Cơ khí chế tạo máy - Mechanical elements – shafts

Ahmed Kovacevic, City University London Design web 1 Mechanical Elements – Shafts Prof Ahmed Kovacevic School of Engineering and Mathematical Sciences Room CG25, Phone: 8780, E-Mail: a.kovacevic@city.ac.uk www.staff.city.ac.uk/~ra600/intro.htm Engineering Drawing and Design - Lecture 15 ME 1110 – Engineering Practice 1 Ahmed Kovacevic, City University London Design web 2 Introduction
Shaft – a rotating element used to transmit power or motion. It provides axis of rotation for rotating elements and controls their motion.
Axle – a non-rotating element that carries no torque and is used to support rotating elements.
Spindle - a short shaft
There are two aspects of shaft design: » Deflection and rigidity (bending and torsional deflection) » Stress and strength
To design a shaft, other elements: gears, pulleys, bearings should be located and specified.
Design objective necessary to check if a shaft diameter is sufficient to sustain loads Ahmed Kovacevic, City University London Design web 3 Ahmed Kovacevic, City University London Design web 4 Common shaft loading mechanisms Ahmed Kovacevic, City University London Design web 5 Shaft design characteristics
Shaft is usually of circular cross section
Deflections are function of the geometry and load.
Steps in the shaft design are: » Define shaft topology » Specify driving elements » Free body diagram » Select bearings » Consider shaft deflection and stress » Specify connections » Dimensions Ahmed Kovacevic, City University London Design web 6 Shaft design
Shaft topology a) Choose a shaft configuration to support and locate the two gears + two bearings b) Solution uses an integral pinion, shaft with three shoulders, key, keyway and sleeve. Bearings are located in the housing c) Choose fan shaft configuration d) Solution uses sleeve bearings, a straight through shaft, locating collars, setscrews, pulley and fan • Driving elements 1. Driving elements (gears, pulleys, sprockets ) have to be selected and calculated 2. Minimum diameter of a rotating element and forces acting on it are relevant for a shaft design Ahmed Kovacevic, City University London Design web 7 Shaft design
Free body diagram Free body diagram is calculated such that the system of interest is separated from the surrounding and connections are replaced by forces a) Reactions in bearings & force diagram b) Bending moment c) Torsional moment ( P=ω T )
Bearing selection a) Equivalent load (forces) b) Bearing rating life based on the size c) Positioning and lubrication Ahmed Kovacevic, City University London Design web 8 Shaft design – bearing positioning Ahmed Kovacevic, City University London Design web 9 3 3 2 23 32 16 32 3 4 z zy s y M M c M Z I d T T c T S J d fd M T S σ pi τ pi pi = = = = = = = + Shaft design
Shaft deflection and stress – minimum diameter Difficult to calculate exactly. Reasons for complexity: a) Variable shaft diameter b) Undercuts and grooves – stress concentration points c) Type of load – axial, radial, torsional, bending, static, dynamic - In this course we will calculate a minimum shaft diameter without considering stress concentration points. - Calculations will be based on the maximum static load. - Diameter will be estimated for allowable stress which depends on the shaft material. Bending stress Torsional stress Minimum diameter distortion energy theory c=d/2 - maximum span I=pid4/64 - second moment of area Z=c/I - section modulus J=pid4/32 - second polar moment of area S=c/J - polar section modulus fs - factor of safety Ahmed Kovacevic, City University London Design web 10 Ahmed Kovacevic, City University London Design web 11 Shaft diameter vs Torque Shaft Dia Pure Torque Power (P=ωT) (at 100 rpm) mm Nm kW 30 132 1.4 40 313 3.3 50 612 6.4 60 1058 10.6 75 2068 21.6 80 2510 26 100 4900 51.3 Ahmed Kovacevic, City University London Design web 12 How to connect elements to the shaft?
Interference fits
Keys & Keyseats
Pins
Hubs
Integral shaft
Splines Ahmed Kovacevic, City University London Design web 13 Limits and Fits Tolerance difference between the maximum and minimum size limits of a part. International Tolerance Grade Numbers are used to specify the size of a tolerance zone. Ahmed Kovacevic, City University London Design web 14 International tolerance grade numbers Ahmed Kovacevic, City University London Design web 15 Preferred fits in the Basic-Hole System To differentiate between holes and shafts, upper and lower case letters are used H – Holes; h - Shafts Ahmed Kovacevic, City University London Design web 16 Preferred Hole Basis System of Fits Standardised by BS4500: ISO Units and Fits Ahmed Kovacevic, City University London Design web 17 Fundamental deviations for shafts Ahmed Kovacevic, City University London Design web 18 Selected fits – Hole basis Ahmed Kovacevic, City University London Design web 19 Force Fit - example Determine the “force fit” for a shaft and bearing hole that have basic diameter 32 mm and pressure fit H7/s6 Hole Shaft Tolerance Grade 0.025 mm 0.016 mm Upper deviation 0.025 mm 0.059 mm Lower deviation 0.000 mm 0.043 mm Max Diameter 32.025 mm 32.059 mm Min Diameter 32.000 mm 32.043 mm Average Diameter 32.013 mm 32.051 mm Max Clearance Cmax = Dmax- dmin = 0.051 mm Min Clearance Cmin = Dmin - dmax = 0.030 mm Hole Shaft 0.025 0.00032 + + 0.059 0.04332 + + Ahmed Kovacevic, City University London Design web 20 Keys and pins Used on shafts to secure rotating elements; gears, pulleys, wheels. Keys – transmit torque between the shaft and the rotating element Pins – axial positioning, transfer of torque or/and thrust. Ahmed Kovacevic, City University London Design web 21 Strength of a key Allowed torque on the key:4 ; 2 2 4 y s DW DT F A WL DWL SF T T A DWL fτ ≈ = = = = ⇒ = Ahmed Kovacevic, City University London Design web 22 Drawing and dimensioning Ahmed Kovacevic, City University London Design web 23 Example Determine the diameter for the solid round shaft 450 mm long, as shown in Figure. The shaft is supported by self-aligning bearings at the ends. Mounted upon the shaft are a V-belt pulley, which contributes a radial load of F1=8kN to the shaft, and a gear which contributes a radial load of F2=3kN. The two loads are in the same plane and have the same directions. The allowable bending stress (strength) is S=70 MPa. F1=8 kN F2=3 kN a=450 mm b=150 mm c=200 mm S=70 MPa d=? SOLUTION: Assumptions - the weight of the shaft is neglected - the shaft is designed for the normal bending stress in the location of max. bending moment Ahmed Kovacevic, City University London Design web 24 Solution 1 1 2 1 2 max 1 ( ) ( ) 6 5 900 c A M a R a b F a b c F R kN R kN M M b R Nm = − + − + − − = = = = ⋅ = ∑ 4 3 3 64 2 0.1 32 d dI c I dZ d c pi pi = = = = ≈ Second moment of area (moment of inertia) Section modulus =--------------------------------------------------------------- max span 6 3 3 5 900 70 10 0.1 900 0.050 50 70 10 Mc MS I Z d d m mm σ= = = = = ⋅ ⋅ = = = ⋅ Stress = Strain = Bending moment / section modulus