Background

My MAE 3 project during COVID lockdown turned an interesting physics problem into a practical design challenge. The goal was to design a clock that reliably ticks through pendulum motion. The interesting part was that we would use two different physics models to predict its timing.

CAD

First, I CADed my own ticking pendulum clock, giving the pendulum the shape of a clawed robot. I then added bolts to add weight.

The Problem

A pendulum clock needs to balance several factors:

1. Tick rate determined by pendulum length and mass distribution

2. Enough energy in each swing to overcome friction

3. Reliable transfer of motion through the escapement mechanism

Rather than just pick dimensions arbitrarily, we approached this by modeling the system two ways and comparing the results.

Model 1: Point Mass Simplification

The simpler approach treats the pendulum as a point mass at its center of mass. The governing equation is:

ω_rad = √(g/L_com)

Where:

• ω_rad is natural frequency in radians

• g is gravitational acceleration

• L_com is length to center of mass

Converting to actual timing:

1. ω_hz = ω_rad/2π

2. T = 1/ω_hz

3. Timing = T × N_teeth

Using dimensions from my Fusion 360 model, this predicted a 7.704 second cycle.

Model 2: Full Rigid Body Analysis

The complete model accounts for mass distribution. The governing equation becomes:

ω_rad = √(M_t × g × L_com / I_total)

Where:

• M_t is total mass

• I_total combines all moments of inertia

Computing I_total required:

1. Acrylic body inertia from CAD (I_ac)

2. Bolt inertias: I_bn = M_b × L_bn²

3. Total: I_total = I_ac + ΣI_bn

This predicted a 7.985 second cycle - a 3.647% longer period than the point mass model.

Results

The two models giving similar results (within ~4%) suggests either would work well for basic design. However, the rigid body analysis revealed how mass distribution affects timing which could be useful for fine-tuning the final design.Unfortunately, COVID meant I couldn't build and test it.

Overall the project taught me about tradeoffs in modeling. Sometimes the simpler model is good enough and other times you need detailed analysis. It's important to understand the fundamentals of the problem you're solving to choose the best approach.