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Mechanisms for Lowering Tethered Payloads:
Design and Analysis.
~mike gradziel.

Between 2005 and 2007 I designed a lowering device for JPL. My objective was to come up with a compact, lightweight mechanism that could lower NASA's next Mars Rover, a monstrous vehicle weighing 3 kN, by a distance of 7 meters to leave it hanging on its tether beneath the rocket-powered descent craft that would then set the rover down on Mars like a helicopter delivering cargo. This approach to landing a spacecraft would work on rough terrain and would deliver the rover alone - without any heavy and costly landing gear - to the surface of Mars.

We already have lowering devices that can do this sort of thing. Almost every crane or hoist serves regularly as a lowering device, mountain climbers and rescue teams employ lowering devices to climb down steep walls, scientific equipment lifted high into the atmosphere by balloon is sometimes deployed by lowering device, and people harnessed to lowering devices leap from buildings for film stunts or sport. NASA even sent lowering devices to Mars on its last three landing missions, and they all worked. My problem was that the new Mars project needed something special: a lowering device so predictable it would deploy within 15% or less of its targeted time, while delivering a precisely controlled descent speed profile that would end with a nice soft stop at full tether extension.

drawing of a Mars Lander deployed by a Lowering Device

Fifteen percent uncertainty seems ample at first but it turns out that nearly every kind of lowering device we already have uses intelligent control, friction, or both. Intelligent control - whether by human or computer - was out of the question, and friction is uncertain on the order of 100% given the conditions between manufacturing and Mars, so achieving fifteen percent uncertainty with a friction brake was going to be a challenge. With no fuel to spare due to the immense cost of launching extra fuel to Mars, deployment would need to be complete in the absolute minimum time possible.

In the pursuit of this requirement and others that limited loads, speeds, mass, volume, and materials, I acquired a great deal of engineering knowledge and, even after the Project switched to a different type of brake to reduce reliance on friction, I prepared some useful designs on my own just for the sake of completeness. JPL and NASA maintain technical documents relating to lessons learned from past work and and design specifics of spacecraft and research projects, but for the greater good I've presented here some general engineering lessons I've come to understand after becoming interested in this topic.

Drag Device Basics and Viscous Fluid Dampers

Running a rope through a friction block or around a capstan post is a fine way to control the lowering of a weight. If something more reliable but equally simple is needed, one can wrap the rope on a spool that drives a paddle wheel or fan - a big one, a meter or more across for lowering the weight of an average person. This has been done for decades, evidently at first to help teach paratroopers how to land following a jump, and since air is the working fluid to which energy is rejected as heat, in an open area these devices - called "fan descenders" are very reliable and predictable. If the spool is wrapped with a single layer of rope and is cylindrical in shape, lowering will proceed at constant speed with the fan brake.

If the spool has a single layer of rope and tapers from a larger starting diameter to a smaller ending diameter, the lowering speed will decrease through the course of deployment as a function of the taper, the load inertia, and the spool and brake inertias. Unfortunately, fan descenders are big and heavy and they aren't the best solution for many projects - especially those that need to work where the air is very thin, like at high altitude or on Mars. Fortunately the basic concept of a tapered spool used with fan descenders to slow the rate of descent towards the end for a soft stop will work with any speed-dependent drag brake. Like fans that stir the air, much smaller viscous fluid dampers instead stir oil or a similar fluid in a closed chamber.

There are many types of viscous fluid dampers, ranging from fan-like vane dampers to concentric shells that shear the oil as they rotate; all of these types suffer from the changes in drag that result from heating of the fluid. For some projects such changes might be good to have, but in general the performance of viscous fluid dampers is difficult to predict and difficult to maintain in changing environmental conditions. Viscous fluid dampers are used in many bicycle trainers and other exercise equipment. Some devices also have magnetic particles suspended in fluid, allowing direct control over the level of drag by adjusting a magnetic field (magnetic particle brakes, widely used to test motors and engines, dispense with the fluid entirely and create drag by friction from an agitator sweeping through a chamber full of magnetic particles which are stuck together with a force dependent on the applied magnetic field). High-tech viscous fluid dampers control unfolding solar arrays on sattelites to keep them from swinging through their stops, or lower scientific instruments beneath atmospheric balloons. Some can lower heavy loads (weighing many kilonewtons) or pay out great lengths of cord (many kilometers). But the equations used to predict viscous drag are complicated and while detailed properties of viscous materials are readily available from many manufacturers, precise determination of how a design will perform is really only possible by doing a comprehensive development and test program.

Centrifugal Friction Brakes

Brakes that use friction in the traditional solid-against-solid sense (after all, viscous fluid dampers dissipate energy by friction) don't offer much improvement in predictability - if any - compared to fluid dampers but they are easy to design and build, and for many applications the reliability and stopping power of a friction brake will far surpass non-friction alternatives that meet mass and cost targets. For a passively operable friction brake to have strong speed dependence, the design must use centrifugal force. Centrifugal brakes, fly-weight brakes, and centrifugal clutches appear in weed trimmer drives, go-kart transmissions, model helicopter gearboxes, fishing reels, chain saw drives, and many other machines. Used as brakes, these mechanisms provide a braking force commensurate with the rate of speed; used as clutches, they allow idling at low speed, slip at excessive torque, and effective power transmission under ordinary conditions. A great deal of research and development has gone into friction materials used in all sorts of industrial, automotive, and aerospace applications, but one must still have a sound understanding of how friction materials behave before designing a mechanism reliant on their properties.

