# Gyro monorail

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Gyro monorail

The gyro monorail, gyroscopic monorail, gyro-stabilized monorail, or gyrocar all denote a single rail land vehicle, road or rail, which uses the gyroscopic action of a spinning wheel, which is forced to precess, to overcome the inherent inverted pendulum instability of balancing on top of a single rail.

The monorail is associated with the names Louis Brennan, August Scherl and Piotr Schilovski, who each built full scale working prototypes during the early part of the twentieth century. An improved version was developed by Ernest F. Swinney, Harry Ferreira and Louis E. Swinney in the USA in 1962. This system is called the [http://www.monorails.org/tMspages/Gyro-Dynamics.html Gyro-Dynamics monorail] .

The gyro monorail has never developed beyond the prototype stage. Schilovski, in the introduction to his book [Schilovsky P P — The Gyroscope, Its construction and Practical Application, E Spon Publications 1922, Preface] , scathingly attributes this failure to widespread ignorance within the engineering profession and vested interest of the railway community.

The principal advantage of the monorail cited by Schilovski is the suppression of the Hunting oscillation, which was a speed limitation encountered by conventional railways at the time. Also, significantly reduced radii of turn are possible compared with the 7km typical of modern high-speed trains, "e.g." TGV. This is possible because the vehicle will bank automatically on bends, like an aircraft, [ Graham R — Brennan, His Helicopter and other Inventions, Aeronautical Journal, Feb. 1973 ] so that no lateral centrifugal acceleration is experienced on board.

Unlike more obvious means of maintaining balance, such as lateral shifting of the centre of gravity, or the use of reaction wheels, the gyroscopic balancing system is statically stable, so that the control system serves only to impart dynamic stability. The active part of the balancing system is therefore more accurately described as a roll damper.

Historical background

Brennan's monorail

The image in the leader section depicts the 22 tonne (unladen weight) prototype vehicle developed by Louis Philip Brennan CB [Tomlinson, N - Louis Brennan, Inventor Extraordinaire. John Hallewell Publications. 1980. ISBN 0-905540-18-2] . Brennan filed his first monorail patent in 1903.

His first demonstration model was just a 2ft 6in by 12inch (762mm by 300mm) box containing the balancing system. However, this was sufficient for the Army Council to recommend a sum of £10,000 for the development of a full size vehicle. This was vetoed by their Financial Department. However, the Army found £2000 from various sources to fund Brennan's work.

Within this budget Brennan produced a larger model, 6ft (1.83m) long by 1ft 6in (0.46m) wide, kept in balance by two 5inch (127mm) diameter gyroscope rotors. This model is still in existence in the London Science Museum. The track for the vehicle was laid in the grounds of Brennan's house in Gillingham, Kent. It consisted of ordinary gas piping laid on wooden sleepers, with a fifty foot wire rope bridge, sharp corners and slopes up to one in five.

Brennan's reduced scale railway largely vindicated the War Department's initial enthusiasm. However, the election in 1906 of a Liberal government, with policies of financial retrenchment, effectively stopped the funding from the Army. However,the India Office voted an advance of £6000.0 in 1907, to develop the monorail for the North West Frontier region and a further £5000.0 was advanced by the Durbar of Kashmir in 1908. This money was almost spent by January 1909, when the India Office advanced a further £2000.0.

On 15 October 1909, the railcar ran under its own power for the first time, carrying 32 people around the factory. The vehicle was 40ft (12.2m) long and 10ft (3m) wide, and with a 20hp (15kW) petrol engine, had a speed of 22mph (35km/h). The transmission was electric, with the petrol engine driving a generator, and electric motors located on both bogies. This generator also supplied power to the gyro motors and the air compressor. The balancing system used a pneumatic servo, rather than the friction wheels used in the earlier model.

The gyros were located in the cab, although Brennan planned to re-site them under the floor of the vehicle before displaying the vehicle in public, but the unveiling of Scherl's machine forced him to bring forward the first public demonstration to 10 November 1909. There was insufficient time to re-position the gyros before the monorail's public debut.

