On mechanical resonances and anti-resonances
To study resonance and near-resonance phenomena, a symbolic (complex) method was used, which makes it possible to significantly increase productivity, simplify and formalize mathematical transformations. Parallel and sequential connections of elements of a mechanical system with a source of force or a source of speed as a source of external mechanical harmonic action are considered. Four modes are described — resonances and antiresonances of forces and velocities. 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cumbersome and time-consuming operations associated with the preparation and solution of differential equations have been replaced by simple algebraic transformations.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, susceptance, conductance and admittance.