Vibration of High-Speed Compliant Gear Pairs

2016 ◽  
Author(s):  
Christopher G. Cooley ◽  
Robert G. Parker
Keyword(s):  
High Speed ◽  
Rotation Speed ◽  
Equations Of Motion ◽  
System Structure ◽  
Vibration Modes ◽  
Mesh Stiffness ◽  
Nodal Diameter ◽  
Gear Teeth ◽  
Wide Range ◽  
System Coupling

This study analytically investigates the vibration of high-speed, compliant gear pairs using a model consisting of coupled, spinning, elastic rings. The gears are elastically coupled by a space-fixed, discrete stiffness element that represents the contacting gear teeth. Hamilton’s principle is used to derive the nonlinear governing equations of motion and boundary conditions. These equations are linearized for small vibrations about the steady equilibrium due to rotation. The equations are cast in operator form, which exemplifies their gyroscopic system structure. The eigenvalue problem is discretized using Galerkin’s method. The natural frequencies and vibration modes for an example aerospace gear pair are numerically calculated for a wide-range of rotation speeds. The system coupling leads to multiple eigenvalue veering regions as the gear rotation speed varies. Highly coupled vibration modes that have meaningful deflection in the discrete mesh stiffness occur within a set frequency band. The vibration modes within this band have distinct nodal diameter components that evolve with rotation speed.

scholarly journals Vibration of Rotating and Revolving Planet Rings with Discrete and Partially Distributed Stiffnesses

2020 ◽  
Vol 11 (1) ◽  
pp. 127
Author(s):  
Fuchun Yang ◽  
Dianrui Wang
Keyword(s):  
High Speed ◽  
Differential Operators ◽  
Natural Frequencies ◽  
Rotation Speed ◽  
Distribution Area ◽  
Vibration Modes ◽  
Governing Equations ◽  
Vibration Properties ◽  
Wide Range ◽  
Forced Responses

Vibration properties of high-speed rotating and revolving planet rings with discrete and partially distributed stiffnesses were studied. The governing equations were obtained by Hamilton’s principle based on a rotating frame on the ring. The governing equations were cast in matrix differential operators and discretized, using Galerkin’s method. The eigenvalue problem was dealt with state space matrix, and the natural frequencies and vibration modes were computed in a wide range of rotation speed. The properties of natural frequencies and vibration modes with rotation speed were studied for free planet rings and planet rings with discrete and partially distributed stiffnesses. The influences of several parameters on the vibration properties of planet rings were also investigated. Finally, the forced responses of planet rings resulted from the excitation of rotating and revolving movement were studied. The results show that the revolving movement not only affects the free vibration of planet rings but results in excitation to the rings. Partially distributed stiffness changes the vibration modes heavily compared to the free planet ring. Each vibration mode comprises several nodal diameter components instead of a single component for a free planet ring. The distribution area and the number of partially distributed stiffnesses mainly affect the high-order frequencies. The forced responses caused by revolving movement are nonlinear and vary with a quasi-period of rotating speed, and the responses in the regions supported by partially distributed stiffnesses are suppressed.

Numerical Simulation of Nonlinear Spur Gear Dynamics

1999 ◽  
Author(s):  
Nagaraj K. Arakere ◽  
C. Nataraj
Keyword(s):  
High Speed ◽  
Equations Of Motion ◽  
Spur Gear ◽  
Journal Bearings ◽  
Accurate Estimation ◽  
Gear Pair ◽  
Mesh Stiffness ◽  
Gear Teeth ◽  
Gear Mesh ◽  
Gear Mesh Stiffness

Abstract An analytical investigation of the nonlinear dynamics of a high-speed spur-gear pair supported on journal bearings is presented. Dynamic tooth loads result from the interaction between periodic variation of gear mesh stiffness, involute tooth profile errors and gear rotor dynamics. Accurate estimation of dynamic tooth loads, as the gear teeth engage and disengage, is critical for fatigue life estimation. Load-deflection characteristics of a spur gear mesh and the periodically varying gear mesh stiffness is developed using a finite element model. Relative displacement between the gear teeth (transmission error) due to tooth deflection along the line of action is evaluated. The coupled torsional-lateral vibrations of a spur-gear pair supported on journal bearings is modeled as a six degree of freedom system. The time dependent radial and tangential forces acting on the gear shaft supported on journal bearings is evaluated. Short bearing theory is used for modeling the journal bearing dynamics. The resulting nonlinear equations of motion are numerically integrated to obtain gear and pinion whirl orbits due to unbalance excitation and dynamic tooth load variation. Dynamic tooth loads are compared with the mean load due to torque transmission.

