To derive the expression for the pressure distribution when an axial load presses a solid shaft into a flexible elastic surface, we need to consider the nature of the contact pressure and how it varies across the surface. Given that the pressure decreases parabolically from a maximum value \( p_0 \) at the center to zero at the edge, we can model this scenario mathematically.
Understanding the Problem
We have a solid shaft with a radius \( R \) that is pressed into a flexible surface. The contact pressure \( p \) at any radial distance \( r \) from the center of the shaft can be expressed as a function of \( r \). The problem states that the pressure decreases parabolically, which suggests a quadratic relationship.
Formulating the Pressure Distribution
To express the pressure distribution mathematically, we can assume a parabolic form for the pressure as follows:
\( p(r) = p_0 \left(1 - \frac{r^2}{R^2}\right) \)
In this equation:
- \( p(r) \) is the pressure at a distance \( r \) from the center.
- \( p_0 \) is the maximum pressure at the center of the shaft.
- \( R \) is the radius of the shaft.
- The term \( \frac{r^2}{R^2} \) represents the parabolic decrease in pressure as \( r \) approaches \( R \).
Deriving the Expression
To derive this expression, we start by recognizing that at the center of the shaft (where \( r = 0 \)), the pressure is maximum, \( p(0) = p_0 \). As we move outward to the edge of the shaft (where \( r = R \)), the pressure must drop to zero, \( p(R) = 0 \). This behavior is characteristic of a parabolic function.
To ensure that our function satisfies these boundary conditions, we can analyze it step by step:
- At \( r = 0 \):
- Substituting \( r = 0 \) into the equation gives \( p(0) = p_0 \), which is correct.
- At \( r = R \):
- Substituting \( r = R \) gives \( p(R) = p_0 \left(1 - \frac{R^2}{R^2}\right) = p_0(1 - 1) = 0 \), confirming our boundary condition.
Visualizing the Pressure Distribution
To visualize this distribution, imagine a graph where the x-axis represents the radial distance \( r \) from the center of the shaft, and the y-axis represents the pressure \( p(r) \). The curve would start at \( p_0 \) when \( r = 0 \) and smoothly curve down to zero at \( r = R \), forming a parabolic shape.
Conclusion
In summary, the expression for the pressure distribution when an axial load presses a solid shaft into a flexible elastic surface, resulting in a parabolic decrease in pressure from the center to the edge, is given by:
\( p(r) = p_0 \left(1 - \frac{r^2}{R^2}\right) \)
This equation effectively captures the behavior of the pressure distribution across the contact area, providing a clear understanding of how the pressure varies with distance from the center of the shaft.
