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)

3. Inertial frame and set of objects Collection of frames, with some dynamics Position and of the orientation of a point. (6 DOF) Basic structure

4. 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.

5. 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));

6. 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

7. 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

8. 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:

9. 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

10. 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…..

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

12. 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:

13. 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

14. 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!

15. Wire and internal modes The low frequency approximation in the frequency domain: simple pendulum. • Order 0: stiffness effects only • Order 2: stiffness & mass effects

16. “Finite element” type approach • Wire = many Low-Frequency wires connected together. • Additional degrees of freedom in the time domain

17. 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.

18. Example: LF facility (1) Actuation: between mirror and reference mass

19. Example: LFF (2) Transfer function from the top

20. 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)