Parametric Instability in a Gear Train System Due to Stiffness Variation

2013 ◽  
Author(s):  
S. Y. Wang ◽  
S. C. Sinha
Keyword(s):  
Differential Equations ◽  
Parametric Instability ◽  
Gear Train ◽  
Elastic Model ◽  
Time Varying ◽  
Ring Gear ◽  
Mesh Stiffness ◽  
Varying Coefficients ◽  
Wide Range ◽  
Stiffness Variation

The excitation from mesh stiffness variation in a tunnel gear driving system can cause excessive noise and vibrations. Since the stiffness variation may induce parametric instability, the system could be damaged on a permanent-basis. Therefore, the study of parametric instability in such system is of paramount importance. In this work, a rigid-elastic model is developed using the energy method, where the ring gear is treated as a rotating thin ring having radial and tangential deflections, whereas the pinions are assumed to be rigid bodies having translational motion relative to the radial directions of the ring gear as well as rotational motions around their centers. All gear meshes are modeled as interactions caused by time-varying springs, and the supports of the pinions are modeled as linear springs in the radial direction relative to the ring gear. The modeling leads to a set of partial-ordinary linear differential equations with time-varying coefficients. For an N planet system, the discretization process yields 2N+2 ordinary differential equations. Stability boundaries are determined using Floque’t theory for a wide range of parameter values. Specifically, the effects of mesh stiffness on the parametric instability are examined. The results show that the instability behaviors are closely related to the basic parameters when considering the time-varying excitation. This could be a serious consideration in the preliminary design of such systems.

Numerical simulation of instability conditions in multiple pinion drives

2011 ◽  
Vol 225 (6) ◽  
pp. 1319-1327 ◽  
Author(s):  
K-Z Zhang ◽  
H-D Yu ◽  
X-X Zeng ◽  
X-M Lai
Keyword(s):  
Parametric Instability ◽  
Industrial Applications ◽  
Lyapunov Theory ◽  
Gear Train ◽  
Heavy Duty ◽  
Mesh Stiffness ◽  
Torque Transmission ◽  
Stiffness Variation ◽  
Parametric Instabilities ◽  
Mesh Phasing

Multiple pinion drives, parallel arrangements of the pinions for large torque transmission, are widely utilized in various heavy-duty industrial applications. For such multi-mesh gear systems, periodic mesh stiffnesses could possibly cause parametric instabilities and server vibrations. Based on the Floquet–Lyapunov theory, numerical simulations are conducted to determine the parametric instability status. For rectangular waveforms assumption of the mesh stiffness variations, the primary, secondary, and combination instabilities of the multiple pinion drives are studied. The effects of mesh stiffness parameters, including mesh frequencies, stiffness variation amplitudes, and mesh phasing, on these instabilities are yielded. Unstable regions are also indicated for different gear pair configurations. Instability conditions of three-pinion drives are obtained and compared with those of the three-stage gear train.

Parametric Instability in Planetary Gears With Frequency-Modulated Time-Varying Mesh Stiffness

2014 ◽  
Author(s):  
Xinghui Qiu ◽  
Qinkai Han ◽  
Fulei Chu
Keyword(s):  
Parametric Instability ◽  
Operating Conditions ◽  
Time Varying ◽  
Mesh Stiffness ◽  
Rotational Model ◽  
Planetary Gears ◽  
Gear Mesh ◽  
Stiffness Variation ◽  
Speed Fluctuations ◽  
Gear Mesh Stiffness

A rotational model of planetary gears is developed which incorporates mesh stiffness variation and input speed fluctuations. Gear mesh stiffness is approximated by rectangle wave and different harmonic orders are considered. Because of speed fluctuations, the mesh stiffness is frequency modulated. The parametric instability associated with frequency-modulated time-varying stiffness is numerically investigated. The operating conditions leading to parametric instability are identified using Floquet theory and numerical integration. Whether the general laws derived for steady speed to suppress particular instabilities are applicable for fluctuating speed is verified. The effects of speed fluctuations on parametric instability are examined.

Parametric Instability of Planetary Gear Transmission Mounted on Discrete Support Struts With Minimum Ring Gear Rim Thickness Design

2018 ◽  
Author(s):  
Peng Guan ◽  
Hans DeSmidt
Keyword(s):  
Parametric Instability ◽  
Planetary Gear ◽  
Gear Transmission ◽  
Ring Gear ◽  
Mesh Stiffness ◽  
Gear Mesh ◽  
Stiffness Variation ◽  
Parametric Instabilities ◽  
Gear Mesh Stiffness ◽  
Flexible Ring

This research explores parametric instabilities of the PGT driveline system and a stability-based method for ring gear rim thickness design. Parametric excitation of a planetary gear transmission (PGT) driveline system arises from two sources: 1) gear mesh stiffness variation, 2) Interaction between moving planets, flexible ring gear and boundary struts. Many researchers have studied the parametric instability of planetary gear transmissions due to gear mesh stiffness variation, however, the effect of interaction between moving planets, flexible ring and discrete boundary struts on parametric instabilities has not been fully studied before. Especially, for sufficiently thin ring gears, this kind of effect becomes even more significant. To illustrate the novel PGT rim design proposal, firstly, a structural dynamics model of a complete PGT driveline system with elastic ring gear supported by discrete boundary struts is established. Secondly, by applying Floquet method, the parametric instability behavior due to the second parametric excitation source is fully investigated. Lastly, the design guidelines for planetary gear transmission ring gear rim thickness are proposed based on system stability from a dynamical viewpoint. The analysis and results provide new and important insights into dynamics and design of lightweight planetary gear transmission ring gear rim.

Nonlinear Dynamic Analysis of a Multi-Gear Train With Time-Varying Mesh Stiffness Including Modification Coefficient Effect

2011 ◽  
Author(s):  
T. N. Shiau ◽  
J. R. Chang ◽  
K. H. Huang ◽  
C. J. Cheng ◽  
C. R. Wang
Keyword(s):  
Dynamic Analysis ◽  
Nonlinear Dynamic ◽  
Nonlinear Dynamic Analysis ◽  
Gear Train ◽  
Work Piece ◽  
Time Varying ◽  
Gear Pair ◽  
Mesh Stiffness ◽  
Harmonic Motion ◽  
Gear Profile

The nonlinear dynamic analysis of a multi-gear train with time-varying mesh stiffness on account of the modification coefficient effect is in vestigated in this paper. The proposed application of the modification coefficient will revise the center distance of the gear pair, avoid undercut and raise the mesh stiffness of the designed gear system. In this study, the gear profile is generated from the relationship between the rack cutter and the gear work piece by using the envelope theory. The rack cutter with the modification coefficient increases the mesh stiffness and thus enhances the strength of the gear tooth. Then the time-varying mesh stiffness at the contact position of the gear pair is calculated from the tooth deflection analysis using the generated gear profile. With the obtained time-varying mesh stiffness, the nonlinear dynamic behavior of multi-gear train is investigated by using Runge-Kutta integration method. The numerical results of the studied examples show the harmonic motion, sub-harmonic motion, chaotic motion and bifurcation phenomenon of the gear train.

Parametric Instability of a Rotating Circular Ring With Moving, Time-Varying Springs

2005 ◽  
Vol 128 (2) ◽  
pp. 231-243 ◽  
Author(s):  
Sripathi Vangipuram Canchi ◽  
Robert G. Parker
Keyword(s):  
Parametric Instability ◽  
Circular Ring ◽  
Time Varying ◽  
Ring Gear ◽  
Geometric Description ◽  
Bending Vibrations ◽  
System Parameters ◽  
Ring Rotation ◽  
Special Cases ◽  
Instability Regions

Parametric instabilities of in-plane bending vibrations of a rotating ring coupled to multiple, discrete, rotating, time-varying stiffness spring-sets of general geometric description are investigated in this work. Instability boundaries are identified analytically using perturbation analysis and given as closed-form expressions in the system parameters. Ring rotation and time-varying stiffness significantly affect instability regions. Different configurations with a rotating and nonrotating ring, and rotating spring-sets are examined. Simple relations governing the occurrence and suppression of instabilities are discussed for special cases with symmetric circumferential spacing of spring-sets. These results are applied to identify possible conditions of ring gear instability in example planetary gears.

