I need to model the system in figure, where turbine, shaft1 and gear1 are a unique rigid body, while shaft2 can be modeled like a torsional spring. Using as coordinates Theta1, clockwise angular displacement of the turbine, and Theta2, anticlockwise displacement of the generator, i obtain these equations of motion

differential equation

where tau is the transmission ratio of the gears and J0 the equivalent moment of inertia of the turbine-shaft-gears block.

The stiffness matrix is not symmetric and it should be impossible, where am I wrong?


1 Answer 1


One way to get this right is to start by ignoring the gearbox and modelling the system with three DOFs, i.e. the angular position of Shaft 1, and the positions of the two ends of Shaft 2. That gives the equations of motion as $$\begin{bmatrix}J_0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & J_1 \end{bmatrix} \begin{bmatrix} \ddot\theta_1 \\ \ddot\theta_g \\ \ddot\theta_2 \end{bmatrix} + \begin{bmatrix} 0 & 0 & 0 \\ 0 & k_{t2} & -k_{t2} \\ 0 & -k_{t2} & k_{t2} \end{bmatrix} \begin{bmatrix} \theta_1 \\ \theta_g \\ \theta_2 \end{bmatrix} = 0$$ or $$M_{3\times3}\,\ddot\theta_{3\times3} + K_{3\times3}\,\theta_{3\times3} = 0.$$

Then, apply the gearbox constraint, i.e. $\theta_g = -\tau\theta_1$. That is equivalent to the transformation matrix $$\begin{bmatrix} \theta_1 \\ \theta_g \\ \theta_2 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ -\tau & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} \theta_1 \\ \theta_2 \end{bmatrix}$$ or $$\theta_{3\times3} = L\,\theta_{2\times2}.$$

So the $2\times2$ stiffness and mass matrices are $$\begin{align} M_{2\times2} &= L^T M_{3\times 3}\, L \\ K_{2\times2} &= L^T K_{3\times 3}\, L \end{align}$$ which are symmetric.

Following a systematic procedure might seem more longer than trying to write the answer down "by inspection," but it's more likely to avoid mistakes.

For example, one of the diagonal terms in the correct stiffness matrix is $\tau^2 k_{t2}$, which doesn't appear anywhere in the OP's equations of motion. That suggests the OP's mistake was getting confused about the units being used to measure the various torques and rotations in the system in terms of $\theta_1$, $\theta_2$, and $\tau$.

  • $\begingroup$ your answer is correct and my answer was incorrect. Somewhere I got confused by the wording and the diagram. Thanks for pointing it out. I will delete my incorrect answer. $\endgroup$
    – Daniel K
    Commented Jun 1, 2017 at 1:37

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