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Message ID: 6195     Entry time: Fri Jan 13 00:51:40 2012
Author: Leo Singer 
Type: Update 
Category: Stewart platform 
Subject: Frequency-dependent requirements for Stewart platform 

Below are revised design parameters for the Stewart platform based on ground motion measurements.

The goal is that the actuators should be able to exceed ground motion by a healthy factor (say, two decades in amplitude) across a range from about .1 Hz to 500 Hz.  I would like to stitch together data from at least two seismometers, an accelerometer, and (if one is available) a microphone, but since today this week I was only able to retrieve data from one of the Guralps, I will use just that for now.

The spectra below, spanning GPS times 1010311450--1010321450, show the x, y, and z axes of one of the Guralps.  Since the Guralp's sensitivity cuts off at 50 Hz or so, I assumed that the ground velocity continues to fall as f-1, but eventually flattens at acoustic frequencies.  The black line shows a very coarse, visual, piecewise linear fit to these spectra.  The corner frequencies are at 0.1, 0.4, 10, 100, and 500 Hz.  From 0.1 to 0.4 Hz, the dependence is f-2, covering the upper edge of what I presume is the microseismic peak.  From 0.4 to 10 Hz, the fit is flat at 2x10-7 m/s/sqrt(Hz).  Then, the fit is f-1 up to 100 Hz.  Finally, the fit remains flat out to 500 Hz.

ground_velocity_spectrum.png

Outside this band of interest, I chose the velocity requirement based on practical considerations.  At high frequencies, the force requirement should go to zero, so the velocity requirement should go as f--2 or faster at high frequencies.  At low frequencies, the displacement requirement should be finite, so the velocity requirement should go as f or faster.

The figure below shows the velocity spectrum extended to DC and infinite frequency using these considerations, and the derived acceleration and displacement requirements.

requirements.png

As a starting point for the design of the platform and the selection of the actuators, let's assume a payload of ~12 kg.  Let's multiply this by 1.5 as a guess for the additional mass of the top platform itself, to make 18 kg.  For the acceleration, let's take the maximum value at any frequency for the acceleration requirement, ~6x10-5 m/s2, which occurs at 500 Hz.  From my previous investigations, I know that for the optimal Stewart platform geometry the actuator force requirement is (2+sqrt(3))/(3 sqrt(2))~0.88 of the net force requirement.  Finally, let's throw in as factor of 100 so that the platform beats ground motion by a factor of 100.  Altogether, the actuator force requirement, which is always of the same order of magnitude as the force requirement, is

(12)(1.5)(6x10-5)(0.88)(100) ~ 10 mN.

Next, the torque requirement.  According to <http://www.iris.edu/hq/instrumentation_meeting/files/pdfs/rotation_iris_igel.pdf>, for a plane shear wave traveling in a medium with phase velocity c, the acceleration a(x, t) is related to the angular rate W'(x, t) through

a(x, t) / W'(x, t) = -2 c.

This implies that |W''(f)| = |a(f)| pi f / c,

where W''(f) is the amplitude spectral density of the angular acceleration and a(f) of the transverse linear acceleration.  I assume that the medium is cement, which according to Wolfram Alpha has a shear modulus of mu = 2.2 GPa and about the density of water: rho ~ 1000 kg/m3.  The shear wave speed in concrete is c = sqrt(mu / rho) ~ 1500 m/s.

The maximum of the acceleration requirement graph is, again, 6x10-5 m/s2 at 500 Hz..  According to Janeen's SolidWorks drawing, the largest principal moment of inertia of the SOS is about 0.26 kg m2.  Including the same fudge factor of (1.5)(100), the net torque requirement is

(0.26) (1.5) (6x10-5) (100) pi (500) / (1500) N m ~ 2.5x10-3 N m.

The quotient of the torque and force requirements is about 0.25 m, so, using some of my previous results, the dimensions of the platform should be as follows:

radius of top plate = 0.25 m,

radius of bottom plate = 2 * 0.25 m = 0.5 m, and

plate separation in home position = sqrt(3) * 0.25 m = 0.43 m.

 

One last thing: perhaps the static load should be taken up directly by the piezos.  If this is the case, then we might rather take the force requirement as being

(10 m/s2)(1.5)(12 kg) = 180 N.

An actuator that can exert a dynamic force of 180 N would easily meet the ground motion requirements by a huge margin.  The dimensions of the platform could also be reduced.  The alternative, I suppose, would be for each piezo to be mechanically in parallel with some sort of passive component to take up some of the static load.

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