Vibration of High-Speed Spur Gear Webs

1998 ◽  
Vol 120 (3) ◽  
pp. 791-800 ◽  
Author(s):  
N. K. Arakere ◽  
C. Nataraj
Keyword(s):  
Weight Reduction ◽  
High Speed ◽  
High Cycle Fatigue ◽  
Equations Of Motion ◽  
Fatigue Loading ◽  
Periodic Variation ◽  
Cycle Fatigue ◽  
Periodic Excitation ◽  
Mesh Stiffness ◽  
Plane Vibration

High cycle fatigue loading of gear webs due to in-plane stresses, caused by forced excitation resulting from centrifugal loading and dynamic tooth loads, has been known to cause radial fatigue cracks. This is especially prevalent in high-speed gears used in aerospace applications, with small web thickness, for weight reduction. Radial cracks have also been observed to originate at the outer edge of lightening holes machined in gear webs for weight reduction. This paper presents an analytical treatment of the in-plane vibration of high-speed gear webs resulting from rotational effects and periodic excitation from dynamic tooth loading. Dynamic tooth loads result from the combined effect of inertia forces of gear wheels which are significant at high speeds, the periodic variation of gear mesh stiffness, and involute tooth profile errors. The gear web is modeled as a thin rotating disc and the governing differential equations of motion and the associated boundary conditions are derived from first principles. The equations are then nondimensionalized which leads to some essential nondimensional parameters. A comprehensive tooth stiffness model for spur gears is used that accounts for periodic variation of mesh stiffness. The dynamic tooth loads are obtained by solving the pertinent equations of motion, using a collocation method, that yields a closed-form expression for the periodic excitation, that is used as an input for the in-plane vibration problem. The in-plane vibration equations are solved by an approximate method of weighted residuals. It is found that the displacement fields and the resulting stresses can be significant under certain speeds and loading conditions. The interaction between the forcing frequencies due to gear teeth dynamics and the in-plane vibration natural frequencies can result in resonances that induce high fatigue stresses in the gear web. The in-plane stresses leading to high cycle fatigue loading, and frequency components of the resulting response are discussed in detail.

Modal Analysis of Ballooning Strings With Small Sag

1999 ◽  
Author(s):  
Rongjun Fan ◽  
Sushil K. Singh ◽  
Christopher D. Rahn
Keyword(s):  
Steady State ◽  
High Speed ◽  
Natural Frequencies ◽  
Rotation Speed ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Mode Shapes ◽  
Theoretical Predictions ◽  
Nonlinear Equations Of Motion

Abstract During the manufacture and transport of textile products, yarns are rotated at high speed and form balloons. The dynamic response of the balloon to varying rotation speed, boundary excitation, and disturbance forces governs the quality of the associated process. Resonance, in particular, can cause large tension variations that reduce product quality and may cause yarn breakage. In this paper, the natural frequencies and mode shapes of a single loop balloon are calculated to predict resonance. The three dimensional nonlinear equations of motion are simplified via small steady state displacement (sag) and vibration assumptions. Axial vibration is assumed to propagate instantaneously or in a quasistatic manner. Galerkin’s method is used to calculate the mode shapes and natural frequencies of the linearized equations. Experimental measurements of the steady state balloon shape and the first two natural frequencies and mode shapes are compared with theoretical predictions.

Vibration Properties of High-Speed Planetary Gears With Gyroscopic Effects

2012 ◽  
Vol 134 (6) ◽  
Author(s):  
Christopher G. Cooley ◽  
Robert G. Parker
Keyword(s):  
High Speed ◽  
Eigenvalue Problems ◽  
Planetary Gear ◽  
Vibration Modes ◽  
Planetary Gears ◽  
High Speeds ◽  
Wide Range ◽  
Modal Property ◽  
Gyroscopic Effects ◽  
Complex Valued

This study investigates the modal property structure of high-speed planetary gears with gyroscopic effects. The vibration modes of these systems are complex-valued and speed-dependent. Equally-spaced and diametrically-opposed planet spacing are considered. Three mode types exist, and these are classified as planet, rotational, and translational modes. The properties of each mode type and that these three types are the only possible types are mathematically proven. Reduced eigenvalue problems are determined for each mode type. The eigenvalues for an example high-speed planetary gear are determined over a wide range of carrier speeds. Divergence and flutter instabilities are observed at extremely high speeds.

scholarly journals The Dynamic Modeling of Multiple Pairs of Spur Gears in Mesh, Including Friction and Geometrical Errors

2003 ◽  
Vol 9 (6) ◽  
pp. 437-442 ◽  
Author(s):  
Shengxiang Jia ◽  
Ian Howard ◽  
Jiande Wang
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Spur Gears ◽  
Mesh Stiffness ◽  
Frequency Spectra ◽  
Gear Teeth ◽  
Geometrical Errors ◽  
Tooth Stiffness ◽  
The Common ◽  
Profile Errors

This article presents a dynamic model of three shafts and two pair of gears in mesh, with 26 degrees of freedom, including the effects of variable tooth stiffness, pitch and profile errors, friction, and a localized tooth crack on one of the gears. The article also details howgeometrical errors in teeth can be included in a model. The model incorporates the effects of variations in torsional mesh stiffness in gear teeth by using a common formula to describe stiffness that occurs as the gears mesh together. The comparison between the presence and absence of geometrical errors in teeth was made by using Matlab and Simulink models, which were developed from the equations of motion. The effects of pitch and profile errors on the resultant input pinion angular velocity coherent-signal of the input pinion's average are discussed by investigating some of the common diagnostic functions and changes to the frequency spectra results.

