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NASA NTRS2008

Development of Eddy Current Techniques for Detection of Deep Fatigue Cracks in Multi-Layer Airframe Components

Thick, multi-layer aluminum structure has been widely used in aircraft design in critical wing splice areas. The multi-layer structure generally consists of three or four aluminum layers with different geometry and varying thickness, which are held together with fasteners. The detection of cracks under fasteners with ultrasonic techniques in subsurface layers away from the skin is impeded primarily by interlayer bonds and faying sealant condition. Further, assessment of such sealant condition is extremely challenging in terms of complexity of structure, limited access, and inspection cost. Although Eddy current techniques can be applied on in-service aircraft from the exterior of the skin without knowing sealant condition, the current eddy current techniques are not able to detect defects with wanted sensitivity. In this work a series of low frequency eddy current probes have been designed, fabricated and tested for this application. A probe design incorporating a shielded magnetic field sensor concentrically located in the interior of a drive coil has been employed to enable a localized deep diffusion of the electromagnetic field into the part under test. Due to the required low frequency inspections, probes have been testing using a variety of magnetic field sensors (pickup coil, giant magneto-resistive, anisotropic magneto-resistive, and spin-dependent tunneling). The probe designs as well as capabilities based upon a target inspection for sub-layer cracking in an airframe wing spar joint is presented.

Wincheski, Russell A.↗

NASA NTRS2008

Making Superconducting Welds between Superconducting Wires

A technique for making superconducting joints between wires made of dissimilar superconducting metals has been devised. The technique is especially suitable for fabrication of superconducting circuits needed to support persistent electric currents in electromagnets in diverse cryogenic applications. Examples of such electromagnets include those in nuclear magnetic resonance (NMR) and magnetic resonance imaging (MRI) systems and in superconducting quantum interference devices (SQUIDs). Sometimes, it is desirable to fabricate different parts of a persistent-current-supporting superconducting loop from different metals. For example, a sensory coil in a SQUID might be made of Pb, a Pb/Sn alloy, or a Cu wire plated with Pb/Sn, while the connections to the sensory coil might be made via Nb or Nb/Ti wires. Conventional wire-bonding techniques, including resistance spot welding and pressed contact, are not workable because of large differences between the hardnesses and melting temperatures of the different metals. The present technique is not subject to this limitation. The present technique involves the use (1) of a cheap, miniature, easy-to-operate, capacitor-discharging welding apparatus that has an Nb or Nb/Ti tip and operates with a continuous local flow of gaseous helium and (2) preparation of a joint in a special spark-discharge welding geometry. In a typical application, a piece of Nb foil about 25 m thick is rolled to form a tube, into which is inserted a wire that one seeks to weld to the tube (see figure). The tube can be slightly crimped for mechanical stability. Then a spark weld is made by use of the aforementioned apparatus with energy and time settings chosen to melt a small section of the niobium foil. The energy setting corresponds to the setting of a voltage to which the capacitor is charged. In an experiment, the technique was used to weld an Nb foil to a copper wire coated with a Pb/Sn soft solder, which is superconducting. The joint was evaluated as part of a persistent-current circuit having an inductance of 1 mH. A current was induced in a loop, and no attenuation of the current after a time interval 1,000 s was discernible in a measurement having a fractional accuracy of 10(exp -4): This observation supports the conclusion that the weld had an electrical resistance <10(exp -10) omega.

Penanen, Konstantin I.↗

NASA NTRS2008

Magnetostrictive valve assembly

A magnetostrictive valve assembly includes a housing that defines a passage with a seat being formed therein. A magnetically-biased and axially-compressed magnetostrictive assembly slidingly fitted in the passage is configured as a hollow and open-ended conduit adapted to support a flow of a fluid therethrough. Current-carrying coil(s) disposed about the passage in the region of the magnetostrictive assembly generate a magnetic field in the passage when current flows through the coil(s). A hollow valve body with side ports is coupled on one end thereof to an axial end of the magnetostrictive assembly. The other end of the valve body is designed to seal with the seat formed in the housing's passage when brought into contact therewith.

Richard, James A.↗

NASA NTRS2009

Room-Temperature-Cured Copolymers for Lithium Battery Gel Electrolytes

Polyimide-PEO copolymers (PEO signifies polyethylene oxide) that have branched rod-coil molecular structures and that can be cured into film form at room temperature have been invented for use as gel electrolytes for lithium-ion electric-power cells. These copolymers offer an alternative to previously patented branched rod-coil polyimides that have been considered for use as polymer electrolytes and that must be cured at a temperature of 200 C. In order to obtain sufficient conductivity for lithium ions in practical applications at and below room temperature, it is necessary to imbibe such a polymer with a suitable carbonate solvent or ionic liquid, but the high-temperature cure makes it impossible to incorporate and retain such a liquid within the polymer molecular framework. By eliminating the high-temperature cure, the present invention makes it possible to incorporate the required liquid.

Meador, Mary Ann B.↗

NASA NTRS2009

Radial Halbach Magnetic Bearings

Radial Halbach magnetic bearings have been investigated as part of an effort to develop increasingly reliable noncontact bearings for future high-speed rotary machines that may be used in such applications as aircraft, industrial, and land-vehicle power systems and in some medical and scientific instrumentation systems. Radial Halbach magnetic bearings are based on the same principle as that of axial Halbach magnetic bearings, differing in geometry as the names of these two types of bearings suggest. Both radial and axial Halbach magnetic bearings are passive in the sense that unlike most other magnetic bearings that have been developed in recent years, they effect stable magnetic levitation without need for complex active control. Axial Halbach magnetic bearings were described in Axial Halbach Magnetic Bearings (LEW-18066-1), NASA Tech Briefs, Vol. 32, No. 7 (July 2008), page 85. In the remainder of this article, the description of the principle of operation from the cited prior article is recapitulated and updated to incorporate the present radial geometry. In simplest terms, the basic principle of levitation in an axial or radial Halbach magnetic bearing is that of the repulsive electromagnetic force between (1) a moving permanent magnet and (2) an electric current induced in a stationary electrical conductor by the motion of the magnetic field. An axial or radial Halbach bearing includes multiple permanent magnets arranged in a Halbach array ("Halbach array" is defined below) in a rotor and multiple conductors in the form of wire coils in a stator, all arranged so the rotary motion produces an axial or radial repulsion that is sufficient to levitate the rotor. A basic Halbach array (see Figure 1) consists of a row of permanent magnets, each oriented so that its magnetic field is at a right angle to that of the adjacent magnet, and the right-angle turns are sequenced so as to maximize the magnitude of the magnetic flux density on one side of the row while minimizing it on the opposite side. The advantage of this configuration is that it makes it possible to approach the theoretical maximum force per unit area that could be exerted by a given amount of permanent-magnet material. The configuration is named after physicist Klaus Halbach, who conceived it for use in particle accelerators. Halbach arrays have also been studied for use in magnetic-levitation ("maglev") railroad trains. In a radial Halbach magnetic bearing, the basic Halbach arrangement is modified into a symmetrical arrangement of sector-shaped permanent magnets mounted on the outer cylindrical surface of a drum rotor (see Figure 2). The magnets are oriented to concentrate the magnetic field on their radially outermost surface. The stator coils are mounted in a stator shell surrounding the rotor.

Eichenberg, Dennis J.↗

NASA NTRS2009

Numerical Investigation of Near-Field Plasma Flows in Magnetic Nozzles

The development and application of a multidimensional numerical simulation code for investigating near-field plasma processes in magnetic nozzles are presented. The code calculates the time-dependent evolution of all three spatial components of both the magnetic field and velocity in a plasma flow, and includes physical models of relevant transport phenomena. It has been applied to an investigation of the behavior of plasma flows found in high-power thrusters, employing a realistic magnetic nozzle configuration. Simulation of a channel-flow case where the flow was super-Alfvenic has demonstrated that such a flow produces adequate back-emf to significantly alter the shape of the total magnetic field, preventing the flow from curving back to the magnetic field coil in the near-field region. Results from this simulation can be insightful in predicting far-field behavior and can be used as a set of self-consistent boundary conditions for far-field simulations. Future investigations will focus on cases where the inlet flow is sub-Alfvenic and where the flow is allowed to freely expand in the radial direction once it is downstream of the coil.

