Contact Stresses and Deformations ME EN 7960 Precision Machine - PDF document

Contact Stresses and Deformations ME EN 7960 – Precision Machine Design Topic 7 ME EN 7960 – Precision Machine Design – Contact Stresses and Deformations 7-1 Curved Surfaces in Contact • The theoretical contact area of two spheres is a point (= 0-dimensional) • The theoretical contact area of two parallel cylinders is a line (= 1-dimensional) → As a result, the pressure between two curved surfaces should be infinite → The infinite pressure at the contact should cause immediate yielding of both surfaces • In reality, a small contact area is being created through elastic deformation, thereby limiting the stresses considerably • These contact stresses are called Hertz contact stresses ME EN 7960 – Precision Machine Design – Contact Stresses and Deformations 7-2 1

Curved Surfaces in Contact – Examples Linear bearings (ball and rollers) Rotary ball bearing Rotary roller bearing ME EN 7960 – Precision Machine Design – Contact Stresses and Deformations 7-3 Curved Surfaces in Contact – Examples (contd.) Gears Ball screw ME EN 7960 – Precision Machine Design – Contact Stresses and Deformations 7-4 2

Spheres in Contact The radius of the contact area is z given by: ⎡ − ν − ν ⎤ 1 2 1 2 F + 3 1 2 ⎢ ⎥ F y ⎣ ⎦ E E Z = 0: = 1 2 a Sphere 1 3 R 1 ⎛ ⎞ 1 1 x E 1 , 1 ⎜ + ⎟ 4 ⎜ ⎟ p max ⎝ R R ⎠ 2 a 1 2 2 a Circular contact area, y Where E 1 and E 2 are the moduli of resulting in a semi-elliptic p max pressure distribution elasticity for spheres 1 and 2 and ν 1 R 2 and ν 2 are the Poisson’s ratios, E 2 , 2 respectively The maximum contact pressure at the center of the circular contact Sphere 2 area is: x 3 F = p max π 2 2 a F ME EN 7960 – Precision Machine Design – Contact Stresses and Deformations 7-5 Spheres in Contact (contd.) • The equations for two spheres in contact are also valid for: – Sphere on a flat plate (a flat plate is a sphere with an infinitely large radius) – Sphere in a spherical groove (a spherical groove is a sphere with a negative radius) z y Z = 0: F x p max 2 a R 1 E 1 , 1 2 a y E 2 , 2 Flat plate ( R 2 = ∞ ) x ME EN 7960 – Precision Machine Design – Contact Stresses and Deformations 7-6 3

Spheres in Contact – Principal Stresses The principal stresses σ 1 , σ 2 , and σ 3 are generated on the z-axis: ⎡ ⎤ ⎢ ⎥ ⎛ − ⎞ 1 ( ) ⎢ z a ⎥ σ = σ = σ = σ = − + ν ⎜ ⎟ − 1 1 arctan p ⎜ ⎟ ⎢ ⎥ 1 2 max ⎛ ⎞ x y 2 ⎝ ⎠ a z z ⎜ + ⎟ ⎢ 2 1 ⎥ ⎜ ⎟ 2 ⎝ a ⎠ ⎣ ⎦ − 1 ⎛ 2 ⎞ z ⎜ ⎟ σ = σ = − + 1 p ⎜ ⎟ 3 max 2 z ⎝ ⎠ a The principal shear stresses are found as: σ − σ τ 3 = 0 τ = τ = τ = 1 3 1 2 max 2 ME EN 7960 – Precision Machine Design – Contact Stresses and Deformations 7-7 Spheres in Contact – Vertical Stress Distribution at Center of Contact Area σ , τ 1 Von Mises 0.8 Ratio of stress to p max σ z Plot shows 0.6 material with Poisson’s σ X, σ y ratio ν = 0.3 0.4 τ max 0.2 0 z 0 0.5 a a 1.5 a 2a 2.5a 3a Depth below contact area • The maximum shear and Von Mises stress are reached below the contact area • This causes pitting where little pieces of material break out of the surface ME EN 7960 – Precision Machine Design – Contact Stresses and Deformations 7-8 4

Cylinders in Contact The half-width b of the rectangular contact area of two parallel cylinders is F found as: E 1 , ν 1 z ⎡ ⎤ − ν 2 − ν 2 1 1 + 4 1 2 F ⎢ ⎥ ⎣ ⎦ E E = 1 2 b ⎛ ⎞ 1 1 ⎜ ⎟ π + E 2 , ν 2 L ⎜ ⎟ ⎝ R R ⎠ p max R 1 1 2 L 2 b Where E 1 and E 2 are the moduli of elasticity for cylinders 1 and 2 and ν 1 and ν 2 are the Poisson’s ratios, x y respectively. L is the length of contact. R 2 F The maximum contact pressure along the center line of the rectangular contact area is: Rectangular contact area 2 F with semi-elliptical pressure max = p π distribution bL ME EN 7960 – Precision Machine Design – Contact Stresses and Deformations 7-9 Cylinders in Contact (contd.) • The equations for two cylinders in contact are also valid for: – Cylinder on a flat plate (a flat plate is a cylinder with an infinitely large radius) – Cylinder in a cylindrical groove (a cylindrical groove is a cylinder with a negative radius) Rectangular contact F E 1 , ν 1 Rectangular contact area with semi - z area with semi -elliptical elliptical pressure F E 1 , ν 1 pressure distribution distribution z R 1 E 2 , ν 2 p max R g L E 2 , ν 2 R 1 2 b L p max x y 2 b Cylindrical groove Flat plate ( R 2 = - R g ) ( R 2 = ? ) x F F y ME EN 7960 – Precision Machine Design – Contact Stresses and Deformations 7-10 5

