I NTRODUCTION O BJECTIVES O NE DOF SYSTEM T APE SPRING C ONCLUSIONS Importance of structural damping in the dynamic analysis of compliant deployable structures Florence Dewalque, Pierre Rochus, Olivier Brüls Department of Aerospace and Mechanical Engineering University of Liège, Belgium 65th International Astronautical Congress Toronto, 30 September 2014 1 / 17
I NTRODUCTION O BJECTIVES O NE DOF SYSTEM T APE SPRING C ONCLUSIONS O UTLINE I NTRODUCTION Tape springs Types of damping O BJECTIVES O NE DEGREE OF FREEDOM SYSTEM T APE SPRING - D YNAMIC ANALYSIS Without structural damping With structural damping Comparison C ONCLUSIONS 2 / 17
I NTRODUCTION O BJECTIVES O NE DOF SYSTEM T APE SPRING C ONCLUSIONS I NTRODUCTION - T APE SPRINGS Definition: Thin strip curved along is width used as a compliant mechanism General characteristics: ◮ Elastic energy ◮ Structural deformation ◮ No external energy sources ◮ Cheap, simple, reliable ◮ Space applications S. Hoffait et al. 3 / 17
I NTRODUCTION O BJECTIVES O NE DOF SYSTEM T APE SPRING C ONCLUSIONS I NTRODUCTION - T APE SPRINGS ◮ Highly nonlinear 600 Opposite sense - Loading Opposite sense - Unloading Equal sense - Loading 500 ◮ Buckling, hysteresis Equal sense - Unloading 400 and self-locking Nmm] 300 ◮ Senses of bending Bending moment [ 200 100 Opposite sense bending Equal sense bending M < 0 0 −100 −200 ����� −300 −15 −10 −5 0 5 10 15 Bending angle [ deg] ����� M > 0 4 / 17
I NTRODUCTION O BJECTIVES O NE DOF SYSTEM T APE SPRING C ONCLUSIONS I NTRODUCTION - T YPES OF DAMPING Structural damping: ◮ Property of the material ◮ Simple rheological models: Maxwell, Kelvin-Voigt, ... ◮ Advanced models: Prony series, Rayleigh damping, ... Numerical damping: ◮ Property of the solver ◮ Examples: Newmark, HHT, generalized- α , Runge Kutta, ... ◮ Role: convergence, filter spurious modes, ... 5 / 17
I NTRODUCTION O BJECTIVES O NE DOF SYSTEM T APE SPRING C ONCLUSIONS O BJECTIVES State of the art: ◮ In the majority of the previous works, F. E. analyses with numerical damping ◮ Structural damping rarely represented ◮ Notable exceptions: Kwok & Pellegrino (2011) and Mobrem & Adams (2009) Objectives: ◮ Determine the impact of the two types of damping ◮ Introduce some structural damping ◮ Reduce the dependence to numerical damping Simulation without structural damping ( Hoffait et al. ). 6 / 17
I NTRODUCTION O BJECTIVES O NE DOF SYSTEM T APE SPRING C ONCLUSIONS O NE DEGREE OF FREEDOM SYSTEM Case study: c q m k Equation of motion: q n + 1 + ω 2 q n + 1 = 0 ¨ q n + 1 + 2 εω ˙ System to be solved: (with the update formulae of the solver) E ( ω h , ε ) x n + 1 = B ( ω h , ε ) x n Amplification matrix: A ( ω h , ε ) = E ( ω h , ε ) − 1 B ( ω h , ε ) 7 / 17
I NTRODUCTION O BJECTIVES O NE DOF SYSTEM T APE SPRING C ONCLUSIONS O NE DEGREE OF FREEDOM SYSTEM Spectral radius: ρ ( ω h , ε ) = max ( | λ 1 | , | λ 2 | , | λ 3 | ) to assess the level of dissipation in the model. For a valid numerical solution: Low frequencies High frequencies ω h � 0 . 5 ω h � 2 Accuracy Good representation of Convergence the physical behaviour Good approximation of Filtering of high the real damping frequency modes 8 / 17
