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The image you've provided is a PTC Mathcad worksheet titled "Involute Gear Design" by Dave Martin. It demonstrates how to calculate the geometry of an involute gear, which is commonly used in mechanical engineering for power transmission. Below is a detailed explanation and solution based on the content of the two pages.

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🧩 Objective

To design an involute gear using fundamental parameters like module, pressure angle, number of teeth, and profile shift coefficient. The goal is to compute key dimensions and generate the involute curve that defines the tooth profile.

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## 🔍 Page 1: Input Parameters & Basic Calculations

✔ Given Inputs:

- Module (m) = 4 mm
→ This determines the size of the gear teeth.
- Pressure Angle (α) = 20°
→ Standard pressure angle for most gears; affects tooth shape and strength.
- Number of Teeth (z) = 30
→ Total number of teeth on the gear.
- Profile Shift Coefficient (x) = 0
→ No profile shift (standard gear).

---

📐 Key Geometric Definitions:

#### 1. Pitch (P):
\[
P = \pi \cdot \text{module} = \pi \cdot 4 = 12.566\,\text{mm}
\]

> This is the distance between corresponding points on adjacent teeth along the pitch circle.

#### 2. Addendum (a):
\[
\text{addendum} = \text{module} = 4\,\text{mm}
\]
→ Height of the tooth above the pitch circle.

#### 3. Dedendum (d):
\[
\text{dedendum} = 1.25 \cdot \text{module} = 1.25 \cdot 4 = 5\,\text{mm}
\]
→ Depth of the tooth below the pitch circle.

#### 4. Total Tooth Height (h):
\[
h = \text{addendum} + \text{dedendum} = 4 + 5 = 9\,\text{mm}
\]

#### 5. Base Diameter (φ_b):
\[
\phi_b = \phi \cdot \cos(\alpha)
\]
where:
- \( \phi = \text{module} \cdot \text{teeth} = 4 \cdot 30 = 120\,\text{mm} \) → Pitch Diameter

So,
\[
\phi_b = 120 \cdot \cos(20^\circ) = 120 \cdot 0.9397 \approx 112.764\,\text{mm}
\]

But wait — note in the worksheet, it shows:
\[
\phi_b = \frac{\phi_b}{2} = 56.382\,\text{mm} \quad \Rightarrow \quad \text{Base Radius} = r_b = 56.382\,\text{mm}
\]
Thus,
\[
r_b = \frac{120}{2} \cdot \cos(20^\circ) = 60 \cdot 0.9397 = 56.382\,\text{mm}
\]
✔️ Correct.

#### 6. Tip Diameter (φ_t):
\[
\phi_t = \phi + 2 \cdot \text{module} \cdot (1 + x)
= 120 + 2 \cdot 4 \cdot (1 + 0) = 128\,\text{mm}
\]
So,
\[
r_t = \frac{128}{2} = 64\,\text{mm} \quad \text{(Tip Radius)}
\]

---

📊 Summary of Key Radii:

| Parameter | Value |
|---------|-------|
| Reference Radius (r) | 60.000 mm |
| Base Radius (r_b) | 56.382 mm |
| Tip Radius (r_t) | 64.000 mm |

These are derived from:
- Reference radius: \( r = \frac{\phi}{2} = \frac{120}{2} = 60\,\text{mm} \)
- Base radius: \( r_b = r \cdot \cos(\alpha) = 60 \cdot \cos(20^\circ) = 56.382\,\text{mm} \)
- Tip radius: \( r_t = r + \text{addendum} = 60 + 4 = 64\,\text{mm} \)

---

## 🔁 Page 2: Involute Curve Generation

The involute curve is generated mathematically as the path traced by a point on a string unwound from a base circle.

🌀 Involute Angle (in radians):

\[
\text{inv}_\alpha = \tan(\alpha) - \alpha
\]
With \( \alpha = 20^\circ = 0.3491\,\text{rad} \), then:
\[
\tan(20^\circ) \approx 0.3640
\]
\[
\text{inv}_\alpha = 0.3640 - 0.3491 = 0.0149 \approx 0.015\,\text{rad}
\]
Converted to degrees:
\[
\text{inv}_\alpha = 0.015 \times \frac{180}{\pi} \approx 0.854^\circ
\]
✔️ Matches the value shown.

---

📈 Coordinate Functions of the Involute Curve

For any radius \( r \) between \( r_b \) and \( r_t \), the coordinates of the involute curve are given by:

\[
\begin{aligned}
X(r) &= r \cdot \cos\left( \tan^{-1}\left( \frac{r}{r_b} \right) - \cos^{-1}\left( \frac{r_b}{r} \right) \right) \\
Y(r) &= r \cdot \sin\left( \tan^{-1}\left( \frac{r}{r_b} \right) - \cos^{-1}\left( \frac{r_b}{r} \right) \right)
\end{aligned}
\]

Wait — actually, the correct form is:

Let \( \theta = \tan^{-1}\left( \frac{r}{r_b} \right) - \cos^{-1}\left( \frac{r_b}{r} \right) \)? ✘ That's not standard.

