Presentation for the course in Multibody Dynamics

Flavia Buonanno

February 2012

• Techniques for the efficient analysis and simulation of large models, reducing

computational requirements without damaging the dynamical properties of the model �

suitable with Finite Elements spatial discretization

• Departing from the linear equation of motion

• The idea is to find a lower dimension space

• Such that the state vector can be approximated as

• The projection versus the lower dimension yields

• With the reduced

system matrices

• The reduced model mass and stiffness matrices can be derived energetically by

substituting the coordinate change into kinectic and potential energy expressions of the

original model

• The transformation T will take different forms depending on the transformation

technique utilized

The equation of

motion for the

undamped system

can be partitioned

Where:

• Main assumption: for the lowest frequency modes, inertia forces on the slave dof are much

less important than the elastic forces transmitted by the master dofs � slave dofs are

assumed to move quasi-statically wrt the motion of the master dofs

• In order to obtain a relationship between xs and xm, inertia terms are neglected (static

nature) and it is assumed that no force is applied on the slave dofs

• From the lower partition:

Expressing the state vector of the full

model in terms of the master dof, the

transformation matrix is obtained as:

� Is a good approximation (exact for static analyses) for the low eigenvalue

spectrum of a FE structure, the quality of the eigenvalue approximation

decreases as the mode number increases

� In general, when high frequency motion is considered, the influence of the inertia

terms is significant � the basic assumption of static reduction is contradicted

and the method becomes inaccurate.

� The quality of the eigenvalue approximations is highly dependent on the location

of the preserved dof (master)

� The mass of the reduced system is not effectively preserved and generally

reduced model frequencies are higher than those of the original full space model

• The structure of the transformation matrix is “similar” to the Guyan reduction, with the

difference that, this time, the inertia information of the model is considered

• Applying the Laplace transformation to the equation of motion of the undamped system,

we obtain the description in frequency domain:

• The system can be rewritten and partitioned in master and slave as in the Guyan

reduction, with

for undamped system

B(w) is called the dynamic

stiffness matrix

• Assuming again that no forces are applied

on the slave dof, and following the same

steps as for the static case, the

transformation matrix is obtained

� For w=0, it reduces to the Guyan reduction

� With respect to its static counterpart, it is a better candidate for approximating

high frequency motion, though, T depends of the choice of an initial frequency

� As the transformation depends on w, the choice of this value should not be

conducted randomly, but it should constitute a value out of the model’s

frequency spectrum (the free vibration equation with the dynamic stiffness

matrix involved is not satisfied)

� The accuracy of the method is limited to a frequency range around the initial

value selected for initializing the method � has high accuracy for systems under

periodical excitations

• As its name indicates, this method is an effort for improving the already presented

reduction methods, by including the inertia terms of the original model in the definition

of the transformation matrix

• From the dynamic equation of a

free vibrated Guyan reduced model:

• From the definition of the

Guyan transformation:

• Both relations are substituted into

the equation for the lower partition

• For obtaining the following transformation matrix

• The transformation obtained can be expanded,

evidentiating its direct dependence on the static

transformation matrix obtained from Guyan

• Taking the Guyan condensation as basis, the static information is perturbed by introducing

the inertia terms as pseudo-static forces

• The generated reduced model capture the dynamics of the original model better than the

Guyan condensation, as expected from the introduction of a summand (that adds

information to the static matrix) containing the inertia info from the original model

• A dynamic counterpart can be found by taking as basis the dynamic reduction, and, with

analogous passages, we obtain the similar transformation matrix for the dynamic variant:

• The IRS variants shown rely on the reduced models of Guyan or dynamic condensation.

Once the transformation is computed, an improved estimate for the transformation matrix

can be obtained by applying an iteration scheme given by:

• Also known as substructuring � division of a structure into components

• Reduced-order models of the components are obtained and then assembled into a

reduced order model of the entire structure

• The individual substructure models are transformed from physical to component modal

coordinates using a set of chosen basis functions � normal modes (eigenproblem),

constraint modes, attachment modes

• One of the most diffused CMS methods in structural analysis is the Craig – Bampton

Method, based on normal modes and constraint modes

• Primary uses of dynamic substructuring:

- Couple reduced order models of moderately complex structures

- Test verification of FE models of components

- Implement computation of dynamics of very large FE models

• The physical dof’s for a singular component can be partitioned into a set of interior dofs

and a set of interface or boundary dof’s :

