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:calOfAngle1
LabelCalculation of angle of first rod
Input\({L_{1}}\), \({L_{2}}\), \({m_{1}}\), \({m_{2}}\), \({θ_{1}}\), \({θ_{2}}\)
Output\({θ_{1}}\)
Input Constraints\[{L_{1}}\gt{}0\]\[{L_{2}}\gt{}0\]\[{m_{1}}\gt{}0\]\[{m_{2}}\gt{}0\]
Output Constraints
Equation\[{α_{1}}\left({θ_{1}},{θ_{2}},{w_{1}},{w_{2}}\right)=\frac{-g\,\left(2\,{m_{1}}+{m_{2}}\right)\,\sin\left({θ_{1}}\right)-{m_{2}}\,g\,\sin\left({θ_{1}}-2\,{θ_{2}}\right)-2\,\sin\left({θ_{1}}-{θ_{2}}\right)\,{m_{2}}\,\left({w_{2}}^{2}\,{L_{2}}+{w_{1}}^{2}\,{L_{1}}\,\cos\left({θ_{1}}-{θ_{2}}\right)\right)}{{L_{1}}\,\left(2\,{m_{1}}+{m_{2}}-{m_{2}}\,\cos\left(2\,{θ_{1}}-2\,{θ_{2}}\right)\right)}\]
Description
  • \({α_{1}}\) is the angular acceleration of the first object (\(\frac{\text{rad}}{\text{s}^{2}}\))
  • \({θ_{1}}\) is the angle of the first rod (\({\text{rad}}\))
  • \({θ_{2}}\) is the angle of the second rod (\({\text{rad}}\))
  • \({w_{1}}\) is the angular velocity of the first object (\(\frac{\text{rad}}{\text{s}}\))
  • \({w_{2}}\) is the angular velocity of the second object (\(\frac{\text{rad}}{\text{s}}\))
  • \(g\) is the magnitude of gravitational acceleration (\(\frac{\text{m}}{\text{s}^{2}}\))
  • \({m_{1}}\) is the mass of the first object (\({\text{kg}}\))
  • \({m_{2}}\) is the mass of the second object (\({\text{kg}}\))
  • \({L_{2}}\) is the length of the second rod (\({\text{m}}\))
  • \({L_{1}}\) is the length of the first rod (\({\text{m}}\))
Notes
  • \({θ_{1}}\) is calculated by solving the ODE here together with the initial conditions and IM:calOfAngle2.
Source–
RefByFR:Output-Values, FR:Calculate-Angle-Of-Rod, and IM:calOfAngle2
RefnameIM:calOfAngle2
LabelCalculation of angle of second rod
Input\({L_{1}}\), \({L_{2}}\), \({m_{1}}\), \({m_{2}}\), \({θ_{1}}\), \({θ_{2}}\)
Output\({θ_{2}}\)
Input Constraints\[{L_{1}}\gt{}0\]\[{L_{2}}\gt{}0\]\[{m_{1}}\gt{}0\]\[{m_{2}}\gt{}0\]
Output Constraints
Equation\[{α_{2}}\left({θ_{1}},{θ_{2}},{w_{1}},{w_{2}}\right)=\frac{2\,\sin\left({θ_{1}}-{θ_{2}}\right)\,\left({w_{1}}^{2}\,{L_{1}}\,\left({m_{1}}+{m_{2}}\right)+g\,\left({m_{1}}+{m_{2}}\right)\,\cos\left({θ_{1}}\right)+{w_{2}}^{2}\,{L_{2}}\,{m_{2}}\,\cos\left({θ_{1}}-{θ_{2}}\right)\right)}{{L_{2}}\,\left(2\,{m_{1}}+{m_{2}}-{m_{2}}\,\cos\left(2\,{θ_{1}}-2\,{θ_{2}}\right)\right)}\]
Description
  • \({α_{2}}\) is the angular acceleration of the second object (\(\frac{\text{rad}}{\text{s}^{2}}\))
  • \({θ_{1}}\) is the angle of the first rod (\({\text{rad}}\))
  • \({θ_{2}}\) is the angle of the second rod (\({\text{rad}}\))
  • \({w_{1}}\) is the angular velocity of the first object (\(\frac{\text{rad}}{\text{s}}\))
  • \({w_{2}}\) is the angular velocity of the second object (\(\frac{\text{rad}}{\text{s}}\))
  • \({L_{1}}\) is the length of the first rod (\({\text{m}}\))
  • \({m_{1}}\) is the mass of the first object (\({\text{kg}}\))
  • \({m_{2}}\) is the mass of the second object (\({\text{kg}}\))
  • \(g\) is the magnitude of gravitational acceleration (\(\frac{\text{m}}{\text{s}^{2}}\))
  • \({L_{2}}\) is the length of the second rod (\({\text{m}}\))
Notes
  • \({θ_{2}}\) is calculated by solving the ODE here together with the initial conditions and IM:calOfAngle1.
Source–
RefByFR:Output-Values, FR:Calculate-Angle-Of-Rod, IM:calOfAngle2, and IM:calOfAngle1