Actually, the correct parametric equations for the involute curve are:

Let \( \theta \) be the angle of rotation of the generating line, then:

\[
\begin{aligned}
x(r) &= r \cdot \cos\left( \theta \right) \\
y(r) &= r \cdot \sin\left( \theta \right)
\end{aligned}
\]
Where:
\[
\theta = \tan(\theta_r) - \theta_r, \quad \theta_r = \cos^{-1}(r_b / r)
\]

But more accurately, the involute function is defined as:

\[
\text{inv}(\theta) = \tan(\theta) - \theta
\]

And the angle of the involute at radius \( r \) is:
\[
\theta_{\text{inv}} = \tan^{-1}\left( \frac{r}{r_b} \right) - \cos^{-1}\left( \frac{r_b}{r} \right)
\]

But this isn't quite right either.

✔ Correct Formula (Standard Involute Parametric Equations):

Let \( r \) be the current radius, \( r_b \) the base radius.

Then the angle of the involute at radius \( r \) is:
\[
\theta = \sqrt{ \left( \frac{r}{r_b} \right)^2 - 1 } - \cos^{-1}\left( \frac{r_b}{r} \right)
\]

But again, no — better to use:

Define:
\[
\theta = \tan(\theta_b) - \theta_b, \quad \text{but } \theta_b = \cos^{-1}(r_b / r)
\]

No — here’s the correct derivation:

Let \( \theta \) be the angle subtended by the unwound string at the center. Then:
- The length of the string is \( s = r_b \cdot \theta \)
- The tangent to the base circle has length \( s \), so the angle of the radial vector is:
\[
\phi = \theta - \frac{\pi}{2} + \text{something}
\]

Actually, standard formula:

Let \( \theta \) be the parameter (angle in radians from base circle). Then:

\[
\begin{aligned}
x &= r_b \cdot (\cos(\theta) + \theta \cdot \sin(\theta)) \\
y &= r_b \cdot (\sin(\theta) - \theta \cdot \cos(\theta))
\end{aligned}
\]

But in this worksheet, they're expressing the curve in terms of radius, not parameter θ.

So let's analyze the given functions:

```math
Curve_X(radius) := radius \cdot \cos\left( \tan\left( \acos\left( \frac{r_b}{radius} \right) \right) - \acos\left( \frac{r_b}{radius} \right) \right)
```

This is unusual.

Let me simplify:

Let:
\[
\theta = \acos\left( \frac{r_b}{r} \right)
\]
Then:
\[
\tan(\theta) = \frac{\sqrt{r^2 - r_b^2}}{r_b}
\]

So:
\[
\tan(\theta) - \theta = \text{involute function}
\]

But the expression inside is:
\[
\tan(\theta) - \theta
\]

So the total angle of the involute curve at radius \( r \) is:
\[
\phi = \tan(\theta) - \theta
\]

Wait — but the argument of cos/sin is:
\[
\tan(\theta) - \theta
\]

That means:
\[
\text{Angle of the involute point} = \tan(\theta) - \theta
\]

But that’s just the involute function, not the actual angular position.

Wait — actually, the angular displacement of the involute from the base circle is:
\[
\psi = \tan(\theta) - \theta
\]
where \( \theta = \cos^{-1}(r_b / r) \)

Then the total polar angle of the point on the involute is:
\[
\phi = \psi = \tan(\theta) - \theta
\]

But that doesn’t make sense dimensionally.

Let’s go back.

✔ Correct Parametric Form of Involute Curve:

Let \( \theta \) be the angle of rotation of the generating line (in radians). Then:

\[
\begin{aligned}
x &= r_b \cdot (\cos(\theta) + \theta \cdot \sin(\theta)) \\
y &= r_b \cdot (\sin(\theta) - \theta \cdot \cos(\theta))
\end{aligned}
\]

But in this worksheet, they are defining the curve as a function of radius.

So instead, we can express:

Let \( r \) be the current radius, then:
\[
\theta = \sqrt{ \left( \frac{r}{r_b} \right)^2 - 1 }
\]
Because the arc length is \( r_b \cdot \theta \), and the straight-line length is \( r \), so:
\[
r = \sqrt{ r_b^2 + (r_b \theta)^2 } = r_b \cdot \sqrt{1 + \theta^2}
\Rightarrow \theta = \sqrt{ \left( \frac{r}{r_b} \right)^2 - 1 }
\]

Then the polar angle of the point is:
\[
\phi = \theta - \arctan(\theta)
\]

Wait — no.

Actually, the angle between the radial line and the tangent is \( \theta \), and the total angle swept is \( \theta \), so the polar angle of the point is:

\[
\phi = \theta - \arcsin\left( \frac{r_b}{r} \right)
\]

I think the formulas in the worksheet are incorrectly written or misinterpreted.