Mesh Stiffness Variation Instabilities in Two-Stage Gear Systems

2001 ◽  
Author(s):  
Jian Lin ◽  
Robert G. Parker
Keyword(s):  
Numerical Solutions ◽  
Parametric Instability ◽  
Operating Conditions ◽  
Mesh Stiffness ◽  
Two Stage ◽  
Instability Regions ◽  
Stiffness Variation ◽  
Parametric Instabilities ◽  
Gear Systems ◽  
Mesh Phasing

Abstract Mesh stiffness variation, the change in stiffness of meshing teeth as the number of teeth in contact changes, causes parametric instabilities and severe vibration in gear systems. The operating conditions leading to parametric instability are investigated for two-stage gear chains, including idler gear and countershaft configurations. Interactions between the stiffness variations at the two meshes are examined. Primary, secondary, and combination instabilities are studied. The effects of mesh stiffness parameters, including stiffness variation amplitudes, mesh frequencies, contact ratios, and mesh phasing, on these instabilities are analytically identified. For mesh stiffness variation with rectangular waveforms, simple design formulae are derived to control the instability regions by adjusting the contact ratios and mesh phasing. The analytical results are compared to numerical solutions.

Mesh Stiffness Variation Instabilities in Two-Stage Gear Systems

2001 ◽  
Vol 124 (1) ◽  
pp. 68-76 ◽  
Author(s):  
Jian Lin ◽  
Robert G. Parker
Keyword(s):  
Numerical Solutions ◽  
Parametric Instability ◽  
Operating Conditions ◽  
Mesh Stiffness ◽  
Two Stage ◽  
Instability Regions ◽  
Stiffness Variation ◽  
Parametric Instabilities ◽  
Gear Systems ◽  
Mesh Phasing

Mesh stiffness variation, the change in stiffness of meshing teeth as the number of teeth in contact changes, causes parametric instabilities and severe vibration in gear systems. The operating conditions leading to parametric instability are investigated for two-stage gear chains, including idler gear and countershaft configurations. Interactions between the stiffness variations at the two meshes are examined. Primary, secondary, and combination instabilities are studied. The effects of mesh stiffness parameters, including stiffness variation amplitudes, mesh frequencies, contact ratios, and mesh phasing, on these instabilities are analytically identified. For mesh stiffness variation with rectangular waveforms, simple design formulas are derived to control the instability regions by adjusting the contact ratios and mesh phasing. The analytical results are compared to numerical solutions.

Non-Linear Vibration Modeling and Simulation of a Gear Pair Based on ADAMS and Simulink

2013 ◽  
Vol 681 ◽  
pp. 219-223 ◽  
Author(s):  
Chun Jing Huo ◽  
Hui Liu ◽  
Zhong Chang Cai ◽  
Ming Zheng Wang
Keyword(s):  
Gear Train ◽  
Transmission Model ◽  
Time Varying ◽  
Gear Pair ◽  
Mesh Stiffness ◽  
Feedback Model ◽  
Non Linear ◽  
Speed Fluctuation ◽  
Set Up ◽  
Multi Body

To set up the virtual prototype of a gear train system in the dynamic analysis software ADAMS, the torsional vibration model of a gear pair was transformed into an equivalent transmission model in which a multi-body model was established in ADAMS and its meshing force solution model was established in Simulink. The time-varying mesh stiffness, gear clearance, meshing errors and other non-linear factors can be included in the gear meshing feedback model, more importantly, the influence of gear speed fluctuation on the time-varying mesh stiffness was taken into consideration. The simulation results contrastively prove the feasibility of co-simulation for obtaining the dynamic characteristics of gear meshing process.

Steady-State Response of Periodically Time-Varying Linear Systems, With Application to an Elastic Mechanism

1995 ◽  
Vol 117 (4) ◽  
pp. 633-639 ◽  
Author(s):  
K. Farhang ◽  
A. Midha
Keyword(s):  
Steady State ◽  
Differential Equations ◽  
Closed Form ◽  
Linear Systems ◽  
Direct Method ◽  
Time Varying ◽  
Steady State Response ◽  
Varying Coefficients ◽  
State Response ◽  
Elastic Mechanism

This paper presents the development of an efficient and direct method for evaluating the steady-state response of periodically time-varying linear systems. The method is general, and its efficacy is demonstrated in its application to a high-speed elastic mechanism. The dynamics of a mechanism comprised of elastic members may be described by a system of coupled, inhomogeneous, nonlinear, second-order partial differential equations with periodically time-varying coefficients. More often than not, these governing equations may be linearized and, facilitated by separation of time and space variables, reduced to a system of linear ordinary differential equations with variable coefficients. Closed-form, numerical expressions for response are derived by dividing the fundamental time period of solution into subintervals, and establishing an equal number of continuity constraints at the intermediate time nodes, and a single periodicity constraint at the end time nodes of the period. The symbolic solution of these constraint equations yields the closed-form numerical expression for the response. The method is exemplified by its application to problems involving a slider-crank mechanism with an elastic coupler link.

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