Detailed derivation of angle of the second rod:

By solving equations GD:xForce2 and GD:yForce2 for \({\boldsymbol{T}_{2}}\,\sin\left({θ_{2}}\right)\) and \({\boldsymbol{T}_{2}}\,\cos\left({θ_{2}}\right)\) and then substituting into equation GD:xForce1 and GD:yForce1 , we can get equations 1 and 2:

\[{m_{1}}\,{a_{\text{x}1}}=-{\boldsymbol{T}_{1}}\,\sin\left({θ_{1}}\right)-{m_{2}}\,{a_{\text{x}2}}\]

\[{m_{1}}\,{a_{\text{y}1}}={\boldsymbol{T}_{1}}\,\cos\left({θ_{1}}\right)-{m_{2}}\,{a_{\text{y}2}}-{m_{2}}\,g-{m_{1}}\,g\]

Multiply the equation 1 by \(\cos\left({θ_{1}}\right)\) and the equation 2 by \(\sin\left({θ_{1}}\right)\) and rearrange to get:

\[{\boldsymbol{T}_{1}}\,\sin\left({θ_{1}}\right)\,\cos\left({θ_{1}}\right)=-\cos\left({θ_{1}}\right)\,\left({m_{1}}\,{a_{\text{x}1}}+{m_{2}}\,{a_{\text{x}2}}\right)\]

\[{\boldsymbol{T}_{1}}\,\sin\left({θ_{1}}\right)\,\cos\left({θ_{1}}\right)=\sin\left({θ_{1}}\right)\,\left({m_{1}}\,{a_{\text{y}1}}+{m_{2}}\,{a_{\text{y}2}}+{m_{2}}\,g+{m_{1}}\,g\right)\]

This leads to the equation 3

\[\sin\left({θ_{1}}\right)\,\left({m_{1}}\,{a_{\text{y}1}}+{m_{2}}\,{a_{\text{y}2}}+{m_{2}}\,g+{m_{1}}\,g\right)=-\cos\left({θ_{1}}\right)\,\left({m_{1}}\,{a_{\text{x}1}}+{m_{2}}\,{a_{\text{x}2}}\right)\]

Next, multiply equation GD:xForce2 by \(\cos\left({θ_{2}}\right)\) and equation GD:yForce2 by \(\sin\left({θ_{2}}\right)\) and rearrange to get:

\[{\boldsymbol{T}_{2}}\,\sin\left({θ_{2}}\right)\,\cos\left({θ_{2}}\right)=-\cos\left({θ_{2}}\right)\,{m_{2}}\,{a_{\text{x}2}}\]

\[{\boldsymbol{T}_{1}}\,\sin\left({θ_{2}}\right)\,\cos\left({θ_{2}}\right)=\sin\left({θ_{2}}\right)\,\left({m_{2}}\,{a_{\text{y}2}}+{m_{2}}\,g\right)\]

which leads to equation 4

\[\sin\left({θ_{2}}\right)\,\left({m_{2}}\,{a_{\text{y}2}}+{m_{2}}\,g\right)=-\cos\left({θ_{2}}\right)\,{m_{2}}\,{a_{\text{x}2}}\]

By giving equations GD:accelerationX1 and GD:accelerationX2 and GD:accelerationY1 and GD:accelerationY2 plus additional two equations, 3 and 4, we can get IM:calOfAngle1 and IM:calOfAngle2 via a computer algebra program: