The Mechanical Simulation Engine library

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16 Νοε 2013 (πριν από 3 χρόνια και 6 μήνες)

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The Mechanical Simulation Engine library

An Introduction and a Tutorial

G. Cella

General principles

It is a fully tridimensional simulation. In this way it
is possible to give extimates on cross couplings
connected to system asymmetries

It is a modular environment. The system is
partitioned in subunities, and each of them can be
modeled internally in an arbitrary way

The equilibrium working point for the system is
automatically calculated.

“Exact” modelization of internal modes is available
(at least in the frequency domain)

It is (hopefully) easy to use

Developers: G.C. & Virginio Sannibale (Caltech)

Position and of the
orientation of a point.

(6 DOF)

Collection of frames, with
some dynamics

Inertial frame and set of

Basic structure

Simple example: suspended mirror.

System pendulum;

RigidBody mirror;

Wire wire1,wire2;

ForceActuator coil1,coil2,coil3,coil4;

PositionSensor sensor;

We declare the relevant objects:

And we set the relevant
parameters (mass, inertia tensor
components, wire diameter etc.)

Now the system can be constructed. This is obtained clamping frames

Simple mirror: construction















Simulation: structure of the system

The system is partitioned in a collection of
connected frames group

A reference frame is choosen in each group.
This is optimized for numerical accuracy

Each reference frame represent six
independent degrees of freedom. In the
mirror case:

Group 1: fixed inertial frame and frames
attached to it

Group 2: mirror and frames attached to it

Simulation: logical diagram

A prerequisite is the search for
the correct working point

We apply external actions using

Time domain: the action
change at each time step

Frequency domain: phase
and amplitude of the action at
each frequency

We measure system response
using sensors

Time domain: a
measurement at each time

Frequency domain: phase
and amplitude of response at
a given frequency

Simulation: system description

A way to calculate the static forces on the
frames, given their positions. This is used in
working point search

A linearized motion equation

Frequency domain:

Time domain:

Linear relations between and I/O
variables (for actuators and sensors)

Each Object must be able to provide:

Working point search

Why it is important to find the correct working point?

Because the linearized dynamics depends from it:

Tensions (more generally, prestressed elements)

Large deformations

The algorithm can be schematized in the following way:

Fix consistently the position of each frame

Ask each Object to compute its energy, (optionally with
derivatives up to the second order)

Compose these quantities to find the ones associated with the DOF

Update DOF (and frames) using some appropriate algorithm

Go to the point 2 until equilibrium is found

Linear models (1)

The basic principle: linear dynamics is described by a quadratic action, which
can be written as a function of the boundary conditions only.

Example: Longitudinal dynamics of a wire:

The general solution:

Substituting we find the effective action…..

Linear models (2)

All the information is contained in the array K:

In the low frequency regime:

Linear model A

Can be used for:

Longitudinal dynamics of a wire

Transverse dynamics of a wire (tension dominated)

Torsional dynamics of a wire

Result: a 2x2 array which couple the two boundary conditions:

Linear model B

Can be used for the transverse dynamics of a beam

Result: a 4x4 array which couple four boundary conditions:

These effective arrays contains a complete description of the effect of
internal modes (through their dependence on the frequency)

The frequency dependence is NOT polynomial. So it cannot be written in
the time domain as a sum of a finite number of differential operators

Low frequency approximation

The effective arrays works well in frequency domain

What we can do in the frequency domain?

Idea: expand in powers of the frequency:

Stiffness effects

Viscous effects

Mass effects

Now we can interpretate these terms as differential operators, and
write the motion equations of our system in the time domain.

There is something lost?

Yes, the internal modes!

Wire and internal modes

The low
in the frequency
domain: simple

Order 0:
stiffness effects

Order 2:
stiffness & mass

“Finite element” type approach

Wire = many Low
Frequency wires connected together.

Additional degrees of freedom in the time domain

Comparison with FE techniques

The method is better than the traditional FE

Good convergence

No need for adaptive gridding

When the solution of is a good approximation
apart from a region near the attachment point.

This singular behavior is well described by the low frequency
approximation: generally NOT in a “generic” finite element.

Example: LF facility (1)

Actuation: between mirror and reference mass

Example: LFF (2)

Transfer function from the top

Further developments

Extensive validation, in particular for

Time domain dynamics

Object decomposition

Automatic evaluation of thermal noise

Accurate modeling of structural damping in
the time domain

Internal modes of massive bodies (mirrors)