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Flexures

date2024-03-24tags:meche:

Idea: Gravity compensated robotic arm using flexures, all made from acrylic

Laser-cut flexure stages

Flexure stages allow accurate linear motion with zero backlash and no moving parts. Professional stages are wire-EDM-machined from solid metal, but simple versions can be laser-cut from plywood or acrylic.

A central block attaches to the frame with four thin arms that constrain motion to a straight line. Two M3 nuts glued into the frame reduce screw backlash. A screw bears on the central block. Hot glue holds everything in place. Overall dimensions are roughly 30x45mm.

https://www.youtube.com/watch?v=YEz-r8KDY-0 https://www.youtube.com/watch?v=H2gJuuhCV7w https://web.mit.edu/mact/www/Blog/Flexures/FlexureIndex.html https://www.youtube.com/c/TheFACTsofMechanicalDesign/videos

https://www.youtube.com/watch?v=rUH3cDmeXC4 https://www.youtube.com/watch?v=ahpqYAT90MA https://www.youtube.com/watch?v=VtxsKUgqP8M&pp=ygUSZmxleHVyZSBtZWNoYW5pc21z

https://www.youtube.com/watch?v=i8Ad-qi9q7A https://lasercutlikeaboss.weebly.com/uploads/2/7/8/8/27883957/advancedjoinery_master_web.pdf https://www.youtube.com/watch?v=lxlXXPj0LJs

https://reprap.org/wiki/Compliant_Linear_Motion_Mechanism_1

http://micro.seas.harvard.edu/publications.html http://robotics.eecs.berkeley.edu/~ronf/PAPERS/ahoover-icra06.pdf http://www-personal.umich.edu/~btrease/share/trease-design-of-large-disp-compliant-joints.pdf http://contentdm.lib.byu.edu/cdm4/item_viewer.php?CISOROOT=/ETD&CISOPTR=1024&CISOBOX=1&REC=5 http://research.et.byu.edu/llhwww/downloads/wyrd.htm http://research.et.byu.edu/llhwww/powerpack.htm http://robotics.eecs.berkeley.edu/~ronf/Prototype/ http://www.seas.gwu.edu/~kjlu/gallery.html http://dev.forums.reprap.org/read.php?1,31371

Introduction

Flexures are bearings that allow motion by bending beams. The arrangement of beams can be designed to be compliant in its degree(s) of freedom (DOF), but relatively stiff in its degree(s) of constraint (DOC). This provides motion in desired directions and constraint in others. Flexures allow stiction-less, controlled, limited-range motion in precision machines. This entry covers the most important parameters. Flexures require: desired kinematics, range of motion, stiffness, load capacity, repeatability, mode shapes/frequencies, and an error budget.

Designing flexures involves modeling, FEA, and experimental verification. A hand calculation shows which variables matter most and gets you near the right dimensions. How short can I make this beam so that it can be at 30% of the yield stress? The model should answer questions like that, keeping real-world constraints in mind.

After applying these equations, test the system in a MATLAB script or Excel spreadsheet. Change one variable at a time to see how it affects the entire system. Flexures are typically modeled using FEA, which is the subject of another entry.

Desired Kinematics

First, identify the degrees of freedom in flexure design. These are the desired directions of travel. You can also find constraints first by identifying which directions should not move. A rigid body has six degrees of freedom: three for translation and three for rotation. Well-designed flexures are orders of magnitude stiffer in constrained directions than in free directions.

Characterize the degrees of freedom next. Specify the required range of motion to aid the choice of flexure topology.

Concept Generation

Generate flexure concepts once kinematics are identified. Methods include: using old solutions, constraint-based design, topological synthesis, FACT (Freedom, Actuation, and Constraint Topologies), and Transmission Theory.

Using an old solution

Several flexure designs exist already. A list appears in "The Handbook of Compliant Mechanisms." Linkages can transform into flexures using the pseudo-rigid body model (PRBM) by Larry Howell.

Constraint-Based Design

Constraint-based design (CBD) generates compliant mechanisms through flexural building blocks. Flexural beams with intersecting action lines form an instant center for rotation. CBD is limited to simple motions — pure translations and rotations. It cannot generate compliant screw mechanisms where translations and rotations are coupled.

Topological Synthesis

Topological synthesis uses computers to iteratively find flexure topologies that meet displacement and force requirements. Optimal solutions are mesh-dependent and not unique. The user may not understand why the flexure has a particular shape or how to modify it.

FACT

FACT (Freedom, Actuation and Constraint Topologies) is a flexure design method created by Jon Hopkins.

The image shows topologies for flexures in series or parallel. Before FACT, flexure design relied on experience and creativity.

First identify the desired degrees of freedom and pick the chart column. Then find a topology for your specific degrees of freedom. A topology under the "parallel pyramid" boundary exhibits all degrees of freedom in one arrangement, which is usually the simplest and most compact design. Designing parallel flexures are covered in these two papers: part 1 and part 2. Part 1 covers FACT theory and part 2 covers implementing FACT in practice. A serial flexure design is necessary when there is no parallel flexure arrangement that meets the desired degrees of freedom (e.g. 3DOF translation). Serial flexures are covered here.

Red lines indicate freedom and blue lines indicate constraint. Green lines indicate screws and orange lines (not shown) indicate wrenches. The details of screws and wrenches are beyond the scope of this entry; for more information, see Hopkins' Master's thesis.

FACT theory does not always work in practice. Real-world constraints on size, frequency, and manufacturability guide the flexure selection. Symmetry is one constraint: redundant flexures may be needed to prevent thermal expansion from affecting output position.

Transmission Theory

Transmission theory is something that I developed based on a paper by Jon Hopkins and Bob Panas for designing flexures that transform an input motion into an output motion of a different type, transmission ratio, or axis. For more information, see Chapter 2 of my Master's thesis. 4 Range of motion

Flexures have a finite range of motion set by their length and maximum allowable stress. Relate critical stress to bending moment, bending moment to applied load, and load to displacement via stiffness. This leads to an equation for range of motion for simple cases of translation: δmaxL∼(σmaxE)(Lh),

where δmax is the maximum displacement, σmax is the maximum allowable stress, E is the Young's Modulus of the material L is the beam length, and h is the beam thickness in the bending direction. The exact equation includes a constant of proportionality that depends on the specific arrangement of beams. For most metals, σmax/E∼10−3, and for most plastics σmax/E∼10−2. Typically L/h∼100. If L/h<∼10, beam theory no longer applies, and if L/h>100, then buckling becomes a more significant issue. The thickness to width ration is typically about 10 for good blade flexure constraint. Thus, for metal flexures δmax∼0.1L. This result can be used to generate dimensions that are in the right ballpark. Take an example of a translation flexure with a desired range of motion of ±

0.25 inches. For a 4-bar translation flexure, a good place to start would be with metal flexures that are about 2.5 inches long, 0.025 inches thick made out of stock that is about 0.25 inches thick. This is not optimal, but is in the right order of magnitude. Also, it's always a good idea to implement hard-stops so that the flexure is ensured to not travel too far. 5 Stiffness

The actuator provides a limited load, thus a maximum DOF stiffness is required to achieve the desired range of motion. Flexures are designed to minimize stiffness in their DOF and maximize stiffness in DOC. Translational stiffness ratios are often represented by a stiffness ellipsoid. Large networks of beams may have their stiffness represented by stiffness matrices. They may also be represented by homogenous transformation matrices (HTMs). More generally, the stiffness of a structure may be found by the Principle of Virtual Work. There are also many beam bending shortcuts that may be used, including putting springs in series or parallel. The stiffness of flexure modules have been explored in a paper by Shorya Awtar et. al. There are also closed-form nonlinear models of flexure stiffness.

Beam behavior is well understood. Key equations cover stiffness, buckling, and stress. k=CEIL3, where I is the second moment of area given by I=∬Ay2dA

And of course, σ=McI.

The critical buckling load is given by Pcr=Ï€2EIL2e Use the chart below to find the effective length, Le=KL

. This chart was adapted from the Steel Construction Manual from the American Institute of Steel Construction.

DOC stiffness changes dramatically with displacement, covered in Design Principles for Precision Mechanisms and Shorya Awtar's PhD thesis. The stiffness of an axially loaded straight beam is ka0=EAL, where A is the cross-sectional area. The axial stiffness changes when the beam deflects by an amount Δx, becoming ka(Δx)=ka011+3175(Δxh)2.

Axial stiffness drops off dramatically after about 1 beam thickness and equals transverse stiffness after 7 thicknesses. Measure constraint stiffness over the full displacement range.

Load Capacity

Flexure load capacity is limited by strength and stability. Stress must stay below a critical level under maximum load, and beams must not buckle. Model both with FEA and implement hard-stops.

