ME751 Advanced Computational Multibody Dynamics · Antonio Recuero University of Wisconsin-Madison....
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ME751 Advanced Computational Multibody Dynamics October 31, 2016 Antonio Recuero University of Wisconsin-Madison
Transcript of 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