The real public debut for Brennan's monorail was the Japan-British Exhibition at the White City, London in 1910. The monorail car carried 50 passengers at a time around a circular track at 20mph. Passengers included Winston Churchill, who showed considerable enthusiasm. Although a viable means of transport, the monorail failed to attract further investment. Of the two vehicles built, one was sold as scrap, and the other was used as a park shelter until 1930.

cherl's car

Just as Brennan completed testing his vehicle, August Scherl, a German publisher and philanthropist, announced a public demonstration of the gyro monorail which he had developed in Germany. The demonstration was to take place on Wednesday 10 November 1909 at the Berlin Zoological Gardens.

Scherl's machine [Anon - The Scherl Gyroscopic Monorail Car. Scientific American. January 22, 1910] , also a full size vehicle, was somewhat smaller than Brennan's, with a length of only 17ft (5.2m). It could accommodate four passengers on a pair of transverse bench seats. The gyros were located under the seats, and had vertical axes, while Brennan used a pair of horizontal axis gyros. The servomechanism was hydraulic, and propulsion electric. Strictly speaking, August Scherl merely provided the financial backing. The righting mechanism was invented by Paul Fröhlich, and the car designed by Emil Falcke.

Although well received and performing perfectly during its public demonstrations, the car failed to attract significant financial support, and Scherl wrote off his investment in it.

chilovski's work

Following the failure of Brennan and Scherl to attract the necessary investment, the practical development of the gyro-monorail after 1910 continued with the work of Piotr Schilovski [Anon - The Schilowski Gyroscopic Monorail System. The Engineer, Jan 23, 1913.

] , a Russian aristocrat residing in London. His balancing system was based on slightly different principles to those of Brennan and Scherl, and permitted the use of a smaller, more slowly spinning gyroscope. Rather than developing a rail vehicle, he designed a gyrocar which was built by the Wolseley Motor Company and tested on the streets of London in 1913. Since it used a single gyro, rather than the counter-rotating pair favoured by Brennan and Scherl, it exhibited asymmetry in its behaviour, and became unstable during sharp left hand turns. It attracted interest but no serious funding.

Post-World War I developments

In 1922 the Soviet government began construction of a Schilovsky monorail between Leningrad and Tsarskoe Selo, but funds ran out shortly after the project was begun.

In 1929, at the age of 74, Brennan also developed a gyrocar. This was turned down by a consortium of Austin/Morris/Rover, on the basis that they could sell all the conventional cars they built.

Principles of operation

Basic idea

The vehicle runs on a single conventional rail, so that without the balancing system it would topple over.

A spinning wheel is mounted in a gimbal frame whose axis of rotation (the precession axis) is perpendicular to the spin axis. The assembly is mounted on the vehicle chassis such that, at equilibrium, the spin axis, precession axis and vehicle roll axis are mutually perpendicular.

Forcing the gimbal to rotate causes the wheel to precess resulting in gyroscopic torques about the roll axis, so that the mechanism has the potential to right the vehicle when tilted from the vertical. The wheel shows a tendency to align its spin axis with the axis of rotation (the gimbal axis), and it is this action which rotates the entire vehicle about its roll axis.

This fundamental behaviour is best illustrated by a toy gyro.

Ideally, the mechanism applying control torques to the gimbal ought to be passive (an arrangement of springs, dampers and levers), but the fundamental nature of the problem indicates that this would be impossible. The equilibrium position is with the vehicle upright, so that any disturbance from this position reduces the height of the centre of gravity, lowering the potential energy of the system. Whatever returns the vehicle to equilibrium must be capable of restoring this potential energy, and hence cannot consist of passive elements alone. The system must contain an active servo of some kind.

In the interests of safety, the actuation would be designed such that failures cause the servo to ‘float’ and apply minimal forces to the gimbal, giving time to stop the vehicle before the situation becomes serious.