Cylinders in Contact – Principal Stresses The principal stresses σ 1 , σ 2 , and σ 3 are generated on the z-axis: ⎡ ⎤ 2 z z σ = σ = − ν + − ⎢ ⎥ 2 1 p 1 max x 2 ⎢ ⎥ b b ⎣ ⎦ ⎡ ⎤ ⎛ − ⎞ 1 ⎛ ⎞ 2 2 ⎜ z ⎟ z z ⎢ ⎜ ⎟ ⎥ σ = σ = − − + + − 2 1 1 2 p ⎜ ⎟ ⎜ ⎟ 2 max 2 2 y ⎢ ⎥ ⎝ ⎠ b b b ⎝ ⎠ ⎣ ⎦ − 1 ⎡ ⎤ 2 z σ = σ = − + ⎢ 1 ⎥ p 3 max 2 z ⎢ b ⎥ ⎣ ⎦ σ − σ σ − σ σ − σ τ = τ = τ = 2 3 , 1 3 , 1 2 1 2 3 2 2 2 ME EN 7960 – Precision Machine Design – Contact Stresses and Deformations 7-11 Cylinders in Contact – Vertical Stress Distribution along Centerline of Contact Area σ , τ 1 Von Mises σ y 0.8 Ratio of stress to p max Plot shows σ z material with 0.6 σ x Poisson’s ratio ν = 0.3 0.4 0.2 τ 1 0 z 0 0.5 b 1.5 b 2 b 2.5 b 3 b b Depth below contact area • The maximum shear and Von Mises stress are reached below the contact area • This causes pitting where little pieces of material break out of the surface ME EN 7960 – Precision Machine Design – Contact Stresses and Deformations 7-12 6

Sphere vs. Cylinder – Von Mises Stress Sphere vs. Cylinder - Von Mises Stress 2.5 . 10 9 Dia 10 mm sphere (steel) on flat plate (steel) Dia 10 mm sphere (steel) on flat plate (steel) Dia 10 mm x 0.5 mm cylinder on flat plate (steel) Dia 10 mm x 0.5 mm cylinder on flat plate (steel) 2 . 10 9 Von Mises Stress [Pa] 1.5 . 10 9 1 . 10 9 5 . 10 8 0 0 20 40 60 80 100 Contact Force [N] • The Von Mises stress does not increase linearly with the contact force • The point contact of a sphere creates significantly larger stresses than the line contact of a cylinder ME EN 7960 – Precision Machine Design – Contact Stresses and Deformations 7-13 Effects of Contact Stresses - Fatigue ME EN 7960 – Precision Machine Design – Contact Stresses and Deformations 7-14 7

Elastic Deformation of Curved Surfaces The displacement of the centers of two spheres is given by: 2 1 ⎡ ⎤ ⎡ ⎤ ⎛ ⎞ 3 ⎛ ⎞ 3 − ν 2 − ν 2 1 1 1 1 1 δ = ⎜ + ⎟ ⎜ + ⎟ 1 2 1 . 04 ⎢ ⎥ ⎢ ⎥ F ⎜ ⎟ ⎜ ⎟ s 2 ⎝ ⎠ ⎝ ⎠ ⎣ E E ⎦ ⎣ R R ⎦ 1 2 1 2 The displacement of the centers of two cylinders is given by: With ν 1 = ν 2 = ν , and E 1 = E 2 = E : ( ) − ν 2 ⎛ ⎞ 2 1 2 4 4 F R R δ = + + ⎜ ln 1 ln 2 ⎟ π c ⎝ 3 ⎠ LE b b Note that the center displacements are highly nonlinear functions of the load ME EN 7960 – Precision Machine Design – Contact Stresses and Deformations 7-15 Sphere vs. Cylinder – Center Displacement Sphere vs. Cylinder - Center Displacement 5 . 10 6 Dia 10 mm sphere (steel) on flat plate (steel) Dia 10 mm sphere (steel) on flat plate (steel) Dia 10 mm x 0.5 mm cylinder (steel) on flat plate (steel) Dia 10 mm x 0.5 mm cylinder (steel) on flat plate (steel) 4 . 10 6 Center Displacement [m] 3 . 10 6 2 . 10 6 1 . 10 6 0 0 20 40 60 80 100 Contact Force [N] • The point contact of a sphere creates significantly larger center displacements than the line contact of a cylinder ME EN 7960 – Precision Machine Design – Contact Stresses and Deformations 7-16 8

Sphere vs. Cylinder – Stiffness Sphere vs. Cylinder - Stiffness 4 . 10 7 Dia 10 mm sphere (steel) on flat plate (steel) Dia 10 mm sphere (steel) on flat plate (steel) Dia 10 mm x 0.5 mm cylinder (steel) on flat plate (steel) Dia 10 mm x 0.5 mm cylinder (steel) on flat plate (steel) 3 . 10 7 Stiffness [N/m] 2 . 10 7 1 . 10 7 0 0 20 40 60 80 100 Contact Force [N] • The point contact of a sphere creates significantly lower stiffness than the line contact of a cylinder ME EN 7960 – Precision Machine Design – Contact Stresses and Deformations 7-17 Effects of Material Combinations • The maximum contact pressure between two curved surfaces depends on: – Type of curvature (sphere vs. cylinder) – Radius of curvature – Magnitude of contact force – Elastic modulus and Poisson’s ratio of contact surfaces • Through careful material pairing, contact stresses may be lowered ME EN 7960 – Precision Machine Design – Contact Stresses and Deformations 7-18 9