Chapter 20
The BCU–Exit Point Method
20.1 INTRODUCTION
We present in this chapter an extended BCU method, termed the BCU–exit point method, to compute accurate critical energy values for direct stability analysis when the BCU method fails. The BCU–exit point method is built on the BCU method. Recall that a controlling unstable equilibrium point (UEP) computed by the BCU method is said to satisfy the boundary property if the controlling UEP lies on the stability boundary of the original system. If the controlling UEP computed by the BCU method satisfies the boundary property, then its energy value can be used as the critical energy. On the other hand, if the boundary property is not satisfied, then the computed controlling UEP is not correct and the energy value at the controlling UEP cannot be used as the critical energy.
In this chapter, we first present a verification scheme to check whether the boundary property of a computed controlling UEP is satisfied instead of checking the one-parameter transversality condition. This verification scheme will then be integrated into the BCU–exit point method to compute accurate critical energy values when the boundary property is not satisfied. Hence, the role of the BCU–exit point method is to determine an accurate critical energy value for direct stability analysis when the BCU method fails. The BCU–exit point method is “degenerated” into the BCU method when the boundary property of a computed controlling UEP by the BCU method is ...
