MODELLING OF THE GEAR BACKLASH

Công Nghệ Thông Tin, it, phầm mềm, website, web, mobile app, trí tuệ nhân tạo, blockchain, AI, machine learning - Công Nghệ Thông Tin, it, phầm mềm, website, web, mobile app, trí tuệ nhân tạo, blockchain, AI, machine learning - Cơ khí - Vật liệu 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. Variables used in the model https:www.academia.edu88067809Modellingofthegearbacklash In this chapter, the necessary variables required for the model are calculated Gear properties for the model: Number of teeth on the pinion gear ̗ Z1 14 Number of teeth on the pinion gear ̗ Z2 85 Module̗ m1 5 mm Face width of the gear̗ b1 10 mm Number of teeth gear 1̗ D1 Έm1 Z1 =D1 70 mm Number of teeth gear 2̗ D2 Έm1 Z2 =D2 425 mm Face width of the gear̗ b1 10 mm Module̗ m1 5 mm Maximum rotation speed̗ nr 1500 rpm =nr 157.08 ƀƀ rad s Material properties: Young modulus̗ Emodulus 200 GPa Poisson ratio̗ ʼn1 0.3 Inertia and weight of the pinions Density of steel̗ Ņsteel 7850 ƀƀ kg m3 Weight gear wheel 1̗ mass1 ΈΈΈΈƀ 1 4 ń іјD1љћ2 b1 Ņsteel =mass1 0.302 kg Weight gear wheel 2̗ mass2 ΈΈΈΈƀ 1 4 ń іјD2љћ2 b1 Ņsteel =mass2 11.136 kg Inertia of teeth gear 1̗ J1gear ΈΈƀ 1 2 mass1 і ї јƀƀ D1 2 љ њ ћ 2 =J1gear 0.000185 Έkg m2 Inertia of teeth gear 2̗ J2gear ΈΈƀ 1 2 mass2 і ї јƀƀ D2 2 љ њ ћ 2 =J2gear 0.251435 Έkg m2 load moment ̗ M0 1000 External drive torque ̗ Mn 5 calculate the reduced modulus (W) using the following formula: Reduced modulus̗ Wreduced Έƀƀƀƀƀ іј Έń Emodulusљћ і ј -1 ʼn1 2 љ ћ m1 =Wreduced іј Έ3.452 109 љћ ƀ N m Calculate the mesh stiffness (kmesh) using the formula: Mesh stiffness̗ kmesh ƀƀƀƀƀ іј ΈΈ2 Wreduced b1љћ D1 =kmesh іј Έ9.864 108 љћ ƀ N m Parameters 1DOF Mass of inertia 1 pinion̗ J1motor 0.00074 Mass of inertia 1 gearbox̗ J1gearbox 0.002 Mass of inertia 1 gearbox̗ J1gear ΈJ1gear ƀƀƀ 1 Έkg m2 =J1gear Έ1.85 10-4 Pitch diameter pinion 1̗ R1 ƀƀ D1 Έ2 m =R1 0.035 Gear ratio̗ Gr1 2 Mass of inertia 2 pinion̗ J1total ++J1motor ΈJ1gearbox іјGr1љћ2 J1gear =J1total 0.009࠙ ΈJ1total ƕƕļ1 ((t)) -Mn іј -ΈΈR1 bz іј --ΈR1 ƕļ1 ((t)) іј ΈR2 ƕļ2 ((t))љћ ƕe ((t))љћ ΈΈR1 cz іј --ΈR1 ļ1 ((t)) іј ΈR2 ļ2 ((t))љћ e ((t))љћљћ࠙ Mn ++ΈJ1total ƕƕļ1 ((t)) ΈΈR1 bz іј --ΈR1 ƕļ1 ((t)) іј ΈR2 ƕļ2 ((t))љћ ƕe ((t))љћ ΈΈR1 cz іј --ΈR1 ļ1 ((t)) іј ΈR2 ļ2 ((t))љћ e ((t))љћ࠙ ƕƕļ1 ((t)) ƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀ +--іј ΈΈR1 bz іј --ΈR1 ƕļ1 ((t)) іј ΈR2 ƕļ2 ((t))љћ e2 ((t))љћљћ ΈΈΈR1 cz іј --ΈR1 ļ1 ((t)) іј ΈR2 ļ2 ((t))љћ e ((t))љћ f іјu1љћ Mn J1total Parameters 2DOF Mass of inertia 2 pinion̗ J2motor 0.074 Mass of inertia 2 gearbox̗ J2gearbox 0.002 Mass of inertia 2 gearbox̗ J2gear ΈJ2gear ƀƀƀ 1 Έkg m2 =J2gear 0.251 Pitch radius pinion 2̗ R2 ƀƀ D1 Έ2 m =R2 0.035 Gravitational constant̗ g1 9.81 Gear ratio̗ Gr2 2 Mass of inertia 2 pinion̗ J2total ++J2motor ΈJ2gearbox іјGr2љћ2 J2gear =J2total 0.333࠙ ΈJ2total ƕƕļ2 ((t)) -+ΈΈR2 bz іј --ΈR1 ƕļ1 ((t)) іј ΈR2 ƕļ2 ((t))љћ ƕe ((t))љћ ΈΈR2 cz іј --ΈR1 ļ1 ((t)) іј ΈR2 ļ2 ((t))љћ e ((t))љћ M0࠙ M0 ---ΈJ2total ƕƕļ2 ((t)) ΈΈR2 bz іј --ΈR1 ƕļ1 ((t)) іј ΈR2 ƕļ2 ((t))љћ ƕe ((t))љћ ΈΈΈR2 cz іј --ΈR1 ļ1 ((t)) іј ΈR2 ļ2 ((t))љћ e ((t))љћ f іјu1љћ M0࠙ ƕƕļ2 ((t)) ƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀ -+ΈΈR2 bz іј --ΈR1 ƕļ1 ((t)) іј ΈR2 ƕļ2 ((t))љћ e2 ((t))љћ ΈΈΈR2 cz іј --ΈR1 ļ1 ((t)) іј ΈR2 ļ2 ((t))љћ e ((t))љћ f іјu1љћ M0 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 element̗ cz 0.002 Spring displacement element̗ bz Έkmesh ƀ m N =bz Έ9.864 108 Backlash value̗ zbacklash 0.005 Is a constant value half of the backlash value.̗ Lz ƀƀƀ zbacklash 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 f...

