Gibbs-Appell & Kane Equations

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Chapter 14 Gibbs-Appell & Kane Equations

In this chapter, we continue our treatment of analytical dynamics by presenting two methods: Gibbs-Appell¹ and Kane ² equations.

We saw in Chapter 13 that the calculation of the virtual power of inertial forces from the system’s kinetic energy requires a separate treatment whether variables \((\bq,\dbq,t)\) are independent or not. We shall see that in the Gibbs-Appell formalism, this separate treatment is avoided by considering the energy of acceleration instead of the kinetic energy. All systems whether holonomic or non-holonomic can be treated uniformly. Unfortunately, these equations require the rather burdensome determination of the energy of acceleration.

One of the drawbacks of both Lagrange or Gibbs-Appell equations is the determination of partial derivatives \(\frac{\partial}{\partial q_i}\) and \(\frac{\partial}{\partial \dq_i}\) of the kinetic energy (for Lagrange equations) and \(\frac{\partial}{\partial \ddq_i}\) of the energy of acceleration. This determination requires tedious algebraic manipulations and does not lead to differential systems of equations in a form amenable to numerical integration by standard techniques. These methods also require the use of physical coordinates and their time-derivatives which can be problematic for coordinates such as Euler angles. To address these issues, Kane introduced generalized speeds which need not be defined as the derivatives of coordinates. We will see that Kane equations are obtained by projection (in the sense of screws) of the Fundamental Theorem of Dynamics on the generalized partial kinematic screws. The corresponding generalized inertial forces are, in general, not determined from the system’s kinetic energy or other scalar function. Instead, they require the determination of the dynamic screw of each body.