It has been decided to reduce the maximum pressure of a cam-follower system to 90%
of its current value by changing the geometry of the contact (all other parameters remaining
the same). The contact currently consists of
two cylinders with parallel axes; one of the cylinders has a radius of 80 mm,
while the other has a radius of 25 mm. To what value would the radius of the smaller cylinder have to be
changed to achieve the desired pressure?

The goal is to reduce the maximum pressure of a cam-follower system to 90% of its current value by altering the geometry of the contact while keeping other parameters constant. This requires modifying the radius of the smaller cylinder, given that the current configuration involves two cylinders with parallel axes: one with a radius of 80 mm and the other with a radius of 25 mm.

### Step 1: Understanding the Contact Mechanics
The maximum pressure in a cam-follower system involving two cylindrical surfaces depends on their radii due to Hertzian contact theory. The contact pressure \( p \) is inversely proportional to the square root of the effective radius of curvature \( R \) of the system:

\[
p \propto \frac{1}{\sqrt{R}}
\]

The effective radius \( R \) for two cylinders with radii \( R_1 \) and \( R_2 \) (where \( R_1 \) and \( R_2 \) are positive if the cylinders are convex and negative if concave) is given by:

\[
\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2}
\]

### Step 2: Current Configuration
Given:
- \( R_1 = 80 \, \text{mm} \)
- \( R_2 = 25 \, \text{mm} \)

The effective radius of curvature is:

\[
\frac{1}{R} = \frac{1}{80} + \frac{1}{25}
\]

Calculating \( R \):

\[
\frac{1}{R} = \frac{1}{80} + \frac{1}{25} = \frac{25 + 80}{80 \times 25} = \frac{105}{2000}
\]

\[
R = \frac{2000}{105} \approx 19.05 \, \text{mm}
\]

### Step 3: New Pressure and New Effective Radius
The new maximum pressure \( p' \) needs to be 90% of the current pressure \( p \):

\[
p' = 0.9p
\]

Since \( p \propto \frac{1}{\sqrt{R}} \), this implies:

\[
\frac{1}{\sqrt{R'}} = 0.9 \times \frac{1}{\sqrt{R}}
\]

\[
\sqrt{R'} = \frac{\sqrt{R}}{0.9}
\]

Squaring both sides:

\[
R' = \frac{R}{0.81}
\]

Substituting the current effective radius:

\[
R' = \frac{19.05}{0.81} \approx 23.52 \, \text{mm}
\]

### Step 4: Solving for the New Radius of the Smaller Cylinder
Let the new radius of the smaller cylinder be \( R_2' \). The effective radius with the new configuration is:

\[
\frac{1}{R'} = \frac{1}{R_1} + \frac{1}{R_2'}
\]

Substituting \( R_1 = 80 \, \text{mm} \) and the calculated \( R' \):

\[
\frac{1}{23.52} = \frac{1}{80} + \frac{1}{R_2'}
\]

\[
\frac{1}{R_2'} = \frac{1}{23.52} - \frac{1}{80}
\]

Calculating \( \frac{1}{R_2'} \):

\[
\frac{1}{R_2'} = \frac{80 - 23.52}{80 \times 23.52} = \frac{56.48}{1881.6}
\]

\[
R_2' \approx \frac{1881.6}{56.48} \approx 33.31 \, \text{mm}
\]

### Final Answer
The radius of the smaller cylinder needs to be changed to approximately **33.31 mm** to achieve the desired reduction in maximum pressure.

Would you like more details on the steps, or do you have any questions?

### Related Questions:
1. How does changing the geometry of the cam-follower system affect other performance characteristics?
2. What other factors, besides geometry, can be adjusted to modify the maximum pressure in such a system?
3. How would the maximum pressure change if the larger cylinder’s radius were altered instead?
4. What are the practical limitations of altering cylinder radii in real-world applications?
5. How is the Hertzian contact theory applied in other mechanical systems involving contact pressure?

### Tip:
Always double-check units and assumptions when applying contact mechanics formulas, as incorrect parameters can significantly affect results.

It has been decided to reduce the maximum pressure of a cam-follower system to 90% of its current value by changing the geometry of the contact (all other parameters remaining the same). The contact currently consists of two cylinders with parallel axes; one of the cylinders has a radius of 80 mm, while the other has a radius of 25 mm. To what value would the radius of the smaller cylinder have to be changed to achieve the desired pressure?

This page explains how to reduce the maximum pressure of a cam-follower system by 10% through modifying the radius of one cylinder. Using Hertzian contact theory and contact mechanics, the new radius of the smaller cylinder is calculated to be approximately 33.31 mm, ensuring a reduction in maximum pressure. Suitable for advanced studies in mechanical engineering.

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