Friction – the drag between materials due to interaction at microscopic levels – depends on so many factors that it's difficult to predict during transient events especially when they happen on Mars after a period of disuse or exposure to varying levels of vibration, humidity, temperature, and atmospheric pressure and composition. Virtually every molecule present at a friction interface has some effect on the friction level, whether it is a gas in the surrounding atmosphere, a smudge of oil, a hydrocarbon in a brake pad, or a haze of oxide formed on a brake drum. Removal of water from friction lining containing graphitic carbon may lead to an increase in friction, and exposure of the braking interface to water may lead to reduced friction by causing metallurgical changes in the iron components of a brake system: hydrogen from water (or from the friction material) may diffuse into the iron and chemically change iron carbide particles, leading to the formation of metal foils that degrade brake performance. Oxidation of exposed metal surfaces, or the loss of the inevitable film of oxide that appears when metal is exposed to air, is known to have significant effect on the friction between metals and the accumulation of a stable “transfer layer” of wear debris has great effect on both the level of friction and the rate of wear.

Common automotive brake materials have been carefully fomulated to be self-renewing and to balance tolerance of water, dirt, and salt with effectiveness in providing drag even when used by unknowing operators. The manner and frequency of use is very important to friction brakes. It cleans the friction surfaces, renews the fine layers of material that provide gripping power, and gives an indication of the health of the brake. Use also alters brakes in unpredictable ways: heat generated by braking feeds chemical reactions in the cocktail of materials present at the friction interface, changing alloys and polymers and expelling moisture and other volatile materials.

Most industrial and automotive brakes use a resin-bound pad containing various metal and mineral powders, chips, and fibers run against a cast iron brake drum or rotor. Cast iron is widely used because it has excellent vibration damping properties, good thermal conductivity, a high melting point, good wear and abrasion resistance, fair corrosion resistance, and low cost. High performance brakes like those used on jet aircraft, the space shuttle, and racecars often have ceramic brake rotors, brake pads, or both; metal matrix composites, sintered metals, or carbon-carbon composites. These high performance materials often have high wear rates and may be highly sensitive to surface preparation. Carbon-carbon composite brakes in particular are said to exhibit widely varying performance dependent on fine characteristics of brake rotor surface finish, brake temperature, and antioxidant coatings applied to various parts of the brake.

Research of friction materials used in space has indicated strong effects from drying and from the accumulation of wear debris during extended use in deep vacuum. In the 1980s, anomalous behavior of friction clutches and brakes was observed in the space shuttle Remote Manipulator System (SRMS) and, after a series of investigations, was attributed to a loss of moisture and to the altered properties of the dry wear debris layer at the friction interface. Other studies of friction materials have investigated stability over time of materials run against different metals (cast iron, for example, showed good stability while chromium plating showed persistent instabiltiy) and of materials run at different operating temperatures. Most friction brake materials are products of years or even decades of development and are unique to the extent that no study of their composition can explain differing performance among similar materials from different manufacturers or even among different runs from the same manufacturer.

test data strung together to show cumulative effects of use and power on drag
Centrifugal friction brake drag test data strung together to show effects of use on drag coefficient.

Standardized tests for measuring the friction coefficients of industrial and automotive friction materials include specific procedures for cleaning and conditioning friction surfaces. SAE-J661, “Brake Linings Quality Control Test Procedure,” specifies a series of preparations, conditioning periods, and test runs all with precise temperature controls and geometric constraints to ensure uniform results.

Many automotive enthusiasts are aware that brake pads and rotors must be “bedded in” following a new installation and periodically thereafter, depending on the nature of use, to maintain good brake performance. Friction brakes used in a lowering device should be prepared following such procedures to increase their chance of operating reliably. If a lowering device is allowed to have a wide performance range, the advantages of centrifugal friction brakes can be very attractive.

Informed selection of friction material and extensive testing and preparation of the brake can improve performance stability enough to support many lowering device applications. An optimization of the brake design will bring further benefits. Care must be taken at all times to control handling of mechanism parts, cleaning of brake drums, and exposure to dirty environments containing things like oil vapors or dust.

The design of a centrifugal friction brake should start with mass and volume targets, drag coefficient, and the amount of uncertainty tolerable in the context of expected conditions and the stability of friction materials that will perform well in the planned use. Some friction materials need to heat up before they become grippy; these would not work for a short duration braking event. In general, the larger in diameter a friction brake is, the more mass efficient it will be but its inertia will also be larger and this affects how fast it spins up and slows down, what load spikes appear in the tether at changes of speed, and what its angular momentum does to the vehicle that carries it. The smallest, lightest brake one can get away with is usually the best for many reasons.

a leading shoe centrifugal friction brake design

a leading shoe centrifugal friction brake design

Illustrations of a leading shoe centrifugal brake design (brake drum not shown).

