Post on 21-May-2015
April 12, 2023
1
Theory and Design of
Statically Balanced Tensegrity Mechanisms
Graduation Colloquium
Mark Schenk
Department of BioMechanical EngineeringFaculty of Mechanical, Maritime and Materials Engineering
Exam Committee
prof. dr. ir. Peter Wieringa
dr. ir. Just Herder
dr. Simon Guest
dr. ir. Arend Schwab
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Contents
Introduction• tensegrity structures• static balancingMSc. thesis • objectivesZero Stiffness Tensegrity Structures• (zero) stiffnessPrototypeConclusions
statically balanced tensegrity mechanisms
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Introduction – MSc. thesis – Zero Stiffness – Prototype – Conclusions
Needle Tower II, 1969Kröller-Müller Museum
Tensegrity Structures– introduction & examples –
Special class of bar framework• pin-jointed• prestressed
pure compression, pure tension -> cables and bars
pretension required for stiffness -> tensile integrity = tensegrity-> delicate balance
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Introduction – MSc. thesis – Zero Stiffness – Prototype – Conclusions
Tensegrity Structures– engineering applications –
Engineering applications:• obvious architectural / design appeal• light-weight• stowable / deployable• biologically inspired
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Introduction – MSc. thesis – Zero Stiffness – Prototype – Conclusions
Static Balancing– introduction & applications –
Static Balancing= continuous equilibrium= neutral stability= zero stiffness (Herder 2001)
Energy-free / energy-efficient design• e.g. rehabilitation, robotics
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Introduction – MSc. thesis – Zero Stiffness – Prototype – Conclusions
Static Balancing– zero-free-length springs –
Zero-free-length springs• tension is proportional to length
normal pretensioned zero-free-length
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Introduction – MSc. thesis – Zero Stiffness – Prototype – Conclusions
Static Balancing– design methodology & limitations –
Current design methodology• basic designs are extended
by means of modification rules• complex results possible
Limitations• ad hoc solutions, no real generic understanding• bottom-up vs. top-down• “limited” to 2D
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Introduction – MSc. thesis – Zero Stiffness – Prototype – Conclusions
Statically Balanced Tensegrity Mechanisms– a combination of two fields –
tensegrity structures + static balancing
= zero stiffness tensegrity structures
= statically balanced tensegrity mechanisms
Previous examples:
(Herder, 2001)
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Introduction – MSc. thesis – Zero Stiffness – Prototype – Conclusions
Statically Balanced Tensegrity Mechanisms– applications & relevance –
Totally new class of structures / mechanisms• robotics, rehabilitation, deployable structures
Academic interest• new insights for static balancing -> generic theory?• tensegrities <-> 3D pin-jointed structures
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Introduction – MSc. thesis – Zero Stiffness – Prototype – Conclusions
Graduation project– title & objective –
Title“Theory and Design of Statically Balanced Tensegrity
Mechanisms.”
Objectives1. fundamental understanding zero stiffness
tensegrities-> generic description of static balancing-> develop design guidelines
2. build demonstration model
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Introduction – MSc. thesis – Zero Stiffness – Prototype – Conclusions
Graduation project– research approach –
Theoretical research : zero stiffness
Stiffness of tensegrity structures• engineers
structural engineering• mathematicians
rigidity theory
combination
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Introduction – MSc. thesis – Zero Stiffness – Prototype – Conclusions
Stiffness of Structures– tangent stiffness matrix –
Stiffness of a structure• counting rules (e.g. Maxwell’s rule)• tensegrity is prestressed• tangent stiffness matrix
Tangent stiffness matrix• displacements d and forces F
• formulations -> zero-free-length springs
FdK t
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Introduction – MSc. thesis – Zero Stiffness – Prototype – Conclusions
Tangent Stiffness Matrix- zero stiffness -
Zero stiffness
Two components:
• both components are zero• components cancel out
t material geometrical K K K ΩKK t
~ˆ
0FdK t
0dΩdKdK t ~ˆ
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Introduction – MSc. thesis – Zero Stiffness – Prototype – Conclusions
Zero Stiffness Tensegrity– conventional structures –
Conventional interpretation• internal mechanisms (undesirable)
• stabilized by self-stress
P
δ
L/2 L/2
3
3
8L
AEP
0dK ˆ 0dΩ ~
0dK t
Lto
4
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Introduction – MSc. thesis – Zero Stiffness – Prototype – Conclusions
Zero Stiffness Tensegrity– structures with zero-free-length springs –
Zero stiffness tensegrity structures• affine transformations (scaling/shear)• preserve length of conventional members
0dΩ ~
0dK ˆ 0dK t
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Introduction – MSc. thesis – Zero Stiffness – Prototype – Conclusions
Zero Stiffness Tensegrity- example structure -
top side
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Introduction – MSc. thesis – Zero Stiffness – Prototype – Conclusions
Zero Stiffness Tensegrity– length-preserving affine & conic –
Length-preserving affine transformation• iff all bar directions lie on conic• infinitesimal -> finite• only zero-stiffness modes
Number of zero-stiffness modes• no. of unique bar directions k on conic
k ≥ 6 : zero-stiffness modes = 1k < 6 : zero-stiffness modes = 6 - k
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Introduction – MSc. thesis – Zero Stiffness – Prototype – Conclusions
Zero Stiffness Babytoy
Babytoy • 6 bars• 3 bar directions• direction lie on conic
k ≥ 6 : zero-stiffness modes = 1k < 6 : zero-stiffness modes = 6 - k
• 3 zero-stiffness modes
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Introduction – MSc. thesis – Zero Stiffness – Prototype – Conclusions
Prototype Structure- description -
Prototype structure• demonstrate properties• from structure to mechanism
Prototype design• classic tensegrity structure• no zero-free-length springs• 3 bars, 9 springs
-> 3 springs on each bar
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Introduction – MSc. thesis – Zero Stiffness – Prototype – Conclusions
Prototype Structure- final model -
… but, there is a lot of friction
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Introduction – MSc. thesis – Zero Stiffness – Prototype – Conclusions
Conclusions
Theory and Design of Statically Balanced Tensegrity Mechanisms
• theory of statically balanced tensegrity mechanisms• length-preserving affine transformations• conventional member directions on conic• finite mechanisms
• design of a statically balanced tensegrity mechanism• working prototype
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Introduction – MSc. thesis – Zero Stiffness – Prototype – Conclusions
Questions