperities and plastically deformed asperities to the total tangential load of rough surface. The method combining the Mindlin partial slip solution with the Hertz solution is used to model the contact formulation of elastically deformed asperities, and for the plastically deformed asperities, the solution combining the fully plastic theory of normal contact with the bilinear relation between the tangential load and deformation developed by Fujimoto is implemented. The total tangential contact load is obtained by Greenwood and Williamson statistical analysis procedure. The proposed model is first compared to the model considering only elastically deformed asperities, and the effect of mean separation and plasticity index on the relationship between the tangential load and deformation is also investigated. It is shown that the present model can be used to describe the stick–slip behavior of the rough surface, and it is a more realistic-based model for the tangential rough contact. A comparison with published experimental results is also made. The proposed model agrees very well with the experimental results when the normal load is small, and shows an error when the normal load is large.

On Direct Methods for Constructing Nonlinear Normal Modes of Continuous Systems

1995 ◽  
Vol 1 (4) ◽  
pp. 389-430 ◽  
Author(s):  
Ali H. Nayfeh
Keyword(s):  
Normal Forms ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Direct Method ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Continuous Systems ◽  
Internal Resonances ◽  
Relief Valve ◽  
Nonlinear Normal

A direct method based on the method of normal forms is proposed for constructing the nonlinear normal modes of continuous systems. The proposed method is compared with the method of multiple scales and the methods of Shaw and Pierre and King and Vakakis by applying them to three conservative systems with cubic nonlinearities: (a) a hinged-hinged beam resting on a nonlinear elastic foundation, (b) a model of a relief valve (linear elastic spring attached to a nonlinear spring with a mass), and (c) a simply supported linear beam with nonlinear torsional springs at both ends. In the absence of internal resonance, the constructed nonlinear modes with all four methods are the same. The method of multiple scales seems to be the simplest and the least computationally demanding. The methods of multiple scales and normal forms are applicable to problems with and without internal resonances, whereas the present forms of the methods of Shaw and Pierre and King and Vakakis are not applicable to problems with internal resonances.

Perfect Mechanical Sealing in Rough Elastic Contacts: Is It Possible?

2012 ◽  
Vol 80 (1) ◽  
Author(s):  
IIya I. Kudish ◽  
Donald K. Cohen ◽  
Brenda Vyletel
Keyword(s):  
Elastic Modulus ◽  
Rough Surface ◽  
Applied Load ◽  
Contact Region ◽  
Effective Elastic Modulus ◽  
Necessary Condition ◽  
Surface Irregularity ◽  
Normal Contact ◽  
Multiply Connected ◽  
Sharp Edges

Generally, it is assumed that under any applied force there will always be some gap between the surfaces in a contact of rough elastic surfaces, resulting in a discontinuous (i.e., multiply connected) contact. The presence of gaps along the line contact relates to the ability to form an adequate mechanical seal across an interface. This paper will demonstrate that for a twice continuously differentiable rough surface with sufficiently small asperity amplitude and/or sufficiently large applied load and/or sufficiently low material elastic modulus, singly connected contacts exist. The solution of a contact problem for a rough elastic half-plane and a perfectly smooth rigid indenter with sharp edges is considered. First, a problem with artificially created surface irregularity is considered and it is shown that, for such a surface, the contact region is always multiply connected. An exact solution of the problem for an indenter with sharp edges resulting in a singly connected contact region is considered and it is conveniently expressed in the form of a series in Chebyshev polynomials. A sufficient (not necessary) condition for a contact of an indenter with sharp edges and a rough elastic surface to be singly connected is derived. The singly connected contact condition depends on the surface microtopography, material effective elastic modulus, and applied load. It is determined that, in most cases, a normal contact of a twice continuously differentiable rough surface with sufficiently small asperity amplitude, sufficiently low material elastic modulus, and/or sufficiently large applied load is singly connected.

On the modulation of water waves on shear flows

1976 ◽  
Vol 347 (1651) ◽  
pp. 537-546 ◽  
Keyword(s):  
Water Waves ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Vries Equation ◽  
Harmonic Wave ◽  
Short Wave ◽  
Long Wave ◽  
Stokes Waves ◽  
The Stability ◽  
Wave Limit

The method of multiple scales is used to examine the slow modulation of a harmonic wave moving over the surface of a two dimensional channel. The flow is assumed inviscid and incompressible, but the basic flow takes the form of an arbitrary shear. The appropriate nonlinear Schrödinger equation is derived with coefficients that depend, in a complicated way, on the shear. It is shown that this equation agrees with previous work for the case of no shear; it also agrees in the long wave limit with the appropriate short wave limit of the Korteweg-de Vries equation, the shear being arbitrary. Finally, it is remarked that the stability of Stokes waves over any shear can be examined by using the results derived here.

scholarly journals Dynamic analysis of a parametrically excited golden Muga silk embedded pneumatic artificial muscle

2018 ◽  
Vol 211 ◽  
pp. 02008 ◽  
Author(s):  
Bhaben Kalita ◽  
S. K. Dwivedy
Keyword(s):  
Dynamic Analysis ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Quadratic Function ◽  
Artificial Muscle ◽  
Equation Of Motion ◽  
Time Response ◽  
Phase Portraits ◽  
Pneumatic Artificial Muscle ◽  
Pneumatic Muscle

In this work a novel pneumatic artificial muscle is fabricated using golden muga silk and silicon rubber. It is assumed that the muscle force is a quadratic function of pressure. Here a single degree of freedom system is considered where a mass is supported by a spring-damper-and pneumatically actuated muscle. While the spring-mass damper is a passive system, the addition of pneumatic muscle makes the system active. The dynamic analysis of this system is carried out by developing the equation of motion which contains multi-frequency excitations with both forced and parametric excitations. Using method of multiple scales the reduced equations are developed for simple and principal parametric resonance conditions. The time response obtained using method of multiple scales have been compared with those obtained by solving the original equation of motion numerically. Using both time response and phase portraits, variation of few systems parameters have been carried out. This work may find application in developing wearable device and robotic device for rehabilitation purpose.

Sign in / Sign up

Export Citation Format

Share Document