MODELLING OF THE GEAR BACKLASH

Introduction

This calculation sheet demonstrates how to modulate backlash between two gears.
The model is constructed using the two rotation laws of Newton. The calculation is
based on a model created by Jerzy Margielewicz. My goal is to make his model
operational in Mathcad and then apply this method to the rack and pinion model I
had previously developed, which did not account for backlash. The intention is to
make Jerzy Margielewicz's model work in Mathcad and subsequently implement it in
the rack and pinion model.

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Variables used in the model
In this chapter, the necessary variables required for the
model are calculated

Gear properties for the model: Z1 ̗ 14 D1 = 70 mm
Number of teeth on the pinion gear Z2 ̗ 85 D2 = 425 mm
Number of teeth on the pinion gear m1 ̗ 5 mm
Module b1 ̗ 10 mm
Face width of the gear D1 ̗ m1 Έ Z1
Number of teeth gear 1 D2 ̗ m1 Έ Z2
Number of teeth gear 2 b1 ̗ 10 mm
Face width of the gear m1 ̗ 5 mm
Module

Inertia and weight of the pinions

Ņsteel ̗ 7850 kg
Density of steel ƀƀ

3

m

Weight gear wheel 1 mass1 ̗ 1 Έ іјD1љћ2 Έ b1 Έ Ņsteel mass1 = 0.302 kg
ƀΈ ń
4 mass2 = 11.136 kg
J1gear = 0.000185 kg Έ m2
Weight gear wheel 2 mass2 ̗ 1 Έ іјD2љћ2 Έ b1 Έ Ņsteel J2gear = 0.251435 kg Έ m2
Inertia of teeth gear 1 ƀΈ ń
Inertia of teeth gear 2 4 M0 ̗ 1000
load moment Mn ̗ 5
1 і D1 љ2
J1gear ̗ ƀΈ mass1 Έ їƀƀ њ
2 ј2ћ

1 і D2 љ2
J2gear ̗ ƀΈ mass2 Έ їƀƀ њ
2 ј2ћ

External drive torque

Maximum rotation speed nr ̗ 1500 rpm rad
Material properties: nr = 157.08 ƀƀ
Young modulus
Poisson ratio s

Emodulus ̗ 200 GPa
ʼn1 ̗ 0.3

calculate the reduced modulus (W) using the following formula:

Wr ̗ іјń Έ Emodulusљћ Wreduced = іј3.452 Έ 109 љћ N
Reduced modulus ƀ ƀƀ ƀƀ Έ m ƀ
e du ce d 1
2
іј1 - ʼn1 љ m
ћ

Calculate the mesh stiffness (k_mesh) using the formula:

Mesh stiffness іј2 Έ Wreduced Έ b1љћ kmesh = іј9.864 Έ 108 љћ N
kmesh ̗ ƀƀƀƀƀ ƀ
m
D1

Parameters 1DOF J1motor ̗ 0.00074
Mass of inertia 1 pinion
Mass of inertia 1 gearbox J1gearbox ̗ 0.002
Mass of inertia 1 gearbox
J1 ̗ J1 Έ 1 J1gear = 1.85 Έ 10-4
Pitch diameter pinion 1 ƀƀƀ
Gear ratio g e ar g e a r
Mass of inertia 2 pinion 2
kg Έ m

D1 R1 = 0.035
R1 ̗ ƀƀ

2Έm
Gr1 ̗ 2

J1total ̗ J1motor + J1gearbox Έ іјGr1љћ2 + J1gear J1total = 0.009

J1total Έ ļ1ƕƕ ((t)) ࠙Mn - іјR1 Έ bz Έ іјR1 Έ ļ1ƕ ((t)) - іјR2 Έ ļ2ƕ ((t))љћ - eƕ ((t))љћ - R1 Έ cz Έ іјR1 Έ ļ1 ((t)) - іјR2 Έ ļ2 ((t))љћ - e ((t))љћљћ
Mn࠙J1total Έ ļ1ƕƕ ((t)) + R1 Έ bz Έ іјR1 Έ ļ1ƕ ((t)) - іјR2 Έ ļ2ƕ ((t))љћ - eƕ ((t))љћ + R1 Έ cz Έ іјR1 Έ ļ1 ((t)) - іјR2 Έ ļ2 ((t))љћ - e ((t))љћ

-іјR1 Έ bz Έ іјR1 Έ ļ1ƕ ((t)) - іјR2 Έ ļ2ƕ ((t))љћ - e2 ((t))љћљћ - R1 Έ cz Έ іјR1 Έ ļ1 ((t)) - іјR2 Έ ļ2 ((t))љћ - e ((t))љћ Έ f іјu1љћ + Mn
ļ1ƕƕ ((t))࠙ƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀ

J1total

Parameters 2DOF J2motor ̗ 0.074
Mass of inertia 2 pinion
Mass of inertia 2 gearbox J2gearbox ̗ 0.002
Mass of inertia 2 gearbox
J2 ̗ J2 Έ 1 J2gear = 0.251
Pitch radius pinion 2 ƀƀƀ
Gravitational constant g e ar g e a r
Gear ratio 2
kg Έ m
Mass of inertia 2 pinion
D1 R2 = 0.035
R2 ̗ ƀƀ

