How a Thrust Stand Can Help Improve Drone Performance

By Lauren Nagel

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In the world of drones, UAVs and eVTOL aircraft, having an optimized motor-propeller configuration not only allows your aircraft to fly, but to perform optimally.

Manufacturers&#; data can give you an idea which motors and propellers will work in your design, but testing is not standardized, so it is impossible to compare parts across manufacturers. 

Here are a few ways that a thrust stand can help you improve drone performance:

Increase Your Flight Time

A major reason for testing your motors and propellers is to increase your drone&#;s flight time. Increasing your vehicle&#;s air time will allow you to shoot longer videos, collect more data, maintain visual contact on a target, and fly farther on a single charge. Simple tests and modifications can add precious minutes to your flight time, giving you a competitive advantage over competitors. A great example comes from two mechanical engineering Masters students at the University of Ottawa who were able to more than double the flight time of their reconnaissance helicopter drone by testing various motor-propeller combinations.

Throughout the testing process, the students were able to determine that lighter electrical components would perform equally well and found a more efficient motor than the one they were previously using. The result was that they increased their helicopter&#;s flight time from 3 minutes to 7 minutes without compromising on noise or payload (and they came up with an idea for a great aerospace company). This example goes to show that flight time can be improved through basic modifications to your design that centre around increasing efficiency.

Further reading: How to Increase Your Drone's Flight Time

Increase Your Payload

Many up-and-coming drone applications require vehicles to carry all kinds of payloads longer and farther than ever before. Meeting demands for payload capacity often requires testing multiple motors and propellers, but the initial investment will almost always pay off due to the improved operation of your UAV. Maximizing a drone&#;s payload capacity is important for industries such as eVTOL design, shipping and delivery, videography, cargo carrying, human transport, and more.

Hobby drones frequently have a payload up to 2kg, while drones in the &#;heavy lift&#; category may carry hundreds of kilograms of cargo. Whatever your payload requirements, testing multiple motor/ propeller configurations can ensure you&#;re getting the most bang for your buck. SkyDrive, a Tokyo-based aerospace company, makes use of such testing to optimize the geometry, size, and components of their drones. Most recently, they were able to build a heavy-lift drone capable of flying 15 minutes with a 30kg payload. Their final product, &#;CargoDrone&#;, contains 4 coaxial rotors for a total of 8 propellers and motors.

Increase Your Range

Improved antennas and range extenders have greatly enhanced our ability to fly UAVs into uncharted territory. The limiting factor is no longer how far we can communicate with the drone, but how long the vehicle can stay airborne on a single charge. Testing a drone&#;s propulsion system can help to extend its range by maximizing powertrain efficiency, contributing to a longer air time. This is especially important when flying missions into inaccessible terrain or over water.

If a drone cannot muster the power for the return trip, it could be lost completely. Ambitious videography projects and reconnaissance flights require a guarantee that the vehicle and footage collected will not be lost. With this guarantee, fantastic footage can be collected with confidence, such as this stunning tour of the Matterhorn in the Swiss Alps. Testing and optimizing your propulsion system can make these flights possible with the added benefit of knowing exactly what to expect from your UAV.

Check Your Noise Levels

While many eagerly await the future of drones and eVTOL aircraft, one of the biggest societal concerns remains the increased noise levels and their impact on our environment. With the potential for drones to be flying overhead delivering packages, inspecting buildings and taking people to work, this is a fair and valid concern. For many drone applications, the amount of sound produced will be an important factor in deciding whether or not they are put to use. This is true not only for everyday applications, but especially for surveillance and reconnaissance operations that demand silence. Testing your drone&#;s propulsion system allows you to anticipate the noise levels produced and resolve any issues before it takes its first flight. This insight can ultimately lead to a more effective and competitive UAV solution.

An inspiring example of this technology at work is in wildlife monitoring and conservation efforts. Ocean Alliance&#;s &#;SnotBot&#; program utilizes modified consumer drones to collect organic samples from whales in order to better understand the health of the population. The drones used in these expeditions are exceptionally silent so as to not disturb or frighten the whales, a paramount requirement for the sustainability of their research. Motors and propellers produce the majority of the noise in a UAV, so testing and comparing motors and props is your best bet for conceiving the quietest version of your design.