I NTRODUCTION O BJECTIVES O NE DOF SYSTEM T APE SPRING C ONCLUSIONS O NE DEGREE OF FREEDOM SYSTEM Structural damping and numerical damping: ε = 0 0 ≤ ρ ∞ ≤ 1 1.4 1.2 1 Spectral radius ρ [ − ] 0.8 0.6 Gen.- α ρ ∞ = 0 0.4 Gen.- α ρ ∞ = 0 . 2 Gen.- α ρ ∞ = 0 . 4 Gen.- α ρ ∞ = 0 . 6 0.2 Gen.- α ρ ∞ = 0 . 8 Gen.- α ρ ∞ = 1 Analytical 0 −3 −2 −1 0 1 2 3 10 10 10 10 10 10 10 ωh [ − ] 9 / 17
I NTRODUCTION O BJECTIVES O NE DOF SYSTEM T APE SPRING C ONCLUSIONS O NE DEGREE OF FREEDOM SYSTEM Structural damping and numerical damping: ε = 0 . 33 0 ≤ ρ ∞ ≤ 1 1.4 1.2 1 Spectral radius ρ [ − ] 0.8 0.6 Gen.- α ρ ∞ = 0 0.4 Gen.- α ρ ∞ = 0 . 2 Gen.- α ρ ∞ = 0 . 4 Gen.- α ρ ∞ = 0 . 6 0.2 Gen.- α ρ ∞ = 0 . 8 Gen.- α ρ ∞ = 1 Analytical 0 −3 −2 −1 0 1 2 3 10 10 10 10 10 10 10 ωh [ − ] 10 / 17
I NTRODUCTION O BJECTIVES O NE DOF SYSTEM T APE SPRING C ONCLUSIONS T APE SPRING - D YNAMIC ANALYSIS Case study: z y x Rigid 400 mm connection Lumped 200 mm mass Folding: in the opposite sense with a bending angle of 60 ◦ Deployment: dynamic analysis for 110 s 11 / 17
I NTRODUCTION O BJECTIVES O NE DOF SYSTEM T APE SPRING C ONCLUSIONS T APE SPRING - D YNAMIC ANALYSIS Without structural damping: 800 M M m ax + 333.6 M m ax [ Nmm ] − 600 333.4 Peak M max 333.2 Bending moment [N mm] moment + 400 333 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Buckling 24 200 + [ Nmm ] Residual 23.5 moment M ∗ 0 23 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 16 − 200 15 Time [ s ] 14 − 400 13 0 20 40 60 80 100 120 140 160 Time [s] 12 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Folding Deployment ρ ∞ [ − ] 12 / 17
I NTRODUCTION O BJECTIVES O NE DOF SYSTEM T APE SPRING C ONCLUSIONS T APE SPRING - D YNAMIC ANALYSIS With structural damping: 500 M Peak M m ax + 333.6 moment M m ax η = 0 s 400 − 333.5 η = 10 − 3 s [ Nmm ] 333.4 300 Bending moment [N mm] 333.3 M max Buckling + 333.2 200 333.1 100 333 Residual 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 moment 0 24 23.8 − 100 + [ Nmm ] 23.6 − 200 23.4 M ∗ 23.2 − 300 0 20 40 60 80 100 120 140 160 Time [s] 23 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Folding Deployment ρ ∞ [ − ] 13 / 17
I NTRODUCTION O BJECTIVES O NE DOF SYSTEM T APE SPRING C ONCLUSIONS T APE SPRING - D YNAMIC ANALYSIS With structural damping: Zone 1 500 M Time (after buckling) [ s ] M m ax 16 + M m ax 400 − 15 14 η = 0 s 300 Bending moment [N mm] 13 η = 10 − 3 s 12 200 11 Zone 2 10 100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Time (after deployment) [ s ] 0 120 118 − 100 116 114 − 200 112 − 300 110 0 20 40 60 80 100 120 140 160 Time [s] 108 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Folding Deployment ρ ∞ [ − ] 14 / 17
I NTRODUCTION O BJECTIVES O NE DOF SYSTEM T APE SPRING C ONCLUSIONS T APE SPRING - D YNAMIC ANALYSIS Comparison of the displacements: With structural damping Without structural damping 15 / 17
I NTRODUCTION O BJECTIVES O NE DOF SYSTEM T APE SPRING C ONCLUSIONS C ONCLUSIONS ◮ The two types of damping are required for a valid numerical solution ◮ Adding some structural damping: ◮ reduces the dependence to numerical damping ◮ ensures a correct representation of the damping of the oscillations after deployment ◮ permits to model the self-locking phenomenon 16 / 17
I NTRODUCTION O BJECTIVES O NE DOF SYSTEM T APE SPRING C ONCLUSIONS T HANK YOU FOR YOUR ATTENTION 17 / 17
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