Centrifugal brakes come in two flavors: leading shoe and trailing shoe. Inside a non-rotating brake drum, a hub rotates and shoes attached to it are flung outward to press against the drum, creating drag. The brake pads usually go on the shoes. If the shoes slide radially outward in slots or an equivalent guiding feature, they are considered trailing shoes. Similarly, if the shoes are pivoted about pins through the hub and the hub rotates such that the shoes trail behind their pivots, the brake has trailing shoe configuration.

If the shoes are pivoted or otherwise supported such that they are pushed around the drum ahead of their pivots, the brake has a leading shoe design. In a leading shoe design, wedging between the shoes and drum directs a portion of applied torque radially into the drum, increasing brake pad loading beyond what is achievable from centrifugal force alone.

This produces higher drag from a brake of particular mass and dimensions, but the wedging also causes the brake to be susceptible to jamming at elevated friction and in general makes it more sensitive to change in friction. The degree of wedging can range, by design, from almost none to so high that the mechanism will physically break before it releases and starts to rotate.

Wedging action can increase brake drag by 2 times or more without presenting much threat of jamming and with very little change in the mechanism mass, if a stable friction material is used. The equations that follow are specifically for leading shoe centrifugal brakes, but if a trailing shoe design is desired the equations may be easily rearranged to study it. A design must start with selection of a brake drum diameter, shoe mass, and number of shoes:

P_radius (m, brake drum diameter)
shoe_mass (kg, per shoe including the brake pad)
n_shoes (at least 3 on a single hub, for dynamic stability)

Next, a pair of friction materials for the brake pad and drum should be selected and, considering their stability, a degree of wedging should be chosen as a starting point:

pressure_angle (greater than zero, less than pi/2; maximum wedge action at pi/2 radians)
shoe_angle (around pi/2 radians is best; more than that increases wedging up to the theoretical maximum of pi radians)

Other dimensions are calculated from these initial parameters:

contact_angle = pi/2 - pressure_angle
P_theta = pi/2 - contact_angle - shoe_angle
pivot_r = P_radius*sin(contact_angle)/sin(shoe_angle)
Lshoe = sqrt(pivot_r^2 + P_radius^2 - 2*pivot_r*P_radius*cos(P_theta))

These values will establish the basic dimensions of the shoe. Next, mass must be placed as efficiently as possible: the shoe mass center should be as far radially out as possible, and should be as close as possible to the pad center if it cannot be even farther ahead on the shoe, in front of the brake pad. Mass close to the shoe pivot point will pull on the shoe pivot rather than push on the brake pad and so will be of no help to braking. The mass center can even be placed beyond the brake drum radius, if the shoes are given wings or side plates that extend radially outward beyond the brake drum. An optimization can be run to determine the best angular place for the mass center for maximum pad loading and tolerable shoe pivot loading (a mass center forward of the pad will increase pivot loading by a lever effect). The effects of brake inertia must be kept in mind when placing shoe mass.

C_theta (radians, angle between a line through the hub center and the shoe pivot, and a line through the hub center and the shoe mass center)
C_radius (m, radial distance between the hub center and the shoe mass center)

diagram showing the dimensions of a leading shoe of the type described in the equations
Drag coefficient can now be determined by the following equation:

c_brake = friction*shoe_mass*C_radius*sin(C_theta)*sin(shoe_angle)*pivot_r*n_shoes/(sin(P_theta)*(sin(contact_angle) - friction*cos(contact_angle)))

c_brake has the units N*m*s^2/rad^2. Drag torque at a particular speed is c_brake*speed^2 where speed is in rad/s^2.
An expansion of the math behind this equation, with some illustrations showing these parameters, is linked here. These equations may be modified to calculate loading at the pivot and within the shoe, or to start with different parameters like pivot_r and P_theta.

a leading shoe centrifugal friction brake design with winged shoes a leading shoe centrifugal friction brake design with winged shoes

Illustrations of a leading shoe centrifugal brake design with lightweight aluminum shoes and heavy tungsten wings to push shoe mass centers beyond the drum surface.

The equations predict drag for an idealized line contact at the center of the brake pad. A real brake will have short-lived contacts at asperities distributed across the pad, so its drag will be somewhat different depending on how long the angular arc of the pad is. Toward the trailing edge of the pad, wedging action is greater, drag is higher, and the risk of jamming or chattering is higher, while toward the leading edge of the pad the opposite is true. A well designed brake will have pads of minimal arc length, but the pads must be wide enough (in the spin-axis direction) to have a total area adequate for the expected rate of energy dissipation. Manufacturers should specify a maximum power density (W/m^2) and a maximum pressure (N/m^2) for which a friction material is rated, and these should be used to size the brake design. Sometimes a maximum sliding speed (m/s) is also specified.

Brake pads should not be too wide either: it is better to put multiple brake hubs side-by-side in a drum than it is to make a single shoe wider, because a wide pad will still only contact a small area at a time and the contact point, as it jumps back and forth on the wide pad, will twist the shoe back an forth leading to noise, poor drag performance, and premature failure. Brake pads must have a narrow enough width (in the spin axis direction) relative to the stance of the shoe pivot to prevent shoe chatter from changing points of contact with the drum. Shoes should be given as wide a stance as possible relative to the pads to help avoid chattering.