2Έm
g1 ̗ 9.81

Gr2 ̗ 2

J2total ̗ J2motor + J2gearbox Έ іјGr2љћ2 + J2gear J2total = 0.333

J2total Έ ļ2ƕƕ ((t)) ࠙R2 Έ bz Έ іјR1 Έ ļ1ƕ ((t)) - іјR2 Έ ļ2ƕ ((t))љћ - eƕ ((t))љћ + R2 Έ cz Έ іјR1 Έ ļ1 ((t)) - іјR2 Έ ļ2 ((t))љћ - e ((t))љћ - M0
M0࠙ J2total Έ ļ2ƕƕ ((t)) - R2 Έ bz Έ іјR1 Έ ļ1ƕ ((t)) - іјR2 Έ ļ2ƕ ((t))љћ - eƕ ((t))љћ - R2 Έ cz Έ іјR1 Έ ļ1 ((t)) - іјR2 Έ ļ2 ((t))љћ - e ((t))љћ Έ f іјu1љћ - M0

R2 Έ bz Έ іјR1 Έ ļ1ƕ ((t)) - іјR2 Έ ļ2ƕ ((t))љћ - e2 ((t))љћ + R2 Έ cz Έ іјR1 Έ ļ1 ((t)) - іјR2 Έ ļ2 ((t))љћ - e ((t))љћ Έ f іјu1љћ - M0
ļ2ƕƕ ((t))࠙ƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀ

J2total

Influence of Backlash on Dynamics

The differential equations above are stated for the case when the gear mesh is operating
without backlash. When a backlash phenomenon occurs, the drive system separates from
the load.

Damping displacement element cz ̗ 0.002

Spring displacement element m bz = 9.864 Έ 108
bz ̗ kmesh Έ ƀ

N

Backlash value zbacklash ̗ 0.005
Is a constant value half of the backlash value. zbacklash

Lz ̗ ƀƀƀ
2

The reduction mathematical model of the gear transmission

The most efficient approach for analyzing the quantitative and qualitative assessment of

the phenomena that occur during the interaction of cooperating wheels is through the
use of a simplified model with a single degree of freedom. Differential equations of
motion can be derived using the classic formalism of LagrangeThe initial step in deriving
this model is to begin with the system of differential equations.
Following various transformations and the introduction of a new coordinate

q1࠙R1 Έ ļ1 ((t)) - R2 Έ ļ2 ((t))

The mathematical model is reduced to the form of theta 1 and theta 2 ode is:

mred Έ ĸ࠙mred Έ qƕƕ ((t)) + bz Έ ((qƕ ((t)) - eƕ ((t)))) + cz Έ ((q - eƕ ((t)))) Έ f ((u))

mre ̗ J1total Έ J2total mred = 7.096
ƀƀ ƀƀ ƀƀ ƀƀ
d
2 2
R1 Έ J2total + R2 Έ J1total

Rotation speed motor ōs ̗ 153
The frequency of meshing ōz ̗ Z1 Έ ōs
Error in gear wheel cooperation e1 ̗ 1 Έ 10-5
Angular velocity of the rotor of the drive e ((t)) ̗ e1 Έ cos іјōz Έ ((t))љћ
Average meshing stiffness c0 ̗ 5.03 Έ 108
Amplitude of dynamic component of meshing stiffness c1 ̗ 5.03 Έ 108

The characteristics of periodically variable meshing stiffness cZ ((t)) ̗ c0 + c1 Έ cos іјōz Έ ((t))љћ
M1 ̗ 6

M0
ŀ1 ̗ ƀƀ

Mn
і R1 R2 Έ ŀ1 љ
ĸ1 ̗ їƀƀ+ ƀƀƀ њ Έ M1
ј J1total J1total ћ

Considering the properties of periodically varying meshing stiffness

cZ ((t))࠙c0 + c1 Έ cos іјōz Έ ((t))љћ
and using the substitution q࠙u + e ((t))

Equation (2) can be expressed as follows:

mred Έ ĸ + mred Έ ōz2 Έ e1 Έ cos іјōz Έ ((t))љћ࠙mred Έ uƕƕ ((t)) + bz Έ uƕ ((t)) - іјc0 + c1 Έ cos іјōz Έ ((t))љћљћ Έ f ((u))

Up until this point, the gear system has been analyzed without considering backlash.
Mathematically, introducing backlash involves replacing the displacement u with a suitable
function f(u) that preserves the displacement characteristics. However, within the so-called
dead zone, this function takes on a value of zero. It's important to emphasize that the
physical interpretation of the function f(u) is identical to that of the displacement itself.

mred Έ ĸ + mred Έ ōz2 Έ e1 Έ cos іјōz Έ ((t))љћ࠙mred Έ uƕƕ ((t)) + bz Έ uƕ ((t)) - іјc0 + c1 Έ cos іјōz Έ ((t))љћљћ Έ f ((u))

