Derivation of Kinematics for the Fluid Film Between Thrust Collar and Bull Gear. Original work by Dr. Karl Wygant [1] Reproduce by Travis Cable

Transcription

1 Derivation of Kinematics for the Fluid Film Between Thrust Collar and Bull Gear Original work by Dr. Karl Wygant [1] Reproduce by Travis Cable

2 Depiction of Area Under Investigation ω P pinion gear ω b bull gear Bull gear pinion gear bull gear Thrust washers Figure 1. Integrally geared compressor utilizing thrust collars [2] Thrust washers Pinion gear Figure 2. Helical gear pair utilizing thrust washers in a wind turbine gear box [2]

3 Nomenclature A = Center of pinion gear B = Center of bull gear C = Center of finite element within film a = Radial coordinate measured from pinion center (m) b = Distance from Bull Gear center to finite element (m) d = Distance between gear centers (m) θ = Angular position (radians) ωb = Bull gear rotational speed (rad/sec) ωp = Pinion Gear/T.C. rotational speed (rad/sec) Δθ = Small change in angular position (radians) Δr = Small change in radial position (m) N P = Number of teeth on pinion gear N B = Number of teeth on bull gear PCD = Pitch Center Diameter

4 Bull Gear Thrust Collar ωb ωp Δr B a Δθ d C b θ A Pinion PCD Bull Gear PCD

5 Finite Element in Film Thickness The film thickness is the width of the finite element represented by the rectangular prism from C to C C C Film Thickness

6 Reference Frame for Pinion Surface The point C on the thrust collar surface is the initial reference frame for the analysis Point C is fixed and has velocity components in the i and j directions i C j ωp a The velocity component in the i direction is zero because the point is fixed θ A The velocity component in the j direction is V j = -a* ωp

7 Reference Frame on Bull Gear Surface The point C is the reference frame on the bull gear surface j i Point C is fixed and has velocity components in the i and j directions The velocity component in the i direction is zero because the point is fixed The velocity component in the j direction is: V j = -b* ωb B b ωb C

8 Coordinate Transformation V N A,C V T A,C = cos(θ) sin(θ) sin(θ) cos(θ) V N B,C V T B,C The velocity derived at point B in C is transferred to the reference frame A by a coordinate transformation matrix The relative velocity at the point A is the difference between the velocity at point A in C and the point A in C V N A,C = b ω B sin(θ) V T A,C = a ω p b ω B cos(θ)

9 Reference Triangle on Gear Pair C b a B d θ A From the law of cosines, express the length b as a function of the radial coordinate a, circumferential coordinate θ, and the distance between gear centers d b = a 2 + d 2 2 a d cos θ 1/2

10 Relative Velocity The relationship between the angular velocity of the bull gear and the pinion is a function of the number of teeth in each gear ω B = N P N B ω P From this relationship and the law of cosines, express the relative velocity in the normal and tangential directions as a function of the radial and circumferential coordinate V N A,C V N A,C = a 2 + d 2 2 a d cos θ 1/2 N P N B ω P sin(θ) V T A,C V T A,C = a ω P a 2 + d 2 2 a d cos θ 1/2 N P N B ω P sin(θ)

11 What to do From Here? Derive the Reynolds equation for this geometry including Thrust Collar Tapper Derive the Energy Equation for solution of fluid temperature and viscosity Look into EHL to evaluate if elastic deformation of Thrust Collar Surface and Bull Gear Surface have a significant effect on performance Figure 3. Predicted elastic deformation of a flat faced thrust washer [3]

12 Project Goals Develop an analytical tool (Fortran code with Excel GUI) for the prediction of the performance characteristics of the Thrust Collar Compare analytical predictions against real time data measured by Dr. Childs to ensure accuracy

13 References 1. Wygant, Karl, 2012, Notes on the derivation of thrust washer film thickness 2. San Andres, Luis, 2012, Thrust Collar Analytic Tool Development, Proposal to Samsung Techwin 3. Jackson, Robert and Green, Itzhak, 2008, The Thermoelastic Behavior of Thrust Washer Bearings Considering Mixed Lubrication, Asperity Contact and Thermoviscous Effects, Tribology Transactions, vol. 51, pp