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Journal articles on the topic 'Stiffness and damping coefficients'

Author: Grafiati
Published: 4 June 2021
Last updated: 16 February 2022

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1

Iasa Abdulhadi, M. "Stiffness and Damping Coefficients of *Rubber." Shock and Vibration Digest 17, no. 5 (May 1, 1985): 3–9. http://dx.doi.org/10.1177/058310248501700502.

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2

Abdulhadi, M. "Stiffness and damping coefficients of rubber." Ingenieur-Archiv 55, no. 6 (1985): 421–27. http://dx.doi.org/10.1007/bf00537649.

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3

Ghosh, M. K., and N. S. Viswanath. "Recess Volume Fluid Compressibility Effect on the Dynamic Characteristics of Multirecess Hydrostatic Journal Bearings With Journal Rotation." Journal of Tribology 109, no. 3 (July 1, 1987): 417–26. http://dx.doi.org/10.1115/1.3261462.

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An analysis using finite difference method and small amplitude perturbation technique has been presented to evaluate the frequency dependent stiffness and damping coefficients of multirecess hydrostatic journal bearings including the effect of shaft rotation that results in hybrid operation. Recess volume fluid compressibility effect that results in the frequency dependent dynamic coefficients have been taken into account. Results for direct and cross stiffness coefficients, direct and cross damping coefficients are presented for a capillary compensated bearing with deep recesses. Frequency effects are shown in terms of dimensionless squeeze number (σ) for various recess volume parameters (γ) for different eccentricity ratios (ε0) and dimensionless speed parameter (Λ). It has been found that the dynamic coefficients are very drastically altered within a very useful range of frequency parameter (σ) and recess volume parameter (γ) resulting in increased direct stiffness coefficient and a substantially decreased direct damping coefficients. Journal speed parameter (Λ) results in substantial magnitude of cross stiffness and cross damping parameters. However, the cross damping coefficient is usually small in comparison to direct damping coefficient.
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4

Xu, Chun Dong, Hui Hui Feng, and Feng Feng Wang. "Analysis of the Dynamic Characteristics of the Aerostatic Journal Bearings." Applied Mechanics and Materials 607 (July 2014): 600–603. http://dx.doi.org/10.4028/www.scientific.net/amm.607.600.

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This paper investigates the dynamic characteristics of the aerostatic journal bearing, the rotation center of which is not the center of the journal length. The Finite Difference Method (FDM) and the perturbation method are employed to calculate the stiffness and damping coefficients. Results show that the coupled stiffness and damping coefficients cannot be neglected due to the rotation center being not the center of the journal length. Furthermore, with the increase of the distance between the rotation center and the center of the journal length, the coupled stiffness and damping coefficients increase.
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5

Xiang, Guo, Yijia Wang, Cheng Wang, and Zhongliang Lv. "Numerical study on the dynamic characteristics of water-lubricated rubber bearing under asperity contact." Industrial Lubrication and Tribology 73, no. 4 (April 2, 2021): 572–80. http://dx.doi.org/10.1108/ilt-12-2020-0453.

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Purpose In this study, the dynamic characteristics of the water-lubricated rubber bearing considering asperity contact are numerically studied, including water-film stiffness and damping coefficients and plastic-elastic contact stiffness coefficient. Design/methodology/approach The Kogut-Etsion elastic-plastic contact model is applied to calculate the contact stiffness coefficient at the bearing-bush interface and the perturbed method is used to calculate the stiffness and damping coefficients of water-film. In addition, the rubber deformation is determined by the finite element method (FEM) during the simulation. Parametric studies are conducted to assess the effects of the radial clearance, rubber thickness and elastic modulus on the dynamic characteristic of water-lubricated rubber bearing. Findings Numerical results indicate that stiffness and damping coefficients of water film and the contact stiffness of asperity are increased with the decreasing of the radial clearance and the dynamic coefficients are less sensitive to the rubber thickness compared with the elastic modulus of rubber. Furthermore, due to the existed groove, a sudden change of the water film direct stiffness and damping coefficients is observed when the eccentricity ratio ranges from 0.6 to 1.0. Originality/value It is expected this study can provide more information to establish a dynamic equation of water-lubricated rubber bearings exposed to mixed lubrication conditions.
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6

Al-Bender, Farid, Federico Colombo, Dominiek Reynaerts, Rodrigo Villavicencio, and Tobias Waumans. "Dynamic Characterization of Rubber O-Rings: Squeeze and Size Effects." Advances in Tribology 2017 (2017): 1–12. http://dx.doi.org/10.1155/2017/2509879.

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This paper concerns the dynamic characterization of rubber O-rings used to introduce damping in high speed gas bearing systems. O-shaped rubber rings composed of high temperature rubber compounds are characterized in terms of stiffness and damping coefficients in the frequency range 100–800 Hz. Simple formulas with frequency independent coefficients were identified to express the viscoelastic properties of the O-rings. The formulas proposed approximate the stiffness and damping coefficients of O-rings of general size.
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7

Jagadish, H. P., and L. Ravikumar. "Calibration of the Stiffness and Damping Characteristics of a Magnetorheological Fluid Long Squeeze Film Damper in Terms of Reynolds Number." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 224, no. 10 (April 22, 2010): 2121–28. http://dx.doi.org/10.1243/09544062jmes2031.

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Magnetorheological (MR) fluids are suspensions of fine micron-sized magnetizable particles in a suitable carrier liquid. The rheological properties of the fluid can be controlled by application of a suitable magnetic field and can be used in a variety of applications where variable damping and stiffness characteristics are required, based on the requirements of the rotor dynamic system. In this work, the stiffness and damping characteristics of MR fluid long squeeze film damper operating at low eccentricity ratios are calibrated in terms of Reynolds number of the squeeze film for different clearance and L/D ratios. A theoretical constant magnetic field viscosity model is developed, based on the literature, and is subsequently used to evaluate the theoretical stiffness and damping coefficients, at a particular excitation frequency. The results indicate that the stiffness and damping coefficients decrease with increasing Reynolds number of the squeeze film and is found to be abysmally low, indicating that the flow has ceased and the film has solidified. This is in accordance with the literature that predicts the formation of chain-like semi-solid structures, restricting the flow, and consequently increasing the viscosity, under the influence of the magnetic field. This change in viscosity, in turn, influences the stiffness and damping coefficients and the Reynolds number of the squeeze film. The stiffness and damping coefficients are found to increase with decreasing clearance, increasing L/D ratio, and eccentricity ratio. The results of the investigation assist the designer in obtaining the stiffness and damping characteristics of the squeeze film damper based on the Reynolds number of the squeeze film. Conversely, the stiffness and damping characteristics of the squeeze film damper are calibrated in terms of the Reynolds number of the squeeze film for different damper configurations.
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8

Peng, J. P., and M. Carpino. "Finite Element Approach to the Prediction of Foil Bearing Rotor Dynamic Coefficients." Journal of Tribology 119, no. 1 (January 1, 1997): 85–90. http://dx.doi.org/10.1115/1.2832484.

