Parametric Rack and Pinion

 The sketch construction began with establishing the base circle, reference circle, and addendum circle as fundamental geometric references. Construction lines were then created to define the pressure angle and establish the involute curve starting point. The actual tooth profile was constructed using a combination of arcs and splines, with each element parametrically constrained to maintain proper relationships as the input parameters changed. 

   The parametric framework needed to handle the interdependencies between dozens of calculated dimensions. For instance, the reference diameter is calculated as d = m × z, while the base diameter requires the formula db = d × cos(α). The addendum height follows ha = 1.0 × m, and the dedendum incorporates clearance with hf = 1.25 × m. Each of these relationships had to be properly encoded in SolidWorks' global variables system, ensuring that changes to any input parameter would cascade through all dependent calculations without breaking the geometric relationships. 

 The tooth profile construction required careful attention to the transition regions between the involute working surface and the root fillet. The root fillet radius was calculated as R = 1.5 × clearance, providing a smooth transition that reduces stress concentrations while maintaining adequate clearance for the mating gear's tooth tips. This fillet had to be properly integrated into the parametric system to ensure it scaled appropriately with changes in module size. 

This same profile can then be matched to the rack part

  The circular pattern implementation needed to be fully parametric, automatically adjusting the number of pattern instances based on the input number of teeth. This required linking the pattern count directly to the "z" variable, ensuring that changes to the number of teeth would automatically update the pattern without manual intervention. The pattern also needed to maintain proper angular relationships and ensure that the teeth were evenly distributed around the gear circumference. 

 A critical aspect of the parametric system was ensuring that the generated gear and rack components would properly mesh together. This required implementing geometric relationships that guaranteed correct center distances and tooth engagement characteristics. The center distance for a rack and pinion system is determined by the pinion's reference radius, as the rack's reference line is tangent to the pinion's reference circle. 

The Final Assembly then used global variables for the module, pressure angle, number of teeth, and desired rack displacement integrated into the local variables for the rack, the pinion, and the temporary housing mechanism. 

 The implementation of the parametric system required careful organization of the variable hierarchy within SolidWorks. The excel document served as the foundation that SolidWorks calculations could be checked against. Once the calculations were confirmed, I could move away from using this.

The reference diameter calculation (d = m × z) formed the basis for most other dimensional calculations. From this, the base diameter could be calculated using db = d × cos(α), which is crucial for proper involute curve generation. The tip diameter calculation (da = d + 2m) and root diameter calculation (df = d - 2.5m) established the outer and inner boundaries of the gear teeth, while the tooth depth (h = 2.25m) and addendum (ha = m) defined the vertical tooth proportions.

Additional calculated parameters included the circular pitch (p = π × d / z), which determines the spacing between adjacent teeth, and the tooth thickness (t = p/2), which defines the width of each tooth at the reference circle. The clearance value (c = 0.25m) was incorporated to prevent interference between the tooth tips of one gear and the root areas of the mating gear.

The parametric system proved capable of generating a wide range of gear configurations while maintaining geometric accuracy and meshing compatibility. Testing with various input combinations demonstrated the system's robustness across different module sizes, tooth counts, and pressure angles. For example, changing the module from 2mm to 5mm automatically scaled all related dimensions proportionally, maintaining proper tooth proportions and clearances throughout the size range.

The number of teeth parameter showed excellent scalability, with successful generation of gears ranging from 8 teeth (minimum practical for avoiding undercutting) to over 100 teeth for large, fine-pitch applications. Each configuration maintained proper involute geometry and appropriate dimensional relationships, demonstrating the mathematical rigor of the underlying parametric framework.

Pressure angle variations were successfully accommodated, with the system properly adjusting tooth profiles for 14.5°, 20°, and 25° pressure angles. Each pressure angle produced distinctly different tooth shapes while maintaining proper meshing characteristics and load distribution properties. This flexibility allows the system to be adapted for different application requirements, from high-speed precision mechanisms to heavy-duty power transmission systems.