Vibration Structure of Gyroscopic Planetary Gears

2011 ◽  
Author(s):  
Christopher G. Cooley ◽  
Robert G. Parker
Keyword(s):  
High Speed ◽  
Eigenvalue Problems ◽  
Planetary Gear ◽  
Vibration Modes ◽  
Planetary Gears ◽  
High Speeds ◽  
Wide Range ◽  
Mode Method ◽  
Complex Valued

This study investigates the vibration structure of high-speed, gyroscopic planetary gears. The vibration modes of these systems are complex-valued and speed dependent. Three mode types exist, and these are classified as planet, rotational, and translational modes. Each mode type is mathematically proven by the use of a candidate mode method. Reduced eigenvalue problems are determined for each mode type. The eigenvalues for an example high-speed planetary gear are determined over a wide range of carrier speeds. Divergence and flutter instabilities are observed at extremely high speeds.

Vibration of Spinning Cantilever Beams Undergoing Coupled Bending and Torsional Motion

2013 ◽  
Author(s):  
Christopher G. Cooley ◽  
Robert G. Parker
Keyword(s):  
Original Problem ◽  
Natural Frequencies ◽  
Rotation Axis ◽  
Equations Of Motion ◽  
Vibration Modes ◽  
Cantilever Beams ◽  
Torsional Motion ◽  
Governing Equations ◽  
Helicopter Blade ◽  
Wide Range

This study investigates the vibration of a spinning cantilever beam with a rigid body attached to its free end undergoing coupled bending and torsional motion. The rotation axis is perpendicular to the beam (like a helicopter blade). The governing equations of motion are cast in a structured way using extended variables and extended operators. With this structure the equations represent a classical gyroscopic system and Galerkin discretization is readily applied where it is not for the original problem. The natural frequencies and vibration modes are investigated over a wide range of rotation speeds.

In-Plane Vibration of High-Speed Spur Gear Webs

1995 ◽  
Author(s):  
Nagaraj K. Arakere ◽  
C. Nataraj
Keyword(s):  
Weight Reduction ◽  
High Speed ◽  
High Cycle Fatigue ◽  
Equations Of Motion ◽  
Fatigue Loading ◽  
Periodic Variation ◽  
Cycle Fatigue ◽  
Periodic Excitation ◽  
Mesh Stiffness ◽  
Plane Vibration

Abstract High cycle fatigue loading of gear webs due to in-plane stresses, caused by forced excitation resulting from centrifugal loading and dynamic tooth loads, has been known to cause radial fatigue cracks. This is especially prevalent in high-speed gears used in aerospace applications, with small web thickness, for weight reduction. Radial cracks have also been observed to originate at the outer edge of lightening holes machined in gear webs for weight reduction. This paper presents an analytical treatment of the in-plane vibration of high-speed gear webs resulting from rotational effects and periodic excitation from dynamic tooth loading. Dynamic tooth loads result from the combined effect of inertia forces of gear wheels which are significant at high speeds, the periodic variation of gear mesh stiffness, and involute tooth profile errors. The gear web is modeled as a thin rotating disc and the governing differential equations of motion and the associated boundary conditions are derived from first principles. A comprehensive tooth stiffness model for spur gears is used that accounts for periodic variation of mesh stiffness. The dynamic tooth loads are obtained by solving the pertinent equations of motion, using a collocation method, that yields a closed-form expression for the periodic excitation, that is used as an input for the in-plane vibration problem. The in-plane vibration equations are solved by an approximate method of weighted residuals. It is found that the displacement fields and the resulting stresses can be significant under certain speeds and loading conditions. The in-plane stresses leading to high cycle fatigue loading, and frequency components of the resulting response are discussed in detail.

Increase in the Growth Rate by Rotating the Seed Crystal at High Speed during the Solution Growth of SiC

2014 ◽  
Vol 778-780 ◽  
pp. 63-66 ◽  
Author(s):  
Tomonori Umezaki ◽  
Daiki Koike ◽  
Atsushi Horio ◽  
S. Harada ◽  
Toru Ujihara
Keyword(s):  
Numerical Simulation ◽  
Growth Rate ◽  
Carbon Concentration ◽  
Concentration Gradient ◽  
High Speed ◽  
Rotation Speed ◽  
Seed Crystal ◽  
Solution Growth ◽  
Growth Interface ◽  
Wide Range

We studied the effect of rotation speed of seed crystal on the growth rate during the solution growth of SiC. The growth rate increased with increasing rotation speed of the seed crystal. The increase in the growth rate was observed in relatively wide range of carbon concentration. According to the numerical simulation, the carbon concentration gradient near the growth interface under 150 rpm condition is larger than 20 rpm (ACRT) condition. This indicates that increase in the growth rate is caused by the increase in the carbon concentration gradient of the diffusion layer.

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