scholarly journals Nonlinear vibration of crowned gear pairs considering the effect of Hertzian contact stiffness

2019 ◽  
Vol 1 (5) ◽  
Author(s):  
Moslem Molaie ◽  
Farhad S. Samani ◽  
Habibollah Motahar
Keyword(s):  
Nonlinear Vibration ◽  
Contact Stiffness ◽  
Hertzian Contact

Nonlinear Vibrations in Contact Atomic Force Microscopy

2003 ◽  
Author(s):  
Joseph A. Turner
Keyword(s):  
Nonlinear Vibration ◽  
Multiple Scales ◽  
Contact Stiffness ◽  
Vibration Response ◽  
Hertzian Contact ◽  
Force Microscopy ◽  
Softening Behavior ◽  
Atomic Force ◽  
Frequency Relation ◽  
Frequency Curves

The nonlinear vibration response of an atomic force microscope cantilever in contact with a vibrating sample is investigated. The tip-sample contact is modeled using Hertzian contact mechanics. The method of multiple scales is used to analyze this problem in which it is assumed that the beam remains in contact with the moving surface at all times. The primary result from this analysis is the amplitude-frequency relation for the various flexural modes. The amplitude-frequency curves exhibit softening behavior as expected. The amount of softening is shown to depend on the linear contact stiffness as well as the specific mode. The modal sensitivity to nonlinearity is the result of the nonlinearity being restricted to a single position. The mode shape greatly affects the degree to which the nonlinearity influences the frequency response. The Hertzian restriction is then loosened slightly such that variations in nonlinear contact stiffness are examined. These results depend on the linear contact stiffness and mode number as well. The nonlinear vibration response is expected to provide new insight on the nonlinear tip mechanics present in these systems.

The Effect of Thermo-Hydrodynamics on Manual Automotive Transmissions Gear Rattle

2009 ◽  
Author(s):  
Miguel De la Cruz ◽  
Stephanos Theodossiades ◽  
Homer Rahnejat ◽  
Patrick Kelly
Keyword(s):  
Contact Stiffness ◽  
Hertzian Contact ◽  
Transmission Model ◽  
System Response ◽  
Common Denominator ◽  
Gear Teeth ◽  
Lubricated Contacts ◽  
Engine Order ◽  
Dynamic Transmission Model ◽  
Gear Rattle

Manual transmission gear rattle is the result of repetitive impacts of gear meshing teeth within their backlash. This NVH phenomenon is a major industrial concern and can occur under various loaded or unloaded conditions. It fundamentally differs from other transient NVH phenomena, such as clonk or thud, which are due to impulsive actions. However, they all have their lowest common denominator in the action of contact/impact forces through lubricated contacts. Various forms of rattle have, therefore, been defined: idle rattle, drive rattle, creep rattle and over-run rattle. This paper presents a dynamic transmission model for creep rattle conditions (engaged gear at low engine RPM). The model takes into account the lubricated impact force between a gear teeth pair during a meshing cycle as well as the friction between their flanks. Hertzian contact conditions are applied to the gear pair along the torque path. Additionally, isoviscous hydrodynamic regime of lubrication is assumed for unselected (loose gear pairs) with lightly loaded impact conditions. The highly non-linear impacts induce a range of system response frequencies. These include engine order harmonics, harmonics of meshing frequency and natural frequencies related to contact stiffness. The last of these are dependent on the contact geometry and lubricant rheology. The analysis includes lubricant viscosity variation due to generated contact pressures as well as temperature. For loose gears, subject to oscillations on their retaining bearings, bearing friction is also considered.

scholarly journals Modeling and Dynamic Analysis of Spherical Roller Bearing with Localized Defects: Analytical Formulation to Calculate Defect Depth and Stiffness