But looking closely:

They define:
\[
\text{Curve\_X}(r) = r \cdot \cos\left( \tan\left( \acos\left( \frac{r_b}{r} \right) \right) - \acos\left( \frac{r_b}{r} \right) \right)
\]

Let’s denote:
\[
\theta = \acos\left( \frac{r_b}{r} \right)
\]
Then:
\[
\tan(\theta) = \frac{\sqrt{r^2 - r_b^2}}{r_b}
\]

So:
\[
\tan(\theta) - \theta = \text{involute function}
\]

But this is not the polar angle of the point.

In fact, the correct polar angle of the involute point is:
\[
\phi = \theta + \tan(\theta) - \theta = \tan(\theta)
\]

No — let’s look at a known source.

✔ Standard Formula:

Let \( \theta \) be the angle of rotation of the string. Then:
- The point on the involute has:
\[
x = r_b (\cos(\theta) + \theta \sin(\theta)) \\
y = r_b (\sin(\theta) - \theta \cos(\theta))
\]

But to express in terms of radius, solve for \( \theta \) such that:
\[
r = \sqrt{x^2 + y^2} = r_b \sqrt{1 + \theta^2}
\Rightarrow \theta = \sqrt{ \left( \frac{r}{r_b} \right)^2 - 1 }
\]

Then:
\[
\phi = \theta \quad \text{(the polar angle)}
\]

Wait — no, the polar angle is:
\[
\phi = \theta - \arctan(\theta)
\]

This is getting messy.

Let’s accept that the worksheet uses a simplified model where:

\[
\text{Angle} = \tan(\theta) - \theta
\]

But that’s not the angle of the point — it's the involute function.

So likely, the graph shows the involute angle vs. radius, not the actual tooth profile.

Looking at the graph:
- X-axis: Radius (mm) from 56.382 to 64.000
- Y-axis: Angle (degrees)
- Curve starts at 0° at base radius, increases to ~1.5° at tip

This matches the involute function:
\[
\text{inv}(\alpha) = \tan(\alpha) - \alpha
\]

At \( \alpha = 20^\circ \), inv(α) ≈ 0.854°

But the graph shows up to 1.5° — why?

Ah! Because the maximum involute angle at the tip is:
\[
\theta = \tan^{-1}\left( \frac{r_t}{r_b} \right) - \cos^{-1}\left( \frac{r_b}{r_t} \right)
\]

Wait — no.

Let’s compute:

Let \( \theta = \cos^{-1}(r_b / r_t) = \cos^{-1}(56.382 / 64) = \cos^{-1}(0.8841) \approx 27.7^\circ \)

Then:
\[
\tan(\theta) = \tan(27.7^\circ) \approx 0.523
\]
\[
\text{inv}(\theta) = 0.523 - 0.483 \approx 0.04 \text{ rad} = 2.3^\circ
\]

But the graph shows only up to ~1.5°.

Wait — the Y-axis says "Angle (Degrees)", and the curve goes from 0 to ~1.5°.

But earlier we had:
\[
\text{inv}_\alpha = 0.854^\circ
\]

So perhaps the graph is showing the variation of the involute function with radius, but it's not clear.

Alternatively, the coordinate functions might be incorrect.

Let’s re-express correctly.

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## ✔ Correct Approach: Generate Involute Profile

To generate the involute curve:

1. Define parameter \( \theta \) from 0 to \( \theta_{\max} \), where:
\[
\theta_{\max} = \sqrt{ \left( \frac{r_t}{r_b} \right)^2 - 1 } = \sqrt{ \left( \frac{64}{56.382} \right)^2 - 1 } = \sqrt{1.25^2 - 1} = \sqrt{1.5625 - 1} = \sqrt{0.5625} = 0.75 \text{ rad}
\]

2. For each \( \theta \), compute:
\[
x = r_b \cdot (\cos(\theta) + \theta \cdot \sin(\theta)) \\
y = r_b \cdot (\sin(\theta) - \theta \cdot \cos(\theta))
\]

3. This gives the parametric involute curve.

Then, the tooth profile is bounded by:
- From base circle to tip circle
- With addendum and dedendum

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## 🎯 Conclusion

The PTC Mathcad worksheet provides:
- A method to compute gear dimensions (pitch diameter, base diameter, etc.)
- An attempt to generate the involute curve using a radius-based formulation

However, the formulas for Curve_X and Curve_Y appear to be incorrect or mislabeled.

🔧 Recommended Fix:

Use parametric equations:
\[
\begin{aligned}
x(\theta) &= r_b (\cos(\theta) + \theta \sin(\theta)) \\
y(\theta) &= r_b (\sin(\theta) - \theta \cos(\theta))
\end{aligned}
\]
for \( \theta \in [0, \theta_{\max}] \), where \( \theta_{\max} = \sqrt{(r_t/r_b)^2 - 1} \)

Then plot \( x(\theta) \) vs \( y(\theta) \) to get the true involute curve.

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## ✔ Final Answer

The worksheet computes:
- Base radius: 56.382 mm
- Tip radius: 64.000 mm
- Involute angle at 20° pressure angle: 0.854°
- Involute curve via radius-based function (likely flawed)

To improve accuracy, use parametric involute equations with angle parameter \( \theta \).

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If you want, I can generate the correct Mathcad-style code or plot for the involute curve. Let me know!