Reduce Vibration

All powertrains generate some degree of vibration, but excessive reverberation can cause damage to your components and is generally indicative of a lack of efficiency. Running a vibration test for your propulsion system is a great way to balance your propeller, detect inefficiencies and streamline your design. In doing so, you will likely notice that parts last longer and you get more performance out of a single battery charge.

Reducing vibration is especially important in the world of drone videography, where vibration can cause shaky or blurry videos, symptoms of the Jello effect. Stabilizers and post-production editing can improve the quality of your videos, but reducing the amount of vibration you are contending with in the first place can save time and money. Realizing these smoother flights and videos can easily be achieved with a bit of testing.

Increase Reliability

There is great incentive to increase reliability in the drone industry as the drone failure rate is about two orders of magnitude higher than that observed in commercial aviation. For UAVs to take over functions currently occupied by manned aircraft, their reliability, or mean time between failures (MTBF), must increase. Testing your propulsion system can help to prevent and predict failures as the data can provide insight into the state of your components.

Performing a Reliability, Availability, Maintainability, and Safety (RAMS) assessment, for example, is a great way to prove the reliability of our drone as it is an industry-recognized test that consumers recognize and trust. Once a system has been optimized, data from reliability tests can be a useful resource to reference or even publish as part of a marketing strategy.

Prevent In-flight Icing

Testing your motors and propellers can help understand how environmental factors affect your drone, such as the risk and impact of in-flight icing. In-flight icing or &#;atmospheric icing&#; can be a major hindrance to drone operations, as ice build-up changes the aerodynamic properties of the aircraft. Ice accumulation results in increased weight and drag leading to loss of lift, thus inhibiting the flight capabilities of your drone. The NRC&#;s wind tunnel experiments also demonstrated how temperature and propeller type impact ice accretion, and that torque increases linearly with time exposed to icing conditions. Cold temperature resistance is key for drones operating in cold climates, high altitudes, and in challenging weather conditions.

Knowing how a drone responds to cold temperatures can help to design it accordingly and determine whether its operations require an additional investment in de-icing products. UBIQ Aerospace, a world leader in de-icing technology, studies the patterns and effects of ice accumulation on UAVs by testing the aerodynamics of propellers in wind tunnel experiments. Their tests have allowed them to refine their D&#;ICE technology, which mitigates ice accumulation for fixed-wing UAVs. For any cold weather drone operations, propeller testing can be an invaluable source of information for anticipating performance.

Further reading: What Conditions Cause Drone Icing

Preventative Maintenance

One of the best ways to save time and money on drone maintenance, and any vehicle&#;s maintenance for that matter, is to repair wear and tear before it becomes a problem. It is infinitely better to invest in replacement parts than to have to replace an entire drone due to a fatal failure.

Testing your motors and propellers is an important part of this protocol as damage is not always evident. Minor erosion from water damage or debris can affect motor efficiency, thus limiting the drone&#;s performance. Propellers can also become unbalanced over time and take on increased vibration, wearing down the entire propulsion system. Testing your powertrain as part of your preventative maintenance schedule can help detect these inefficiencies, resulting in improved drone performance and great savings in time and money.

Acquire Diagnostics

In addition to the design phase, motor and propeller data can be useful throughout your vehicle&#;s lifetime. Recording diagnostics at scheduled intervals, every 50 flight hours, for example, can help to monitor a drone&#;s performance over time. Such tests are useful for detecting wear and tear as well as lost efficiencies.

Diagnostic motor and propeller testing can also provide insight as to why a UAV hasn&#;t been performing optimally, or give warning that it may not achieve the performance you are expecting due to unforeseen damage. Drones operating in environments with high humidity, high temperatures, or dust and debris may wear out faster than expected and surprise you with shorter-than-expected flight times or less-than-expected max throttle performance. Diagnostic testing of the propulsion system can detect these weaknesses before a failure occurs, preventing uncomfortable or potentially dangerous situations.

Improve Safety

Safety is perhaps the number one concern of investors and regulators in the drone and electric vehicle industry. Without certain guarantees, aerial vehicles will not be permitted to enter the market and serve their purpose. Achieving a safer design can be accomplished relatively painlessly by testing the system&#;s components, especially the motors and propellers that complete thousands of revolutions per minute.

