Gear Design – Part 1 These slides are prepared by Suat Kadıoğlu and modified by Sezer Özerinç Design of Helical and Spur Gears • Gear tooth fails due to bending (yielding or fatigue) or due to wear. • Wear is used as a general term for surface failure, which is usually caused by pitting. • Pitting: Formation of pits due to repetitive contact stresses (surface fatigue). • Abrasion (wear due to foreign particles) and scoring (due to lubrication failure) can also result in failure. Photo-elastic analysis of two gear teeth in contact. Note that due to bending of the tooth two types of stress occurs at the tooth flank: • On the right side tensile stresses • On the left side compressive stresses. These stress states repeat themselves every revolution, and may result in fatigue failure. Moreover, contact stresses occur at the contact point, which may result in surface fatigue. Design Method • A method developed by American Gear Manufacturers Association (AGMA) will be followed. It is called AGMA method. • Underlying principles of AGMA method, which historically provided the background for this method, will be reviewed first. • These principles can be used for a rough estimate of gear dimensions, power rating, and similar. Lewis Bending Equation • Provides the basis for gear design against bending failure. • Proposed by Wilfred Lewis in 1892. • In calculating the bending stress at the root of the teeth, the tooth form is taken into account. Face width, 8mnF16mn max Ft Section modulus: I/c=(Ft3/12)/(t/2)=Ft2/6 max a t/2 = x t/2 b a b M = I /c 6Wt = Ft 2 t2 x= 4 tangent to fillet max 6Wt Wt 4 Wt = = = 2 2 Ft 4F / 6 t 2 3 Fx max Wt p = 2 3 Fxp 2x let y = 3p then multiply both numerator and denominator by circular pitch, p. max Wt = Fpy Original form of Lewis equation Lewis Form factor • Lewis form factor, y, can be obtained by graphical layout of the gear tooth or by numerical computation. • Noting that p = p/P = pm (where P is the diametral pitch) Y = py can be defined and bending stress can be expressed as: Wt Wt P max = or max = FmY FY Assumptions in Using Lewis Equation • Only the bending stress due to load Wt is considered; hence, the compression due to Wr is neglected. In reality, the radial force causes compressive stresses over the base cross-section. • The teeth do not share the load and the highest force is exerted at the tip of the tooth. In reality, the tip condition is not the worst case, as another tooth will be in contact (contact ratio > 1), and the load will be shared. Assumptions in Using Lewis Equation • When the contact proceeds towards the middle of the tooth due to the rotation of the gear, only one tooth will start supporting the load, and therefore maximum load occurs near the middle of the tooth. But in this case the moment arm will be smaller. More detailed analysis shows that maximum stress tends to occur when a single pair of teeth is in contact. • Stress concentration is not taken into account. • Loads are static, dynamic effects are not considered. Dynamic Effects [MPa] • Dynamic effects increases the bending stress. They are taken into account by the dynamic (velocity) factor, Kv. • In the 19th century, Carl G. Barth first introduced the velocity factor. • Kv expressions are empirically derived (based on experiments). max K vWt = FmY [N] [mm] 3.05 + V Kv = cast iron, cast profile 3.05 6 .1 + V Kv = cut or milled profile 6 .1 3.56 + V Kv = 3.56 5.56 + V Kv = 5.56 hobbed or shaped profile shaved or ground profile V is the pitch line velocity in [m/s] Surface Durability • Types of surface failure for gear tooth: – pitting (formation of pits due to repetitive high contact stresses) – scoring (lubrication failure) – abrasion (due to presence of foreign materials) • Scoring and abrasion should be prevented by maintaining the lubrication well and keeping the gears free from hard abrasive particles. • But pitting is a type of failure that goes into the design of gears. • Hertz theory for contact stresses is utilized for analysis. Hertz Contact Stresses Surface Compressive stress c = pmax 2F = pb 1/ 2 2F 1 + 2 b= ( ) ( ) p 1 / d + 1 / d 1 2 For gears F = Wt / cos = Face width, F d1, 2 = 2r1, 2 1 − 2 = E Instantaneous radii of curvature at the contact point of the tooth profile c 2 Wt (1 / r1 ) + (1 / r2 ) = pF cos 1 + 2 c can be calculated at any point on the tooth profile by considering the load sharing effect on Wt. Along pressure line, only pure rolling exists. But since contact point has a finite width, sliding will also take place. The Hertz contact approach does no take friction into account. Experimentally it is observed that wear occurs first near the pitch line, and at pitch point P. The radius of curvature there: r1 = d p sin 2 r2 = d g sin 2 Define an elastic coefficient Cp as follows (by AGMA): 1 Cp = p ( p + g ) Adding dynamic factor Kv as a multiplier