Plane motion of the spring pendulum is considered. The mathematical model of the system is transformed into dimensionless complex form and then the analytic approach using multiple time scale method is applied to solve the obtained initial value problem. The approximate analytical solution gives possibility to analyse steady-state and transient motion of the system for various parameters. Especially non stationary processes are discussed. The nonlinear transition of the dynamics and intersections of the tori are presented. For steady-state vibration the frequency response functions are derived. All analytical results are fully confirmed by numerical analysis

Dynamics behavior of the micromechanical gyroscope designed for measuring one component of the angular velocity is studied in the paper. The Cardan suspension is applied to connect the sensing plate with the substrate whose angular velocity is measured. The gimbal and the plate with sensors are connected via torsional joints. Vibrating motion of the sensing plate is excited mainly by a torque resulting from the Coriolis effect. The mathematical model equations have been derived using the Lagrange equation of the second kind. Both nonlinear effects of the geometrical nature and the nonlinear characteristics of the torsional joints are taken into account. The governing equations are solved with help of the method of multiple scales in time domain that belongs to the broad class of asymptotic methods. The approximate solution of analytical form has been obtained for non-resonant vibration as well as for the case of the main and internal resonances that occur simultaneously. Analytical form of solution allows for extensive analysis of the behavior of the system. The desirable state of the gyroscope work is steady-state vibration in resonance that is discussed in detail.