Practical Information on gears 8 GEAR FORCES In designing a gear, it is important to analyze the magnitude Table 8.1 presents the equations for tangential (circumferential) and direction of the forces acting upon the gear teeth, shafts, force Ft (kgf), axial (thrust) force Fx(kgf), and radial force bearings, etc. In analyzing these forces, an idealized assumption Fr in relation to the transmission force Fn acting upon the is made that the tooth forces are acting upon the central part of central part of the tooh flank. the tooth flank. T and T1 shown therein represent input torque (kgf·m). Table 8.1 Forces acting upon a gear Types of gears Ft :Tangential force Fx : Axial force Fr : Radial force Spur gear —————— Ft tanα 2000T Ft = tanα n Helical gear d Ft tan β Ft cos β Straight bevel gear Ft tanα sinδ Ft tanα cosδ 2000T Ft = When convex surface is working: dm Ft Ft (tanα n sinδ – sinβ m cosδ ) (tanα n cosδ + sinβ m sinδ ) cos βm cos β m Spiral bevel gear m is the central reference diameter d When concave surface is working: dm = d – b sinδ Ft Ft (tanαn sin δ + sinβm cos δ ) (tanα n cosδ – sinβ m sinδ ) cos βm cos β m α γ μ γ Worm 2000T1 cos n cos – sin Worm Ft Ft (Driver) = cosα n sin γ + μ cos γ d1 sinα n gear Ft Worm cosα n sin γ + μ cos γ cosα n cos γ – μ sin γ pair Wheel Ft Ft α γ μ γ (Driven) cos n sin + cos Screw gear Driver 2000T1 cosα n sin β – μ cos β Ft = Ft gear d1 cosα n cos β + μ sin β Σ = 90° sinα n Ft cosα n cos β + μ sin β β = 45° Driven cosα n sin β – μ cos β Ft Ft gear cosα n cos β + μ sin β 8.1 Forces in a Spur Gear Mesh The Spur Gear's transmission force Fn , which is normal to the Figure 8.2. The tangential component of the drive gear, Ft1 is tooth surface , as in Figure 8.1, can be resolved into a tangential equal to the driven gear's tangential component, Ft2, but the component, Ft , and a radial component, Fr . Refer to Equation directions are opposite. Similarly, the same is true of the radial (8.1). components. Table 7.8 expresses the reference value of viscosity of lubricant used in the equations for the strength of worm gears in JGMA 405-01. Ft = Fn cosα' (8.1) Fr = Fn sinα' Drive gear There will be no axial force, Fx . The direction of the forces acting on the gears are shown in Fr1 Fr1 Ft2 Ft1 Ft1 Ft2 Fr2 Fr2 Ft Fn Fr Driven gear α' Fig.8.1 Forces acting on a spur gear mesh Fig.8.2 Directions of forces acting on a spur gear mesh 455 Practical Information on gears 8.2 Forces in a Helical Gear Mesh The directions of forces acting on a helical gear mesh are shown in Figure 8.4. The helical gear's transmission force, Fn, which is normal to the The axial thrust sub-component from drive gear, Fx1, equals the tooth surface, can be resolved into a tangential component, F1, driven gear's, Fx2, but their directions are opposite. and a radial component, Fr, as shown in Figure 8.3. Again, this case is the same as tangential components and radial components. F1 = Fn cosα n (8.2) Fr = Fn sinα n Fx β The tangential component, F1, can be further resolved into circular subcomponent, Ft , and axial thrust subcomponent, Fx. F1 Ft Ft = F1 cos β (8.3) Fx = F1 sin β Substituting and manipulating the above equations result in: Fr Fx = Ft tan β α (8.4) F1 αn tan n Fr = Ft Fn cos β Fig.8.3 Forces acting on a helical gear mesh Right-hand pinion as drive gear Left-hand pinion as drive gear Left-hand gear as driven gear Right-hand gear as driven gear Fr1 Fr1 Fr1 Fr1 Ft2 Ft1 Ft2 Ft1 Fx1 Fx2 Fx2 Fx1 F Fx2 Fx1 x1 Fx2 Ft1 Ft1 Ft2 Ft2 Fr2 Fr2 Fr2 Fr2 Fig.8.4 Directions of forces acting on a helical gear mesh 8.3 Forces in a Straight Bevel Gear Mesh The forces acting on a straight bevel gear are shown in Figure 8.5. The force which is normal to the central part of the tooth face, Fn , can be split into tangential component, Ft , and F1 radial component, F1, in the normal plane of the tooth. Ft Fn αn α Ft = Fn cos n (8.5) F1 = Fn sinα n Fx Fr F1 Again, the radial