Sankaran, Kamesh↗

NASA NTRS2009

Miniature Scroll Pumps Fabricated by LIGA

Miniature scroll pumps have been proposed as roughing pumps (low - vacuum pumps) for miniature scientific instruments (e.g., portable mass spectrometers and gas analyzers) that depend on vacuum. The larger scroll pumps used as roughing pumps in some older vacuum systems are fabricated by conventional machining. Typically, such an older scroll pump includes (1) an electric motor with an eccentric shaft to generate orbital motion of a scroll and (2) conventional bearings to restrict the orbital motion to a circle. The proposed miniature scroll pumps would differ from the prior, larger ones in both design and fabrication. A miniature scroll pump would include two scrolls: one mounted on a stationary baseplate and one on a flexure stage (see figure). An electromagnetic actuator in the form of two pairs of voice coils in a push-pull configuration would make the flexure stage move in the desired circular orbit. The capacitance between the scrolls would be monitored to provide position (gap) feedback to a control system that would adjust the drive signals applied to the voice coils to maintain the circular orbit as needed for precise sealing of the scrolls. To minimize power consumption and maximize precision of control, the flexure stage would be driven at the frequency of its mechanical resonance. The miniaturization of these pumps would entail both operational and manufacturing tolerances of <1 m. Such tight tolerances cannot be achieved easily by conventional machining of high-aspect-ratio structures like those of scroll-pump components. In addition, the vibrations of conventional motors and ball bearings exceed these tight tolerances by an order of magnitude. Therefore, the proposed pumps would be fabricated by the microfabrication method known by the German acronym LIGA ( lithographie, galvanoformung, abformung, which means lithography, electroforming, molding) because LIGA has been shown to be capable of providing the required tolerances at large aspect ratios.

Wiberg, Dean↗

NASA NTRS2009

Apparatus and method for enhancing tissue repair in mammals

An apparatus is introduced for the use of enhancing tissue repair in mammals. The apparatus includes a sleeve; an electrically conductive coil; a sleeve support; an electrical circuit configured to supply the coil with a square wave time varying electrical current sufficient to create approximately 0.05 gauss to 0.5 gauss. When in use, the sleeve of the apparatus is placed on a mammalian body part and the time varying electromagnetic force of from approximately 0.05 gauss to 0.5 gauss is generated on the mammalian body for an extended period of time so that the tissue is encouraged to be regenerated in the mammalian body part at a rate in excess of the normal tissue regeneration rate relative to regeneration without application of the time varying electromagnetic force.

Goodwin, Thomas J.↗

NASA NTRS2010

Effect of a Second, Parallel Capacitor on the Performance of a Pulse Inductive Plasma Thruster

Pulsed inductive plasma accelerators are electrodeless space propulsion devices where a capacitor is charged to an initial voltage and is then discharged through an inductive coil that couples energy into the propellant, ionizing and accelerating it to produce thrust. A model that employs a set of circuit equations (as illustrated in Fig. 1a) coupled to a one-dimensional momentum equation has been previously used by Lovberg and Dailey [1] and Polzin et al. [2-4] to model the plasma acceleration process in pulsed inductive thrusters. In this paper an extra capacitor, inductor, and resistor are added to the system in the manner illustrated in the schematic shown in Fig. 1b. If the second capacitor has a smaller value than the initially charged capacitor, it can serve to increase the current rise rate through the inductive coil. Increasing the current rise rate should serve to better ionize the propellant. The equation of motion is solved to find the effect of an increased current rise rate on the acceleration process. We examine the tradeoffs between enhancing the breakdown process (increasing current rise rate) and altering the plasma acceleration process. These results provide insight into the performance of modified circuits in an inductive thruster, revealing how this design permutation can affect an inductive thruster's performance.

