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  3. Volume 57, 2023
  4. Model of two-mass oscillatory system with parallel elastic and damping elements

Model of two-mass oscillatory system with parallel elastic and damping elements

ISTCIPA.
2023;
Volume 57
: pp. 5 - 12
https://doi.org/10.23939/istcipa2023.57.005
Authors:
  1. Vasyl Dmytriv
  2. Stepan Berehuliak
  3. Maksym Somak
1
Lviv Polytechnic National University
2
Lviv Polytechnic National University, Ukraine
3
Lviv Polytechnic National University, Ukraine

Aim. Development of a method of analytical research of a two-mass oscillating system with parallel elastic and damping elements, which makes it possible to expand the design of such systems in various tasks of the functioning of machines and equipment. Method. We will conduct a parametric study of the dynamic oscillation system to assess the effect of the elasticity coefficient and damping on the change in the natural frequency, using the Laplace transform method. A mathematical model of the system with two masses connected by elastic and damping elements placed in parallel is presented. Disturbances are transmitted to the masses through an elastic-damping system. Results. The solution of the system of differential equations through the Laplace transform for two variants of the characteristic equation is given. The biquadratic characteristic equation is solved by the Ferrari method. Both for the roots with complex numbers and for the roots with real numbers, the values of the roots λ1 ... λi are obtained. The value of the roots of the biquadratic equation for the general solution of a homogeneous system of differential equations enables stability for the values of the six roots of the characteristic equation. Unlike previous studies, where applied disturbances were described in the form of harmonic oscillations, we proposed a solution for disturbances in the form of discrete single pulses of different durations and different tracking frequencies, from one pulse to n pulses. Scientific novelty. The analytical solution of the system of differential equations that describe the mutual oscillation of masses connected in parallel by elastic and viscous elements refers to a subspecies of the classical Kelvin-Voigt body. In comparison with other mathematical models, the analytical solution of the system of differential equations describing the movement of the masses of the system will allow to study the system with structural parameters in a wide numerical range of their numerical values. Practical value. The analytical model allows modeling technical systems that work according to this principle. For example, suspensions of cars and other vehicles for various purposes. The mathematical model is analytically solved and allows to optimize suspension designs.

two-mass system
elasticity
damper
Laplace transform
frequency square
damping coefficient
characteristic equation
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