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

Variables used in the model

https://www.academia.edu/88067809/Modelling_of_the_gear_backlash

In this chapter, the necessary variables required for the

model are calculated

Gear properties for the model:

Maximum rotation speed n r̗1500 rpm n r=157.08 ƀƀrad

s

Material properties:

Inertia and weight of the pinions

m3

4 ń іјD 1љћ2 b 1 Ņ steel mass 1=0.302 kg

4 ń іјD 2љћ2 b 1 Ņ steel mass 2=11.136 kg

Inertia of teeth gear 1 J 1gear̗ƀ1Έ Έ

2 mass 1

і ї

D 1

2

љ њ ћ

2

=

J 1gear 0.000185 kg mΈ 2

Inertia of teeth gear 2 J 2gear̗ƀ1Έ Έ

2 mass 2

і ї

D 2

2

љ њ ћ

2

=

J 2gear 0.251435 kg mΈ 2

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

Reduced modulus W reduced̗ƀƀƀƀƀіј Έń E modulusљћΈ

іј -1 ʼn 12љћ m 1 W reduced=іј3.452 10Έ

9љћ ƀN

m

Calculate the mesh stiffness (k_mesh) using the formula:

Mesh stiffness k mesh̗ƀƀƀƀƀіј2 WΈ reducedΈb 1љћ

D 1 k mesh=іј9.864 10Έ 8љћ ƀN

m

Parameters 1DOF

Mass of inertia 1 gearbox J 1gearbox̗0.002

Mass of inertia 1 gearbox J 1gear̗J 1gearΈƀƀƀ1

Έ

kg m2 J 1gear=1.85 10Έ -4

Έ

Mass of inertia 2 pinion J 1total̗J 1motor+J 1gearboxΈіјG r1љћ2+J 1gear J 1total=0.009

࠙

Έ

J 1total ļ 1ƕƕ((t)) M n- іјR 1Έb zΈ іјR 1Έļ 1ƕ((t)) іј Έ- R 2 ļ 2ƕ((t))љћ-e ((t))љћƕ -R 1Έc zΈ іјR 1Έļ 1 ((t)) іј Έ- R 2 ļ 2 ((t))љћ e ((t))љћљћ