Passive behavior

In order to assess how the vehicle would respond in the event of servo failure, we need to consider how the balancing loop behaves without a servo.

The vehicle/gimbal/gyro system described above is essentially a spinning top with toppling moment applied in one plane but not the other. The motion is described as two oscillations — a high frequency nutation mode, and a low frequency precession mode. With no feedback at all to the gimbal, the gyro is just dead weight, so the vehicle would simply topple over.

The toppling moment on a spinning top increases as the top displaces from the vertical and tends to increase the displacement further. The rolling moment on the monorail varies with roll angle in a similar fashion, but a free gimbal has no moment applied to it.

In order to emulate the spinning top, the moment on the gimbal must also tend to force the gimbal further away from the equilibrium position — the gimbal mounting must be unstable. This could be arranged using a toggle mechanism, such as is typically used in a bathroom light switch. The action of this unstable mounting is referred to as ‘acceleration of the precession’ [Schilovsky P P — The Gyroscope, Its construction and Practical Application, E Spon Publications 1922. Chapter 2] .

A spring mechanism is to be preferred to a top-heavy gimbal arrangement, as the latter would be affected by acceleration and braking.

The effect of a stable gimbal mounting, which engineering intuition might prefer, would be catastrophic. The resulting gyroscopic moment would act in the same sense as the toppling moment, rather than opposing it, so the toppling would take place more rapidly than if no feedback at all were applied.

With this feedback, the system behaves a lot like a sleeping top. After a while however, it will start to exhibit an oscillation in roll, which gradually increases in amplitude with time. The friction in the system (gimbal pivots, etc.) dissipates energy — and consequently there must be a tendency to reduce the time-averaged height of the centre of gravity.

By careful design, the time taken for the roll oscillation to build up may be sufficient to render the vehicle safe by other means. It seems possible to design a balancing system, which in the event of servo failure degrades to a system which is dynamically unstable, but remains statically stable.

It is this residual static stability which distinguishes the gyroscopic balancing system from alternatives based on, for example, reaction wheels and/or lateral payload shift, which would lose both static and dynamic stability if the servo were to fail.

Damping the precession

The application of mechanical feedback to the gimbal achieves the main design objective of getting the vehicle to stand up at all. However, the divergent oscillation which will always be present using a passive system, is clearly unsatisfactory, and some means of countering it is required. It is in the methods of damping out this oscillation that the differences between the Brennan and the Schilovsky balancing systems become apparent.

Considering first the Scherl/Brennan solution [Cousins H — The Stability of Gyroscopic Single Track Vehicles, Engineer Nov 21, Nov 28, Dec. 12 1913] : The cause of the divergence in the vehicle roll angle was traced to friction in the gimbals, so the cure would appear to provide positive velocity feedback (effectively negative friction) to the gimbal. After all, positive static feedback appeared to render the system statically stable.

As already mentioned, the motion is characterised by a high frequency (nutation) oscillation and a low frequency (precession) oscillation. Usually,we only notice the slow precession motion, as the vehicle rolls from side to side, and the gimbal also swings to and fro, but superimposed on this obvious motion is a very rapid oscillation of both vehicle and gimbal, which is usually damped out very quickly.

This rapid motion is called nutation, and is usually insignificant. However, if the friction were somehow reversed in sign ("i.e". tending to act in the same direction as the motion, rather than opposing it), the nutation would soon become painfully apparent. Instead of damping out, the nutation motion would diverge, and since the nutation is a very rapid motion, the vehicle would topple very quickly indeed.

At first sight it would appear that positive rate feedback would have a catastrophic effect. The trick used by Brennan and by Scherl, however, exploited the wide separation in frequency of the two motions characterising the vehicle response. If the actuator response were too slow to influence the nutation, but fast enough to affect the precession, positive rate feedback could be used and would damp the precession, whilst the nutation would still be damped out by friction.

Schilovsky, and quite a few modern authorities, considered the Brennan design intrinsically flawed. His approach followed a different line of reasoning. He reasoned that a system designed to cancel roll angle must include roll angle explicitly as a feedback.

All Schilovsky designs used explicit feedback of roll, usually through various pendulum mechanisms. Unlike Scherl and Brennan, who each used linear-proportional actuation (either hydraulic or pneumatic), Schilovsky always arranged for the actuation energy to be extracted from the gyro, and actuation was consequently intermittent. The resulting vehicle motion would inevitably include a steady state limit cycle, ‘wobble’, which may not be acceptable. There is no reason in principle why linear proportional actuation cannot be used with roll angle feedback.

Explicit roll angle feedback reduces the requirement for a wide separation between the nutation and precession frequencies, which the Brennan design evidently requires.

This separation is determined by the size and spin rate of the gyro, hence the Schilovsky design can use smaller gyros than the Brennan design. The system does require a more rapid, and consequently more expensive, servo. Also, when subjected to a contact side load (e.g. cross-wind or laterally offset load), the gimbal would exhibit a steady state deflection at equilibrium. With the bang-bang mechanisms which Schilovsky proposed, this would manifest itself as an increase in limit cycle amplitude.

If constant side forces were resisted by gyroscopic action alone, the gimbal would rotate quickly on to the stops, and the vehicle would topple. In fact, the mechanism causes the vehicle to lean into the disturbance, resisting it with a component of weight, with the gyro near its undeflected position.

Inertial side forces, arising from cornering, cause the vehicle to lean into the corner. A single gyro introduces an asymmetry which will cause the vehicle to lean too far, or not far enough for the net force to remain in the plane of symmetry, so side forces will still be experienced on board.

In order to ensure that the vehicle banks correctly on corners, it is necessary to remove the gyroscopic torque arising from the vehicle rate of turn.

A free gyro keeps its orientation with respect to inertial space, and gyroscopic moments are generated by rotating it about an axis perpendicular to the spin axis. But the control system deflects the gyro with respect to the chassis, and not with respect to the fixed stars. It follows that the pitch and yaw motion of the vehicle with respect to inertial space will introduce additional unwanted, gyroscopic torques. These give rise to unsatisfactory equilibria, but more seriously, cause a loss of static stability when turning in one direction, and an increase in static stability in the opposite direction. Schilovsky encountered this problem with his road vehicle, which consequently could not make sharp left hand turns.

Brennan and Scherl were fully aware of this problem, and implemented their balancing systems with pairs of counter rotating gyros, precessing in opposite directions. With this arrangement, all motion of the vehicle with respect to inertial space causes equal and opposite torques on the two gyros, and are consequently cancelled out. With the double gyro system, the instability on bends is eliminated and the vehicle will bank to the correct angle, so that no net side force is experienced on board.

Schilovsky claimed to have difficulty ensuring stability with double-gyro systems, although the reason why this should be so is not clear. His solution was to vary the control loop parameters with turn rate, to maintain similar response in turns of either direction.

Offset loads similarly cause the vehicle to lean until the centre of gravity lies above the support point. Side winds cause the vehicle to tilt into them, to resist them with a component of weight. These contact forces are likely to cause more discomfort than cornering forces, because they will result in net side forces being experienced on board.

The contact side forces result in a gimbal deflection bias in a Schilovsky loop. This may be used as an input to a slower loop to shift the centre of gravity laterally, so that the vehicle remains upright in the presence of sustained non-inertial forces. This combination of gyro and lateral cg shift is the subject of a 1962 patent. A vehicle using a gyro/lateral payload shift was built by Ernest F. Swinney, Harry Ferreira and Louis E. Swinney in the USA in 1962. This system is called the Gyro-Dynamics monorail.

The advantages of the monorail over conventional railways were summarised by Schilovski. In general, they follow from the behaviour described above. Regarding the limitations of conventional railways, the reader is referred to Hunting oscillation and Rail adhesion. The following have been claimed.

Universal gauge tracks

Different countries use different gauges (widths) of tracks, so the logistics get rather problematic for trains that travel to different countries with different gauges, i.e trains need to transfer cargo, change axles, or some similar time and money-consuming task must be performed. A single-rail track should eliminate these problems and hence simplify international rail transport.

Reduced right-of-way problem

The close association of the vehicle with its single rail, its inherent ability to bank on bends, and the reduced reliance on adhesion forces are all factors which are pertinent to the development of surface travel. In principle, steeper gradients and sharper corners may be negotiated compared with a conventional adhesion railway. Typical high speed train designs have radius of turn of 7km, with consequently few options for new routes within developed countries, where almost all of the land is under individual or corporate ownership.

Reduced total system cost

While the individual vehicles are likely to be expensive, the greatest cost arises from the construction and maintenance of the permanent way, which, for a single rail at ground level must be cheaper.

Benign failure modes

It is in the area of safety where myths are most prevalent. The angular momentum in the gyros is so high that loss of power will not present a hazard for a good half hour in a well designed system. Servos, being force, rather than position, demand can be easily designed to prevent locking in the event of failure, and sufficient residual stability is available to bring the vehicle safely to a halt even following servo failure. In any case, duplexing, and even a manual back up, should not present a major problem.

When considering safety, it is important to compare against a familiar baseline, in this case the lateral stability of two track vehicles must be the starting point. When the side force builds up, there are no cues that the situation is becoming dangerous, until at a critical value, the vehicle starts to topple. The resisting moment of the two track vehicle then actually decreases as toppling proceeds, and the vehicle cannot recover.

With the gyrotrain, the more the side force builds up, the further the vehicle leans into it, giving obvious early warning of a dangerous situation. The resisting force always increases with toppling, to resist the toppling moment further. This is gracious degradation, while the two track response is brittle failure, which through familiarity, is usually overlooked.

In a conventional tilt test, a gyrotrain would stay upright regardless of the angle of tilt of the ramp.

Reduced weight

Considering weight, Schilovski pointed out that his designs were actually lighter than the equivalent duo-rail vehicles. The gyro mass, according to Brennan, accounts for 3-5% of the vehicle weight, which is comparable to the bogie weight saved in using a single track design.

Potential for high speed

High speed conventionally requires straight track, introducing a right of way problem in developed countries. Wheel profiles which permit sharp cornering tend to encounter the classical hunting oscillation at low speeds. Running on a single rail is an effective means to suppress hunting.

Mathematical analysis

Introduction

The quantitative analysis of the behaviour of the gyro monorail may not be accessible to a wide readership, hence the article so far has been restricted to a qualitative description. It has been assumed that those who are able to understand the theoretical analysis ought to be able to demonstrate the stated behaviour from first principles.

This analysis is presented for the sake of completeness and traceability, to justify the qualitative statements of the remainder of the article. The non-mathematically inclined may safely ignore it.

Equations of motion

Stability is examined by deriving the characteristic equation from the equations of motion.

Denoting the roll angle with respect to the vertical $phi$, and the deflection of the gimbal $heta$, the gyroscopic torque acting about the vehicle roll axis is::::$Iomegafrac\left\{d heta\right\}\left\{dt\right\}$where I is the moment of inertia of the gyro about the spin axis, and $omega$ is the spin rate. For compactness these will be combined into a single angular momentum term::::$H=Iomega$The moment causing the vehicle to topple is::::$Wh sin\left(phi\right)$where W is the vehicle weight, h is the height of the centre of gravity. The small perturbation equation of motion about the roll axis is, therefore::::$Afrac\left\{d^2phi\right\}\left\{dt^2\right\}+H frac\left\{d heta\right\}\left\{dt\right\}=Wh phi$where A is the moment of inertia of the vehicle about the rail.

