The dynamic properties of primary-secondary structural systems are examined using analytical expressions for the modal properties. The analysis begins with a Lagrange multiplier formulation to develop a characteristic equation in terms of primary system mobilities and secondary system impedances. The complexity of the characteristic equation by developing new method, frequency window method. It is shown that the reduction of complexity can only be obtained by a reduction of accuracy, but by retaining the dominant effects of the dynamics problem, the loss of accuracy is not excessive. The reduced problem is examined further to develop simple expressions for the modal properties which provide insight into the resonance characteristics of the primary-secondary system problem. The results are useful as a complement to existing computational techniques for understanding and interpreting dynamic analysis results.