For most materials, static friction is greater than kinetic friction. For leading shoe centrifugal brakes this presents a real threat, because their self-energizing design will cause jamming that's impossible to overcome if friction is high enough. If static friction becomes unexpectedly high, the brake may never start rotating at all when it needs to do so. One way to reduce this risk is to ensure that pads do not normally contact the drum. If they are held inward by low force springs, first motion will occur before the pads touch the drum. However, these springs reduce drag from a brake of a particular mass. A spring feature can be used to finely tune a brake to a particular drag coefficient if speedy adjustment is more important than mass.

Brake rotational inertia must also be considered in the context of the lowering task. If a lengthy deployment at a constant speed is planned, a brake having high rotational inertia may work well but the same brake may have unsatisfactory performance if a varying speed deployment is planned, because the brake must slow itself down as well as its payload. Similarly, if initial spin-up is gradual, a high inertia brake may work fine but if there is even a small amount of slack in the payload tether, an undesirable spike in tether tension may occur due to rapid acceleration of a high inertia brake.

This spike will occur whenever descent speed is changes (such as by a change in tether spool taper) and whenever there is a disturbance to the lowered load (such as when the Mars Rover unlatches its wheels and suspension arms and lets them fall into place in preparation for landing). A high inertia brake will help maintain smooth lowering in the event of disturbances affecting the lowered load, while a low inertia brake will store less energy and so may be stopped quickly, with minimal loads, at the end of its operation. In some designs, brake shoe mass centers may be positioned with respect to the shoe pivots so the shoes are flung radially inward during brake spin-up. This could possibly reduce the chance of jamming, reduce load spikes at spin-up, and shorten the time required to reach a sustained speed.

Brake friction is strongly affected by temperature. Some materials are more stable than others, but while some become more grippy at first as temperature rises, all eventually fade as their materials break down and friction decreases. Brake drag will therefore be affected by starting temperature, and probably more so by heating that results during use. Supplying thermal mass or even phase-change materials such as paraffin wax surrounding the brake drum will help stabilize performance (though it would be challenging to use wax that expands when melted and make absolutely sure it can't leak out and lubricate the friction surface). With so much mass needed in the brake shoes to provide centrifugal force, it's logical to make the shoes from a material having high thermal capacitance for its mass; however since many friction materials are not nearly as thermally conductive as metals, most of the heat from braking may go into the brake drum. Inverting the materials so the friction lining is on the drum and the brake shoes are bare metal may be a good solution for some mass-constrained designs, but after a period of use the drum will wear out of round and brake performance will be degraded.

Wearing in ("bedding in") the brake pads at an appropriate power level is a critical part of preparing a centrifugal friction brake for use. With many friction materials, friction will increase after a period of intense use but subsequent low-intensity use will wear away the grippy films of materials created at the friction interface and drag will decline. Repeated brake operation at a constant load, with cooling-off periods as required, will condition the brake pads and drum; after conditioning, the brake will be very sensitive to contamination, cleaning activities, and reaction with atmospheric gases.

plots of drag coefficient vs. other parameters showing their effects on drag
These plots show some sensitivities and trends of a centrifugal brake design, with all but one variable held constant for each plot. Note that the nominal design isn't necessarily a good one. It is evident from the plots that the brake jams at a friction coefficient of 0.7, and that pad placement relates to sensitivity in a similar way that friction does.

Electromagnetic Brakes

Electromagnetic brakes fall into another class of damper useful for lowering devices. They convert kinetic energy to heat by driving electric eddy currents through resistive metal, by heating magnetically permeable material using an alternating magnetic field (magnetic hysteresis), and by driving electric current through circuits with resistive elements. Most brakes dissipate some energy in all three ways, but they generally fall into two classes: brakes that heat up during use, and brakes that stay cool by rejecting most of their energy as heat in a remotely located component. By staying cool, generator brakes avoid drift in drag coefficient that would result from increasing temperature, which would change winding resistance and current flow. Eddy current or magnetic hysteresis brakes must be fitted with cooling systems to dissipate energy without change in their drag characteristics. While eddy current and generator brake drag is dependent on speed, hysteresis brake drag is not.

Electromagnetic brakes are widely used to control yarn tension in the textile industry, to test motors and engines, and to recover energy from decelerating vehicles. In the course of normal operation, many hoists and winches function as generator brakes when lowering loads. Lowering devices using generator brakes have been built and operated in the past. Their precision components (gears, bearings, electrical wires and insulators) operate at high speed and must be protected from damage, perhaps more so than other types of drag brake, but electromagnetic brakes produce reliable drag that is easily adjusted. Simple circuitry can even be added for open-loop or feedback-driven control of lowering speed.

The magnetic term from the Lorentz Force Equation gives the force on a charged particle moving through a non-parallel magnetic field. In a simplistic sense, this force F=B*I*L where B is the magnetic flux density, I is the current in a conductor, and L is the conductor length. Predicting the torque or drag of electric motor or generator requires much more detailed calculations that consider, among other things, the permeability and configuration of flux-carrying rotor and stator cores, magnetic field shape and density, the interaction of fields within the device, demagnetizing effects and inductance, and the properties of the conductive windings.

A device designed to be an electric motor may not be a very good generator or generator brake. Also, gearboxes designed to slow down an input rotation may not work as well when driven in reverse, as speed-increasers. Even so, many off-the-shelf generators will work well when used to make descent devices. There are many types of motors and generators including coreless, axial gap, brushed and brushless, permanent magnet and induction motors. The drag of an induction generator used as a drag brake can be adjusted by changing the current in the field coils, but passive operation (i.e. no computer control) of such a system requires a carefully tuned design. All generator brake systems require careful design considering electrical and thermal characteristics, material capacities, and loads resulting from operation.

Other Options for Energy Dissipation

Other options for dissipating energy from a lowering device include things like ripping textile absorbers (rip-stitching, etc.) and plastic deformation (crushing, extrusion, swaging, crumpling, shearing, stretching) of sacrificial absorber elements. Energy storage in springs, flywheels, or pressure systems may also be useful in some applications. Speed-maintaining mechanisms are also worthy of mention in context with lowering devices: sometimes a deployment may be driven by a spring, gas pressure, or some other form of stored energy that declines during operation.

If constant speed is needed, a mechanism like the wax cylinder music player speed control pictured below may be useful to store energy in angular momentum and recover it as a function of speed. Fly-weights mounted on springs will expand outward at higher speeds, and the momentum they acquire in doing so will be recovered as speed declines and they are pulled inward by the springs. The mechanism pictured is part of a Kasten Puck wax cylinder player, dated 1905.

a speed maintaining flyball mechanism on spring powered wax cylinder music player from 1905 a spring powered wax cylinder music player from 1905
A speed maintaining flyball mechanism on a spring powered wax cylinder music player from 1905.

Continuing with this for a moment, consider the mechanical speed governor pictured below from the Navesink Lighthouse, circa 1828. Its construction is nearly identical to the Kasten Puck speed regulator, but in this installation the flyballs are arranged to pull a lever down, thereby commanding a change perhaps in the power source.

a flyball governor from the Navesink Lighthouse, circa 1828.

Tether Design

A lowering device needs a tether system as well as a drag brake. The tether may be a single cord or several cords in parallel, unwound from a spool or pulled out of a box. This tether may pull tight at full extension, or it may hand off load to a heavier webbing or cord deployed in parallel to take the high loads at end-of-travel or loads from subsequent maneuvering of the hanging equipment. Getting cord off a spool without it snarling or tangling is a challenge itself; if cord is wrapped over itself like a common spool of string, when tension is applied the cords will typically sink down between lower layers and become tangled unless all the layers have been wound at high tension comparable to the weight of the payload. However when this is done, the innermost layers experience tremendous crushing pressure that is often high enough to damage them and the spool core and side flanges are highly loaded as well. Metal wire fares better in this case than synthetic fiber.

Synthetic fiber, however, is often a better choice because it is more elastic. This limits loads in the tether and lowers the natural frequency of the system so it is less likely to couple with a disturbance from the lowering mechanism or another machine element. High strength aramid fibers offer amazing strength for their weight and are available in many different sizes and lengths suitable for almost any application. Some are very sensitive to ultraviolet light and particulate contamination, while others are suitable for long term use exposed to the weather - on a sailboat, for example.

A single layer of tether cord wrapped on a spool is probably the best approach for a lowering device to be used for a relatively short deployment. An option for longer deployments is to store tether cord on a spool under low tension and then wrap multiple turns of the cord around a capstan wheel to transfer tension from the cord to the wheel. The equation for load sharing between the two ends of the tether and the capstan wheel is:

T2 = T1*exp(friction*wrap_angle)
where T2>T1 (T2 is the tension due to the suspended payload)
T2 and T1 are tensions, in Newtons

friction is the coefficient of static friction between the capstan wheel and the tether (making the wheel surface rough will help, but it may fray the cord; another option is to make the roller from a grippy polymer like urethane)

lightly tensioned payout spool for use with a capstan capstan device like those used on Mars

From a payload weight and friction coefficient, one can calculate the number of turns required for a particular pre-wound tension T1. Some sort of drag device must restrain the tether feed spool to create T1; in this case it is usually safe to use a plain spring-energized friction brake. T1 can be much less than T2. This lets the overall system be lightweight and compact compared to a pre-tensioned tether spool capable of taking full tether loads.

Descent Speed Profiles

If the radius of the tether spool decreases over the course of deployment, either from payout of tether from an over-wrapped spool or from the taper of a spool having a single layer of cord, the relationship between tether tension, lowering speed, and drag torque will be changed. This can be very useful if a lowering speed profile is needed, for example to create a quick deployment with a soft stop at full tether extension. Normally, a speed dependent drag brake will control lowering speed at a constant rate if the brake driving torque from the hanging weight remains constant. However if the spool radius or the gear ratio between the spool or capstan and the drag brake is changed through the course of lowering, descent speed will change. The inertia of both the mechanism and the descending payload, and compliance and damping in the payload tether and in mechanism components, affect how responsive the mechanism is.

A zero mass brake, spool, and tether system subject to an applied force on the tether will rotate as a function of the spool radius, drag coefficient (c_brake), and applied load:
drag_torque = c_brake*omega for a first-order speed dependent brake
driving_torque = load*radius
lowering_speed = omega*radius
drag_torque = driving_torque
lowering_speed = load*radius^2/c_brake (for a first-order brake)
Or, if drag_torque = c_brake*omega^2 (for a second-order brake, like a centrifugal friction brake)
lowering_speed = (load*radius^3/c_brake)^0.5

plot of idealized performance and real performance for two types of brake
The plots above compare a steady-state curve (dashed green line) that would occur in a massless system with the real system. The idealized descent curves are for a conically tapered spool used with two different types of brake sized to deploy a weight in the same amount of time. The real system seeks the steady-state curve.

Tapered spools are also used in tool counterweights, in drive mechanisms for some powered sliding car doors, and in mine hoists (where the tapered spool increases torque to compensate for the weight of heavy steel cable thousands of feet long at full extension). Designing a spool's taper is rather like designing a cam-and-follower system in that abrupt changes in spool slope will produce large accelerations and high loads and torques. The problem lends itself to computational optimization, which allows designs to be located that offer short-duration deployments with minimal loads and the desired ending speed. However, equations for deployed cord length as a function of spool rotation are rather complicated for all but a couple of simple shapes.

The tapered spool pictured at right compensates for changing spring torque and pendant cable weight so a bridge crane control pendant can be positioned freely.

a tapered spool used with a bridge crane pendant suspension line
The equations for a helix in an [xyz] coordinate frame are:
x = r*cos(theta)
y = r*sin(theta)
z = b*theta

(dS/dtheta)^2 = (dx/dtheta)^2 + (dy/dtheta)^2 + (dz/dtheta)^2
This reduces to dS = (r^2+b^2)^(1/2) dtheta; length with respect to theta is therefore sqrt((r*theta)^2 + (b*theta)^2)

where r is the helix radius, b is the axial lead in meters per radian, and theta is the rotation in radians For preliminary, inexact studies of a spool wound with a single layer of cord it is usually acceptable to use the function Length = radius*theta, ignoring the very small length that is added by the cord being wrapped in successive turns down the length of the spool.

For modeling a tapered spool, a slightly more involved formula is needed to capture the change of radius as a function of theta. The following equation is for a conically tapered helix or a flat Archimedes' spiral; if the axial lead "b" is a nonlinear function of theta, the equation will be different.

The equations for the polar curve in an [xyz] coordinate frame are:
x = a*theta*cos(theta)
y = a*theta*sin(theta)
z = b*theta
(dS/dtheta)^2 = (dx/dtheta)^2 + (dy/dtheta)^2 + (dz/dtheta)^2

with some math, this takes a form available in common integral tables and the formula for length as a function of theta is obtained:

Length(theta) = a/2*(theta*sqrt(1 + b^2/a^2 + theta^2) + (1 + b^2/a^2)*ln(theta + sqrt(1 + b^2/a^2 + theta^2)))

If b = 0 the spiral is flat (an Archimedes' spiral). If a = 0 the formula presented earlier, for a straight helix, must be used. This formula gives the length of a conical spiral from the apex of the cone to the rotation "theta" along the cone. A real spool will have the shape of a frustum of a cone, so an offset must be added to the rotation term and a length representing the curve from the apex to the small end of the spool must be subtracted.

It turns out that conical spools are almost ideal, compared to spools having football or bottle shapes with sinusoidal or hyperbolic tangent curvature, for creating fast initial spin-up and gradually slowing lowering speed for a soft stop after minimum elapsed time. Conical spools are also easy to manufacture as their CNC programming would be similar to pipe-threading - if they have grooves; if the taper isn't steep and friction between the cord and spool is good, grooves may not be necessary. Other spool shapes, like power curve tapers (horn shaped), may be useful in special applications.

a tapered tether spool a tapered spool with tether

There are other options besides tapered spools for creating a descent speed profile. One design used in industry and during the 1997 landing of NASA's Mars Pathfinder spacecraft uses a thin metal tape as the tether, avoiding the problem of cord sinking into inner layers on an over-wrapped spool while still taking advantage of the speed profile that results from the shrinking radius of the tape spool. Metal belts move things ranging in size from disk drive heads to giant industrial machines. Some elevators in high-rise buildings hang on reinforced rubber belts instead of wire rope because sheaves can be much smaller diameter, the motors that drive them can have much smaller gearboxes, and the resulting systems can be more compact, cheaper, and more efficient. When metal tape is wound up on a sheave rather than just bent around it to transfer power, the tape tension as a function of drive torque changes through the course of winding and unwinding.

