The Mach-Effect thruster is a propellantless propulsion concept that has been in development by J.F. Woodward for more than two decades. It consists of a piezo stack that produces mass fluctuations, which in turn can lead to net time-averaged thrusts. So far, thrusts predictions had to use an efficiency factor to explain some two orders of magnitude discrepancy between model and observations. Here (M Tajmar) presents a detailed 1D analytical model that takes piezo material parameters and geometry dimensions into account leading to correct thrust predictions in line with experimental measurements. Scaling laws can now be derived to improve thrust range and efficiency. An important difference in this study is that only the mechanical power developed by the piezo stack is considered to be responsible for the mass fluctuations, whereas prior works focused on the electrical energy into the system. This may explain why some previous designs did not work as expected. The good match between this new mathematical formulation and experiments should boost confidence in the Mach effect thruster concept to stimulate further developments.
Mach-Effect thruster model (PDF Download Available). Available from: https://www.researchgate.net/publication/319974638_Mach-Effect_thruster_model [accessed Oct 17 2017].
Woodward devised a method to use these mass fluctuations for a novel propulsion scheme: Push the mass when it is heavy and pull it back when it is lighter. This cycle can create a time-averaged net linear impulse in one direction that satisfies the definition of a propellantless thruster. Apart from Woodward’s own thrust measurements in 2016 Buldrini independently replicated this effect. Recently, it has been shown explicitly that such a scheme does not violate conservation of momentum.
Of course, energy must still be spent to vary the mass and accelerate it. The power-to-thrust ratio is an important figure of merit to compare it against photon (P/F=3⋅10^5 W/mN) and other electric thrusters (P/F=20-60 W/mN). At present, typical experimental values for the Mach-Effect thruster are an order of magnitude better than the photon rocket (P/F=3⋅10^4 W/mN). Woodward is using Piezo crystals both as capacitors and actuators to oscillate their energy and to push/pull them. Both processes must appear at a proper phase between them to produce thrust.
After significant improvements of the experimental techniques, the observed thrusts are in the sub-µN – µN range, which requires micro thrust balances with high resolution. Proper analysis and shielding is necessary to rule out possible artifacts such as thermal effects, outgassing or magnetic interactions as demonstrated by Woodward and coworker.
Tajmar has a fully analytical model of the Mach-Effect thruster is presented whose predictions match experimental data and allows the design of optimized thrusters based on mass fluctuations by taking both design and material properties into account. The model gives an important insight into how mass fluctuations appear and why the present design works but other designs failed.
The current embodiment of the Mach-Effect thruster consists of a stack of piezo discs that is similar in design to typical actuators using ferroelectric (PZT=Lead Zirconate Titanate) materials, which are sold by many suppliers e.g. for ultrasonic applications. In general, if an electric field is applied across such PZT discs, they expand and contract depending on the field strength and direction of the field. The piezo/PZT stack is made of several discs that are mechanically connected in series but electrically connected in parallel (i.e. all discs have the same electric potential applied between their electrodes). This is achieved by always switching the polarity from disc to disc such that every electrode faces another electrode with the same polarity to avoid electric short circuits. Woodward uses brass electrodes which are glued with epoxy between each disc. The whole assembly is clamped with stainless steel screws between two end caps, a larger one made from brass with threaded holes and a smaller one made from aluminum. The screws are tightened to ensure that the piezo stack is well compressed between the stiff end caps.
In contrast to prior Mach thruster analysis, the assertion in this analysis is that only the mechanical (inertial) energy contributions to the Mach fluctuations, whereas the prior interpretations focused on the electrical energy in the capacitors (or coils). This makes sense as Sciama’s model describes inertial and hence only inertial (=mechanical) energy. In some previous experiments, mechanical oscillation was replaced by ion/lattice movements that were thought to be much more efficient because they can oscillate at much higher frequencies. However, although early papers reported thrusts up to the mN range, no net thrusts were seen when proper electrical shielding and setups were used in subsequent measurements. As a result, it was thought the bulk acceleration is necessary for the effect to occur, however as we will see, it is not only bulk acceleration but pure mechanical energy that is responsible for the correct thrust values observed.
Qm is the mechanical quality factor of the stack. Although Qm can be high for individual PZT discs, it is quite low for a stack with epoxy and electrode material in between. The values are determined by spectrum analysis and are typically around 60. Again, using our example, we get an effective power of 63 W and a power loss of 2.6 W and a total capacity of 14 nF.
Tajmar has developed a 1D analytical model that can accurately predict the thrust from Mach- Effect thrusters taking design and materials parameters into account. It compares well to experimental data and allows for further optimization to obtain higher thrusts and efficiencies. Apart from the well-known voltage and frequency scaling, it predicts higher thrusts e.g. for larger disc diameters and higher stiffness. For example, if the PZT discs are increased to a diameter of 25 mm, the 2nd resonance frequency should rise to 51 kHz. Both should lead to an increase in thrust to 12 µN at an amplitude of 200 V. Of course, there are several shortcomings and simplifications that may be corrected in future iterations such as implementing resonances 22 into thrust model, use of electric field dependent piezo material parameters, include temperature degradation effects and adding the influence of clamping torque from screws.
The model is flexible enough to be modified for different geometries (e.g. piezo rings instead of discs with one single screw in the middle). One of the main conclusions of this analysis is that the thrust is only accurately calculated if only the mechanical power is used in the transient mass equation. This can explain why some previous designs (Mach-Lorentz thrusters) did not work as expected. It is hoped that the model and its fit to experimental results adds further confidence into Mach effect thrusters and stimulates further research in that area.
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