The mathematical model above can be further reduced to a dimensionless form:

Fav + Fe Έ ō2 Έ cos ((ō Έ ((t))))࠙x1ƕƕ + 2 Έ h1 Έ x1ƕ + іј1 + ĵ1 Έ іјwx Έ ňљћљћ Έ f ((x))

Own frequency system ō0 ̗ ƠƠcƠ0 ƠƠ ō0 = 8.419 Έ 103
Damping factor system ƀƀ h1 = 8.255 Έ 103
mred

bz
h1 ̗ ƀƀƀƀ

2 Έ ƠmƠrƠedƠΈƠcƠ0

Mesh ratio c1 ĵ1 = 1
ĵ1 ̗ ƀ ōx = 0.254
Fav = 0.027
c0 Fe = 0.004
ōz a1 ̗ 10
ōx ̗ ƀ
ō0
mred Έ ĸ1
Fav ̗ ƀƀƀ

Lz Έ c0
e1
Fe ̗ ƀ
Lz

Mathematical models of gear backlash
The introduction of a new variable x, dependent on the dimensionless time τ = ω0t, affects the width

of the dead zone of the tooth gap, now falling within the range
limited by the values − 1 and 1

This sheet focuses on assessing how the approximation of gear backlash characteristics affects
the dynamic properties of the gear. We obtained numerical results using a discontinuous
function that models the backlash.

f іјu1љћ ̗ Ɓ if u1 ̧ -Lz u f іјx1љћ ̗ Ɓ if x1 ̧ -1
Ɓ x࠙ƀ Ɓ
Ɓ u1 + Lz Ɓ x1 + 1
ƁƁ Ɓ Lz ƁƁ Ɓ
Ɓ if -Lz ̧ u1 ̧ Lz Ɓ if -1 ̧ x1 ̧ 1
Ɓ Ɓ0 Ɓ Ɓ0
ƁƁ ƁƁ
Ɓ if u1 ̨ Lz Ɓ if x1 - 1 ̨ 1
ƁƁ ƁƁ
Ɓ Ɓ u1 - Lz Ɓ Ɓ x1 - 1
Ɓ Ɓ

From the mathematical point of view, the best results are achieved using the logarithmic
function proposed in the

1 і 1 + ea1 Έ ((x - 1)) љ
f ((x)) ̗ ƀΈ ln їƀƀƀ ( ƀ) њ

a1 ј 1 - ea1 Έ (x - 1) ћ

Variable check before placing it in the solve bock

-іјR1 Έ bz Έ іјR1 Έ ļ1ƕ ((t)) - іјR2 Έ ļ2ƕ ((t))љћ - e2 ((t))љћљћ - R1 Έ cz Έ іјR1 Έ ļ1 ((t)) - іјR2 Έ ļ2 ((t))љћ - e ((t))љћ Έ f іјu1љћ + Mn
ļ1ƕƕ ((t))࠙ƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀ

J1total

R1 = 0.035 R2 = 0.035 Mn = 5
bz = 9.864 Έ 108 cz = 0.002 J1total = 0.009

R2 Έ bz Έ іјR1 Έ ļ1ƕ (t) - іјR2 Έ ļ2ƕ (t)љћ - e2 (t)љћ + R2 Έ cz Έ іјR1 Έ ļ1 (t) - іјR2 Έ ļ2 (t)љћ - e (t)љћ Έ f іјu1љћ - M0
ļ2ƕƕ ((t))࠙ƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀ

J2total

R2 = 0.035 R1 = 0.035 cz = 0.002 M0 = 1 Έ 103 bz = 9.864 Έ 108

x1ƕƕ ((t))࠙-2 Έ h1 Έ x1ƕ - іј1 + ĵ1 Έ іјōx Έ ō0 Έ ((t))љћљћ Έ f іјx1љћ + Fav + Fe Έ ōx2 Έ cos іјōx Έ ((t))љћ

h1 = 8.255 Έ 103 ĵ1 = 1 ōx = 0.254 ō0 = 8.419 Έ 103 Fav = 0.027
Fe = 0.004

u1ƕƕ ((t))࠙mred Έ ĸ1 + mred Έ ōz2 Έ e1 Έ cos іјōz Έ ((t))љћ - bz Έ u1ƕ ((t)) + іјc0 + c1 Έ cos іјōz Έ ((t))љћљћ Έ f іјu1љћ

mred = 7.096 ĸ1 = 4.729 Έ 103 e1 = 1 Έ 10-5 ōz = 2.142 Έ 103 bz = 9.864 Έ 108
c0 = 5.03 Έ 108 c1 = 5.03 Έ 108



t ̗ 0,0.0005Ɛ2

10

8

6

4

2

0 ļ1 ((t))
ļ2
-2

-4

-6

-8
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1

-10

t

t ̗ 0,0.0005Ɛ0.5

10

8

6

4

2

0 f іјx1љћ

-2

-4

-6

-8
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1

-10

x