Polzin, Kurt A.↗

NASA NTRS2010

Reusable Hot-Wire Cable Cutter

During the early development stage of balloon deployment systems for missions, nichrome wire cable cutters were often used in place of pyro-actuated cutters. Typically, a nichrome wire is wrapped around a bundle of polymer cables with a low melting point and connected to a relay-actuated electric circuit. The heat from the nichrome reduces the strength of the cable bundle, which quickly breaks under a mechanical load and can thus be used as a release mechanism for a deployment system. However, the use of hand-made heated nichrome wire for cutters is not very reliable. Often, the wrapped nichrome wire does not cut through the cable because it either pulls away from its power source or does not stay in contact with the cable being cut. Because nichrome is not readily soldered to copper wire, unreliable mechanical crimps are often made to connect the nichrome to an electric circuit. A self-contained device that is reusable and reliable was developed to sever cables for device release or deployment. The nichrome wire in this new device is housed within an enclosure to prevent it from being damaged by handling. The electric power leads are internally connected within the unit to the nichrome wire using a screw terminal connection. A bayonet plug, a quick and secure method of connecting the cutter to the power source, is used to connect the cutter to the power leads similar to those used in pyro-cutter devices. A small ceramic tube [0.25-in. wide 0.5-in. long (.6.4-mm wide 13-mm long)] houses a spiraled nichrome wire that is heated when a cable release action is required. The wire is formed into a spiral coil by wrapping it around a mandrel. It is then laid inside the ceramic tube so that it fits closely to the inner surface of the tube. The ceramic tube provides some thermal and electrical insulation so that most of the heat generated by the wire is directed toward the cable bundle in the center of the spiral. The ceramic tube is cemented into an aluminum block, which holds it in position. The leads of the nichrome wire are attached to screw terminals that connect them to power leads. A bayonet plug mounted at the bottom of the rectangular block connects the power leads to a relay circuit. A thin aluminum shell encloses the entire structure, leaving access points to attach to the bayonet plug and to feed a cable into the cylinder. The access holes for the deployment cable are a smaller diameter than the nichrome coil to prevent the cable from coming in direct contact with the nichrome when loaded. It uses the same general method of severing a cable with a heated wire as was used previously, but implements it in such a way that it is more reliable and less prone to failure. It creates a mechanism to create repeatability that was nonexistent in the previous method.

Pauken, Michael T.↗

NASA NTRS2011

Advanced Development of a Compact 5-15 lbf Lox/Methane Thruster for an Integrated Reaction Control and Main Engine Propulsion System

This paper describes the advanced development and testing of a compact 5 to 15 lbf LOX/LCH4 thruster for a pressure-fed integrated main engine and RCS propulsion system to be used on a spacecraft "vertical" test bed (VTB). The ability of the RCS thruster and the main engine to operate off the same propellant supply in zero-g reduces mass and improves mission flexibility. This compact RCS engine incorporates several features to dramatically reduce mass and parts count, to ease manufacturing, and to maintain acceptable performance given that specific impulse (Isp) is not the driver. For example, radial injection holes placed on the chamber body for easier drilling, and high temperature Haynes 230 were selected for the chamber over other more expensive options. The valve inlets are rotatable before welding allowing different orientations for vehicle integration. In addition, the engine design effort selected a coil-on-plug ignition system which integrates a relay and coil with the plug electrode, and moves some exciter electronics to avionics driver board. The engine injector design has small dribble volumes to target minimum pulse widths of 20 msec. and an efficient minimum impulse bit of less than 0.05 lbf-sec. The propellants, oxygen and methane, were chosen because together they are a non-toxic, Mars-forward, high density, space storable, and high performance propellant combination that is capable of pressure-fed and pump-fed configurations and integration with life support and power subsystems. This paper will present the results of the advanced development testing to date of the RCS thruster and the integration with a vehicle propulsion system.

Hurlbert, Eric A.↗

NASA NTRS2004

Improved Electromagnetic Brake

A proposed design for an electromagnetic brake would increase the reliability while reducing the number of parts and the weight, relative to a prior commercially available electromagnetic brake. The reductions of weight and the number of parts could also lead to a reduction of cost. A description of the commercial brake is prerequisite to a description of the proposed electromagnetic brake. The commercial brake (see upper part of figure) includes (1) a permanent magnet and an electromagnet coil on a stator and (2) a rotor that includes a steel contact plate mounted, with tension spring loading, on an aluminum hub. The stator is mounted securely on a stationary object, which would ordinarily be the housing of a gear drive or a motor. The rotor is mounted on the shaft of the gear drive or motor. The commercial brake nominally operates in a fail-safe (in the sense of normally braking) mode: In the absence of current in the electromagnet coil, the permanent magnet pulls the contact plate, against the spring tension, into contact with the stator. To release the brake, one excites the electromagnet with a current of the magnitude and polarity chosen to cancel the magnetic flux of the permanent magnet, thereby enabling the spring tension to pull the contact plate out of contact with the stator. The fail-safe operation of the commercial brake depends on careful mounting of the rotor in relation to the stator. The rotor/stator gap must be set with a tolerance between 10 and 15 mils (between about 0.25 and about 0.38 mm). If the gap or the contact pad is thicker than the maximum allowable value, then the permanent magnetic field will not be strong enough to pull the steel plate across the gap. (For this reason, any contact pad between the contact plate and the stator must also be correspondingly thin.) If the gap exceeds the maximum allowable value because of shaft end play, it becomes impossible to set the brake by turning off the electromagnet current. Although it may still be possible to set the brake by applying an electromagnet current to aid the permanent magnetic field instead of canceling it, this action can mask an out-of-tolerance condition in the brake and it does not restore the fail-safe function of setting the brake when current is lost.

Martin, Toby B.↗

NASA NTRS2011

Umbilical Deployment Device

The landing scheme for NASA's next-generation Mars rover will encompass a novel landing technique (see figure). The rover will be lowered from a rocket-powered descent stage and then placed onto the surface while hanging from three bridles. Communication between the rover and descent stage will be maintained through an electrical umbilical cable, which will be deployed in parallel with structural bridles. The -inch (13-mm) umbilical cable contains a Kevlar rope core, around which wires are wrapped to create a cable. This cable is helically coiled between two concentric truncated cones. It is deployed by pulling one end of the cable from the cone. A retractable mechanism maintains tension on the cable after deployment. A break-tie tethers the umbilical end attached to the rover even after the cable is cut after touchdown. This break-tie allows the descent stage to develop some velocity away from the rover prior to the cable releasing from the rover deck, then breaks away once the cable is fully extended. The descent stage pulls the cable up so that recontact is not made. The packaging and deployment technique can store a long length of cable in a relatively small volume while maintaining compliance with the minimum bend radius requirement for the cable being deployed. While the packaging technique could be implemented without the use of break-ties, they were needed in this design due to the vibratory environment and the retraction required by the cable. The break-ties used created a series of load-spikes in the deployment signature. The load spikes during the deployment of the initial three coils of umbilical showed no increase between the different temperature trials. The cold deployment did show an increased load requirement for cable extraction in the region where no break-ties were used. This increase in cable drag was superimposed on the loads required to rupture the last set of break-ties, and as such, these loads saw significant increase when compared to their ambient counterparts. While the loads showed spikes of high magnitude, they were of short duration. Because of this, neither the deployment of the rover, nor the motion of the descent stage, would be adversely affected. In addition, the umbilical was found to have a maximum of 1.2 percent chance for recontact with the ultra-high frequency antenna due to the large margin of safety built in.

Shafer, Michael W.↗

NASA NTRS2011

Comparison Between MAD-IPA Thrust Stand Measurements and Performance Modeling

The high specific impulse values associated with electric propulsion (EP) that allow for higher payload fractions than conventional, chemical propulsion are achieved by the acceleration of ionized propellant (plasma) by electromagnetic body forces. The lifetime of many EP devices is limited by electrode erosion caused by direct plasma-electrode interaction, while the efficiency is often limited by, among other thruster-specific factors, the available power in space. The efficiency can be increased during higher power operation since the amount of power required to ionize the propellant is fixed and decreases as a percentage of the increased input power. Pulsed inductive plasma accelerators [1-3] are a potentially elegant solution to the problems of high power demands and electrode erosion. The former is addressed in the pulsed nature of these devices, which allows for instantaneous operation on the order of megawatts while requiring a continuous supply on the order of only kilowatts, while the latter is addressed by inductive coupling of the thruster to ionized propellant. These devices operate by rapidly (on the order of microseconds) discharging stored energy at a given pulse repetition rate through a coil creating time-varying magnetic and electric fields that cause ionization, current formation and acceleration of propellant away from the coil. While the inductive (contact-less) nature of the energy transfer from the thruster to the propellant alleviates the problem of erosion and enables the use of in-situ and storable propellants incompatible with metallic electrodes, it places high voltage demands (on the order of tens of kilovolts) on the energy storage system to achieve propellant ionization.

Hallock, Ashley K.↗

NASA NTRS2011

Computational Validation of a Two-Dimensional Semi-Empirical Model for Inductive Coupling in a Conical Pulsed Inductive Plasma Thruster

A two-dimensional semi-empirical model of pulsed inductive thrust efficiency is developed to predict the effect of such a geometry on thrust efficiency. The model includes electromagnetic and gas-dynamic forces but excludes energy conversion from radial motion to axial motion, with the intention of characterizing thrust efficiency loss mechanisms that result from a conical versus a at inductive coil geometry. The range of conical pulsed inductive thruster geometries to which this model can be applied is explored with the use of finite element analysis. A semi-empirical relation for inductance as a function of current sheet radial and axial position is the limiting feature of the model, restricting the applicability as a function of half cone angle to a range from ten degrees to about 60 degrees. The model is nondimensionalized, yielding a set of dimensionless performance scaling parameters. Results of the model indicate that radial current sheet motion changes the axial dynamic impedance parameter at which thrust efficiency is maximized. This shift indicates that when radial current sheet motion is permitted in the model longer characteristic circuit timescales are more efficient, which can be attributed to a lower current sheet axial velocity as the plasma more rapidly decouples from the coil through radial motion. Thrust efficiency is shown to increase monotonically for decreasing values of the radial dynamic impedance parameter. This trend indicates that to maximize the radial decoupling timescale should be long compared to the characteristic circuit timescale.