This is useful to create a changing pull force, for example to unfurl solar arrays which, by the geometry of their hinges and actuators, might require lots of torque at first when the first wraps of tape are being pulled onto the spool but later might benefit from faster, lower-force deployment once leverage has improved. Much of the stress in such tapes comes from bending around a small sheave or spool, but this stress can be halved by pre-curving the tape and so it is equally stressed when pulled straight and when wound to its smallest diameter. Laminating fiber-reinforced plastics to the metal tape or using fiber-reinforced polymer belting alone presents options for managing stresses while providing adequate tape thickness for creating the desired speed profile. A smoothly changing tape thickness might also be feasible for additional design flexibility. Arrangements with multiple spools bring additional options for changing the gear ratio or using separate tapes and tethers.

System Dynamic Models

Detailed design of a lowering device requires a complete system model to assess the effects of brake inertia, tether compliance and damping, and characteristics of other parts of the driving mechanism. Assuming a rigid overhead support, the equations for descent of a payload suspended by a rigid tether unwound from an idealized cylindrical spool (ignoring spool length) are:

brake_acceleration = (c_brake*brake_speed^2/rspool^3 + F)/(m + I_brake/rspool^2)
payload_acceleration = brake_acceleration/rspool
payoad_speed = brake_speed/rspool

with variables as defined below:
m (kg, payload mass)
g (m/s^2, acceleration due to gravity)
h (m, lowering distance)
rspool (m, spool radius)
I_brake (kg*m^2, inertia of rotating components about spin axis)
F = m*g (N, weight of payload)

One might set up this differential equation in a solver tool like Matlab as follows:
function ydot=drop_EOM(t,y,m,h,rspool,c_brake,I_brake,F)
ydot(1) = y(2); % m/s, descent speed
ydot(2) = (c_brake*y(2)^2/rspool^3 + F)/(m + I_brake/rspool^2); % m/s^2, descent acceleration

This simple equation may suffice for early design studies seeking to bound lowering time and speed, but the elasticity of all elements must be considered when evaluating loads. It is good practice to include the mass and elasticity of the overhead support, as it affects loads and the behavior of the system immediately after separation of the descending payload. Initial free-fall, no matter how small, should also be included a an initial for a good understanding of transient loads.

For a 2-mass system with tether cord deploying from a spool:
overhead_support_accel = (F2 - F1 + m1*g)/m1
payload_accel = (m2*g - F2)/m2
spool_angular_accel = (F2*spool_radius - k_shaft*(spool_rotation - brake_rotation) - c_shaft*(spool_speed - brake_speed))/I_spool
brake_angular_accel = (k_shaft*(spool_rotation - brake_rotation) - c_shaft*(spool_speed - brake_speed) - c_brake*brake_speed)/I_brake

with variables as defined below:
m1 (kg, mass of the equipment to which the top end of the tether is attached)
m2 (kg, mass of the payload to be lowered)
g (m/s^2, acceleration due to gravity)

F2 = ktether*stretch + ctether*vstretch
ktether (N/m, tether stiffness) is a function of deployed tether length (ktether = AE/length, where AE has the units N*m/m and is analogous to modulus times area, though for a rope or cord this is not meaningful)
stretch (m) is the difference between the separation distance of the two masses and the length of tether unwound from the spool ctether (N*s/m) is the damping within the tether. It's usually quite small.
vstretch (m/s) s the rate of change in stretch

F1 = ksupport*deflection + csupport*deflectionspeed
As for F2, F1 must be calculated using the stiffness and damping of the structure that supports mass "m1"

k_shaft (N*m/rad, angular stiffness of the element - gearbox, etc. - between the spool and the brake)
c_shaft (N*m*s/rad, angular damping of the element between the spool and the brake)
c_brake (N*m*s/rad, angular damping or drag coefficient of the brake device)

The spool radius as a function of rotation may be calculated from the formulas presented earlier, and the speeds and displacements may be found by integration with respect to time.

plots from a dynamic model of a lowering device plots from a dynamic model of a lowering device

If a cord guide is used, a correction must be made to the tether tension term F2 when it appears in the spool acceleration equation, because not all of the tether tension is applied tangent to the spool. Similarly, if more than one tether is used (a set of three in parallel, for example) then a correction must be made to F2 to adjust for the components of tether tension not being used to accelerate the payload upward. Furthermore, if excitation in the lateral direction is expected (pendulum swing, etc.) then additional parameters must be added to the equations.

Real Behavior and Testing

Elasticity, brake drag, and other properties will be non-linear, and other errors or omissions combined with this will cause test results to be inconsistent with model predictions. It's important to keep assumptions well documented especially when so many factors affect how a payload is lowered. Functions can be used to update constants like c_brake once test data is available, or to study sensitivity to change if - for example - c_brake becomes degraded as a function of accumulated energy. At some point, predictions must be made based on test measurements rather than model results. When this is done, it's important that statistical methods be maintained and that sample size and standard deviation always be transmitted with any data.

References and Further Reading

1. Jumper, George Y., Jr.; Kirpa, Geoffrey R.; D’Urso, Anthony A.; Mineau, David A. High Performance Viscous Brake for Balloon Payload Deployment. Proceedings of AIAA Lighter-Than-Air Systems Technology Conference, 11th. Clearwater Beach, FL. May 15-18, 1995.