2016 ◽  
Vol 2016 ◽  
pp. 1-11 ◽  
Author(s):  
Behnam Ghalamchi ◽  
Jussi Sopanen ◽  
Aki Mikkola
Keyword(s):  
Dynamic Model ◽  
Contact Stiffness ◽  
Roller Bearing ◽  
Measurement Data ◽  
Contact Forces ◽  
Hertzian Contact ◽  
Radial Clearance ◽  
Outer Race ◽  
Rolling Elements ◽  
Localized Defects

Since spherical roller bearings can carry high load in both axial and radial direction, they are increasingly used in industrial machineries and it is becoming important to understand the dynamic behavior of SRBs, especially when they are affected by internal imperfections. This paper introduces a dynamic model for an SRB that includes an inner and outer race surface defect. The proposed model shows the behavior of the bearing as a function of defect location and size. The new dynamic model describes the contact forces between bearing rolling elements and race surfaces as nonlinear Hertzian contact deformations, taking radial clearance into account. Two defect cases were simulated: an elliptical surface on the inner and outer races. In elliptical surface concavity, it is assumed that roller-to-race-surface contact is continuous as each roller passes over the defect. Contact stiffness in the defect area varies as a function of the defect contact geometry. Compared to measurement data, the results obtained using the simulation are highly accurate.

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].

Braking Impact of Normal Dither Signals

2005 ◽  
Author(s):  
Jeff Badertscher ◽  
Kenneth A. Cunefare
Keyword(s):  
Contact Stiffness ◽  
Frequency Control ◽  
Primary Resonance ◽  
Theoretical Models ◽  
Friction Reduction ◽  
Hertzian Contact ◽  
Brake Squeal ◽  
Stick Slip ◽  
Friction Oscillator ◽  
Braking Torque

Dither control is a method of introducing high frequency control efforts into a system to suppress a lower frequency disturbance. One application of dither control is the suppression of automotive brake squeal. Brake squeal is a problem that has plagued the automotive industry for years. Placing a piezoceramic stack actuator in the piston of a floating caliper brake creates an experimental normal dither system. Many theoretical models indicate a reduction in the braking torque due to the normal dither signal. Using a Hertzian contact stiffness model the loss in friction is due to lowering the average normal force. There are also theories that the dither signal eliminates the ‘stick-slip’ oscillation causing an effective decrease in the friction force. Yet another theory indicates that the effective contact area is reduced, lowering the mean coefficient of friction. A particular approach considering a single degree of freedom friction oscillator predicts a maximum friction reduction of 10%, occurring at the primary resonance of the system. This paper will concentrate on validating this claim by experimentally determining braking torque reduction for a variety of dither control signals. Several dither control frequencies were chosen at system resonances, while others were chosen at frequencies most likely to provide control of the system. These frequencies were chosen based on previous squeal suppression research. The results indicate that dither control frequencies at system resonances have a greater impact on the braking system’s performance. In general, dither control reduces braking torque by no more than 2%.

The Effect of Hertzian, Bending and Shearing Stiffness on Noise and Vibration

2001 ◽  
Author(s):  
Jamil Abdo ◽  
Kambiz Farhang ◽  
Mousa Mohsen
Keyword(s):  
Mathematical Model ◽  
Contact Stiffness ◽  
Hertzian Contact ◽  
Hertz Contact ◽  
Resonant Frequencies ◽  
Tangential Contact ◽  
Apparent Stiffness ◽  
Fundamental Understanding ◽  
The Mathematical Model ◽  
Friction Induced Vibration

Abstract Since the apparent stiffness due to contact of one surface on another relates directly to the localized resonant frequencies, it is believed that accurate account of this property will lead to the fundamental understanding of causes of friction-induced vibration and noise. The mathematical model of contact is utilized to develop formulae for normal and tangential contact stiffness. The inclusion of a study in which the various modes of elastic deflections of an asperity are also considered, as well as their effects. The bending, shear and Hertz contact modes of elastic deflection are assumed to simultaneously occur for an asperity. Investigation of the combined effect of bending, shear and Hertzian contributions to the contact stiffness is indicating that the equivalent contact stiffness is best represented, among the three types of stiffness, by that due to Hertzian contact.

An Improved Model for Calculating the Mesh Stiffness of Helical Gears

2019 ◽  
Author(s):  
Jing Wei ◽  
Shaoshuai Hou ◽  
Aiqiang Zhang ◽  
Chunpeng Zhang
Keyword(s):  
Analytical Method ◽  
Coupling Effect ◽  
Contact Stiffness ◽  
Helical Gear ◽  
Accurate Solution ◽  
Gear Transmission ◽  
Hertzian Contact ◽  
Mesh Stiffness ◽  
Single Tooth ◽  
Helical Angle

Abstract Time-varying mesh stiffness (TVMS) is one of the important internal excitations of gear transmission systems. Accurate solution of meshing stiffness is the key to research the vibration response of gear transmission system. In the traditional analytical method (TAM), the TVMS of single-teeth engaged region consist of bending, shearing, axial compression deformation stiffness, fillet-foundation stiffness, and Hertzian contact stiffness, the TVMS of double-tooth engaged region is the sum of the single-tooth engaged region, which will lead to repeated calculation of the fillet-foundation stiffness. In order to overcome this shortcoming, considering the coupling effect between two pairs of meshing tooth, an improved method of fillet-foundation is adopted to calculate to TVMS of each slice gear. According to the ‘slicing method’, the helical gear is divided into slice gear. Considering the coupling effect of each slice gear, the TVMS of helical gear can be obtained. The improved analytical method (IAM) is verified by comparing with finite element method (FEM) and TAM. Based on the IAM, the effects of the helical angle, face width, the number of gear, and modification coefficient on the mesh characteristics are analyzed. The results show that the IAM is consistent with the FEM and also consistent with TAM in single-tooth engagement. However, there is obviously error with the TAM in double-tooth or multi-tooth engagement.

Advancement and Application of the Bezgin Method to Estimate Effects of Stiffness Variations along Railways on Wheel Forces

2019 ◽  
Vol 2673 (7) ◽  
pp. 248-264
Author(s):  
Niyazi Özgür Bezgin ◽  
Mohamed Wehbi
Keyword(s):  
Analytical Method ◽  
Contact Stiffness ◽  
Hertzian Contact ◽  
Dynamic Impact ◽  
Transition Zones ◽  
Railway Tracks ◽  
Impact Forces ◽  
Track Stiffness ◽  
Railway Engineering ◽  
New Equation

The need for an analytical method that one can apply manually to estimate dynamic impact forces on railway tracks that occur because of varying track stiffness or track profile initiated a study to develop an analytical method named as the Bezgin Method. The advancement of this method presented in this paper includes an extension of a set of equations developed and introduced by the first author earlier as the Bezgin Equations using the proposed method and development of a new equation. In addition to track stiffness taken into consideration in the equations introduced earlier, the Extended Bezgin Equations presented in this paper take into account bogie stiffness, wheel spring stiffness, Hertzian contact stiffness, and a factor for damping. The new equation takes into account the effect of vertical wheel acceleration as a train transitions to a stiffer structure or transitions along an ascending track profile. The paper unites and applies these equations to estimate wheel forces that develop along stiffness transition zones by considering an array of train speeds for an array of track stiffness ratios and representative values for track profile deviations along the transitions. Final section of the paper includes elaborate finite element analyses of structural track models that involve transitions of soil supported ballasted railway tracks with concrete based ballasted tracks along various transition lengths and compares their estimates for dynamic impact force factors with those estimated by the Extended Bezgin Equations. The paper concludes with a discussion of the potential uses, benefits, and the value of the Bezgin Method for railway engineering.

On the undamped free vibration of a mass interacting with a Hertzian contact stiffness

2011 ◽  
Vol 38 (8) ◽  
pp. 560-564 ◽  
Author(s):  
Huifang Xiao ◽  
M.J. Brennan ◽  
Yimin Shao
Keyword(s):  
Free Vibration ◽  
Contact Stiffness ◽  
Hertzian Contact
Sign in / Sign up

Export Citation Format

Share Document