Instance Models

This section transforms the problem defined in the problem description into one which is expressed in mathematical terms. It uses concrete symbols defined in the data definitions to replace the abstract symbols in the models identified in theoretical models and general definitions.

RefnameIM:calOfAngularDisplacement
LabelCalculation of angular displacement
Input\({L_{\text{rod}}}\), \({θ_{i}}\), \(\boldsymbol{g}\)
Output\({θ_{p}}\)
Input Constraints\[{L_{\text{rod}}}\gt{}0\]\[{θ_{i}}\gt{}0\]\[\boldsymbol{g}\gt{}0\]
Output Constraints\[{θ_{p}}\gt{}0\]
Equation\[{θ_{p}}\left(t\right)={θ_{i}}\,\cos\left(Ω\,t\right)\]
Description
  • \({θ_{p}}\) is the displacement angle of the pendulum (\({\text{rad}}\))
  • \(t\) is the time (\({\text{s}}\))
  • \({θ_{i}}\) is the initial pendulum angle (\({\text{rad}}\))
  • \(Ω\) is the angular frequency (\({\text{s}}\))
Notes
  • The constraint \({θ_{i}}\gt{}0\) is required. The angular frequency is defined in GD:angFrequencyGD.
Source–
RefByFR:Output-Values and FR:Calculate-Angular-Position-Of-Mass

Detailed derivation of angular displacement:

When the pendulum is displaced to an initial angle and released, the pendulum swings back and forth with periodic motion. By applying Newton’s second law for rotational motion, the equation of motion for the pendulum may be obtained:

\[\boldsymbol{τ}=\boldsymbol{I}\,α\]

Where \(\boldsymbol{τ}\) denotes the torque, \(\boldsymbol{I}\) denotes the moment of inertia and \(α\) denotes the angular acceleration. This implies:

\[-m\,\boldsymbol{g}\,\sin\left({θ_{p}}\right)\,{L_{\text{rod}}}=m\,{L_{\text{rod}}}^{2}\,\frac{\,d\frac{\,d{θ_{p}}}{\,dt}}{\,dt}\]

And rearranged as:

\[\frac{\,d\frac{\,d{θ_{p}}}{\,dt}}{\,dt}+\frac{\boldsymbol{g}}{{L_{\text{rod}}}}\,\sin\left({θ_{p}}\right)=0\]

If the amplitude of angular displacement is small enough, we can approximate \(\sin\left({θ_{p}}\right)={θ_{p}}\) for the purpose of a simple pendulum at very small angles. Then the equation of motion reduces to the equation of simple harmonic motion:

\[\frac{\,d\frac{\,d{θ_{p}}}{\,dt}}{\,dt}+\frac{\boldsymbol{g}}{{L_{\text{rod}}}}\,{θ_{p}}=0\]

Thus the simple harmonic motion is:

\[{θ_{p}}\left(t\right)={θ_{i}}\,\cos\left(Ω\,t\right)\]