The Mechanical Simulation Engine library

1
The Mechanical Simulation Engine library

• An Introduction and a Tutorial
• G. Cella

2
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)

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Basic structure
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Simple example suspended mirror.
We declare the relevant objects
System pendulum RigidBody mirror Wire
wire1,wire2 ForceActuator coil1,coil2,coil3,coil4
PositionSensor sensor
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.
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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))
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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

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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

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Simulation system description
Each Object must be able to provide

• 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)

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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

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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..
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Linear models (2)
All the information is contained in the array K
In the low frequency regime
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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
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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

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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!
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Wire and internal modes
The low frequency approximation in the frequency
domain simple pendulum.

• Order 0 stiffness effects only
• Order 2 stiffness mass effects

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Finite element type approach

• Wire many Low-Frequency wires connected
  together.
• Additional degrees of freedom in the time domain

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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.
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Example LF facility (1)
Actuation between mirror and reference mass
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Example LFF (2)
Transfer function from the top
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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)