-࠙

M n J 1totalΈļ 1ƕƕ((t))+R 1Έb zΈ іјR 1Έļ 1ƕ((t)) іј Έ- R 2 ļ 2ƕ((t))љћ-e ((t))љћƕ +R 1Έc zΈ іјR 1Έļ 1 ((t)) іј Έ- R 2 ļ 2 ((t))љћ e ((t))љћ

-࠙

ƕƕ

ļ 1 ((t)) ƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀ-іјR 1Έb zΈіјR 1Έļ 1ƕ((t)) іј Έ- R 2 ļ 2ƕ((t))љћ e- 2 ((t))љћљћ-R 1Έc zΈ іјR 1Έļ 1 ((t)) іј Έ- R 2 ļ 2 ((t))љћ e ((t))љћ f іјu- Έ 1 љћ M+ n

J 1total

Parameters 2DOF

Mass of inertia 2 pinion J 2motor̗0.074

Mass of inertia 2 gearbox J 2gearbox̗0.002

Mass of inertia 2 gearbox J 2gear̗J 2gearΈƀƀƀ1

Έ

kg m2 J 2gear=0.251

Έ

Gravitational constant g 1̗9.81

Mass of inertia 2 pinion J 2total̗J 2motor+J 2gearboxΈіјG r2љћ2+J 2gear J 2total=0.333

࠙

Έ

J 2total ļ 2ƕƕ((t)) R 2Έb zΈ іјR 1Έļ 1ƕ((t)) іј Έ- R 2 ļ 2ƕ((t))љћ-e ((t))љћƕ +R 2Έc zΈ іјR 1Έļ 1 ((t)) іј Έ- R 2 ļ 2 ((t))љћ e ((t))љћ M- - 0

࠙

M 0 J 2totalΈļ 2ƕƕ((t))-R 2Έb zΈ іјR 1Έļ 1ƕ((t)) іј Έ- R 2 ļ 2ƕ((t))љћ-e ((t))љћƕ -R 2Έc zΈ іјR 1Έļ 1 ((t)) іј Έ- R 2 ļ 2 ((t))љћ e ((t))љћ f іјu- Έ 1 љћ M- 0

࠙

ƕƕ

ļ 2 ((t)) ƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀ R 2Έb zΈіјR 1Έļ 1ƕ((t)) іј Έ- R 2 ļ 2ƕ((t))љћ e- 2 ((t))љћ+R 2Έc zΈ іјR 1Έļ 1 ((t)) іј Έ- R 2 ļ 2 ((t))љћ e ((t))љћ f іјu- Έ 1 љћ M- 0

J

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

N b z=9.864 10Έ

8

Is a constant value half of the backlash value L z̗ƀƀƀz backlash

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

࠙

q 1 R 1Έļ 1 ((t))-R 2Έļ 2 ((t))

̗

ō s 153

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

࠙

Έ

m red ĸ m redΈq ((t))ƕƕ +b zΈ((q ((t))ƕ -e ((t))))ƕ +c zΈ(( -q e ((t)))) f ((u))ƕ Έ

̗

m red ƀƀƀƀƀƀƀƀJ 1totalΈJ 2total

+ Έ

R 12 J 2total R 22ΈJ 1total m red=7.096

Rotation speed motor

Angular velocity of the rotor of the drive e ((t))̗e 1Έcos іј Έō z ((t))љћ

Amplitude of dynamic component of meshing stiffness c 1̗5.03 10Έ 8

The characteristics of periodically variable meshing stiffness c Z ((t))̗c 0+c 1Έcos іј Έō z ((t))љћ

̗

̗

ŀ 1 ƀƀM 0

M n

̗

R 1

J 1total ƀƀƀ

Έ

R 2 ŀ 1

J 1total

љ њ

ћ M 1 Considering the properties of periodically varying meshing stiffness

࠙

c Z ((t)) c 0+c 1Έcos іј Έ ō z ((t))љћ

and using the substitution q࠙u e ((t))+

Equation (2) can be expressed as follows:

࠙ +

Έ

m red ĸ m redΈō z2Έe 1Έcos іј Έ ō z ((t))љћ m redΈu ((t))ƕƕ +b zΈu ((t))ƕ -іј +c 0 c 1Έcos іј Έ ō z ((t))љћљћ f((u))Έ