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A finite element perturbation approach to the prediction of foil bearing stiffness and damping coefficients is presented. The fluid lubricant is modeled as a simple barotropic fluid which is described by the Reynolds equation. The structural model includes membrane, bending, and elastic foundation effects in a general geometry. The equivalent viscous damping of the Coulomb friction caused by the foil relative motion is included in the structural calculation. Bearing stiffness and damping coefficients are predicted for an air-lubricated foil bearing with a corrugated sub-foil. The effects of the bearing number, bearing compliance, sub-foil Coulomb friction, and foil membrane stiffness on the bearing dynamic coefficients are discussed.
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9

Ku, C. P. Roger, J. F. Walton, and J. W. Lund. "Dynamic Coefficients of Axial Spline Couplings in High-Speed Rotating Machinery." Journal of Vibration and Acoustics 116, no. 3 (July 1, 1994): 250–56. http://dx.doi.org/10.1115/1.2930421.

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This paper provided the first opportunity to quantify the angular stiffness and equivalent viscous damping coefficients of an axial spline coupling used in highspeed turbomachinery. The bending moments and angular deflections transmitted across an axial spline coupling were measured while a nonrotating shaft was excited by an external shaker. A rotordynamics computer program was used to simulate the test conditions and to correlate the angular stiffness and damping coefficients. The effects of external force and frequency were also investigated. The angular stiffness and damping coefficients were used to perform a linear steady-state rotordynamics stability analysis, and the unstable natural frequency was calculated and compared to the experimental measurements.
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10

Marcinkevičius, Andrejus Henrikas. "Theoretical Analysis of Vibrations of a Mass Connected with a Support through a Chain of Elastic Elements." Solid State Phenomena 164 (June 2010): 303–7. http://dx.doi.org/10.4028/www.scientific.net/ssp.164.303.

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Vibrations of a mass are analyzed when the mass is connected with the support in succession of two elastic elements characterized by stiffness and damping coefficients. It is demonstrated that upon harmonic excitation of the mass and considering different values of stiffness and damping coefficients the system can respond differently in comparison to the case when the mass is connected to the support by means of single elastic element. Reduction of stiffness and damping coefficients of a pair of elements to a single one (as it is proposed in calculations of considered systems) can lead to incorrect results. This is confirmed by presented calculations and dependences.
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11

Murphy, B. T., and M. N. Wagner. "Measurement of Rotordynamic Coefficients for a Hydrostatic Radial Bearing." Journal of Tribology 113, no. 3 (July 1, 1991): 518–25. http://dx.doi.org/10.1115/1.2920654.

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Measurement of rotordynamic coefficients is presented for a pair of hydrostatic radial bearings, including direct and cross-coupled stiffness and damping. Two different hydrostatic configurations were tested: (1) an externally fed bearing 74.7 mm (2.95 in.) in diameter with a nominal direct stiffness of approximately 210 MN/m (1.2 million lb/in.) and (2) an internally fed bearing 54.6 mm (2.15 in.) in diameter with a nominal direct stiffness of approximately 88 MN/m (0.5 million lb/in.). Each bearing had 6 equally spaced hydrostatic pressure pockets, stationary for the externally fed bearing and rotating for the internally fed bearing. Also, both bearings had extended exit regions to provide additional damping. The top rotational speed was 22,700 rpm and the maximum axial Reynolds number was 50,000 using a freon derivative, Freon-113, as the working fluid. The test apparatus was a “synchronous rig” as an intentionally eccentric journal was used as the sole source of excitation. Data reduction was done by performing a matrix solution to separate damping from stiffness. Results show the internally fed bearing to be 20 percent less stiff than predicted, and to have a significant amount of damping which agrees well with predictions. The internally fed bearing was found to be approximately 60 percent less stiff than predicted, and to be roughly neutral in terms of damping, as predicted.
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12

Li, Jiming, David Ransom, Luis San Andre´s, and John Vance. "Comparison of Predictions With Test Results for Rotordynamic Coefficients of a Four-Pocket Gas Damper Seal." Journal of Tribology 121, no. 2 (April 1, 1999): 363–69. http://dx.doi.org/10.1115/1.2833948.

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Experiments and field applications have demonstrated that multiple-pocket gas damper seals effectively eliminate subsynchronous vibration and attenuate imbalance response at the critical speeds in turbomachinery. A one-control volume, turbulent bulk-flow model for the prediction of the seal leakage and rotordynamic force coefficients of centered multiple-pocket damper seals is hereby detailed. Comparisons of numerical predictions with experimental force coefficients for a four-pocket damper seal are presented. The bulk-flow model and experiments indicate the seal direct stiffness and damping force coefficients are insensitive to journal speed while the cross-coupled stiffnesses increase slightly. However, the current model overpredicts the direct damping coefficient and underpredicts the direct stiffness coefficient for increasing test pressure ratios. Computed results show that the force coefficients of multiple-pocket gas damper seals are also functions of the rotor excitation frequency.
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13

Qi, She Miao, Y. S. Ho, Hai Peng Geng, and Lie Yu. "A Theoretical Study on the Influence of Frequency on the Dynamic Characteristics of Air Film." Key Engineering Materials 462-463 (January 2011): 1230–37. http://dx.doi.org/10.4028/www.scientific.net/kem.462-463.1230.

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A general method is developed to deal with the compressibility of air film and to predict the linear dynamic stiffness and damping coefficients of air-lubricated bearings over the whole range of frequency. Numerical results reveal that both stiffness and damping coefficients are functions of not only the static parameters such as eccentricity and attitude angle, but also the frequency of disturbance. The coefficients are continuous and tend to certain values even when the frequency approaches zero or positive infinity. In general, the direct terms of the dynamic stiffness increase with the increase of the frequency, while the cross-coupling terms tend to an equal value when the frequency becomes infinite, and all the damping coefficients decrease in the higher frequency region and vanish when the frequency approaches positive infinity.
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14

Qi, She Miao, Y. S. Ho, Hai Peng Geng, and Lie Yu. "A Theoretical Study on the Influence of Frequency on the Dynamic Characteristics of Air Film." Key Engineering Materials 462-463 (January 2011): 817–24. http://dx.doi.org/10.4028/www.scientific.net/kem.462-463.817.

