Analytical Response of a Mass-Spring Oscillator With Time-Varying Inertia

In this paper, the analytical response of a single-degree-of-freedom system with time-varying inertia is investigated. The system model represents several engineering systems, including resonant biosensors and load-carrying conveyor belts. The resonant biosensing applications, the mass of the resonator changes exponentially with time, while the mass of a load-carrying conveyor belt changes linearly. In general, the system’s inertial can increase or decrease. Considering that a first-order approximation of exponential function yields a linear function, the latter function was used to model the variations of the mass with time with the assumption that the rate of change can be either positive or negative. The equation of motion is derived using newton’s second law of motion. Although the resultant ordinary differential equation has a similar form for both cases of increasing and decreasing mass, it is demonstrated that solving each case requires a different transformation, and the subsequent differential equations have different forms of analytical solutions. Each analytical solution is derived and compared with two corresponding cases: 1) the numerical solution of the differential equation with linearly-varying inertia, and 2) the numerical solution of the differential equation with exponentially-varying mass. Excellent agreement between the analytical solution and numerical solution of the system with linearly-varying inertia is demonstrated. This agreement confirms the validity of the analytical solution. On the other hand, the analytical solution agreed with the numerical solution of the system with exponentially-varying mass only for the initial moment of the integrations. This disagreement suggests that a linear approximation is not an accurate model for resonant biosensing applications where the mass variation is exponential by nature. Nevertheless, the linearly-varying mass model provides insights about how these systems behave differently from those with a constant mass. In particular, in this paper, it is demonstrated that an increasing mass system shows added damping effect. In contrast, a decreasing mass system behaves as if it has a “negative” damping. This finding is particularly significant because, along with other linear or nonlinear effects in the system that are not necessarily considered, it can produce catastrophically unwanted behaviors. Another impressive effect that is observed is the change of free vibration response of the system with time. For the case of increasing mass system, the frequency of oscillation of both linearly-varying and exponentially-varying systems decreases with time. In contrast, the frequency increases for the systems with increasing mass. This phenomenon is expected because the frequency of free vibration is inversely proportional to the mass of the system.