for Wt, and using Cp: c = −C p K vWt 1 1 + F cos r1 r2 negative sign means compressive stress Factor of Safety • Factor of safety for bending is defined as: St loss - of - function - load nt = = imposed - load max St is the gear bending strength • Note that contact stress is not linearly dependent to applied transmitted force Wt, but rather proportional to 𝑊𝑡 . • The factor of safety for contact, nc, is defined as; 2 2 loss - of - function - load Sc = 2 nc = imposed - load c Sc is the gear surface strength AGMA Stress Equations • There are two stress equations, one for bending and the other for contact (pitting resistance). • In AGMA terminology, the stresses are referred to as stress numbers. • The stress equations are based on the previously explained equations but there are many correction factors. AGMA Bending Equation Lewis equation with dynamic correction: 1 KH KB = Wt K o K v K s Fm YJ max K vWt = FmY 2𝜋𝑥 𝑌= 3𝑝 • Wt: tangential transmitted load [N] • Ko: Overload Factor (depends on application, driving vs. driven machinery) • Kv: Dynamic Factor (dynamic effects in load depending on pitch line velocity and tooth form) • Ks: Size Factor The equation given here • F: Face width (here we use the US letter is in SI units. The values in parentheses are U.S. to be consistent with previous slides) units, that is why there are • m: transverse module [mm] two different letters for • KH or (Km): Load distribution factor some of these coefficients. • KB : Rim thickness factor • YJ (J): Geometry factor for bending, which includes root fillet stress concentration factor, Kf (that is, an extended Lewis form factor). AGMA Contact Stress Equation Original form of contact stress equation c = ZE c = −C p K vWt 1 1 + F cos r1 r2 KH ZR Wt K o K v K s d pF ZI 𝐶𝑝 = 𝑍𝐸 = 1 𝜋 𝜅𝑝 + 𝜅𝑔 The equation given here is in SI units. The values in parentheses below are U.S. units, that is why there are two different letters for some of these coefficients. • ZE (Cp): Previously defined elastic coefficient [N/mm2]1/2 • ZR (Cf): Surface condition factor • dp: Pitch diameter of the pinion [mm] width (here we use the US letter to be consistent with previous slides) • ZI (I): Geometry factor for pitting resistance • Other factors are the same as those in bending equation. • In AGMA method, the stresses obtained from these equations are compared with corresponding “AGMA gear strengths” which are also referred to as "allowable stress numbers". AGMA Strength Equations • In general gear strengths (St for bending, Sc for contact) are given in terms of the hardness of the material for a certain number of stress cycles. • St and Sc are also modified by various factors to take into account the different service conditions and requirements. Bending and Contact Strength Equations all St YN = S F Y YZ • St : Bending strength [N/mm2] • YN: Stress cycle (life) factor for bending • YQ (KT): Temperature factor • YZ (KR): Reliability factor • SF: AGMA factor of safety for bending. c,all S c Z N ZW = S H Y YZ • Sc : Contact strength [N/mm2] • ZN: Stress cycle (life) factor for contact • CH or ZW (): Hardness ratio factor for pitting resistance • YQ (KT): Temperature factor • YZ (KR): Reliability factor • SH: AGMA factor of safety for pitting. 1 ksi = 6.89 MPa all St YN = S F Y YZ all St YN = S F Y YZ all St YN = S F Y YZ Sc Curves and Equations c,all S c Z N ZW = S H Y YZ Remarks on Gear Strengths • The use of St and Sc are restricted to gear problems. • They are for – unidirectional loading (gearset driven always in a single direction) – 10 million stress cycles – 99% reliability • When two-way (reversed loading) occurs as with idler gears, AGMA recommends using 70% of St values. Geometry Factors I (ZI) and J (YJ) = Wt K o K v K s 1 KH KB Fm YJ c = Z E Wt K o K v K s KH ZR d pF ZI • These are used to account for tooth form and load sharing. • I and J depend on face contact ratio mF. • mF = F/px (face width / axial pitch). We will consider spur gears (mF = 0) and helical gears (mF > 1). • Helical gears with mF < 1 are "low contact ratio" gears and they are not much different than spur gears as far as noise is concerned. Y J= K f mN obtained from tooth layout for highest point of single-tooth contact m N = 1 .0 for spur gears normal base pitch pN mN 0.95Z for helical gears with mF > 2.0 stress concentration load sharing ratio Lab length of action mN is the ratio of face width to minimum total length of lines of contact. Note that mN pN 0.95Z is an approximation. p N = pn cos n Lab = (r2 + a2 ) 2 − (r2 cos ) + (r1 + a1 ) 2 − (r1 cos ) − (r1 + r2 ) sin 2 2 • The book gives ready-to-use graphs for J, which makes the process easier. • For spur gears with F = 20 degrees and FDI teeth, use Fig. 14-6. • For helical gears with Fn=20 degrees and mF 2, use Fig. 14-7 and 14-8. (the latter is used to correct the values obtained from the former.)