Mathematical models of resonance and antiresonance processes

Abstract

The use of the symbolic (complex) method has significantly simplified the study of resonance and near-resonance phenomena, in particular, it has made it possible to deeply unify and formalize the consideration of various mechanical systems. The cumbersome and time-consuming operations associated with composing and solving differential equations have been replaced by simple algebraic transformations. The method is based on the mechanical analogue of Ohm’s law in a complex representation and the concept of mechanical reactance, resistance, impedance, susseptance, conductance and admittance. Resonances and antiresonances of forces and velocities are determined. Resonances occur when the elements are connected in parallel with a force source, or when the elements are connected in series with a velocity source. Antiresonances occur when a parallel connection and a speed source are combined, or a serial connection and a force source are combined. These concepts are a generalization to mechanics of the concepts of «voltage source» and «current source» from theoretical electrical engineering. The closest to the source of speed in its properties is a crank-rocker (connecting rod) mechanism with a massive flywheel. The source of force corresponds more to the rod of the significantly smaller of the two connected pneumatic cylinders.