TABLE OF CONTENTS Analysis of vibrations and stability of a conservative system is a classical question. It is of great importance due to many applications in physics and mechanics. Determination of frequencies and modes of vibration is a typical requirement in the design of buildings, bridges, and machines. In many cases modification of frequencies and modes by changing design parameters is necessary to avoid resonances and noise. Stability problems for conservative systems appear in studying elastic structures under action of potential forces like stationary loads, gravity forces etc. In many practical problems there is a specific parameter F describing load of the system. The minimal value of the load parameter, at which the system becomes unstable, is called the critical load F cr. Avoiding instability, required in the design of structures, implies that the loads F must be less than F cr. In the presence of several parameters, graph of the critical load in the parameter space represents a boundary of the stability domain. Analysis of the stability boundary allows changing design parameters in order to modify (increase or decrease) the critical load of the system.
Sensitivity analysis of simple and multiple frequencies with application to stability optimization problems was done in [Bratus and Seyranian (1984); Seyranian et al. (1994); Seyranian (1997)]. Multiple eigenvalues of multi-parameter symmetric and Hermitian matrices were studied in [Wigner and von Neumann (1929); Arnold (1978)]. Stability domains for conservative systems linearly dependent on parameters were investigated in [Papkovich (1963); Huseyin (1978)].