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A general method is developed to deal with the compressibility of air film and to predict the linear dynamic stiffness and damping coefficients of air-lubricated bearings over the whole range of frequency. Numerical results reveal that both stiffness and damping coefficients are functions of not only the static parameters such as eccentricity and attitude angle, but also the frequency of disturbance. The coefficients are continuous and tend to certain values even when the frequency approaches zero or positive infinity. In general, the direct terms of the dynamic stiffness increase with the increase of the frequency, while the cross-coupling terms tend to an equal value when the frequency becomes infinite, and all the damping coefficients decrease in the higher frequency region and vanish when the frequency approaches positive infinity.
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15

Radomirovic, Dragi, and Ivana Kovacic. "On the equivalent systems for concurrent springs and dampers – Part 2: Small out-of-plane oscillations." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 228, no. 14 (January 9, 2014): 2532–44. http://dx.doi.org/10.1177/0954406213519617.

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Concurrent linear springs belonging to systems that perform small out-of-plane oscillations around a stable equilibrium position are considered with a view to obtaining equivalent systems of three mutually orthogonal linear springs. Theorems defining their stiffness coefficients as well as their position, i.e. the position of the principal stiffness axes for which the potential energy does not contain mixed terms, are stated and proven. So far unknown invariants related to the sum of original and new stiffness coefficients are provided. In addition, the equivalent system of three mutually orthogonal dampers is obtained for any system of out-of-plane concurrent linear viscous. The theorem defining their damping coefficients and their directions, collinear with the principal damping axes for which the dissipative function does not contain mixed terms, is provided. The corresponding invariant for damping coefficients is presented, too. An ellipsoid of displacement and an ellipsoid of stiffness are discussed. Three illustrated examples are given.
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16

Wang, Li-li, Qing-liang Zeng, and Xin Zhang. "Influence of Spiral Angle on the Performance of Spiral Oil Wedge Sleeve Bearing." International Journal of Rotating Machinery 2018 (June 5, 2018): 1–7. http://dx.doi.org/10.1155/2018/5051794.

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Spiral angel is an important structure parameter of spiral oil wedge sleeve bearing, which produces greater impact on bearing performance. Based on JFO boundary condition, the generalized Reynolds equations considering four slip conditions are established. Using the concept of partial derivatives, stiffness and damping coefficients of sleeve bearing are calculated. The results show that carrying capacity and friction drag of oil film decrease, temperature rise decreases first and then increases, and end leakage rate, stiffness, and damping coefficients generally increase first and then decrease with the increase of spiral angle. The carrying capacity, friction drag, temperature rise, stiffness, and damping coefficients are smaller and the end leakage rate is higher considering wall slip and JFO condition compared with reckoning with no slip and Reynolds boundary condition.
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17

QI, Shemiao. "Dynamic stiffness and dynamic damping coefficients of aerodynamic bearings." Chinese Journal of Mechanical Engineering 43, no. 05 (2007): 91. http://dx.doi.org/10.3901/jme.2007.05.091.

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18

K. P., Lijesh, and Harish Hirani. "Stiffness and damping coefficients for rubber mounted hybrid bearing." Lubrication Science 26, no. 5 (February 26, 2014): 301–14. http://dx.doi.org/10.1002/ls.1252.

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19

Nelson, C. C. "Rotordynamic Coefficients for Compressible Flow in Tapered Annular Seals." Journal of Tribology 107, no. 3 (July 1, 1985): 318–25. http://dx.doi.org/10.1115/1.3261062.

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Derivation of the governing equations for compressible flow in a tapered annular seal is based on Hirs’ turbulent bulk-flow model. Zeroth and first-order perturbation equations are developed by an expansion in the eccentricity ratio. These equations are numerically integrated to obtain the leakage, and the direct and cross-coupled stiffness and damping coefficients. Seal parameters similar to the Space Shuttle Main Engine High Pressure Oxidizer Turbopump are used to demonstrate output from the analysis procedure. The effects of preswirl and seal taper are shown for three different length-to-diameter ratios. Generally the results indicate that prerotating the fluid significantly increases the cross-coupled stiffness but has little effect on the other coefficients, and increasing the convergent taper increases the direct stiffness while decreasing the direct damping and cross-coupled stiffness.
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20

Roberts, J. B., M. D. Ramli, and J. Ellis. "Experimental Determination of Squeeze-Film Dynamic Coefficients by a Frequency Domain Method." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 202, no. 4 (July 1988): 235–44. http://dx.doi.org/10.1243/pime_proc_1988_202_115_02.

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A frequency domain, forced vibration technique for obtaining estimates of the direct fluid damping, inertial and stiffness coefficients of a squeeze-film bearing is presented. The three coefficients are estimated through a linear, least-squares regression analysis performed on data obtained from a sequence of forced vibration tests, conducted using single sinusoids of differing frequencies. The method is applied to experimental data obtained from a squeeze-film damper with a central circumferential groove and no end seals. In addition to accurate estimates of damping coefficients, reliable estimates of inertial and stiffness coefficients are obtained.
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21

Si, Heyong, Lihua Cao, and Pan Li. "Dynamic Characteristics and Stability Prediction of Steam Turbine Rotor Based on Mesh Deformation." Strojniški vestnik – Journal of Mechanical Engineering 66, no. 3 (March 15, 2020): 164–74. http://dx.doi.org/10.5545/sv-jme.2019.6283.

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In order to study the steam flow excited vibration caused by the eccentricity of a rotor, three-dimensional rotor whirl motion is simulated based on mesh deformation. The mechanism of steam flow excited vibration and its influence on the dynamic characteristics of the rotor are investigated. The results show that the exciting forces change with the displacement of the rotor’s centre. Rotor dynamic coefficients are nonlinear when the rotor whirls pass the mesh deformation. The rotor dynamic coefficients and effective damping increase with the increase of whirl frequency. When the whirl frequency is 24.41 Hz, the rotor dynamic coefficients are strongly affected by rotational velocity. The maximum fluctuations of average direct stiffness, cross-coupling stiffness, direct damping and cross-coupling damping are 8.1 %, 113.2 %, 45.8 %, and 121.0 %, respectively. Effective damping fluctuates greatly when both whirl and rotational frequency are 24.41 Hz. The direct stiffness, direct damping, and effective damping increase with the increase of pressure ratio, which can improve rotor stability. The pressure fluctuation on the rotor’s surface is a primary reason for steam flow excited vibration. The stability margin of the rotor can be estimated precisely via effective damping.
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22

Zhanlong, Li, Sun Dagang, Qin Yuan, Zhang Wenjun, and Sun Bao. "Stiffness-Damping Matching Modelling for Vibration Isolation System of Roadheader ECB." International Journal of Acoustics and Vibration 25, no. 1 (March 30, 2020): 54–61. http://dx.doi.org/10.20855/ijav.2020.25.11514.