2. Gilpatrick, A. E.; Fruge, Romain C., Jr.; Beckett, Lloyd S., Jr.; Courtoglous, Paul T. Operation Hardtack. Project 9.2b. Operation of Balloon Carrier for Very- High-Altitude Nuclear Detonation. Air Force Cambridge Research Labs, Hanscom AFB, MA. July 25, 1958.

3. Vermalle, J. C. Centrifugal Regulator for Control of Deployment Rates of Deployable Elements. Proceedings of Aerospace Mechanisms Symposium, 14th. NASA Langley Research Center, Hampton, VA. May 1-2, 1980.

4. Pankow, D.; Wilkes, R.; Besuner, R.; Ullrich, R. The FAST Boom Mechanisms. Proceedings of Aerospace Mechanisms Symposium, 32nd. NASA Kennedy Space Center, FL. May 13-15, 1998.

5. Brake Lining Quality Test Procedure SAE J661. Society of Automotive Engineers, 1997.

6. Urban, I.; Dorfel, I.; Osterle, W.; Gesatzke, W.; Engelhardt, M. Microstructural Characterization of Loaded Brake Materials. Federal Institute of Materials Research and Testing, Berlin, Germany.

7. Ihm, Mark. Introduction to Gray Cast Iron Brake Rotor Metallurgy. Proceedings of Society of Automotive Engineers Brake Colloquium & Exhibition. http://www.sae.org/events/bce/tutorial-ihm.pdf

8. Blau, Peter J. Oak Ridge National Laboratory, Metals and Ceramics Division. Compositions, Functions, and Testing of Friction Brake Materials and Their Additives. Oak Ridge, TN, August 2001.

9. Jiang, J.C.; Huang, B.Y.; Xiong, X. The Relationship of Surface Quality and the Running-in Character of Carbon-Carbon Composite (CCC) Aircraft Brake Disks. New Carbon Materials, 2002, Volume 17, Number 2, Pages 35-40.

10. Hawthone, H. M. Wear Debris Induced Friction Anomalies of Organic Brake Materials in Vacuo. Wear Materials, Volume 1, Pages 381-387. 1987.

11. Hawthone, H. M. Tribomaterial Factors in Space Mechanism Brake Performance. Proceedings of Aerospace Mechanisms Symposium, 24th. NASA Kennedy Space Center, FL. April 18-20, 1990.

12. Hawthorne, H.M.; Kavanagh, J. The Tribology of Space Mechanism Friction Brake Materials. Canadian Aeronautics and Space Journal, Volume 36, Pages 57-61. June, 1990.

13. Holinski, R. Improvement of Comfort of Friction Brakes. Dow Corning Weisbaden, Germany.

14. Holinski, R.; Hesse, D. Changes at Interfaces of Friction Components During Braking. Journal of Automobile Engineering, Volume 217, Number 9, 2003.

15. Ashby, Michael F.; Jones, David R. H. Engineering Materials 1, Third Edition. Elsevier Butterworth Heinemann, 2005. Page 373.

16. Williams, J.A.; Morris, J.H.; Ball, A. The Effect of Transfer Layers on the Surface Contact and Wear of Carbon-Graphite Materials. Triboloy International. Volume 30, Number 9, Pages 663-676. 1997.

17. Osterle, W.; Gesatzke, W.; Griepentrog, M.; Klaffke, D.; Urban, I. Microstructural Aspects Controlling Friction and Wear of Engineering Materials. Proceedings of World Tribology Congress, Vienna. 2001.

18. Skyjump®, Skyjump Ltd. Auckland, New Zealand. http://www.skyjump.co.nz/

19. Hendershot, J. R., Jr.; Miller, TJE. Design of Brushless Permanent-Magnet Motors. Magna Physics Publishing and Oxford University Press, 1994.

20. Rivellini, Tommaso P.; Bickler, Donald B.; Swenson, Bradford L.; Baughman, James A.; Gallon, John C.; Ingle, Jack. California Institute of Technology, Jet Propulsion Laboratory New Technology Report NTR 40109: Descent Rate Limiting Device. 31 March 2003.

21. Maus, Daryl. Metal Band Drives in Spacecraft Mechanisms. Proceedings of Aerospace Mechanisms Symposium, 27th. NASA Ames Research Center, Moffett Field, CA. May 12-14, 1993.

22. Adams, Douglas S. BUD Conical Spiral Length Equations and Numerical Implementation. Jet Propulsion Laboratory internal memorandum IOM 352L-DSA-0606. Rev. A. 31 October 2006.

23. Bash, John F. (editor). Handbook of Oceanographic Winch, Wire and Cable Technology. Third Edition. National Science Foundation, 2001.

24. Whitehill, A.S., Sr.; Huntley, E.W. Cycle History Data and Elongation Characteristics for Polyester and Aramid Wire-Lay Construction Ropes. OCEANS '97. MTS/IEEE Conference Proceedings. Halifax, NS, Canada. October 6-9, 1997.

25. Stenvers, Danielle; Roberts, Phil; Chou, Rafael. Testing of High Strength Synthetic Ropes. Samson Rope Technologies, Inc. http://www.samsonrope.com/site_files/Testing_of_High_Strength_Synthetic_Rope.pdf

26. CRC handbook of fiber finish technology

This research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration.

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