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INTRODUCTION OBJECTIVES ONE DOF SYSTEM TAPE SPRING CONCLUSIONS

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

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INTRODUCTION OBJECTIVES ONE DOF SYSTEM TAPE SPRING CONCLUSIONS

OUTLINE

INTRODUCTION Tape springs Types of damping OBJECTIVES ONE DEGREE OF FREEDOM SYSTEM TAPE SPRING - DYNAMIC ANALYSIS Without structural damping With structural damping Comparison CONCLUSIONS

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INTRODUCTION OBJECTIVES ONE DOF SYSTEM TAPE SPRING CONCLUSIONS

INTRODUCTION - TAPE 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.

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INTRODUCTION OBJECTIVES ONE DOF SYSTEM TAPE SPRING CONCLUSIONS

INTRODUCTION - TAPE SPRINGS

◮ Highly nonlinear ◮ Buckling, hysteresis

and self-locking

◮ Senses of bending

M > 0 M < 0 Opposite sense bending Equal sense bending

• −15

−10 −5 5 10 15 −300 −200 −100 100 200 300 400 500 600

Bending angle [ deg] Bending moment [ Nmm] Opposite sense - Loading Opposite sense - Unloading Equal sense - Loading Equal sense - Unloading

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INTRODUCTION OBJECTIVES ONE DOF SYSTEM TAPE SPRING CONCLUSIONS

INTRODUCTION - TYPES 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, ...

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INTRODUCTION OBJECTIVES ONE DOF SYSTEM TAPE SPRING CONCLUSIONS

OBJECTIVES

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

INTRODUCTION OBJECTIVES ONE DOF SYSTEM TAPE SPRING CONCLUSIONS

ONE DEGREE OF FREEDOM SYSTEM

Case study:

c k m q

Equation of motion: ¨ qn+1 + 2εω˙ qn+1 + ω2qn+1 = 0 System to be solved: (with the update formulae of the solver) E(ωh, ε)xn+1 = B(ωh, ε)xn Amplification matrix: A(ωh, ε) = E(ωh, ε)−1B(ωh, ε)

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INTRODUCTION OBJECTIVES ONE DOF SYSTEM TAPE SPRING CONCLUSIONS

ONE DEGREE OF FREEDOM SYSTEM

Spectral radius: ρ(ωh, ε) = max(|λ1|, |λ2|, |λ3|) to assess the level of dissipation in the model. For a valid numerical solution: Low frequencies High frequencies ωh 0.5 ωh 2 Accuracy Good representation of Convergence the physical behaviour Good approximation of Filtering of high the real damping frequency modes

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INTRODUCTION OBJECTIVES ONE DOF SYSTEM TAPE SPRING CONCLUSIONS

ONE DEGREE OF FREEDOM SYSTEM

Structural damping and numerical damping: ε = 0 0 ≤ ρ∞ ≤ 1

10

−3

10

−2

10

−1

10 10

1

10

2

10

3

0.2 0.4 0.6 0.8 1 1.2 1.4

Spectral radius ρ [−] ωh [−] Gen.-α ρ∞ = 0 Gen.-α ρ∞ = 0.2 Gen.-α ρ∞ = 0.4 Gen.-α ρ∞ = 0.6 Gen.-α ρ∞ = 0.8 Gen.-α ρ∞ = 1 Analytical 9 / 17

INTRODUCTION OBJECTIVES ONE DOF SYSTEM TAPE SPRING CONCLUSIONS

ONE DEGREE OF FREEDOM SYSTEM

Structural damping and numerical damping: ε = 0.33 0 ≤ ρ∞ ≤ 1

10

−3

10

−2

10

−1

10 10

1

10

2

10

3

0.2 0.4 0.6 0.8 1 1.2 1.4

Spectral radius ρ [−] ωh [−] Gen.-α ρ∞ = 0 Gen.-α ρ∞ = 0.2 Gen.-α ρ∞ = 0.4 Gen.-α ρ∞ = 0.6 Gen.-α ρ∞ = 0.8 Gen.-α ρ∞ = 1 Analytical 10 / 17

INTRODUCTION OBJECTIVES ONE DOF SYSTEM TAPE SPRING CONCLUSIONS

TAPE SPRING - DYNAMIC ANALYSIS

Case study:

x y z 200 mm 400 mm

Rigid connection Lumped mass

Folding: in the opposite sense with a bending angle of 60◦ Deployment: dynamic analysis for 110 s

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INTRODUCTION OBJECTIVES ONE DOF SYSTEM TAPE SPRING CONCLUSIONS

TAPE SPRING - DYNAMIC ANALYSIS

Without structural damping:

20 40 60 80 100 120 140 160 −400 −200 200 400 600 800

Time [s] Bending moment [N mm] M M m ax

+

M m ax

−

Folding Deployment Peak moment Residual moment Buckling

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 333 333.2 333.4 333.6

Mmax

+

[Nmm]

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 23 23.5 24

M∗

+ [Nmm] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 12 13 14 15 16

ρ∞ [−] Time [s]

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INTRODUCTION OBJECTIVES ONE DOF SYSTEM TAPE SPRING CONCLUSIONS

TAPE SPRING - DYNAMIC ANALYSIS

With structural damping:

20 40 60 80 100 120 140 160 −300 −200 −100 100 200 300 400 500

Time [s] Bending moment [N mm] M M m ax

+

M m ax

−

Folding Deployment Peak moment Residual moment Buckling

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 333 333.1 333.2 333.3 333.4 333.5 333.6

Mmax

+

[Nmm] η = 0 s η = 10−3 s

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 23 23.2 23.4 23.6 23.8 24

ρ∞ [−] M∗

+ [Nmm]

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INTRODUCTION OBJECTIVES ONE DOF SYSTEM TAPE SPRING CONCLUSIONS

TAPE SPRING - DYNAMIC ANALYSIS

With structural damping:

20 40 60 80 100 120 140 160 −300 −200 −100 100 200 300 400 500

Time [s] Bending moment [N mm] M M m ax

+

M m ax

−

Folding Deployment Zone 1 Zone 2

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 10 11 12 13 14 15 16

Time (after buckling) [s] η = 0 s η = 10−3 s

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 108 110 112 114 116 118 120

ρ∞ [−] Time (after deployment) [s]

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INTRODUCTION OBJECTIVES ONE DOF SYSTEM TAPE SPRING CONCLUSIONS

TAPE SPRING - DYNAMIC ANALYSIS

Comparison of the displacements: With structural damping Without structural damping

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INTRODUCTION OBJECTIVES ONE DOF SYSTEM TAPE SPRING CONCLUSIONS

CONCLUSIONS

◮ 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

• scillations after deployment

◮ permits to model the self-locking phenomenon 16 / 17

INTRODUCTION OBJECTIVES ONE DOF SYSTEM TAPE SPRING CONCLUSIONS

THANK YOU FOR YOUR ATTENTION

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