Damping Models for Shear Beams with Applications to Spacecraft Wiring Harnesses

Abstract

Spacecraft wiring harnesses can fundamentally alter a spacecraft's structural dynamics, necessitating a model to predict the coupled dynamic response of the structure and attached cabling. While a beam model including first-order transverse shear can accurately predict vibration resonance frequencies, current time-domain damping models are inadequate. For example, the common proportional damping model results in modal damping that depends unrealistically on the frequency. Inspired by a geometric rotation-based viscous damping model that provides frequency-independent modal damping in an Euler-Bernoulli formulation, a time-domain viscous damping model with terms associated with the shear and bending angles is presented. This model demonstrates a much weaker dependence on frequency than proportional damping models. Specifically, the model provides modal damping that is approximately constant in the bending-dominated regime (low mode numbers), increasing by at most 6% for a particular selection of bending and shear angle-based damping coefficients. In the shear-dominated regime (high mode numbers), damping values increase linearly with mode number and are proportional to the shear angle-based damping coefficient. A key feature of this model is its ready implementation in a finite element analysis, requiring only the typical mass, stiffness, and geometric stiffness (associated with axial loads) matrices as developed for an Euler-Bernoulli beam. Such an analysis using empirically determined damping coefficients generates damping values that agree well with existing spacecraft cable bundle data. Nomenclature α viscous damping coefficient β shear angle nondimensional shear parameter, EIπ 2 κAGL 2 κ shear coefficient [M ], [K], and [K G ] beam finite element mass, stiffness, and geometric stiffness matrices ω m natural frequency of vibration ρ density ϕ rotation of beam centerline associated with bending ζ m modal damping ratio A cross-sectional area a m modal coordinate E Young's modulus G shear modulus I moment of inertia

Damping Models for Shear-Deformable Beam with Applications to Spacecraft Wiring Harness

“…Furthermore, experimental tests indicated the cable bundles displayed modal damping that was approximately constant across a range of vibration modes. As such, a viscous damping model developed by Lesieutre that produced frequency-independent modal damping in Euler-Bernoulli beams was extended to and modified for the shear beam [5,6]. This model yielded modal damping that was approximately frequency-independent and agreed well with experimental data.…”

“…Lesieutre provided a solution for this unrealistic behavior, developing a "rotation-based" (also called "geometric-based") viscous damping model that results in approximately frequencyindependent modal damping for Euler-Bernoulli beams [5]. Subsequent research extended and modified this model for application to shear beams; that model serves as the starting point for this study of frequency-independent modal damping for Timoshenko beams [6].…”

“…Finally, rotation-(geometric-) based terms are associated with loads that oppose the temporal rates of change of the beam centerline angle (α φ ) and shear angle (α β ). The resulting equations of motion are (6) Note that due to the use of all three coordinates, some of these damping terms will necessarily overlap; in fact, some terms may actually offset the damping introduced by other terms. For example, the equations of motion were presented in terms of the displacement (w) and slope of the beam centerline associated with bending (φ) only; substituting Equation 2into Equation (6) yields: (7) In the first equation, the ' term is multiplied by the difference between α φ and α κβ .…”

“…This approach involves representing the governing equation of motion in terms of differential operators: (17) where γ includes the coordinate(s) describing the system configuration and f includes the external load(s) acting on the system. A damping differential operator that is twice the square root of the product of the mass and stiffness differential operators can be scaled to produce a desired modal damping ς0 that is independent of frequency [6]: (18) In general, the square root of these operators depends on the system boundary conditions; however, approximate results can be found that provide modal damping that is approximately constant, at least over a large range of mode numbers.…”

Multidimensional Vibrations of Cable-Harnessed Beam Structures with Periodic Pattern: Modeling and Experiment

“…Reference [4] models the structure using EB theory and studied the pure transverse modes of cable-harnessed beam. Kauffman et al [6,7] characterized damping effects in pure bending vibrations of standalone cables using Timoshenko model. Spak [8,9] further developed a distributed parameter model in order to predict the dissipation effects.…”