Hallock, Ashley K.↗

NASA NTRS2012

High Temperature, Wireless Seismometer Sensor for Venus

Space agency mission plans state the need to measure the seismic activity on Venus. Because of the high temperature on Venus (462? C average surface temperature) and the difficulty in placing and wiring multiple sensors using robots, a high temperature, wireless sensor using a wide bandgap semiconductor is an attractive option. This paper presents the description and proof of concept measurements of a high temperature, wireless seismometer sensor for Venus. A variation in inductance of a coil caused by the movement of an aluminum probe held in the coil and attached to a balanced leaf-spring seismometer causes a variation of 700 Hz in the transmitted signal from the oscillator/sensor system at 426? C. This result indicates that the concept may be used on Venus.

Ponchak, George E.↗

NASA NTRS2012

Amplitude-Stabilized Oscillator for a Capacitance-Probe Electrometer

A multichannel electrometer voltmeter that employs a mechanical resonator maintained in sustained amplitude-stabilized oscillation has been developed for the space-based measurement of an Internal Electrostatic Discharge Monitor (IESDM) sensor. The IESDM is new sensor technology targeted for integration into a Space Environmental Monitor (SEM) subsystem used for the characterization and monitoring of deep dielectric charging on spacecraft. Creating a stable oscillator from the mechanical resonator was achieved by employing magnetic induction for sensing the resonator s velocity, and forcing a current through a coil embedded in the resonator to produce a Lorentz actuation force that overcomes the resonator s dissipative losses. Control electronics employing an AGC loop provide conditions for stabilized, constant amplitude harmonic oscillation. The prototype resonator was composed of insulating FR4 printed-wireboard (PWB) material containing a flat, embedded, rectangular coil connected through flexure springs to a base PWB, and immersed in a magnetic field having two regions of opposite field direction generated by four neodymium block magnets. In addition to maintaining the mechanical movement needed for the electrometer s capacitor-probe transducer, this oscillator provides a reference signal for synchronous detection of the capacitor probe s output signal current so drift of oscillation frequency due to environmental effects is inconsequential.

Blaes, Brent R.↗

NASA NTRS2012

Optimal Force Control of Vibro-Impact Systems for Autonomous Drilling Applications

The need to maintain optimal energy efficiency is critical during the drilling operations performed on future and current planetary rover missions (see figure). Specifically, this innovation seeks to solve the following problem. Given a spring-loaded percussive drill driven by a voice-coil motor, one needs to determine the optimal input voltage waveform (periodic function) and the optimal hammering period that minimizes the dissipated energy, while ensuring that the hammer-to-rock impacts are made with sufficient (user-defined) impact velocity (or impact energy). To solve this problem, it was first observed that when voice-coil-actuated percussive drills are driven at high power, it is of paramount importance to ensure that the electrical current of the device remains in phase with the velocity of the hammer. Otherwise, negative work is performed and the drill experiences a loss of performance (i.e., reduced impact energy) and an increase in Joule heating (i.e., reduction in energy efficiency). This observation has motivated many drilling products to incorporate the standard bang-bang control approach for driving their percussive drills. However, the bang-bang control approach is significantly less efficient than the optimal energy-efficient control approach solved herein. To obtain this solution, the standard tools of classical optimal control theory were applied. It is worth noting that these tools inherently require the solution of a two-point boundary value problem (TPBVP), i.e., a system of differential equations where half the equations have unknown boundary conditions. Typically, the TPBVP is impossible to solve analytically for high-dimensional dynamic systems. However, for the case of the spring-loaded vibro-impactor, this approach yields the exact optimal control solution as the sum of four analytic functions whose coefficients are determined using a simple, easy-to-implement algorithm. Once the optimal control waveform is determined, it can be used optimally in the context of both open-loop and closed-loop control modes (using standard realtime control hardware).

Aldrich, Jack B.↗