The Chen-Mckenna suspension bridge equation is a nonlinear PDE which is 2nd order in time and is used to model traveling waves on a suspended beam. For certain parameter regimes, it admits multi-pulse traveling wave solutions, which are small perturbations of the stable, primary pulse solution. Linear stability of these multi-pulse solutions is determined by eigenvalues near the origin representing the interaction between the individual pulses. Linearization about these multi-pulse solutions yields a quadratic eigenvalue problem. To study this problem, we use a reformulated version of the Krein matrix, which was presented by Todd Kapitula in a previous talk. Using an appropriate leading order expansion of the Krein matrix, we are able to give analytical criteria for the stability of these multi-pulse solutions. We also present numerical results to support our analysis.