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The electronic control box (ECB) is a key and precise component of a roadheader. The vibration of the ECB is an increasingly prominent issue as the machine capacity grows. In order to promote the isolation effect of the ECB, a whole-body vibration model considering the cutting effect is derived, based upon which the stiffness-damping matching strategy for the ECB isolator is acquired. For engineering application, the tubular constrained damping isolator (TCDI) is developed based on the constrained damping theory and the matching strategy. The theoretical results show that the isolation effect of the ECB isolator strengthens as the stiffness coefficients decline and the damping coefficients increase. The configurations with larger rear stiffness coefficients and larger front damping coefficients could lead to a better vibration control effect. The experiment results indicate that the TCDI exhibits a greater capability of isolating the impact excitation than the traditional E-type isolator, thus verifying the whole-body vibration model for the roadheader and the matching strategy for the ECB isolator. This research can provide theoretical and practical references for the investigation of the dynamic behaviour of complex viscoelastic structures.
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23

Ariaratnam, S. T., and Wei Chau Xie. "Effect of Correlation on the Almost-Sure Asymptotic Stability of Second-Order Linear Stochastic Systems." Journal of Applied Mechanics 56, no. 3 (September 1, 1989): 685–90. http://dx.doi.org/10.1115/1.3176147.

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A method of obtaining a sufficient almost-sure (a.s.) asymptotic stability condition for second-order, linear systems with both ergodic damping and stiffness coefficients is presented. The probabilistic property of the correlation between the damping and stiffness coefficients is taken into account. A sufficient condition for a.s. asymptotic stability is derived and numerical results are presented for the case of Gaussian noise coefficients. Results obtained in some of the previous investigations are included in the present study as special cases.
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24

Falkowski, Krzysztof. "Second-Order Model of the Radial Passive Magnetic Bearing with Halbach's Array." Solid State Phenomena 198 (March 2013): 400–405. http://dx.doi.org/10.4028/www.scientific.net/ssp.198.400.

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In the paper is presented model of the passive magnetic bearing. The response of bearing is approximate by second order model. There are presented the damping and stiffness coefficient of suspension. The coefficients derived from Biot-Savards law, Ohms law and Lorenzs force. There is presented loop with molecular current as a model of magnet, final formula of damping and stiffness coefficients and static characteristic of passive magnetic bearing.
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25

Lindsey, W. Todd, and Dara W. Childs. "The Effects of Converging and Diverging Axial Taper on the Rotordynamic Coefficients of Liquid Annular Pressure Seals: Theory Versus Experiment." Journal of Vibration and Acoustics 122, no. 2 (November 1, 1995): 126–31. http://dx.doi.org/10.1115/1.568457.

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Experimental results are compared to predictions for turbulent flow, short (D=76.2mm,L/D=.17), smooth annular seals with converging and diverging axial taper. Results are presented for four geometries with the same minimum clearances: two convergent, two divergent, and a constant-clearance. Measurements were taken at seal pressure differentials and shaft rotation rates ranging from 1.34 to 3.54 MPa and 10,200 to 24,600 rpm, respectively. Measurements parameters include leakage, direct stiffness, cross-coupled stiffness, and direct damping coefficients. Results show that direct stiffness generally increases with converging axial taper and decreases with diverging axial taper; however, direct stiffness decreases in the first increase in the taper angle, contrary to predictions. Direct damping and cross-coupled stiffness were shown to decrease with increasing convergent or divergent taper. Measured damping values increase with increased running speed and decreasing average clearance. Theoretical predictions for rotordynamic coefficients are in reasonable qualitative agreement with measured results. The theory consistently underpredicts leakage by ranges of 10∼30 percent. The accuracy of predictions for leakage and rotordynamic coefficients was not influenced by running speed. [S0739-3717(00)70102-4]
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26

Shi, Yao Chen, Zhan Guo Li, and Dan Liu. "Research on the Tension Effects on Automobile Damping Synchronous Belt Transmission System Damping and Dynamic Stiffness." Applied Mechanics and Materials 687-691 (November 2014): 407–10. http://dx.doi.org/10.4028/www.scientific.net/amm.687-691.407.

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because of the synchronism belts driving has the advantages of vibration absorption, noise reduction, constant transmission ratio, so it is widely used in automotive engine timing driving system, The stiffness and damping coefficients of the synchronous belt is the main factor affecting the synchronous belt transmission in the process of vibration and noise. In this paper, the model W automotive synchronous belt is simplified as a spring damper system, to solve the stiffness coefficient and damping coefficient of synchronous belt, design of a device for measuring the synchronous belt stiffness coefficient and damping coefficient, measured the belt stiffness coefficient and damping coefficient in the conditions of different tension, and the accuracy of the solution method for synchronous belt stiffness coefficient and damping coefficient was verified as well.
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27

Ferfecki, Petr, Jaroslav Zapoměl, Michal Šofer, František Pochylý, and Simona Fialová. "Numerical computation of the damping and stiffness coefficients of the classical and magnetorheological squeeze film damper." MATEC Web of Conferences 157 (2018): 08001. http://dx.doi.org/10.1051/matecconf/201815708001.

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Technological solution, frequently used to suppress vibrations in rotating machines, consists in adding damping devices between the rotor and its frame. This is enabled by dampers working on the principle of a squeezing thin classical or magnetorheological fluid film. The Navier-Stokes equations, Reynolds equation, and modified Navier-Stokes equations are used to determine the pressure distribution in the thin fluid film. The damping and stiffness coefficients are computed by the developed procedure presented in this paper. The proposed computational approach is based on the perturbation of the synchronous circular whirling motion. The carried-out computational simulations show that the investigated mathematical models of the squeeze film damper and magnetorheological squeeze film damper allowed computation of the damping and stiffness coefficients. It has been found that the stiffness coefficients computed by the proposed mathematical models may be different.
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28

Peng, J. P., and M. Carpino. "Calculation of Stiffness and Damping Coefficients for Elastically Supported Gas Foil Bearings." Journal of Tribology 115, no. 1 (January 1, 1993): 20–27. http://dx.doi.org/10.1115/1.2920982.

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The stiffness and damping coefficients of an elastically supported gas foil bearing are calculated. A perfect gas is used as the lubricant, and its behavior is described by the Reynolds equation. The structural model consists only of an elastic foundation. The fluid equations and the structural equations are coupled. A perturbation method is used to obtain the linearized dynamic coefficient equations. A finite difference formulation has been developed to solve for the four stiffness and the four damping coefficients. The effect of the bearing compliance on the dynamic coefficients is discussed in this paper.
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29

Jin, Chaowu, Yuanping Xu, Jin Zhou, and Changli Cheng. "Active Magnetic Bearings Stiffness and Damping Identification from Frequency Characteristics of Control System." Shock and Vibration 2016 (2016): 1–8. http://dx.doi.org/10.1155/2016/1067506.

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At present, the stiffness and damping identification for active magnetic bearings (AMBs) are still in the stage of theoretical analysis. The theoretical analysis indicates that if the mechanical structure and system parameters are determined, AMBs stiffness and damping are only related to frequency characteristic of control system, ignoring operating condition. More importantly, few verification methods are proposed. Considering the shortcomings of the theoretical identification, this paper obtains these coefficients from the experiment by using the magnetic bearing as a sine exciter. The identification results show that AMBs stiffness and damping have a great relationship with the control system and rotating speed. Specifically, at low rotating speed, the stiffness and damping can be obtained from the rotor static suspension by adding the same excitation frequency. However, at high speed, different from the static suspension situation, the AMBs supporting coefficients are not only related to the frequency characteristics of control system, but also related to the system operating conditions.
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30

Jia, Chenhui, Haijiang Zhang, Shijun Guo, Ming Qiu, Wensuo Ma, and Zhuangya Zhang. "Study on dynamic characteristics of gas films of spherical spiral groove hybrid gas bearings." Proceedings of the Institution of Mechanical Engineers, Part J: Journal of Engineering Tribology 233, no. 8 (January 15, 2019): 1169–81. http://dx.doi.org/10.1177/1350650118823892.

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According to the gas film force variation law, when the bearing axis is slightly displaced from the static equilibrium position, displacement and velocity disturbance relation expressions for the gas film force increment are constructed. Moreover, combined with the bearing rotor system motion equation, calculation model equations for the gas film stiffness and damping coefficients are established. The axial and radial vibration and velocity of the gas bearings during operation are collected. The instantaneous stiffness and damping coefficients of the gas film are calculated by the rolling iteration algorithm using MATLAB. The dynamic changes in the gas film stiffness and damping under different motion states are analyzed, and the mechanism of the gas film vortex and oscillation is studied. The results demonstrate the following: (1) When the gas bearing is running in the linear steady state in cycle 1, the dynamic pressure effect is enhanced and the stability is improved by increasing the eccentricity; when the gas supply pressure is increased, the static pressure effect is enhanced and the gas film vortex is reduced, but the oscillation is strengthened. (2) With the increase in rotational speed, the gas film vortex force gradually exceeds the gas film damping force, and the stability gradually worsens, causing a fluctuation in the gas film stiffness and damping, following which singularity occurs and a half-speed vortex is formed. Meanwhile, the gas film oscillation is intensified, and the rotor enters the nonlinear stable cycle 2 state operation. (3) As the fluctuation of the film force increases, the instantaneous stiffness and damping oscillation of the film intensifies, most of the stiffness and damping coefficients exhibit distortion, and the rotor operation will enter a chaotic or unstable state. Therefore, the gas bearing stiffness and damping variation characteristics can be used to study and predict the gas bearing operating state. Finally, measures for reducing the vortex and oscillation of the gas film and improving the stability of the gas bearing operation are proposed.
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31

Ha, Hyun Cheon, and Seong Heon Yang. "Excitation Frequency Effects on the Stiffness and Damping Coefficients of a Five-Pad Tilting Pad Journal Bearing." Journal of Tribology 121, no. 3 (July 1, 1999): 517–22. http://dx.doi.org/10.1115/1.2834098.

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An experimental study is performed to investigate the frequency effects of the excitation force on the linear stiffness and damping coefficients of a LOP (load on pad) type five-pad tilting pad journal bearing with the diameter of 300.91 mm and the length of 149.80 mm. The main parameter of interest in the present work is excitation frequency to shake the test hearing. The excitation frequency is controlled independently, using orthogonally mounted hydraulic exciters, as follows: 1) excitation frequency ratio in the x-axis direction νx = 0.5, 2) excitation frequency ratio in the y-axis direction νy = 0.6, 0.7, 0.8, 0.9. The magnitude of the excitation force is controlled to make sure that the test hearing has a linear behavior during the test. The relative movement between the bearing and shaft, and the acceleration of the bearing casing are measured as a function of excitation frequency using the different values of bearing load and shaft speed. Measurements show that the variation of excitation, frequency has quite a little effect on both stiffness and damping coefficients. The stiffness coefficients of the five-pad tilting pad journal bearing slightly decrease as the excitation frequency ratio increases, while the damping coefficients slightly increase with excitation frequency ratio, especially in the case of lower speed and higher load. Both direct stiffness and damping coefficients in the direction of bearing load decrease with an increase of shaft speed, but increase with the bearing load.
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32

Kaneko, S., H. Kamei, Y. Yanagisawa, and H. Kawahara. "Experimental Study on Static and Dynamic Characteristics of Annular Plain Seals With Porous Materials." Journal of Tribology 120, no. 2 (April 1, 1998): 165–72. http://dx.doi.org/10.1115/1.2834404.

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Static and dynamic characteristics are experimentally investigated for annular plain seals with porous materials applied to the seal surface by insertion into the middle of the seal. Experimental results show that annular plain seals with porous materials have a higher leakage flow rate, larger main stiffness coefficients, and smaller cross-coupled stiffness coefficients and main damping coefficients than conventional annular plain seals with solid surfaces. In the porous seals, an increase of approximately 30 percent in the leakage flow rate and reduction of the same amount in the main damping coefficients are obtained, whereas the main stiffness coefficients for the porous seals are four to six times as much as those for the solid seals due to the increase in the hydrostatic force induced by a function of the hydrostatic porous bearing. This suggests that the quantitative effects of the porous materials on the main stiffness coefficients are much more significant than the effects on the leakage flow rate and the other dynamic coefficients. The larger main stiffness coefficients for the porous seals yield larger radial reaction force for a small concentric whirling motion, which would contribute to rotor stability from the viewpoint of increasing speed limits due to a stiffer rotor support.
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33

Zhao, Guofeng, Xinwen Wang, Chi Yu, Shucheng Liu, Jun Zhou, and Guohui Zhu. "Research on Static and Dynamic Characteristics of Shear Spring of the Vibrating Flip-Flow Screen." Symmetry 12, no. 10 (October 7, 2020): 1644. http://dx.doi.org/10.3390/sym12101644.

