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