Evidently by controlling $heta$, it may be possible to control the roll angle, in particular, keep it near zero. However, the control must be such that both the gimbal deflection and roll angle tend to zero as time progesses, so the gimbal motion must be considered::::$Jfrac\left\{d^2 heta\right\}\left\{dt^2\right\} -Hfrac\left\{dphi\right\}\left\{dt\right\}=M$where J is the moment of inertia of the gyro/gimbal assembly about the gimbal pivot, M is the moment applied to the gimbal, i. e. the control input to the system. With no controlling moment, the characteristic equation becomes::::$lambda^2\left(lambda^2+\left(frac\left\{H^2\right\}\left\{AJ\right\}-frac\left\{Wh\right\}\left\{A\right\}\right)\right)=0$where $lambda$ is an eigenvalue.

if::::$H^2>WhJ$the system possesses stability of sorts, but there is nothing to prevent the gimbal rotating on to the stops, causing immediate toppling. To impart static stability feedback proportional to the gimbal deflection must be applied::::$M=k heta$where k is the stiffnes of the gimbal mount.the characteristic equation now becomes::::$lambda^4+\left(frac\left\{H^2\right\}\left\{AJ\right\}-frac\left\{Wh\right\}\left\{A\right\}-frac\left\{k\right\}\left\{J\right\}\right)lambda^2+frac\left\{kWh\right\}\left\{AJ\right\}=0$static stability requires the constant term to be positive, so the gimbal 'stiffness' feedback must be positive, and not negative. The mechanism must, to quote Schilovski; 'accelerate the precession'.

Friction

It is assumed that friction is adequately represented as viscous friction. The effective control input becomes::::$M=k heta-ffrac\left\{d heta\right\}\left\{dt\right\}$where f is the viscous friction coefficientresulting in the characteristic equation::::$lambda^4+frac\left\{f\right\}\left\{J\right\}lambda^3+\left(frac\left\{H^2\right\}\left\{AJ\right\}-frac\left\{Wh\right\}\left\{A\right\}-frac\left\{k\right\}\left\{I\right\}\right)lambda^2-frac\left\{fWh\right\}\left\{JA\right\}lambda+frac\left\{kWh\right\}\left\{AJ\right\}=0$The coefficient of $lambda$ is negative, indicating an instability.In order to proceed, the stability quartic must be factorised into a pair of quadratic terms. This is achieved by assuming::::$H^2>>WhJ+kA$This is not a severe restriction, because a wide separation between the modes is needed from considerations of robustness. If the approximate factorisation were not to apply, the stability of the resulting system would be rather precarious. The Brennan/Scherl design requires a 6:1 ratio of the frequencies of the two modes, while the Schilovski design can tolerate a narrower separation. The approximate factorisation is therefore adequate for any practical balancing system. The two factors become:

Nutation::::$lambda^2+frac\left\{f\right\}\left\{J\right\}lambda+frac\left\{H^2\right\}\left\{AJ\right\}=0$Precession::::$lambda^2-frac\left\{fWh\right\}\left\{H^2\right\}lambda+frac\left\{kWh\right\}\left\{H^2\right\}=0$Evidently, the precession is negatively damped. Reversing the sign of f will de-stabilise the nutation, hence the common belief that positive feedback of gimbal rate will not work, despite the fact that this was the very solution used by both Brennan and Scherl.

Brennan/Scherl loop

Whatever applies the positive feedback is likely to be a servo of some kind, which will have a finite response time. To keep things simple, a first order lag response is assumed::::where C is the controlling moment applied to the gimbal by the servo, $au$ is the servo time constant and is a gain. The gimbal moment is now::::$M=k heta-ffrac\left\{d heta\right\}\left\{dt\right\}+C$if the servo time constant is chosen to be in the region of::::$frac\left\{1\right\}\left\{ au\right\}=sqrt\left\{frac\left\{Wh\right\}\left\{A$the characteristic equation may be factorised into nutation, precession and servo modes. Ignoring the servo mode, the stability quartic reduces to::::applying the assumption that the nutation and precession modes are widely separated, the feedback does not affect the nutation appreciably, but has the potential for stabilising the precession. In effect, the servo is too slow to have much effect on the higher frequency mode. The precession mode is given by::::Provided the positive rate feedback gain is greater than the gimbal friction, this will be positively damped.

chilovski loop

The solution proposed by Schilovski, was to include feedback of roll angle. This will introduce a servo mode, which will affect stability, but for the purposes of this analysis, the servo bandwidth will be considered infinite. The control scheme is::::$M=k heta-ffrac\left\{d heta\right\}\left\{dt\right\}+alphaphi$where $alpha$ is the roll feedback gain.The stability quartic takes the form::::$lambda^4+frac\left\{f\right\}\left\{J\right\}lambda^3+\left(frac\left\{H^2\right\}\left\{AJ\right\}-frac\left\{Wh\right\}\left\{A\right\}-frac\left\{k\right\}\left\{J\right\}\right)lambda^2+\left(frac\left\{alpha H\right\}\left\{AJ\right\}-frac\left\{Whf\right\}\left\{AJ\right\}\right)lambda+frac\left\{Whk\right\}\left\{AJ\right\}=0$This does not affect the term in $lambda^2$, so the separation between the modes does not need to be as great as is required for the Brennan loop, implying smaller gyros can be used. When considering a practical servo, this apparent advantage is reduced.The precession mode is now characterised by the term::::$lambda^2+\left(frac\left\{alpha \right\}\left\{H\right\}-frac\left\{Whf\right\}\left\{H^2\right\}\right)lambda+frac\left\{Whk\right\}\left\{H^2\right\}=0$This will also be adequately damped with appropriate choice of gains.

Since the system is stable, the response to a side force is given by the equations of motion with the time derivatives set to zero::::$Whphi=L$where L is the disturbing moment, the vehicle will therefore lean into the disturbance. For a Brennan loop, the equilibrium gimbal deflection will be zero, but for the Schilovski loop, there will be a residual deflection::::$k heta=alpha phi$The tilt angle is detected using an inertial instrument such as a lateral accelerometer or pendulum, so that cornering accelerations would yield no net gimbal deflection. This gimbal deflection would only be observed with contact side forces. Instrument biases also appear as net biases on the gimbal deflection.

Similarly, a constant gimbal moment, as might result from a servo failure, would result in a constant gimbal deflection, provided the acceleration of the precession is implemented passively. The servo authority should be restricted to a level which is insufficient to deflect the gimbal on to the stops against the precession mechanism, in order to accommodate this type of failure.

Turning corners

Considering a vehicle negotiating a horizontal curve, the most serious problems arise if the gyro axis is vertical. There is a component of turn rate $Omega$ acting about the gimbal pivot, so that an additional gyroscopic moment is introduced into the roll equation::::$Afrac\left\{d^2phi\right\}\left\{dt^2\right\}+H\left(frac\left\{d heta\right\}\left\{dt\right\}+Omega phi\right)=Wh phi$This displaces the roll from the correct bank angle for the turn, but more seriously, changes the constant term in the characteristic equation to::::$frac\left\{\left(Wh-HOmega\right)k\right\}\left\{AJ\right\}$Evidently, if the turn rate exceeds a critical value:::$Omega=frac\left\{Wh\right\}\left\{H\right\}$The balancing loop will become unstable.However, an identical gyro spinning in the opposite sense will cancel the roll torque which is causing the instability, and if it is forced to precess in the opposite direction to the first gyro will produce a control torque in the same direction.

In 1972, the Canadian Government's Division of Mechanical Engineering rejected a monorail proposal largely on the basis of this perceived problem. Their analysis [Hamill, P.A. et al. - Comments on a Gyro-Stabilised Monorail Proposal. Canada. Control Systems Laboratory. LTR-CS-77. December 1972] was correct, but restricted in scope to single vertical axes gyro systems, and not universal, as the authors implied.