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The vibrating flip-flow screen (VFFS) is a high-efficiency device currently used for deep screening of moist fine-grained materials. During VFFS operation, the normal operation of the screen is affected by fatigue damage to the shear springs arranged symmetrically on both sides of the screen, leading to equipment failures and disruption production. In this paper, the shear spring’s static and dynamic characteristics in different operation conditions were studied using the INSTRON 8801 fatigue test system and Dynacell dynamic sensors. Using an experimental test of shear spring stiffness and damping coefficients, the effects of some factors, i.e., temperature, hardness, amplitude and frequency, were studied. The results show that the temperature of the shear spring on the left side of the flip-flow screen was higher than that of the right side (driving side). With an increase in temperature, the stiffness of the shear spring decreased. With the increase in amplitude, the dynamic stiffness decreased and the damping coefficients did not change; with the increase in frequency, the dynamic stiffness increased and the damping coefficient decreased. At the same amplitude, with the increase in hardness of the shear spring, the dynamic stiffness increased. Finally, the stiffness and damping coefficients of the shear spring before and after tearing were obviously reduced. These research results reveal the relationship of the characteristics of a shear spring with operational conditions, and could provide a theoretical reference for the design of the VFFS and the selection of the shear spring.
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34

Czołczyński, K. "How to obtain stiffness and damping coefficients of gas bearings." Wear 201, no. 1-2 (December 1996): 265–75. http://dx.doi.org/10.1016/s0043-1648(96)07267-5.

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35

Ransom, David, Jiming Li, Luis San Andre´s, and John Vance. "Experimental Force Coefficients for a Two-Bladed Labyrinth Seal and a Four-Pocket Damper Seal." Journal of Tribology 121, no. 2 (April 1, 1999): 370–76. http://dx.doi.org/10.1115/1.2833949.

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Experiments are presented to identify the stiffness and damping force coefficients of a two-blade, teeth on stator labyrinth seal with diverging clearance and its modified version as a four-pocket gas damper seal. The seals were tested without journal rotation and at rotor speeds of 1500 rpm and 3000 rpm for seal supply to ambient pressure ratios ranging from 1 to 3. Calibrated impact loads excite a flexibly supported housing holding rigidly the test seal. The impact loads and seal displacement and acceleration time responses are measured and recorded as frequency spectra. The instrumental variable filter method is used to identify the seal dynamic force coefficients from the measured transfer functions over a frequency range. The experiments demonstrate the four pocket gas damper seal has large (positive) direct damping coefficients and relatively small (negative) direct stiffness coefficients. The two bladed labyrinth seal exhibits positive direct stiffness and negative damping force coefficients. The leakage performance of both seals is nearly identical. The four pocket damper seal clearly outperforms the labyrinth seal in terms of rotordynamic forces. Both seals show a minimal amount of cross-coupling force effects, well within the experimental uncertainty.
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36

Brito, Geraldo Carvalho, Roberto Dalledone Machado, and Anselmo Chaves Neto. "Experimental Estimation of Journal Bearing Stiffness for Damage Detection in Large Hydrogenerators." Shock and Vibration 2017 (2017): 1–17. http://dx.doi.org/10.1155/2017/4647868.

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Based on experimental pieces of evidence collected in a set of twenty healthy large hydrogenerators, this article shows that the operating conditions of the tilting pad journal bearings of these machines may have unpredictable and significant changes. This behavior prevents the theoretical determination of bearing stiffness and damping coefficients with an adequate accuracy and makes damage detection difficult. Considering that dynamic coefficients have similar sensitivity to damage and considering that it is easier to monitor bearing stiffness than bearing damping, this article discusses a method to estimate experimentally the effective stiffness coefficients of hydrogenerators journal bearings, using only the usually monitored vibrations, with damage detection purposes. Validated using vibration signals synthesized by a simplified mathematical model that simulates the dynamic behavior of large hydrogenerators, the method was applied to a journal bearing of a 700 MW hydrogenerator, using two different excitations, the generator rotor unbalance and the vortices formed in the turbine rotor when this machine operates at partial loads. The experimental bearing stiffnesses obtained using both excitations were similar, but they were also much lower than the theoretical predictions. The article briefly discusses the causes of these discrepancies, the method’s uncertainties, and the possible improvements in its application.
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37

Tsai, N.-C., H.-Y. Li, C.-C. Lin, C.-W. Chiang, and P.-L. Wang. "Magnetic damping and stiffness effects on rod translation by active magnetic bearing." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 225, no. 8 (May 16, 2011): 1950–62. http://dx.doi.org/10.1177/0954406211402904.

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The purpose of this article is to apply the wavelet transform algorithm to identify the magnetic damping and magnetic stiffness coefficients of the drive rod with which a set of 4-pole active magnetic bearing (AMB) is equipped. By taking advantage of time–frequency analysis feature, the ridge curve of rod free response after wavelet transformation can be extracted to find the natural frequency of the rod/AMB system. In other words, due to the influence of magnetized field by AMB, the stiffness of the rod dynamics is not linear any more and can be estimated from the curve of the amplitude versus frequency by wavelet transformation. On the other hand, the non-linear damping coefficients can be estimated from the derivative of amplitude versus amplitude by wavelet transformation of rod free vibration. It is found that the non-linear magnetic damping coefficients are up to second order in polynomial and the stiffness coefficient is mainly of third order, respectively. In addition, the identified second-order damping coefficient is negative and hence implies that under specific rod displacement and speed, the dynamic of rod/AMB system in axial direction is unstable.
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38

Czolczyñski, K., T. Kapitaniak, and K. Marynowski. "Stiffness and damping coefficients of air rings with a chamber feeding system." Proceedings of the Institution of Mechanical Engineers, Part J: Journal of Engineering Tribology 212, no. 2 (February 1, 1998): 131–43. http://dx.doi.org/10.1243/1350650981541958.

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When, during the operation of rotors supported in gas bearings, their rotational velocity reaches a sufficiently high value, loss of steady state stability occurs. This instability is caused by the loss of damping properties of the gas film, which leads to self-excited vibrations. These vibrations are the basic obstacle to a widespread application of gas bearings. The phenomenon of self-excited vibrations can be avoided by introducing an elastic supporting structure between the bearing bushes and the casing, characterized by properly selected stiffness and damping coefficients. In practice, such a structure can have the form of an externally pressurized gas ring. In this paper, on the basis of selected examples, those ranges of the values of stiffness and damping coefficients of the gas ring that make it possible to retain steady state stability at practically any rotational velocity of the rotor are demonstrated. A design of the ring structure, especially of its feeding system, is also shown, which ensures the required values of stiffness and damping coefficients (with regard to the stability). These investigations have been carried out by means of a numerical simulation method with the use of a mathematical model of the gas bearing, which has already been verified many times.
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39

Tian, Zhuxin, and Runchang Chen. "A Derivation of Stiffness and Damping Coefficients for Short Hydrodynamic Journal Bearings with Pseudo-Plastic Lubricants." Journal of Mechanics 36, no. 6 (October 14, 2020): 943–53. http://dx.doi.org/10.1017/jmech.2020.42.

