End-to-end analysis of vibration of time-varying mass systems

doi:10.1007/s10483-026-3414-8

Abstract

Abstract:

Vibration systems with time-varying mass are prevalent in engineering practice, exemplified by rockets with fuel depletion, vehicles with changing mass, and systems with cable-hoisted payloads. However, progress has been constrained by the lack of an end-to-end approach capable of integrating modeling, closed-form analysis, numerically stable calculations, and isolation design. Focusing on a typical system involving rocket fuel combustion with linear mass depletion, we first derive the equations of motion from the momentum theorem. A parameter transformation, constructed via the method of undetermined coefficients, converts the time-varying differential equation into a standard Bessel equation, yielding a closed-form analytical solution. To achieve reliable numerical solutions, a strategy combining variable upper-limit integration with grid-search-optimized lower bounds is deployed to replace indefinite integrals, thereby overcoming non-integrable products of Bessel functions. Benchmark comparisons with harmonic excitations show excellent agreement, validating the formulation and solution scheme. Building on the closed-form response, an end-to-end vibration isolation workflow is established via transmissibility and isolation failure-time-threshold (FTT) metrics, allowing for the selection of stiffness based on the instantaneous frequency ratio throughout the mass variation process. The resulting framework provides a general analytical tool for linear differential equations with time-varying mass and a practical pathway for vibration isolation design in mass-varying structures, with special relevance to aerospace applications.

Key words: vibration isolation, time-varying mass, Bessel equation, variable upper-limit integration

2010 MSC Number:

O322

Kai WANG, Ao CHENG, Tingting CHEN, Jiaxi ZHOU, Zhuang LI, Shengtao ZHANG, Li CHENG. End-to-end analysis of vibration of time-varying mass systems. Applied Mathematics and Mechanics (English Edition), 2026, 47(8): 1723-1746.

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URL: https://www.amm.shu.edu.cn/EN/10.1007/s10483-026-3414-8

https://www.amm.shu.edu.cn/EN/Y2026/V47/I8/1723

Figures/Tables 11

Fig. 1

Fig. 2

Table 1

The system parameters used for calculations"

Parameter	Symbol	Value	Parameter	Symbol	Value
Constant mass	m₀	5 kg	Damping coefficient	c	2 N · s/m
Time-dependent mass	$Δ m$	50 kg	Stiffness	K	10 000 N/m
Combustion duration	T	25 s

Table 1

Fig. 3

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