Dynamics

Design the flexure to achieve appropriate mode shapes and frequencies. Actuator noise, electrical noise, and sudden impulses can excite resonant modes, which must be avoided. Also, flexures are lightly damped (ζ≈0.01 ) which means that the vibrations take a long time to die out, and that they have a large dynamic amplification factor at resonance. The dynamic amplification factor is a ratio of the dynamic displacement (zdynamic) to the static displacement (zstatic), which changes with frequency (ω). This animation may be viewed for a more intuitive understanding of how the displacement changes with frequency. For a 1-DOF underdamped second-order system, zdynamiczstatic=1[1−(ωωn)2]2+[2ζωωn]2−−−−−−−−−−−−−−−−−−−−√, where ζ is the damping ratio and ωn is the natural frequency given by ζ=c2km−−−√,ωn=km−−−√, where m is the mass, c is the dashpot constant, and k is the spring stiffness. The maximum dynamic amplification factor occurs at ωpeak: ωpeak=ωn1−2ζ2−−−−−−√≈ωn, and (zdynamiczstatic)max=12ζ1−ζ2−−−−−√≈12ζ. Meso-scale (10s cm-scale) flexures typically have ζ≈0.01, so when they are actuated at resonance, they have a dynamic amplification factor ≈50

. There are some ways to fix this: actuate it at a much higher frequency (or move the resonant frequency much lower) to make the dynamic amplification factor < 1, and the damping may be increased by using a thin film of bearing grease between the output stage and ground. See Chapter 4 of my Master's thesis for a way to measure the damping ratio by taking the log decrement.

The frequency domain is useful for known noise frequencies. The time response is more useful for sudden excitations like impulses or steps. The step response and impulse response can show how long it takes for noise to die out, and the amount of overshoot.

Also, plastics are viscoelastic, so other dynamic issues to consider are creep and stress relaxation. This behavior can be quite problematic for a micro-positioning stage with a force-controlled actuator, for example. The flexures themselves may not be made from plastic, but viscoelasticity from epoxy can affect the overall dynamics.

MEMS resonators involve many damping mechanisms: thermoelastic damping, phonon-phonon scattering, anchor loss, surface effects, and electrical losses. They may also be thermo-mechanically actuated, so the thermal time-constant is important as well. See papers by Marc Weinberg, Amy Duwel, and Rob Candler on MEMS damping mechanisms, and see papers by Shi-Chi Chen and Martin Culpepper on thermo-mechanical actuation.

Fatigue

Fatigue analysis ensures infinite life. Steel flexures should stay below the endurance limit, preferably stainless to avoid rust. Lab flexures are often aluminum for easy machining, but aluminum has no endurance limit. Commercial devices require fatigue analysis. If σ<σy/3, the flexure lasts >10^6 cycles. See Shigley's Mechanical Engineering Design for more.

Repeatability

Flexures alone may have Å-level repeatability. To maximize precision, keep σ<σy/3. The motion repeatability is limited by the whole system: actuator precision, sensor accuracy, material hysteresis, and bolted joint constraints. If the flexures are assembled with bolted joints, there may be hysteresis caused by micro-slip at the interface. Micro-slip is a friction-based phenomenon where parts move by ~10s of microns at a force < μstaticN

. The surface asperities need to be crushed to mitigate micro-slip, so the compressive stress should be high. This may be achieved by tightening the bolts (high force), and placing the bolted joints on washers (small area). This micro-slip may be measured by applying a required load to the flexure and measuring the displacement of the bolted joint ("ground").

Error Budget

Flexures have errors that must be bounded. Geometric errors come from fabrication imperfections and thermal expansion. Thin flexures have small thermal time constants. Actuator accuracy, sensor accuracy, and rolled material anisotropy also contribute. The material used may not have the average material properties. For example, materials that have been rolled (e.g. sheet metal) are anisotropic (i.e. the stiffness will depend on direction). These errors should be taken into account when modeling the flexures.

Optimization

Optimize flexures for: minimizing DOF stiffness, maximizing DOC stiffness, maximizing load capacity, distributing mass, minimizing stress, and maximizing range of motion. Tools like MATLAB and SolidWorks Design Studies help. The design is fine if it meets functional requirements.

Fabrication

Fabricate flexures by wire EDM (~±1 micron), Waterjet (±0.005 inches), CNC milling, or laser cutting. Wire EDM is preferred for precision. CNC milling is tricky because cutting forces deflect thin beams, giving tolerances similar to Waterjet. Laser cutting can cause beams to stick to sidewalls or break when removing parts. For MEMS, fabrication is complex and error-prone. Stages may be purchased from Motus Mechanical, Riverhawk, Thorlabs, and Bal-Tec.

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