Following calculator has been developed to calculate contact stress of sphere
(ball) in a circular race. This type of situation is generally seen at the
contact region of ball bearings. The schematic representation of the contact is
given in the figure.
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RESULTS |
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Parameter |
Obj.-1 |
Obj.-2 |
Unit |
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Maximum Hertzian contact pressure [pmax] |
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Maximum shear stress [τmax] |
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Rigid distance of approach of contacting bodies [d] |
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Semimajor axis of contact ellipse [a] |
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Semiminor axis of contact ellipse [b] |
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Note: Use dot "." as decimal separator.
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$$a=1.145\cdot { n }_{ a }\cdot { (F\cdot K\cdot \gamma ) }^{ 1/3 }$$ |
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$$b=1.145\cdot { n }_{ b }\cdot { (F\cdot K\cdot \gamma ) }^{ 1/3 }$$ |
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$${ P }_{ max }=0.365\cdot { n }_{ c }\cdot { [F/({ K }^{ 2 }{ \cdot \gamma }^{
2 })] }^{ 1/3 }$$ |
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$${ \tau }_{ max }=\quad {\sigma }_{ c }(0.3906{ k }^{ 5 }-1.1198{ k }^{ 4
}+1.2448{ k }^{ 3 }-0.7177{ k }^{ 2 }+0.2121k+0.3)$$ |
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$$d=0.655\cdot { n }_{ d }\cdot { ({ F }^{ 2 }\cdot { \gamma }^{ 2 }/K) }^{ 1/3
}$$ |
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$$A\quad =\quad \frac { 1 }{ 2 } \cdot (\frac { 1 }{ { R }_{ 1 } } -\frac { 1 }{
{ R }_{ 2 } } )$$ |
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$$B\quad =\quad \frac { 1 }{ 2 } \cdot (\frac { 1 }{ { R }_{ 1 } } +\frac { 1 }{
{ R }_{ 3 } } )$$ |
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$$\gamma =\frac { (1-{ \upsilon }_{ 1 }^{ 2 }) }{ { E }_{ 1 } } +\frac { (1-{
\upsilon }_{ 2 }^{ 2 }) }{ { E }_{ 2 } } $$ |
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$$K=\frac { 1 }{ \frac { 2 }{ { R }_{ 1 } } -\frac { 1 }{ { R }_{ 2 } } +\frac {
1 }{ { R }_{ 3 } } } $$ |
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$$E(e)=\int _{ 0 }^{ \pi /2 }{ \sqrt { 1-{ e }^{ 2 }\cdot \sin ^{ 2 }{ (\varphi
) } } d\varphi } $$ |
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$$K(e)=\int _{ 0 }^{ \pi /2 }{ \frac { d\varphi }{ \sqrt { 1-{ e }^{ 2 }\cdot
\sin ^{ 2 }{ (\varphi ) } } } } $$ |
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$$e=\sqrt { 1-{ (b/a) }^{
2 }} $$ |
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$$k=\frac { b }{ a } $$ |
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$$\frac { A }{ B } =\frac { K(e)-E(e) }{ (1/{ k }^{ 2 })\cdot E(e)-K(e) } $$ |
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$${ n }_{ a }=\frac { 1 }{ k } \cdot { (\frac { 2\cdot k\cdot E(e) }{ \pi } )
}^{ 1/3 }$$ |
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$${ n }_{ b }={ (\frac { 2\cdot k\cdot E(e) }{ \pi } ) }^{ 1/3 }$$ |
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$${ n }_{ c }=\frac { 1 }{ E(e) } \cdot { (\frac { { \pi }^{ 2 }\cdot k\cdot
E(e) }{ 4 } ) }^{ 1/3 }$$ |