Example Planar Pendulum

Exploiting the generalized coordinate i9 and Eq. (2.44), the Lagrangian may be expressed in terms of the generalized coordinate only, 

According to Eq. (2.54) the canonical momentum is therefore given as = mf-d, the generalized tangential force is given by = —mgl sind, and the Euler-Lagrange equation (2.53) may be rearranged to yield 

This is the Newtonian equation of motion for a two-dimensional pendulum of fixed length subject to the gravitational field. 

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Recent mathematical work suggests that—especially for nonlinear phenomena—certain geometric properties can be as important as accuracy and (linear) stability. It has long been known that the flows of Hamiltonian systems posess invariants and symmetries which describe the behavior of groups of nearby trajectories. Consider, for example, a two-dimensional Hamiltonian system such as the planar pendulum H = — cos(g)) or the... 

For many situations of interest elementary Newtonian mechanics is not directly applicable, since the system might be subject to a set of constraints. These constraints and the corresponding a priori unknown forces of constraint hamper both the setting up and the solution of the Newtonian equations of motion. For example, consider a planar pendulum of mass m with fixed length I oscillating in the two-dimensional xy-plane as sketched in Figure 2.2. Experimentally the fixed length of the pendulum may be realized by an iron rod... 

Chapter 9 gives a review on how multibody systems can be modelled by means of multibond graphs. A major contribution of the chapter is a procedure that provides a minimum number of break variables in multibond graphs with ZCPs. For the state variables and these break variables (also called semi-state variables) a DAE system can be formulated that can be solved by means of the backward differentiation formula (BDF) method implemented in the widely used DASSL code. The approach is illustrated by means of a multibond graph with ZCPs of the planar physical pendulum example. 

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