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ABSTRACTA new derivation considering the non-linear terms has been proposed to calculate stiffness and damping coefficients for short hydrodynamic journal bearings lubricated with pseudo-plastic fluids. The proposed method has relaxed the constraint of small perturbation method applicable to only small values of non-Newtonian factor α. An analytical solution is also given. The non-linear Reynolds equation is solved with a more reasonable boundary condition ∂p*/∂z* = 0 at the location of z*=0 while the analytical pressure distribution is obtained by seven-point Gauss-Legendre integral formula. When the non-dimensional non-Newtonian factor α is small, the stiffness and damping coefficients of computed by the proposed method can agree well with those from small perturbation method, which could verify the proposed derivation. As for large non-dimensional non-Newtonian factor α, the stiffness coefficients $K_{XX}^*$ , $K_{XY}^*$ and $K_{YX}^*$ as well as the damping coefficients $C_{XX}^*$ , $C_{XY}^*$ and $C_{YX}^*$ decrease with the increasing of non-dimensional non-Newtonian factor α. The significance of the derivation is that it can relax the constraint of small α and simplify the computation process.
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40

Arumugam, P., S. Swarnamani, and B. S. Prabhu. "An Experimental Investigation on the Static and Dynamic Characteristics of Journal Bearings Under the Influence of Twisting Misalignment." Journal of Tribology 119, no. 1 (January 1, 1997): 188–92. http://dx.doi.org/10.1115/1.2832456.

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The misalignment between the journal and the bearing in a rotor-bearing system may be due to manufacturing error, elastic deflection, thermal expansion etc. In the present work, the eight linearized stiffness and damping coefficients of the cylindrical and three lobe bearings are identified at different levels of bearing misalignment (twisting misalignment) and at different speeds of the rotor. The identification method used here needs FRFs (Frequency Response Functions) obtained by the measurements and the finite element method. The twisting misalignment changes the stiffness and damping coefficients in the vertical and horizontal directions. In the case of three lobe bearings, for 0.7 degree of misalignment, the stiffness in the vertical direction is increased by about 12 percent.
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41

Wang, Chao, Demi Ai, and Wei-Xin Ren. "A wavelet transform and substructure algorithm for tracking the abrupt stiffness degradation of shear structure." Advances in Structural Engineering 22, no. 5 (October 25, 2018): 1136–48. http://dx.doi.org/10.1177/1369433218807690.

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Time-varying parameter identification is an important research topic for structural health monitoring, performance evaluation, damage diagnosis, and maintenance. Practical civil engineering structures usually contain multiple degrees of freedom; however, damage often locally occurs. In this study, a discrete wavelet transform and substructure algorithm is presented for tracking the abrupt stiffness degradation of shear structures. A substructure model is built by the extraction of the local structure which may contain damaged region. Time-varying stiffness and damping are expanded into multi-scales using discrete wavelet analysis. An optimization method based on Akaike information criterion is introduced to select the decomposition scale. The expanded scale coefficients are evaluated using least square method, then the original time-varying stiffness or damping parameter is identified by reconstructing from the scale coefficients. To validate the proposed method, a numerical example of seven-story shear structure with time-varying stiffness and damping is proposed. Experiment for a three-story shear-type structure with abrupt stiffness degradation is also tested in the laboratory. Both numerical and experimental results indicate that the proposed method can effectively identify the abrupt degradation of stiffness parameter with a satisfactory accuracy.
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42

Qi, Feng, Tianshu Wang, and Junfeng Li. "The elastic contact influences on passive walking gaits." Robotica 29, no. 5 (December 2, 2010): 787–96. http://dx.doi.org/10.1017/s0263574710000779.

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SUMMARYThis paper presents a new planar passive dynamic model with contact between the feet and the ground. The Hertz contact law and the approximate Coulomb friction law were introduced into this human-like model. In contrast to McGeer's passive dynamic models, contact stiffness, contact damping, and coefficients of friction were added to characterize the walking model. Through numerical simulation, stable period-one gait and period-two gait cycles were found, and the contact forces were derived from the results. After investigating the effects of the contact parameters on walking gaits, we found that changes in contact stiffness led to changes in the global characteristics of the walking gait, but not in contact damping. The coefficients of friction related to whether the model could walk or not. For the simulation of the routes to chaos, we found that a small contact stiffness value will lead to a delayed point of bifurcation, meaning that a less rigid surface is easier for a passive model to walk on. The effects of contact damping and friction coefficients on routes to chaos were quite small.
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43

Hahn, E. J. "An Energy Approach to Linearizing Squeeze-Film Damper Forces." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 199, no. 1 (January 1985): 57–63. http://dx.doi.org/10.1243/pime_proc_1985_199_091_02.

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Analyses of multi-degree of freedom rotor-bearing systems incorporating non-linear elements, such as squeeze-film dampers, generally necessitate time consuming transient solution. Consequently, it is often too expensive to carry out parametric design studies on such systems. This paper presents a general technique for linearizing the non-linear element forces using equivalent stiffness and damping coefficients with energy dissipation and energy storage-release concepts. The approach is illustrated and tested for both centrally preloaded squeeze-film dampers and for squeeze-film dampers without centralizing springs under a combination of unidirectional and unbalance loading. The results predicted by using such equivalent stiffness and damping coefficients agree quite well with those obtained from the full transient solution, even where the unidirectional load exceeds the dynamic load and the damper is operating at high eccentricity. An iterative procedure is proposed which, with the aid of such stiffness and damping coefficients, should significantly reduce the computation time presently needed to carry out parametric design studies on general multi-degree of freedom systems incorporating non-linear elements such as squeeze-film dampers.
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44

Ting, Leehua, and John Tichy. "Stiffness and Damping of an Eddy Current Magnetic Bearing." Journal of Tribology 114, no. 3 (July 1, 1992): 600–605. http://dx.doi.org/10.1115/1.2920923.

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The perturbation method, aided by symbolic computation software, is employed to solve Maxwell’s equations to obtain theoretical predictions of forces and stiffness and damping coefficients for a simplified eddy current magnetic bearing. The results show that this kind of bearing has low stiffness and extremely low damping compared to conventional bearings. In fact, the damping is probably negative in most cases. In addition, the cross-coupled stiffness is relatively high, further contributing to rotor-dynamic problems. Despite these drawbacks, if advanced high temperature superconducting materials become practical, eddy current magnetic bearings may be useful in many applications.
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45

Chen, S. S., S. Zhu, and J. A. Jendrzejczyk. "Fluid Damping and Fluid Stiffness of a Tube Row in Crossflow." Journal of Pressure Vessel Technology 116, no. 4 (November 1, 1994): 370–83. http://dx.doi.org/10.1115/1.2929604.