Better understanding these elements can help prevent overheating, engine failure, power loss and more, thus preventing accidents and injuries. As our skies become increasingly populated by aerial vehicles, citizens must feel confident that they will perform as prescribed, posing no threat to human activities. Safety tests can significantly increase the credibility of individual vehicles and the industry as a whole, guaranteeing performance and promoting peace of mind for investors, government officials and the public alike.

Get Certification

Special permission is required for many common drone operations in the form of a waiver or exemption from a regulative authority. In the USA, for example, a waiver is required for any operation not included in Part 107 of the FAA&#;s Small UAS Rules. The list of operations requiring a waiver includes flying at night, beyond visual line of sight (BVLOS), over people, more than 400ft above ground level (AGL), or more than 100 miles per hour. This list naturally overlaps with the activities of many hobby and commercial drone operations.

Obtaining a waiver is largely dependent upon the applicant&#;s ability to convince the FAA that their operation is safe, and extensive testing of a UAV&#;s propulsion system is an important part of the evaluation of a safe vehicle. Performing and demonstrating the replicability of propulsion tests can greatly support a waiver application while providing the designer with valuable information. While regulations vary worldwide, many designers will find propulsion data useful or mandatory for drone certifications all over the world. 

Improve Structure Design

Improving a drone&#;s efficiency is a circular process that begins with certain assumptions. These assumptions can include the total weight of the drone and the weight of individual components, as well as its intended use. Once these initial assumptions are made, propulsion testing can help determine whether your design will meet the requirements for its proposed purpose. For racing drones, you can determine if your design will meet speed requirements or for a delivery drone, whether it will achieve flight time minima.

The best part is that if your initial design doesn&#;t meet your needs, you can swap in new motors and propellers to find an ideal configuration. Once you have found your ideal powertrain set-up, you can also go on to try new batteries or modify your frame. The initial assumptions guide the optimization process, but once you have completed one round of review, you can make informed decisions about other modifications to your design.

Further reading: Drone Design Calculations and Assumptions

Quality Assurance

Propulsion testing can strengthen and legitimize the quality assurance offered to potential investors and clients. Backing up a marketing pitch with rigorous performance data is an excellent way to foster confidence in your product and give your sales team a competitive edge. In addition to sales, testing your motors and propellers can provide valuable information for data sheets and allow designers to breathe easy knowing the vehicles will perform as advertised.

Propulsion testing will also allow you to further standardize products, ensuring each unit performs equally. This translates to less time spent on customer support requests as the additional level of security will ensure a lower number of defective products leaving the facility. Whether your operation is large or small, testing motors and propellers is a great way to legitimize your quality guarantee and ensure consistency between products.

Study Throttle Response

It is often necessary to learn how quickly a propulsion system can react to a change in control input. Data from such tests provides insight into how quickly the UAV can react to disturbances such as wind gusts. A typical way to physically test the reactivity of a propulsion system is to subject it to a frequency sweep control signal. A frequency sweep is a sinusoidal signal whose frequency is constantly varied to cover the whole spectrum of frequencies to be tested. With the data collected, the UAV designer can determine how fast the propulsion system can react to sudden changes in throttle.

Another method to learn about the reaction time of a powertrain is to subject it to a step input. While data is recorded at high speed, a sudden change in throttle is applied. After some time, the propeller stabilizes to a new rotation speed. The time it takes to stabilize the speed is known as the settling time. Finally, a proportional integral derivative (PID) test can measure your propulsion system&#;s consistency over an extended period of time by commanding a consistent, targeted thrust. These tests can be performed using a dynamometer with sufficient scripting capabilities, allowing for a complete understanding of a UAV&#;s throttle response.

Learn from Flight Replication

Propulsion testing can contribute to simpler, more efficient flight replication for applications such as agriculture, surveying, research and more. The ability to record flights using propulsion testing equipment allows designers to plan and save a route automatically, then reproduce the throttle data later on. Without even leaving the lab, designers can optimize their flight plan based on data collected on motor and propeller performance. Agricultural applications of autonomous UAV technology are often based on the principle of efficiency, minimizing labour and input costs with the added benefit of protecting human health.