component, F1, can be divided into an axial δ force, Fx, and a radial force, Fr , perpendicular to the axis. δ Fx = F1 sinδ (8.6) Fr = F1 cosδ And the following can be derived: Fig.8.5 Forces acting on a straight bevel gear mesh Fx = Ft tanα n sinδ (8.7) Fr = Ft tanα n cosδ 456 Practical Information on gears Let a pair of straight bevel gears with a shaft angle Σ = 90° , a 8.4 Forces in A Spiral Bevel Gear Mesh pressure angle α n = 20° and tangential force, Ft , to the central part of tooth face be 100. Axial force, Fx , and radial force, Fr, Spiral bevel gear teeth have convex and concave sides. will be as presented in Table 8.2. Depending on which surface the force is acting on, the direction and magnitude changes. They differ depending upon which is the driver and which is the driven. Axial force Fx Table 8.2 Values Radial force Fr (1) Pinion Concave surface Gear ratio z2/z1 Forces on the gear tooth 1.0 1.5 2.0 2.5 3.0 4.0 5.0 Axial force 25.7 20.2 16.3 13.5 11.5 8.8 7.1 Radial force 25.7 30.3 32.6 33.8 34.5 35.3 35.7 Gear tooth Gear tooth Convex surface (2) Gear Right-hand spiral Left-hand spiral Fig.8.7 Convex surface and concave surface Gear ratio z2/z1 of a spiral bevel gear Forces on the gear tooth 1.0 1.5 2.0 2.5 3.0 4.0 5.0 Figure 8.7 presents the profile orientations of right-hand and Axial force 25.7 30.3 32.6 33.8 34.5 35.3 35.7 left-hand spiral teeth. If the profile of the driving gear is Radial force 25.7 20.2 16.3 13.5 11.5 8.8 7.1 convex, then the profile of the driven gear must be concave. Table 8.3 presents the convex/concave relationships. Figure 8.6 contains the directions of forces acting on a straight bevel gear mesh. In the meshing of a pair of straight bevel gears with shaft angle Σ = 90° , the axial force acting on drive gear Fx1 Table 8.3 Concave and convex sides of a spiral bevel gear equals the radial force acting on driven gear Fr2. Similarly, the Right-hand gear as drive gear radial force acting on drive gear Fr1 equals the axial force acting on driven gear Fx2. The tangential force Ft1 equals that of Ft2. Meshing tooth face Rotational direction of drive gear Right-hand drive gearLeft-hand driven gear Clockwise Convex Concave Pinion as drive gear Gear as driven gear Counterclockwise Concave Convex Fr1 Ft1 Ft2 Fr1 Left-hand gear as drive gear Fx1 Fx1 Meshing tooth face Fr2 Fr2 Rotational direction of drive gear Right-hand driven Ft2 Ft1 Left-hand drive gear Fx2 Fx2 gear Clockwise Concave Convex Fig.8.6 Directions of forces acting on a straight bevel gear mesh Counterclockwise Convex Concave All the forces have relations as per Equations (8.8). Ft1 = Ft2 Fr1 = Fx2 (8.8) Fx1 = Fr2 457 Practical Information on gears (1) Forces on Convex Side Profile F1 = Fn cosα n (8.13) F2 = Fn sinα n And F1 can be separated into components Ft and Fs on the pitch Fs F2 βm surface: αn Ft Ft = F1 cosβ m F1 (8.14) Fs = F1 sinβ m F1 Fx Fn So far, the equations are identical to the convex case. However, Fs F2 differences exist in the signs for equation terms. On the Fr axial surface, F2 and Fs can be resolved into axial and radial δ subcomponents. Note the sign differences. Fx = F2 sinδ + Fs cosδ (8.15) δ δ Fig.8.8 When meshing on the convex side of tooth face Fr = F2 cos – Fs sin The above can be manipulated to yield: The transmission force, Fn , can be resolved into components F1 and F2. (See Figure 8.8). Ft Fx = ( tanα n sinδ + sinβ m cosδ ) cos β m F1 = Fn cosα n Ft ( Fr = ( tanα n cosδ – sinβ m sinδ ) 8.9) cos β m F2 = Fn sinα n (8.16) Then F1 can be resolved into components Ft and Fs: Let a pair of spiral bevel gears have a shaft angle Σ = 90° , Ft = F1 cosβ m a pressure angle α n = 20° , and a spiral angle βm = 35° . If the (8.10) t Fs = F1 sinβ m tangential force, F to the central portion of the tooth face is 100, the axial force, Fx, and radial force, Fr, have the relationship On the axial surface, F2 and Fs can be resolved into axial and shown in Table 8.4.