Free interface

Fixed interface

Fixed – Interface Normal Modes:

Mass normalised eigenvectors

of the component with the

interface dofs fixed, which

form the columns of the fixed

interface modal matrix ��; a

subset of k modes are kept

Constraint Modes:

Static displacement of the

component due to a unit

displacement of one interface dof

with all other interface dofs fixed,

where �� is a matrix of

displacements of the interior dofs

and Ψ� is the constraint mode matrix

• CM space includes both fixed interface normal modes and static constraint modes

The interior physical coordinates �� are

transformed into the fixed interface modal

coordinates ��, and the physical interface

coordinates �� are retained, but denoted

as constraint coordinates ��

• After transformation, the equation of motion for the generic component remains:

• The component modal mass and stiffness matrices

Where:

�� ��� � constraint modal mass and

stiffness matrices for component

�� � coupling matrix

�� � diagonal matrix of kept modal

eigenvalues

• For the synthesis of two components and �, compatibility of

displacements at the boundary is given by

• Imposing the coupling conditions through a transformation matrix,

the assembled stiffness and mass matrixes are obtained

• Most applications of CMS employ one from two approaches: - Constraint mode: employ constraint and fixed interface normal modes (Hurty and Craig-Bamptom)

- Attachment mode: attachment and free-interface normal modes (McNeal and Rubin)

• Component models based on the used of fixed-interface plus constraint modes are

essentially «superelements» � all physical boundary coordinates are retained as

independent generalized coordinates, greatly facilitating component coupling

• CMS generates reduced order models which capture very well the dynamics of the

original model with relatively few component modes, no wonder is implemented as a

standard and most reliable reduction algorithm in several FE packages

• The components couple in a quite straightforward way, and the sparsity patterns of the

resulting system matrices allow for faster computation times

• The constrained modes embodying the quasi-static effect of high-level normal modes

can compensate the error caused by normal modes truncation at certain extents and

accelerate convergence. The reduced model depends on the sum of the normal and

constrained modes. The more dof at the interface, the more accurate the description of

structural motion.

• The basis is the modal matrix of the discretized FE structure, Φ��� where n is the

dimension of the model, and q the number of computed eigenvectors

• By making use of the modal transformation

(relation between state vector and modal

coordinates) and partitioning, it holds

• From the first line of the equation, the modal coordinates can be expressed in terms of

the master dof, by using the pseudo-inverse definition

• Then, by substituting into the initial modal transformation, the SEREP transformation

matrix is obtained

• Due to the nature of the transformation, an analytical expression for the reduced mass

and stiffness matrices can be derived The reduced system matrices can be computed

directly by using only the eigenvalue data

• SEREP exactly preserves all the modes selected from the original model, and the

reduction’s quality depends on the selection of the full eigenvectors

• It has two main drawbacks:

- Its application is only feasible if the modal matrix of the original model is

available: sparse algebra techniques or lumped mass approximation can be used

to overcome this problem

- The system is truly equivalent when m=q. Different choices can be made, but

careful must be taken.

• The reduction methods presented so far are based on the selection of master and slave dofs,

where

- master are kept after reduction and therefore constitute the dof of the reduced model

- slave are the unwanted dof to be removed from the ODE of the original model

The selection of master and slave dof sets is model-dependent and requires much experience

• Other reduction techniques non dependent on the selection of master dofs have been

originated within control theory, such as Krylov Subspace Method and Balanced Truncation

approach. Though, those methods were initially developed for LTI first order systems

• It is not advised the direct aplication of KSM or BT , since there is no guarantee for

preservation of reduced model structure � Modifications have been introduced to derive

second order methods

• Krylov Subspace: A is a constant nxn matrix, b a nx1 start vector. The Krylov subspace is the

subspace spanned by the q column vectors as defined below

FEW BASIC CONCEPTS

• Given the set of equations of a MIMO system, and its transfer matrix obtained after Laplace

transformationInput matrices

Output matrix

Input , output and state vectors

• Two important quantities can be defined:

System Moments:

coefficients of the

Taylor expansion of

H around zero

System Markov Parameters: replacing the eigenvalue term s by 1/lambda, the

Taylor expansion about lambda=0 yields a different expansion

Reproduces low frequency range

Reproduces high frequency range

• A reduced q dimension subspace is seeked (q<<n) in order to approximate the state space

vector of the original system �

• For then projecting the system in the subspace spanned

by another coordinate transformation matrix

• The KSM algorithm provides certain Krylov subspaces for the assignment of the projection

matrices, such that the moments of the original and reduced model match � they are