Airboard Agro&#;s agriculture drone is a great example, as it sprays crops on pre-programmed routes in challenging terrain, conducting the laborious and repetitive work previously performed by humans. To truly reap the benefits of this technology, drones should be optimized for the specific task they will carry out, taking into consideration the speed, thrust and stability required. Propulsion testing is a great resource for improving these flight replications and ensuring UAVs will attain their maximum efficiency for autonomous missions.

Prevent Overheating and Thermal Failures

One of the most common causes of drone failure is engine overheating leading to engine failure. Maximum temperature and voltage ratings are often provided with electric motors, but it can be unclear when your motor is approaching these limits. Additionally, despite the fact that engine cooling depends on current, current ratings are not standard in the industry. One way to test a motor&#;s limits is to measure its temperature at various speed intervals using thermal probes, a useful strategy for circumventing failures.

At Kent State University in Ohio, Dr. Blake Stringer&#;s lab is performing such thermal tests with electric motors to investigate their thermal properties and to look at management of sUAS eVTOL motors under high-power conditions. This video produced in their lab shows the unfortunate consequences of thermal runaway, resulting in the overheating and destruction of the motor. These studies are increasingly important as drone operations become longer and move into harsher, hotter climates. Testing motors early in the design process can prevent overheating and ultimately save paying the cost to replace them.

For more information, please visit thrust testing.

Further reading: How to Automate Propulsion Tests for Your Drone

Conclusion

We hope that this article has demonstrated the multitude of benefits that come with testing your drone's propulsion system. We offer several tools that can perform these testing functions and help you take your design to the next level:

• Series

   - measures up to 5 kgf of thrust / 2 Nm of torque

• Flight Stand 15

   - measures up to 15 kgf of thrust / 8 Nm of torque

• Flight Stand 50

   - measures up to 50 kgf of thrust / 30 Nm of torque

• Flight Stand 150

   - measures up to 150 kgf of thrust / 150 Nm of torque

• Flight Stand 500

   

  - measures up to 500 kgf of thrust / Nm of torque
• Windshapers - open air wind tunnels for free flight testing

Our test stands measure thrust, torque, RPM, current, voltage, mechanical power, electrical power, motor efficiency, propeller efficiency and overall efficiency. Check out our online store to get yours or contact our sales team for a quote on our bigger tools.

Calibration serves three primary purposes. First, the calibration process produces the relationship between deflection or sensor output and thrust or impulse. Second, the accuracy, precision, and repeatability of a thrust stand are determined by repeated application of known forces or impulses, so it is an integral part of the error quantification. Finally, comparison of measured response with the expected theoretical response can reveal systematic problems that would bias the measurements if uncorrected.

where x ¯ is the mean of the measured responses. This is a measure of the proportion of total variability in the x i that can be explained by the F i . However, it is only a measure of correlation and a high value does not mean that the data have been well-fit, or that the calibration curve will yield results with low uncertainty [ 38 ]. To illustrate this, Anscombe [ 39 ] generated four hypothetical data sets, each yielding the same summary statistics for a linear least squares fit (including the same value for R 2 ). However, three of the four data sets are clearly not well-represented by the linear fit, deviating from the linear model due to different pathologies. Careful examination of the residuals can help avoid these pitfalls and is recommended for high quality thrust measurements. For example, a lack of linearity or repeatability apparent in the residuals typically indicates a mechanical problem with the thrust stand installation, such as dragging or unintentional contact with something as the thrust stand deflects, which should be corrected prior to making thrust measurements.