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

࠙ +

Έ

m red ĸ m redΈō z2Έe 1Έcos іј Έ ō z ((t))љћ m redΈu ((t))ƕƕ +b zΈu ((t))ƕ -іј +c 0 c 1Έcos іј Έ ō z ((t))љћљћ f((u))Έ The mathematical model above can be further reduced to a dimensionless form:

࠙ +

F av F eΈō2Έcos (( Έ ō ((t)))) x 1ƕƕ+2 hΈ 1Έx 1ƕ+іј +1 ĵ 1Έіј Έw x ňљћљћ f((x))Έ

Own frequency system ō 0̗ ƠƠƠƠƠƀƀc 0

3

Έ

2 ƠƠƠƠƠƠm redΈc 0 h 1=8.255 10Έ

3

Mesh ratio ĵ 1̗ƀc 1

̗

ō x ƀō z

̗

F av ƀƀƀm redΈĸ 1

Έ

̗

F e ƀe 1

̗

a 1 10

Mathematical models of gear backlash

The introduction of a new variable x, dependent on the dimensionless time τ = ω0 t, 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

࠙

L z

̗

f іјu 1љћ ƁƁ

Ɓ

Ɓ

Ɓ

Ɓ

Ɓ

Ɓ

Ɓ

ƁƁ

ifu 1̧-L z

Ɓ

Ɓu 1+L z

if-L z̧u 1̧L z

Ɓ

Ɓ 0

ifu 1̨L z

Ɓ

Ɓu 1-L z

̗

f іјx 1љћ ƁƁ

Ɓ Ɓ Ɓ Ɓ Ɓ Ɓ Ɓ ƁƁ

if x 1̧-1 Ɓ

Ɓx 1+1

if -1 x̧ 1̧1 Ɓ

Ɓ 0

if x 1-1 1̨ Ɓ

Ɓx 1-1

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

̗

f ((x)) ƀ1 Έ

a 1 ln

і ї

+

1 e a 1Έ(( -x 1))

-1 e a 1Έ (( -x 1))

љ њ ћ

Variable check before placing it in the solve bock

࠙

ƕƕ

ļ 1 ((t)) ƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀ-іјR 1Έb zΈіјR 1Έļ 1ƕ((t)) іј Έ- R 2 ļ 2ƕ((t))љћ e- 2 ((t))љћљћ-R 1Έc zΈ іјR 1Έļ 1 ((t)) іј Έ- R 2 ļ 2 ((t))љћ e ((t))љћ f іјu- Έ 1 љћ M+ n

J 1total

=

=

b z 9.864 10Έ 8 c z=0.002 J 1total=0.009

࠙

ƕƕ

ļ 2 ((t)) ƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀƀ R 2Έb zΈіјR 1Έļ 1ƕ((t)) іј Έ- R 2 ļ 2ƕ((t))љћ e- 2 ((t))љћ+R 2Έc zΈ іјR 1Έļ 1 ((t)) іј Έ- R 2 ļ 2 ((t))љћ e ((t))љћ fіјu- Έ 1 љћ M- 0

J 2total

=

R 1 0.035

=

࠙

ƕƕ

x 1 ((t)) -2 hΈ 1Έx 1ƕ-іј +1 ĵ 1Έіјō xΈō 0Έ((t))љћљћ fіјxΈ 1 љћ F+ av+F eΈō x2Έcos іј Έ ō x ((t))љћ

=

h 1 8.255 10Έ 3 ĵ 1=1 ō x=0.254 ō 0=8.419 10Έ 3 F av=0.027

=

F e 0.004

࠙

ƕƕ

u 1 ((t)) m redΈĸ 1+m redΈō z2Έe 1Έcos іј Έ ō z ((t))љћ-b zΈu 1ƕ((t))+іј +c 0 c 1Έcos іј Έ ō z ((t))љћљћ fіјuΈ 1љћ

=

m red 7.096 ĸ 1=4.729 10Έ 3 e 1=1 10Έ -5 ō z=2.142 10Έ 3 b z=9.864 10Έ 8

=

c 0 5.03 10Έ 8 c 1=5.03 10Έ 8

t 0 0.0005 2, Ɛ

-6

-4

-2

0

2

4

6

8

-10

-8

10

t

ļ 1 ((t)) ļ 2

t 0 0.0005 0.5, Ɛ

-6

-4

-2

0

2

4

6

8

-10

-8

10

x

f іјx 1љћ