Participation Factors

The mode shape, as indicated by the right eigenvector, gives the relative phase of each state in a particular mode. However, it does not give the influence of each state on the mode. We would like to be able to obtain the influence of states on modes because then we will know which states (machines) to control in order to increase damping of a certain problem mode. Let's define a new state variable (" xi ") as follows: x q x q T k k T k k & & = ⇒ = ξ ξ Notice that ξ k is a scalar since the transpose of the left eigenvector is a 1×n and the state vector is an n×1. Thus, ξ k is a combination of all of the states, but the manner in which all of the other states is combined is through the left eigenvector elements of the k th mode. An important attribute of ξ k , and the reason why it is of great interest here, is that it is a state which is associated with the k th mode and no other mode. We can prove this as follows. Start with the system state equations: x A x = & Pre-multiply both sides by T k q. x A q x q T k T k = & (P-1) Recall that the left eigenvector is defined as T k k T k q A q λ = (P-2) Notice that the left-hand-side of eq. (P-2), is on the right-hand-side of eq. (P-1). Substituting the right-hand-side of eq. (P-2) into the right-hand-side of eq. (P-1), we obtain: