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Nonlinear Time-Varying Dynamic Analysis of a Multi-Mesh Spur Gear Train

  • Conference paper
  • First Online:
Dynamics of Coupled Structures, Volume 4

Abstract

The nonlinear dynamics of a multi-mesh spur gear train is considered in this study. The gear train consists of three spur gears, with one of the gears in mesh with the other two. Dynamic model includes gear backlash in the form of clearance-type displacement functions and time variation of gear mesh stiffness. The system is reduced to a two-degree-of-freedom definite model by using the relative gear mesh displacements as the coordinates. The equations of motion are solved for periodic steady-state response by using Harmonic Balance Method (HBM). The accuracy of the HBM solutions is demonstrated by comparing them to direct numerical integration solutions. Floquet theory is applied to determine the stability of the steady-state solutions. Two different loading conditions, where the system is driven by the middle gear and driven by one of the end gears, are considered. Phase difference between the two gear meshes is determined under each loading condition and natural modes are predicted for each loading condition. The forced response due to the combination of parametric excitation and static transmission error excitation is obtained and effects of loading conditions and asymmetric positioning on the response are explored.

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Abbreviations

b :

Half of gear backlash

c :

Damping coefficient

e :

Static transmission error

g :

Nonlinear displacement functions

F :

External force

G :

Periodic state matrix

I :

Polar mass moment of inertia

J :

Jacobian matrix

k :

Mesh stiffness

M :

Monodromy matrix

m :

Equivalent mass

p :

Relative gear mesh displacement

Δp :

Perturbation of relative gear mesh displacement

r :

Gear base radius

S :

Nonlinear algebraic equations in matrix form

T :

Torque

t :

Time

U :

Solution vector

u :

Gear mesh displacement harmonic amplitude

Z :

Number of teeth

z :

Perturbation state vector

α :

Phase of static transmission error harmonic

ρ :

Discrete time interval

ϕ :

Discontinuous separation function

θ :

Rotational displacement

Π:

Phase difference between meshes

Ω:

Dimensionless frequency

ω :

Characteristic frequency

ζ :

Damping ratio

ψ :

Angle between the lines connecting the centers of the gears

γ :

Constant angle in mesh phasing calculation

a :

Alternating component

i :

Mesh index

m :

Mean component

i :

Mesh index

rms:

Root-mean-square value

T :

Matrix transpose

. :

Derivative with respect to time

':

Derivative with respect to dimensionless time

-:

Dimensional quantities

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Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Middle East Technical University, Universiteler Mahallesi, Dumlupinar Bulvari No:1, 06800, Cankaya, Ankara, Turkey

    Siar Deniz Yavuz & Ender Cigeroglu

  2. Turkish Aerospace Industries, Inc. (TAI), Fethiye Mahallesi Havacilik Bulvari No:17, 06980, Kazan, Ankara, Turkey

    Siar Deniz Yavuz & Zihni Burcay Saribay

Authors
  1. Siar Deniz Yavuz
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  2. Zihni Burcay Saribay
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  3. Ender Cigeroglu
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Corresponding author

Correspondence to Ender Cigeroglu .

Editor information

Editors and Affiliations

  1. Engg Phys Dept 535 Engg Res Bldg, University of Wisconsin Madison, Madison, Wisconsin, USA

    Matt Allen

  2. MS 0557, Sandia National Laboratories, Albuquerque, New Mexico, USA

    Randall L. Mayes

  3. Lehrstuhl für Angewandte Mechanik, TU München, Garching, Bayern, Germany

    Daniel Rixen

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Yavuz, S.D., Saribay, Z.B., Cigeroglu, E. (2016). Nonlinear Time-Varying Dynamic Analysis of a Multi-Mesh Spur Gear Train. In: Allen, M., Mayes, R., Rixen, D. (eds) Dynamics of Coupled Structures, Volume 4. Conference Proceedings of the Society for Experimental Mechanics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-29763-7_30

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  • DOI: https://doi.org/10.1007/978-3-319-29763-7_30

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-29762-0

  • Online ISBN: 978-3-319-29763-7

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