Strongly Nonlinear Effects of an Unbalanced Mass and the Supply Voltage of the Non-Ideal DC Motor on the Vibration Amplitude of a Thin Rectangular Plate

In this paper, the strongly nonlinear effects of an unbalanced mass and the supply voltage of a non-ideal DC motor on the vibration amplitude of a thin rectangular plate are investigated. The influence of these two parameters on the occurrence of the Sommerfeld phenomenon—specifically in terms of the energy delivered by the motor—as well as on the emergence of the so-called limited energy source phenomenon related to voltage saturation, is thoroughly analyzed using analytical and numerical methods. By considering the interaction between the thin rectangular plate and the non-ideal DC motor, the governing equations of the vibration system are derived using the Lagrange formalism. A perturbation method based on averaging is then employed to obtain the steady-state response. Numerical simulations using the fourth-order Runge–Kutta method are also conducted to validate the analytical results. The effect of the non-ideal loading is strongly nonlinear, leading to a non-monotonic behavior of the vibration amplitude with respect to both the supply voltage and the unbalanced mass of the motor. Moreover, the amplitude jump phenomenon, which indicates the presence of the Sommerfeld effect, is clearly captured across a wide range of parameters. This paper provides useful insights for the design of industrial floor structures that support rotating machinery.