[Paper Review] Frequency sweep for a beam system with local unilateral contact modeling satellite solar arrays

Hamad Hazim, B. Rousselet|ArXiv.org|Oct 1, 2009

Dynamics and Control of Mechanical Systems4 references3 citations

TL;DR

This paper investigates the nonlinear dynamic behavior of a satellite solar array modeled as a clamped-free Bernoulli beam with a unilateral contact spring (snubber) using finite element analysis and frequency sweep simulations. The key finding is that unilateral contact introduces superharmonic and subharmonic frequencies, along with complex resonance phenomena, distinct from linear systems, with non-linear normal modes under development for theoretical validation.

ABSTRACT

In order to save mass of satellite solar arrays, the flexibility of the panels becomes not negligible and they may strike each other; this may damage the structure. To prevent this, rubber snubbers are mounted at well chosen points of the structure and they act as one sided linear spring; as a negative consequence, the dynamic of these panels becomes nonlinear. The finite element approximation is used to solve partial differential equations governing the structural dynamic. Frequency sweep has been performed numerically to study the dynamic behavior. Non linear normal modes are under study

Motivation & Objective

• To model and simulate the nonlinear dynamic response of satellite solar arrays with unilateral contact snubbers to prevent panel-to-panel impact.
• To analyze how unilateral contact introduces nonlinearity into the beam system's dynamics, altering resonance behavior.
• To compare linear and nonlinear frequency responses using numerical frequency sweeps and validate results via non-linear normal modes (NNM).
• To prepare for future experimental validation by simulating dynamic behavior under periodic excitation.
• To develop an alternative approach to non-linear normal modes for systems with unilateral constraints.

Proposed method

• Model the beam-spring system using the Euler-Bernoulli beam equation with unilateral spring force at the free end.
• Apply the classical Hermite cubic finite element method to discretize the partial differential equation into a system of ODEs.
• Formulate the semi-discrete system as M¨q + Kq = k_r (d(t) - q_n)+ e_n, where the nonlinearity arises from the (·)+ function.
• Integrate the resulting ODE system numerically using the stiff ODE solver BDF from ODEPACK in Scilab.
• Perform a frequency sweep by varying the excitation frequency ω and recording the maximum displacement across all nodes for each frequency.
• Compare results with FFT of time-domain signals and validate against analytical linear eigenfrequencies and nonlinear normal modes.

Experimental results

Research questions

• RQ1 How does the introduction of a unilateral contact spring alter the dynamic response of a flexible beam system compared to a linear bilateral spring?
• RQ2 What nonlinear frequency components (e.g., superharmonics, subharmonics) emerge due to the unilateral contact nonlinearity?
• RQ3 How do the resonance frequencies in the nonlinear system compare to those of the linearized (bilateral) system?
• RQ4 Can numerical frequency sweeps accurately capture the complex dynamic behavior, including jump phenomena and multi-modal responses?
• RQ5 To what extent can non-linear normal modes serve as a theoretical framework to validate the numerical simulations?

Key findings

• The linearized system exhibits eigenfrequencies at 196.36 Hz, 472.08 Hz, and 961.52 Hz, corresponding to the first three modes.
• Nonlinear frequency sweeps reveal additional resonance peaks due to superharmonic (e.g., 2×, 3×) and subharmonic (e.g., ½×, ⅓×) components.
• The presence of unilateral contact causes significant deviation from linear behavior, including complex amplitude and frequency responses.
• The simulation results show clear non-linear effects such as jump phenomena and multi-frequency excitation, especially near the linear eigenfrequencies.
• The numerical results are consistent with expected nonlinear dynamics and are being validated against ongoing experimental work.
• The development of non-linear normal modes is underway as a theoretical tool to further validate and interpret the numerical findings.