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
objects
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
together.
Simple mirror: construction
PD.connect(wire1.frame(0));
PD.connect(wire2.frame(0));
PD.connect(coil1.frame(0));
PD.connect(coil2.frame(0));
PD.connect(coil3.frame(0));
PD.connect(coil4.frame(0));
PD.connect(sensor.frame(0));
PD.connect(wire1.frame(1),mirror.frame(0));
PD.connect(wire2.frame(1),mirror.frame(0));
PD.connect(coil1.frame(1),mirror.frame(0));
PD.connect(coil2.frame(1),mirror.frame(0));
PD.connect(coil3.frame(1),mirror.frame(0));
PD.connect(coil4.frame(1),mirror.frame(0));
PD.connect(sensor.frame(1),mirror.frame(0));
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
actuators
•
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
step
•
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:
1.
Fix consistently the position of each frame
2.
Ask each Object to compute its energy, (optionally with
derivatives up to the second order)
3.
Compose these quantities to find the ones associated with the DOF
4.
Update DOF (and frames) using some appropriate algorithm
5.
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
frequency
approximation
in the frequency
domain: simple
pendulum.
•
Order 0:
stiffness effects
only
•
Order 2:
stiffness & mass
effects
“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
approach:
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)
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