Eigenvalue Dynamics of an Operator Pencil in the Gyroscopic Stabilization Problem

The dynamics of eigenvalues of the quadratic pencil $${\cal L} (\lambda) := \lambda^{2} R + \lambda\,\omega\,iB + C,$$ where $$\omega\to \infty$$ is a numerical parameter and $$R$$ , $$B$$ , and $$C$$ are complex symmetric $$n\times n$$ matrices, is studied. The skew-symmetric matrix $$\Gamma = \omega iB$$ expresses the gyroscopic forces in the problem described by the equation $$R\ddot u(t) - \Gamma \dot u(t) + C u(t) =0.$$ The case $$C<0$$ is considered. Let $$\{\kappa_+, \kappa_-, \kappa_0\}$$ be the signature of the matrix $$B$$ (i.e., the numbers of positive and negative eigenvalues and the dimension of the kernel). Explicit expressions for the eigenvalues $$\lambda_k(\omega)$$ of the pencil $${\cal L} (\lambda)$$ as $$\omega \to \infty$$ are obtained. It follows from these expressions that, for any sufficiently large $$\omega > \omega_0$$ , all the $$2n$$ eigenvalues of the pencil are semisimple, $$2(n -\kappa_0) $$ of them are purely imaginary, and $$2\kappa_0$$ are located symmetrically with respect to the imaginary axis in the open left and right half-planes. In particular, the instability index of the problem is exactly $$\kappa_0$$ for large $$\omega$$ . Remarkably, the coefficients in the asymptotic formulas are determined explicitly and coincide with the eigenvalues of the linear pencils $$\lambda C +B$$ , $$\lambda R +B$$ , and $$P(\lambda R +C)P$$ , where $$P$$ is the orthogonal projection onto the kernel of $$B$$ .