INVARIANT parameters

• For an undamped mechanical system, it can be demonstrated that

• The method is implemented by using the Arnoldi algorithm for calculation of basis vectors

and Gram-Schmidt orthonormalization for obtaining the orthonormal basis vectors

• The associated reduced model preserves structure and stability properties of

the original model

• Eigenvalues are well approximated, generally better than the rest of the

reduction approaches already presented � captures very well the dynamics of

the original model

• It is independent from the position of master and slave dof, thus the method

minimizes user intervention and has better numerical properties since the band

diagonal structure of the system matrices is not damaged

• However, the dependence of the Arnoldi algorithm on the definition of a

starting vector leads to the derivation of different projection matrices when

different starting vectors are applied � a suitable algorithm for obtaining a start

vector is essential

• Large dimension FE discretized systems: due to high accuracy required, or geometry

complexity leading to smoother meshes � the efficiency of the reduction is affected by

the system’s large dimension

• The efficiency can be measured through its goodness in capturing the dyamics of the

original model, and the computation time � likewise high for large systems

• For coping with this, the reduction can be accomplished stepwise in two stages:

- In the first stage, a reduction is applied generating a still large reduced order model,

computed relatively fast and able to capture well the dynamics of the original model, up

to certain frequency

- the dimension is still reduced by applying a further reduction method

• Choice of the algorithm for first ROM is determinant for the efficiency of the two step

method � it should be able to create an accurate reduced model with a fairly low

computation time

• The master and slave allocation constitutes an essential aspect for this first stage: the

method should be m non – dependent (to ensure that an inadequate selection of dofs

would not affect negatively the first step) but simultaneously the set of m dof must be

included in the first total set of dof

• A study on a rod was performed by Koutsovasilis and Beitelschmidt, comparing the

different reduction techniques presented

Characteristics of the Model:

- Discretized with FE in ANSYS

- Tetrahedral elements

- A total of 23835 nodes, with UX, UY and UZ

dofs each � n = 71505

- Master nodes are m=10 (black dots)

• For validity check, comparison between full and reduced model is done � it is useful to

compare the eigenfrequencies and eigenvectors of both models by a modal analysis

• For comparison of eigenvectors, the transformation matrix should be available, so as to

compare eigenvectors of the same dimension: expand the reduced eigenvector to the

dimension of the full model or viceversa

• Different Modal Correlation Criteria can be applied in order to make the comparison

• The lower the value, the

better the reduction

method

Modal Correlation Criterion: Eigenfrequency Difference

and Normalized Relative Difference

Modal Correlation Criterion:

Modified Modal Assurance Criterion

• Gives information about the angle of both compared eigenvectors, both mass

normalized and reduced to the same dimension

• 100% means absolute correlation, but a modMAC correlation equal or larger than 80%

indicates a qualitative succesful reduction

• Gives information about the relative vector difference of stiffness-normalized

eigenvectors

• Similar to the normalized relative frequency difference, since both modal stiffnesses

are compared, which are eigenfrequency dependent

• The smaller the value, the best is the comparison

Modal Correlation Criterion:

Stiffness Normalized Vector Difference

Modal Correlation Criterion:

Normalized Modal Difference

• Gives the deviation of single coordinates of compared eigenvectors � isolates the

vector coordinates that result in the worst correlation

• User is able to define at this or near this node position a master dof and reapply the

reduction procedure, by minimizing the bad coordinate correlation

• Based on Modal Scale Factor

• Results are shown for some eigenvectors

Modal Correlation Criterion:

Normalized Modal Difference

Modal Correlation Criterion:

Normalized Modal Difference

• SEREP and Krylov were the techniques delivering better eigenvalues results

• Guyan is the least reliable method for approximation of high frequency motion, due

to its static nature

• IRS has good correlation results for the lower and some higher frequency motions �

by iterative scheme, the IRS transformation matrix can be improved giving good

approximation results

• For CMS, the success depends on the selected number of eigenmodes for the internal

structure (for the rod, 5 eigenmodes were taken) � a selective choice of CB modes

belonging to the whole eigenmodes spectrum (from lower, medium and higher

ranges) seem to improve the results

• In general, the results could have been improved for a different choice of master dofs

� Krylov is a promising reduction method since there is no such dependence, and

the user intervention is minimized, by just deciding the max dimension of the reduced

system

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