thin beam models
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Thin Beam Models
”He is a self-made man and worships his creator,” John Bright
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Motivating Applications
THUNDER: THin layer UNimorph DrivERand sensor
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Motivating Applications
Unimorphs: e.g., actuation for micro-robotics
Energy Harvesting:
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Modeling AssumptionsPhysical Assumptions:
• Consider only motion in the x-direction
• Small displacements which permits linear theory
• Planar cross-sections remain planar during bending
• Negligible shear deformations (thickness is smallcompared with the length)
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Force and Moment BalanceNotation:
Force Balance:
Moment Balance:
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Constitutive and Kinematic Relations
Force and Moment Balance:
Constitutive and Kinematic Relations:
Assumption 1:
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Constitutive and Kinematic Relations
Note:
Moment:
Model (Assumption 1):
Model (Assumption 2):
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Boundary Conditions
1. Cantilever:
2. Pinned at Both Ends:
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Weak Formulation -- Cantilever Beam
State Space:
Space of Test Functions: Cantilever beam
Weak Formulation: Multiply by test functions and integrate by parts to get
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Weak Formulation -- Cantilever Beam
Hamiltonian Formulation: Kinetic and potential energies
Action Integral:
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Analytic Beam Model SolutionConsider:
Separation of Variables:
General Solutions:
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Analytic Beam Model SolutionPinned End Conditions:
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Analytic Beam Model SolutionCantilever End Conditions:
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Galerkin Method for Beam ModelWeak Formulation: See Section 8.3 of Smith, 2005
Basis:
where
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Galerkin Method for Beam ModelApproximate Solution:
System:
where
and