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Motion-dependent fluid forces acting on a tube array were measured as a function of excitation frequency, excitation amplitude, and flow velocity. Fluid-damping and fluid-stiffness coefficients were obtained from measured motion-dependent fluid forces as a function of reduced flow velocity and excitation amplitude. The water channel and test setup provide a sound facility for obtaining key coefficients for fluidelastic instability of tube arrays in crossflow. Once the motion-dependent fluid-force coefficients have been measured, a reliable design guideline, based on the unsteady flow theory, can be developed for fluidelastic instability of tube arrays in crossflow.
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46

Sarangi, M., B. C. Majumdar, and A. S. Sekhar. "On the Dynamics of Elastohydrodynamic Mixed Lubricated Ball Bearings. Part I: Formulation of Stiffness and Damping Coefficients." Proceedings of the Institution of Mechanical Engineers, Part J: Journal of Engineering Tribology 219, no. 6 (June 1, 2005): 411–21. http://dx.doi.org/10.1243/135065005x34071.

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The problems of stiffness and damping characteristics of isothermal elastohydrodynamic mixed lubricated point contact are evaluated numerically considering surface roughness effect including asperity contact load. A set of equations under steady-state and dynamic conditions is derived from the classical Reynolds equation, using linear perturbation method. The elasticity equation and steady-state Reynolds equation are solved simultaneously for finding the steady-state pressure distribution, using finite difference method. Then, the set of perturbed equations is solved for the dynamic pressure distribution in the contact. A Gaussian surface roughness is adopted to model both surface roughness and mixed lubrication. Total load capacity of the contact is calculated from the lubricant film pressure and contact pressure distribution. Results are compared with those of smooth isothermal cases. The stiffness and damping coefficients of the contact are determined using the dynamic pressures. The asperity contact stiffness is calculated separately. Effect of various design parameters on stiffness and damping characteristics of a ball bearing is investigated.
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47

Ertas, B., A. Gamal, and J. Vance. "Rotordynamic Force Coefficients of Pocket Damper Seals." Journal of Turbomachinery 128, no. 4 (February 1, 2006): 725–37. http://dx.doi.org/10.1115/1.2221327.

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This paper presents measured frequency dependent stiffness and damping coefficients for 12-bladed and 8-bladed pocket damper seals (PDS) subdivided into four different seal configurations. Rotating experimental tests are presented for inlet pressures at 69 bar (1000 psi), a frequency excitation range of 20–300 Hz, and rotor speeds up to 20,200 rpm. The testing method used to determine direct and cross-coupled force coefficients was the mechanical impedance method, which required the measurement of external shaker forces, system accelerations, and motion in two orthogonal directions. In addition to the impedance measurements, dynamic pressure responses were measured for individual seal cavities of the eight-bladed PDS. Results of the frequency dependent force coefficients for the four PDS designs are compared. The conclusions of the tests show that the eight-bladed PDS possessed significantly more positive direct damping and negative direct stiffness than the 12-bladed seal. The results from the dynamic pressure response tests show that the diverging clearance design strongly influences the dynamic pressure phase and force density of the seal cavities. The tests also revealed the measurement of same-sign cross-coupled (cross-axis) stiffness coefficients for all seals, which indicate that the seals do not produce a destabilizing influence on rotor-bearing systems.
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48

Hirani, H., K. Athre, and S. Biswas. "Dynamic Analysis of Engine Bearings." International Journal of Rotating Machinery 5, no. 4 (1999): 283–93. http://dx.doi.org/10.1155/s1023621x99000251.

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This paper presents a simple methodology to evaluate the stiffness and damping coefficients of an engine bearing over a load cycle. A rapid technique is used to determine the shaft ‘limit cycle’ under engine dynamic loads. The proposed theoretical model is based on short and long bearing approximations. The results obtained by present approximation are compared with those obtained by numerical method. The influence of thermal effects on the stiffness and damping coefficients is predicted by using a simplified thermal analysis. In order to illustrate the application of the proposed scheme, one engine main bearing and a connecting rod bearing are analysed.
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49

Sarangi, M., B. C. Majumdar, and A. S. Sekhar. "On the Dynamics of Elastohydrodynamic Mixed Lubricated Ball Bearings. Part II: Non-Linear Structural Vibration." Proceedings of the Institution of Mechanical Engineers, Part J: Journal of Engineering Tribology 219, no. 6 (June 1, 2005): 423–33. http://dx.doi.org/10.1243/135065005x34080.

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Equations of motion of a ball bearing are formulated in generalized coordinates, using Lagrange's equation considering the vibrational characteristics of the individual constituents such as inner race, outer race, cage, and balls, in order to investigate the structural vibration of the bearing. This article is the second part of the present study dealing with structural vibration, whereas in the first part, elastohydrodynamic mixed lubricated contact stiffness and damping coefficients are determined. Utilizing these stiffness and damping coefficients, a non-linear load-deflection contact model is developed. This is then used in the equations of motion. The equations of motion are solved using Runge-Kutta integration technique. This work differs from the previous studies in the sense that the model simulates the vibration, considering that both the lubricated contact stiffness and damping correspond to the conservative and dissipative energies, respectively. It is observed that under undamped conditions, all the elements of the bearing actively participate in energy sharing and oscillate periodically, containing more than one frequency. The system vibration, however, died down rapidly in the presence of damping.
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50

Qin, Ming, Ning Xie, Hui Wang, Kai Zhang, Xue Ping Liu, and Dong Xiang. "Gear Rattle Modeling Based on Transient Stiffness and Damping Analysis." Applied Mechanics and Materials 271-272 (December 2012): 936–47. http://dx.doi.org/10.4028/www.scientific.net/amm.271-272.936.

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Structural characters of gear contact have a determinative impact on gear rattle. Contrarily traditional method using average stiffness and damp coefficients which weakens the accuracy on modeling and analysis of gear rattle phenomenon, in this paper, a methodology modeling gear rattle process with transient stiffness and damp coefficients is proposed. Gear rattle process is modeled by considering the physical model of gear contact, gear pair movement, and actually geometrical meshing curve. A study of gear rattle is made by the simulation program and an experiment is also done to verify the method. Results show that this method can effectively analyze the frequency and the relative displacement of gear rattle, etc.
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