The standard deviation s x is a way to characterize the variability in response with fixed input and plays a key role in the error estimate, discussed in more detail in Section VI . The standardized residuals, e i / s x = ( x ^ i &#; x i ) / s x are also plotted in Fig. (12) . Residual plots are a very powerful way to evaluate the quality of a fit [ 38 ]. In general, if the assumed model is correct, the standardized residuals will fall uniformly between &#;2 and +2 and will be randomly distributed around zero. A systematic pattern of variation in the residuals can indicate a deviation from linear behavior or other violations of the assumptions on which least squares analysis rely. Another parameter often used in evaluating fits is the square of the correlation coefficient,

where the x i are the measured responses to input forces F i , S in this case corresponds to the sensitivity of the balance, and the u i are random disturbances. A finite force has to be applied to overcome static friction, so a plot of deflection vs. force will typically exhibit a negative intercept. If the linear fit to the data is forced to go through zero, it will introduce additional error. The intercept β should be estimated from the data. The disturbances are assumed to be normally distributed with a mean of zero and a common variance of σ x 2 . The input forces are assumed to be known without error, so the calibration method must be designed so that the uncertainty of the input loads is much less than that of the stand response. Standard analysis packages can then be used to perform least squares analysis to determine b and S cal , estimates of β and S that minimize the sum of squared residuals:

Figure (12) shows an example of a calibration curve constructed from n = 40 observations of thrust stand response for given input forces similar to those plotted in Fig. (11) . The response is modeled as a linear function of the applied forces,

As in this example, the force from the calibration weights is often stepped up and down incrementally using a remotely operated mechanism. The transducer signal is measured after applying weights and after removing them (the thrust stand zero), as shown by the solid circles in Fig. (11) . These measurements are often averaged over a number of samples to obtain a good estimate of the mean value, particularly if the transducer signal is noisy. The thrust stand response is defined as the difference between the signal with weights and the zero measured immediately afterwards. This approach automatically corrects for any drift in the thrust stand zero. The calibration process should be designed so that the calibration weights span the range of expected thrust values and are evenly distributed across the range.

Calibration forces can be applied in a number of ways. One common approach is to load and unload weights with known masses on a flexible fiber that passes over a pulley and attaches to the pendulum, as illustrated in Fig. (10) . The fiber must be carefully aligned with the thrust vector, and the pulley must be designed with minimum static and dynamic friction so that it transmits all of the force from the weights to the stand. An advantage of null balances is that they are unaffected by dynamic friction. An example of the thrust stand response to this kind of calibration is plotted in Fig. (11) . These data were obtained with an inverted pendulum thrust stand similar to that shown in Fig. (7) . In this case four calibration weights were hung from a fine chain that passed over a pulley and attached to the rear of the moveable upper platform. The other end of the chain was attached to a cylinder mounted behind the pulley that could be turned with a small DC motor to raise the weights on the cylinder-side of the chain loop, removing the force from the thrust stand. The position of the take-up cylinder was measured with a potentiometer. The change in the LVDT signal, which is proportional to the thrust stand displacement, with application of one, two, three and all four weights was measured. The position of the balance is sensitive to the inclination of the base, so this was measured with an inclinometer and actively controlled using a DC motor-driven screw to raise or lower one end of a cantilever beam on which the thrust stand was mounted. The inclinometer output voltage was used as the feedback signal to a software proportional controller that controlled the motor. Application of the weights caused small shifts in the inclination, as shown by the inclinometer signal in Fig. (11) , which were then corrected by the controller.

The uncertainty in the ratio L cal /L t must be included in the total thrust uncertainty estimate. If the calibration forces are applied in the same location as the thrust forces, the ratio L cal /L t = 1 and the thrust force can be determined directly from the calibration curve for a given Δx ss measurement. The sensitivity of null balance thrust stands is calibrated by measuring the force actuator coil current, rather than pendulum displacement, as a function of applied force.

To calculate the steady-state thrust of a thruster from a measured displacement Δx ss , this sensitivity must be scaled by the difference in distances from the pivot at which the calibration and thrust forces are applied:

Laboratory testing of an electric thruster often requires complex interfaces between the thruster and the test facility. Electric current, instrumentation, and various propellants must be provided to the thruster through these interfaces, all of which contribute elastic stiffness and affect the static equilibrium. While these contributions could be characterized individually and summed to calculate the total effective spring constants that determine thrust stand sensitivity [ 15 ], it is much more practical to perform an end-to-end calibration of the entire thrust stand installation, where all dynamic and static forces are characterized simultaneously. The calibration process is typically performed with the entire installation fully prepared to test, under vacuum, and only minutes before start-up of the electric thruster. Calibration involves applying known forces F cal at a point L cal from the pivot and monitoring the change in position at L pm to determine the thrust stand sensitivity,

B. Application of a Known Impulse

Pendulum thrust stands can also be used to make impulsive measurements but must be calibrated to characterize the response over the appropriate range of impulse bits. Thrust stands used in impulse measurements are typically underdamped pendulums which are allowed to oscillate at their natural frequency after an impulse perturbation. Hanging or inverted pendulums can be used but torsional balances offer improved sensitivity. Because the free response of the thrust stand to the impulse is measured, these are not null balances. Active electromagnetic dampers may be used to reduce vibrations from the facility, but they are turned off prior to making impulse measurements [2, 7].

Although thrust stand parameters such as effective spring constant and natural frequency or moment of inertia can be measured to calculate the sensitivity [7, 15], calibration is usually accomplished by applying known impulses Ibit,cal at a point Lcal from the pivot and inferring the sensitivity from a change in the dynamic response of the swinging arm (such as initial velocity, peak amplitude, or full range of motion), similar to the method outlined above for thrust measurements. For example,

Scal=Δx˙(0)Ibit,cal=LpmLcalI, (27)

assuming an impulsive force at t = 0. As in thrust measurements, this sensitivity must be scaled by the difference in distances from the pivot at which the calibration and thrust forces are applied to calculate the impulse from a measured change in velocity Δx˙(0), for instance:

Ibit=(Lcal/Lt)ScalΔx˙(0). (28)

Typical calibration methods can be broken down into either contact or non-contact methods. Several contact calibration methods involve the swinging of known masses [4, 40] or the use of impact pendulums [2] or impact hammers [8]. Non-contact methods include calibration using electrostatic forces between planar electrodes [14], free molecular gas flow from underexpanded orifices [9], electrostatic combs [24], and electromagnetic coils [17]. Of these methods we recommend piezoelectric impact hammers and electrostatic combs, which are common practice and provide precise, controllable impulses over a broad range.

Electrostatic combs (ESCs) or fins are known for their versatility as they can provide both a steady-state force as well as a wide range of impulses. ESCs consists of a set of interlocking non-contacting combs separated by a small gap. One set of the pair is placed on the moving part of the stand (usually the grounded set) and the other is mounted on the fixed part of the stand and aligned with the moving combs. The attractive force provided by the combs is a function of the applied voltage, the geometry, and the number of comb pairs in a set [42]. Unlike electrostatic actuators using planar electrodes [14], this comb geometry is independent of the gap, or engagement distance between the combs. This has two major implications for impulse balance calibration. First, it does not require that the location of the stand, and therefore the engagement distance, be known with great accuracy. Second, even though the engagement distance is changing slightly as the stand oscillates, the force applied by the combs does not change throughout the stand&#;s motion.

By accurately controlling the amount of charge, or voltage, on the combs as well as the time that the charge is applied, a well-known impulse can be created. ESCs can accurately produce forces from 10s of nN [41] to 10s of mN [24] with errors typically less than 1%. By varying the pulse width and amplitude of the applied voltage, Pancotti demonstrated impulses ranging from 0.01 mN-s to 20 mN-s which produced responses of a torsional thrust balance that were linear to within 0.52% [24]. The capacitance of the combs may lead to voltage overshoot at the beginning of the pulse. While this might not be an issue for longer pulse times, voltage overshoot from very short pulses may induce errors [24]. It is important therefore to understand the RLC characteristics of the pulse-forming system and design the circuit to minimize overshoot.

As with any calibration system the calibration mechanism itself must be well calibrated so that the uncertainty in the applied impulses is very low, a key assumption of the linear least squares analysis. The force generated by electrostatic combs as a function of applied voltage is typically measured by mounting the comb assembly on a micro-balance scale which has been calibrated with NIST-traceable standards. The impulse delivered by a pulsed comb system is calculated by integrating the force over the pulse width.

Piezoelectric impactors are commercially available in a variety of shapes and sizes, and are generally pre-calibrated. The typical procedure for impulse balance calibration is to strike the stand with the force transducer while recording the output voltage from the transducer. This output voltage is calibrated to correspond to a force, and integrating over time yields a total impulse. The force transducer can be mounted on a swinging pendulum arm that, when released, will strike the stand. The magnitude of impulses applied in this manner can be varied by adjusting the pendulum release height and the striking material. Typically the cocked angle and the release trigger are controlled by an electromagnetic actuator [8]. Forces ranging from 0.5 N to 100 N with errors of ± 2% can be achieved with this method [43]. Calibration hammers can also be driven with a servo motor, which can provide very accurate control of position and velocity and improve the accuracy of the applied impulse. Servo-controlled swing arms have been used to produce impulses ranging from 10 mN-s to 750 mN-s and generated a thrust stand response that was linear to within 0.5% over that range [24].

For piezoelectric hammers, a proper calibration should be performed either by the manufacturer or preferably in-house under the conditions the force transducer will be used. To calibrate a force transducer, a free suspended reference mass with an attached accelerometer can be used [24]. Beyond the calibration itself, other factors including strike location and digitization and integration all have to be carefully conducted in order to properly track errors and determine the total accuracy of the impulse calibration system.

The known calibration impulse will perturb the natural motion of the stand and cause it to ring or oscillate at its natural frequency. Figure (14) shows a plot of LVDT voltage as a function of time for a torsional balance that has experienced an impulsive perturbation. The impulse caused the stand to deflect with an initial velocity Δx˙(0) and a maximum range of travel, defined as the difference between the first peak and the first valley in the oscillatory response. These parameters can be estimated by fitting a damped sinusoid (for underdamped pendulums) to the data. The analysis of linear fits outlined above can be extended to nonlinear fits such as this. In practice, there is often a small amount of background motion or drift that must be subtracted to determine the change in motion due to the impulse. In this case, a damped sinusoid is also fit to the position history before the impulse and the response is defined as the difference between that and the motion after the impulse [8]. In a careful study of the variance associated with various fitting approaches, Koizumi et al. [2] found that the noise was predominantly at the natural frequency of the stand and could be distinguished from the thruster impulse by fitting the data over a range of (&#;2τn, 2τn) with a function of the form

x(t)={ANsin(ωnt)+x(0)cos(ωnt),t&#;0AFsin(ωnt)+ANsin(ωnt)+x(0)cos(ωnt)t>0, (29)

where An is the amplitude of the noise and Af is the amplitude due to the impulse. Their stand was minimally damped, so they did not include a damping term. An exponential decay representing damping could be added to the fit for stands with greater damping (see Eqn. 10).

Fig. 14.

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Displacement sensor (LVDT) response as a function of time from an applied impulse.

If the impulse applied to the stand is known accurately, a good correlation between either the initial velocity or the maximum range and the applied impulse can be determined. An example calibration curve obtained with data similar to that shown in Fig. (14) is plotted in Fig. (15). Figure (16) displays a calibration curve for the torsional balance shown in Fig. (9) based on measurements of the initial velocity change due to impact hammer impulses. In both of these examples the transducer signal is not translated into actual displacement or velocity. As long as it is linearly proportional to the motion, the transducer signal can serve as the parameter that is correlated with thrust or impulse. Data such as these can be fit using linear least squares analysis to obtain an estimate of the sensitivity Scal. The same recommendations for the design and analysis of linear calibration curves listed above should be followed with impulse calibrations as well.

Fig. 15.

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Calibration curve obtained by measuring maximum LVDT signal range (which is proportional to the pendulum amplitude range) as a function of impact hammer impulse values [23].

Fig. 16.

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Calibration curve obtained by measuring the initial rate of LVDT signal change (which is proportional to the pendulum velocity in the linear range) as a function of impact hammer impulse values.

The thruster impulse must approximate a true impulsive load, F(t) = Ibitδ(t). In practice, a linear response can be obtained for finite pulse lengths, as long as they are much shorter than the natural frequency of the stand. Figure (17) shows a range of travel for varying impulse times τ divided by the stand&#;s natural period τn. As impulse times become greater than about 1/10 the stand&#;s natural period the results become nonlinear. Therefore we recommend that known calibration impulses be applied with pulse widths much less than the period of the stand.

Fig. 17.

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Thrust stand maximum deflection as a function of impulse pulse width normalized by the natural period of the balance [21].

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