me751 advanced computational multibody dynamics · antonio recuero university of wisconsin-madison....

ME751 Advanced Computational

Multibody Dynamics

October 31, 2016

Antonio RecueroUniversity of Wisconsin-Madison

Quotes of the Day

“Everything should be made as simple as possible, but not simpler.” - Albert Einstein

“In mathematics, you don't understand things. You just get used to them." - Johann von Neumann

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Before we get started… On Friday, we learned:

To derive –in general terms- the equations of motion of the FFR Mass matrix, external forces, material forces Form of inertia shape integrals Eigenmode analysis to reduce the order of a FE system

Got some feedback from some you, in short I am going too fast –math hard to follow Difficult to understand the equations without numerical examples

This lecture… We will review the equations presented on Friday (quickly) We will derive, more in detail, key equations of FFR We will go over a numerical example of a 2D beam within the FFR context FFR Intermediate coordinate system –FE analysis

Note: Advanced model order reduction methods has been dropped: It will not be covered –supposed to have been covered last Friday.

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2. Equations of FFR

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Equations of FFR

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Equations of FFR

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Equations of FFR

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Equations of FFR

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Is the mass matrix constant?

TranslationalRotational

Flexible coordinates

Equations of FFR

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Equations of FFR

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Equations of FFR

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Can this force make the body rotate? Translate? Deform?

Equations of FFR

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Generalized force

GF translational coordinates

GF rotational coordinates

GF flexible coordinates: modal participation factors

Equations of FFR

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Can be particularized at the deformed location

Matrix of shape functions is always evaluated at the undeformed configuration

As per discussion with Dan Negrut, the application point of external load was changed to the deformed point location.

Note the following: FFR formulation allows for including the effect of the deformed point location in the generalized rotation coordinates.

Note: The loads in FFR do not follow the material, deformed point, as it is based on small deformation finite elements: Shape functions can only be evaluated in the undeformed configuration, so, in terms of deformation it’d be equivalent to think the load is always applied in the undeformed configuration –this is not true for other flexible MBD formulations

Equations of FFR

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How do these matrices look like?

3. Inertia Shape Integrals

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Inertia Shape Integrals

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Inertia Shape Integrals

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Inertia Shape Integrals

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1. Finite Element FFR

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Source: www.directindustry.com

1. Finite Element FFR

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1. Intermediate Coordinate System

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1. Intermediate Coordinate System

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1. Intermediate Coordinate System

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1. Intermediate Coordinate System

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New shape functions?

1. Intermediate Coordinate System

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2. Connectivity conditions

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New shape functions?

3. Reference conditions (Rev.)

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4a. Kinematics of FE/FFR

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4b. Generalized elastic force

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Only space dependency

Constant elastic coefficients