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Point Mass Analysis
Point Mass Analysis
A point-mass analysis uses measurements taken from Fusion 360 along with a calculated center of mass (COM), but it simplifies the pendulum by assuming it has no rotational inertia. While not highly precise, this method provides a quick and useful estimate that can help verify more detailed calculations.
Definition of point mass oscillation time T in seconds (s), where:
r = distance between the rotation point and location of Center of Gravity (CoG) of the point mass representation of the pendulum clock.
g = gravitational acceleration.
It can be noted that:
Only the distance r is needed for this calculation
r and g must use the same length units to ensure T has the units of seconds
if g = 9.81 m/s^2, then r must be expressed in meters (m) too
Assumptions:
1. Friction can be neglected.
2. The maximum angle of motion a, is relatively small.
3. The mass of the pendulum is concentrated at one point.
4. The pendulum swings freely. (we do not consider the effect of the escapement wheel)
Calculated Center of Mass (without bolts) = 14.478 cm = 0.14478 m
Total mass of pendulum = 98.871 g
Effective Length of Center of Mass = Lcom = 0.09 m
Natural Frequency = 10.40 radians/sec = 1.66 Hz
Predicted Period of Oscillation: 0.60 seconds/cycle
Inertial Analysis
Inertial Analysis
An intertial analysis also relies on measurements from CAD, but unlike the point-mass method, it accounts for the full shape of the pendulum and its rotational inertia. This approach is more detailed and involved, but it typically leads to a more accurate prediction of the pendulum’s motion.
Definition of inertial oscillation time T in seconds (s), where:
I_tot is the total inertia of the pendulum clock around the rotation point, including the pendulum body and the possible n screws/bolts included in your pendulum.
The value of I_tot is computed as the sum of two main components:
The inertial effect of the pendulum body as the pendulum body mass m times the squared distance r between the rotation point and location of Center of Gravity (CoG) of the pendulum body
The inertial effect of the possible n screws/bolts as the screw/bolt mass m_k times the squared distance r_k between the rotation point and location of Center of Gravity (CoG) of each screw/bolt
m_r is the distance weighted mass of the pendulum clock around the rotation point, including the pendulum body and the possible n screws/bolts included in your pendulum.
The value of m_r is also computed as the sum of two main components:
The distance weighted mass of the pendulum body as the pendulum body mass m times the distance r between the rotation point and location of Center of Gravity (CoG) of the pendulum body.
The distance weighted mass of the possible n screws/bolts as the screw/bolt mass m_k times the distance r_k between the rotation point and location of Center of Gravity (CoG) of each screw/bolt
Assumptions:
1. Friction can be neglected.
2. The maximum angle of motion a, is relatively small.
3. The mass of the pendulum is concentrated at one point.
4. The pendulum swings freely. (we do not consider the effect of the escapement wheel)
Lcom = 0.0905115408 m = 9.05115408 cm
Rotational Inertia of the Acrylic = I_a = 6011.289 g/cm²
Rotational Inertia of the Bolts (total from 8 bolts) =
487.261476 + 487.261476 + 493.995076 + 683.927104 + 683.927104 + 816.587776 + 848.5569 + 848.5569 = 5350.073812 g/cm²
Total Rotational Inertia = 6011.289 + 5350.073812 = 11361.362812 g/cm² = 1.136136281 g·m²
Natural Frequency = 9.328659662 radians/sec = 1.48470229763 Hz
Predicted Period of Oscillation: 1 / f = ~0.66 seconds/cycle)
Two-dimensional Simulation Analysis (WM2D)
Two-dimensional Simulation Analysis (WM2D)
The model designed in Autodesk Fusion and AutoCAD was imported into Working Model 2D in order to simulate what the oscillation of the pendulum would look like. Working Model 2D (WM2D) is a two-dimensional physics simulation program that allows for dynamic modeling of rigid bodies using an intuitive, visual interface. It can simulate motion, forces, torques, collisions, gears, springs, and dampers between multiple objects in a system. In this project, WM2D was used to replicate the real-time swinging motion of my pendulum based on the physical properties defined in the CAD model. By simulating how the pendulum would behave before physically building it, I was able to test the mechanical design, analyze its dynamic behavior, and evaluate the accuracy of my theoretical timing predictions.
Comparison
AlexCheng-2ndwm2dPendulum_video.aviIMG_2322.MOVPage updated
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