Session: 08-01-01: General Dynamics, Vibration, and Control I

Paper Number: 165008

Dynamics of a System With Exponentially Decaying Mass and No Viscous Damping 

This paper presents a model for a single-degree-of-freedom mass-spring system with exponentially decaying mass, representing scenarios such as a rocket mounted on a test rig with negligible damping. The governing equations of motion are derived under the assumption that damping effects are insignificant, justified by the nature of the surrounding environment: in conventional mechanical systems, damping arises primarily from viscous interactions with the ambient medium. However, in the case of a rocket operating in a controlled test rig or near-vacuum conditions, the surrounding matter exhibits negligible viscosity, allowing the analysis to focus solely on the dynamic effects of mass variation and elastic restoring forces. To facilitate an analytical solution, a variable transformation is introduced that redefines the time domain in terms of an “exponentially stretched time.” This transformation accounts for the exponential decay of mass by effectively rescaling the temporal axis, allowing the governing differential equation to be expressed in a more tractable form. Specifically, the new time variable incorporates the mass depletion time constant, which modulates the rate at which the system’s inertia diminishes. This approach simplifies the original governing equation into a form resembling Bessel’s differential equation, enabling the use of well-established mathematical techniques to derive closed-form solutions. The resulting analytical solution is expressed in terms of Bessel functions of the first kind and second kind, with their derivatives provided to facilitate the application of initial conditions. The initial conditions were carefully selected to activate both the solution and its derivative, ensuring that the full structure of the analytical formulation is exercised. This approach guarantees that the derivations are internally consistent and that the resulting expressions accurately capture the system’s dynamic behavior from the outset. By invoking both components of the general solution, the validation process rigorously tests the completeness and correctness of the analytical model. To validate the solution, numerical integration is performed using commercial software, and the root mean square error between analytical and numerical results is calculated, demonstrating excellent agreement. The system’s response is further analyzed under varying initial natural frequencies and mass decay time constants. A key parameter in the system’s behavior is the mass depletion time constant, which governs the rate at which the system’s mass decreases exponentially. A shorter time constant implies rapid mass loss, leading to more abrupt changes in inertia and consequently more pronounced transient effects, while a longer time constant results in a more gradual evolution of the system’s natural frequency and amplitude. The analytical formulation offers a significant advantage by allowing direct computation of the system’s response at any time, without requiring continuous numerical integration. This capability is particularly valuable for applications involving real-time control, design optimization, or predictive modeling of systems with variable mass.

Presenting Author: Pezhman Hassanpour California State Polytechnic University, Pomona

Presenting Author Biography: Dr. Pezhman Hassanpour is an Assistant Professor at California State Polytechnic University, Pomona. He is a registered professional engineer with extensive experience in both academia and forensic engineering. Dr. Hassanpour has dedicated several years to teaching and research, while also applying his expertise in real-world forensic engineering cases. His dual roles in education and practice allow him to bring a unique perspective to his work, bridging the gap between theoretical knowledge and practical application.

Authors:

Pezhman Hassanpour California State Polytechnic University, Pomona