Flexure Characterization and Design

Flexure Test

Overview

Flexures are deformable solid bodies used to connect elements in a mechanical system. This flexibility allows for greater freedom of motion of the parts relative to each other than a rigid joint does, but at the cost of complicating the dynamics of the system. As one can imagine, it is important to know the properties of the flexures in order to predict and control the behavior of a system. This project is primarily focused on the flexures used in the PPOD projects in LIMS which are used to connect the linear actuators to the table. In this case the flexures allow the table to move in all six degrees of freedom (three translational and three rotational) which the use of rigid joints would not allow. The goals of this project are to be able to test the performance of the existing flexures and to use this information to design new ones to improve the performance of the PPOD.

Current Design

Current Flexure

The flexures currently on the PPOD are made of a 1/4" Tygon tubing glued to aluminum mounts. Different lengths have been used in the past, varying from 0.5" to 1" of exposed tubing and overall the plate behavior is easier to control, especially at high amplitudes, with shorter flexures.

Hardware

Flexure Test Setup

There are three basic pieces of hardware needed to characterize a flexure: a forcing mechanism that can move in all six degrees of freedom, accelerameters to collect motion data, and a six-axis force sensor to collect the forces and moments. The PPOD was used to force the flexures because it did not require building a separate system, it already collects acceleration data of the plate, and it can control motion in all six degrees of motion at frequencies and amplitudes that flexures will be used at.

Force Sensor mount

Force Sensor Mount

In order to attach the flexure and force sensor to the PPOD, we had to design a detachable mounting system. To make it light enough that the PPOD would not have much difficulty in controlling it, the arms were made from aluminum honeycomb and glued in the center to an aluminum plate where the force sensor attaches. This central plate is to make the mount more rigid as the arms are separate pieces. Three C-clamps are used to attach the mount to the PPOD plate, one at the end of each arm.

Force Sensor

Force Sensor

The force sensor used is an ATI Industrial Automation Mini 40 Force/Torque Sensor, model US-10-20. The data sheet can be directly seen here or found on ATI's website.

Rigid Mount

Rigid Mount for Free End

The free end of the flexure must be mounted to a rigid body in order to measure the force the flexure is applying to the plate. This is currently a large T of steel that fits over the PPOD and the plate that the flexure is mounted to can be clamped to the bar. The height and position of the plate can be adjusted by adding spacers between the plate and the mount.

Data Collection

Data collection is relatively easy and is done through the Flexure_GUI shown below. This graphical user interface shows the controller information on the left side and the force sensor information the right.

• Flexure Frequency Response - This button starts the frequency test for a flexure. After the flexure is mounted and connected to the force sensor, and you have checked all the options, press this to begin. If this is the first test then the controller will ask to perform a frequency sweep, it is recommended to do so.

• Plate Motion - These are the important input parameters for flexure testing.
  • f - A vector of frequencies to test the flexure at. The program sweeps through this range in one direction before moving on the to next direction. The range of usable frequencies depends on the amplitude, but for the default amplitude is 15 to 55 Hz. Above this and the controller will not be able to control plate motions.
  • A - These are two vectors for the linear and angular accelerations of the plate, respectively. These have not been tested above default values for the range of usable frequencies.

• Sampling - Decreasing these will speed up computation time but may cause the program to crash if too low.
  • Samples/second - Adjusts the number of samples per second. Maximum value for current DAQ hardware is around 10000 so it is recommended to stay with the default 9000.
  • Transient cycles - Number of cycles to wait after a controller update before collecting data to allow for the transient response to die out.
  • Logged cycles - Number of cycles to collect data. If this is increased you should also increase the transient and processing cycles to allow for the extra computation time.
  • Computation cycles - Number of cycles to wait before the controller update to allow for computation time.

• Force Signals - These six plots show the data collected from the force sensor during one period. The left column is for forces in the linear direction shown in N and the right column is for the moments about the coordinate axis in Nm. The high frequency noise is expected and this is filtered out by the program when it does computations.

• Controller Information - found on the left half
  • Control Signals - These plots show the input voltage to each of the actuators over one period of the plate motion. Set the saturation voltage level using the text box in the central pane. Default is 4V and should never exceed 8V, especially for high amplitude driving.
  • Plate Signals - These plots show the desired and actual plate motions over one period. The left column displays the linear accelerations in $m/s^{2}$ and the right column is the angular accelerations about the coordinate axes in $rad/s^{2}$. Adjust the "Max Number of Updates" box in the central pane to the allow the controller to have enough time to match the match the desired response at a given frequency.
  • Highest Controller Harmonic - Adjust this so the controller will control the harmonics of the input frequencies. This can help get rid of noise and give better responses.
  • Controller Gain - This is the gain of the controller. Increase this to increase the response of the controller, however this should not be increased along with the saturation voltage.
  • Initial Control Signals - Adjust these for different initial guesses for the control signals.
    • Previous signals - Takes the last used signals as the starting point for the next response it tries to control.
    • Freq response based guesses - The controller will run a sweep of the input frequencies and their harmonics to make guesses for controller signals. This is the recommended setting and will generally control the fastest.
    • Zeros - This starts the controller from nothing so it will take longer to find good controller values.
  • ||e|| tolerance - The error tolerance. When the controller error drops below this number it means the response is "close enough" to the desired response that it can move on to the next response.

• Frequency Response - This allows you to manually generate and view the frequency response of the system. The test allows you to input a range of frequencies to test as well as the actuator voltage.

Transfer Function Fitting

Until now, only a simple approximation has been used to describe the flexures when modeling the system. In reality, a flexure will have a mass, damping, and spring matrix associated with it that maps its motion to the forces it applies.

${\overrightarrow {f}}=M{\ddot {\overrightarrow {x}}}+B{\dot {\overrightarrow {x}}}+K{\overrightarrow {x}}$

Here ${\overrightarrow {f}}=[f_{x},f_{y},f_{z},M_{x},M_{y},M_{z}]^{T}$ is a vector of forces and moments and ${\overrightarrow {x}}=[x_{x},x_{y},x_{z},\theta _{x},\theta _{y},\theta _{z}]^{T}$ is a vector of coordinates. This leads to a transfer function from input acceleration $i$ to output force $j$ given by:

$F_{j}(s)={\frac {M_{ji}s^{2}+B_{ji}s+K_{ji}}{s^{2}}}A_{i}(s)$

We then have to fit this model to each combination of the accelerations and forces for a total of 36 transfer functions, one for each entry in the mass, damping, and spring matrices. For this we used the MATLAB code found below.