Effect of Earth Rotation

The motion of the vehicle with respect to inertial space introduces effects which must be compensated for, typically by using counter-rotating gyros, although Schilovsky claimed success with his system of varying the gimbal feedback with rate of turn.

Even when stationary, however, the vehicle will be rotating in pitch and yaw, because it is located on an Earth rotating at angular velocity $omega$. At a latitude $Theta$, the vertical component of the Earth's angular velocity is $omega sin\left(Theta\right)$, and the horizontal is $omega cos\left(Theta\right)$. If the heading of the vehicle is $psi$ with respect to North, the horizontal components will be $omega cos\left( Theta\right)cos\left( psi\right)$ North and $omega cos\left( Theta\right)sin\left( psi\right)$ East. (Note: the direction is of the axis of rotation associated with the angular velocity).

Evidently, the greatest effect of the horizontal component would be with the vehicle heading East to West or West to East, and this would have its maximum value at the Equator. A vertical axis gyro would be influenced mainly by the horizontal component of the Earths rotation, whilst a horizontal axis gyro would be influenced more by the vertical component, hence a vertical axis gyro would experience maximum unfavourable angular velocity at the Equator, but for a horizontal axis gyro the maximum effect would be experienced at either Pole.

The vehicle is forced to rotate at the same angular velocity as the Earth, this gives rise to a precession rate on the gyro equal to the siderial rotation rate of the Earth ($omega$). However, the balancing system constrains the gimbal near its undeflected condition, so that a steady roll torque equal to $H omega$, where H is the spin angular momentum of the gyro, acts about the vehicle roll axis. The value of H needed to cause loss of stability at such a tiny precession rate,is impractically high, so the net effect of the precession is a change in roll angle equal to:

::$delta phi = frac \left\{H omega\right\}\left\{Wh\right\}$

The Earth's rotation will also influence the gimbal orientation. This effect will be a maximum with the roll axis aligned with the Earth's axis, i.e. at the Equator facing North or South. The equilibrium increment in gimbal deflection will be:

::$delta heta = frac \left\{H omega\right\} \left\{k\right\}$

Where k is the gimbal static feedback (the 'acceleration of the precession')

Consider a 10 tonne vehicle.

Brennan's recommendation for the gyro mass would be about 300kg. Assuming a disc having diameter to thickness of 8:1, the radius would be (assuming steel, of density roughly 8000kg/ $m^3$) 0.36m, the spin moment of inertia is 19.4 kg $m^2$. Assuming a peripheral speed well within the metallurgical limit, say 100m/s, the angular velocity is 278 radians/sec (about 3000rpm), and the spin angular momentum is 19.4×278=5389 kg $m^2$/sec.

The overturning moment for a cg height of 1.5 metres is 10000.0×9.81×1.5=147150Nm per radian.

The maximum precession rate is the rotation rate of the Earth at 0.00007 radians per second, so the gyroscopic torque arising from this precession rate is 0.38Nm.

The increment in roll angle caused by the Earth's rotation will be 0.000026 radians or about one thousandth of a degree. Since the gimbal static feedback is typically similar in magnitude to the overturning moment, the maximum additional gimbal deflection arising from the rotation of the Earth is similarly of the order of one thousandth of a degree.

Even if these assumptions were an order of magnitude in error, the torques arising from the Earth's rotation would still be absurdly small.

For a perfectly matched pair of counter-rotating gyros, these deflections would be identically zero.

Gyro size

Robustness to slow down

The gyro characteristics are chosen specifically so that failure of the power supply will not cause immediate catastrophic failure, but the system will remain stable for sufficient time for it to be rendered safe.

As the gyro spins down, the frequencies of the nutation and precession modes move closer together, until at the point of instability they become equal. Normal operation should be well away from this point, so that the modes may be treated separately.

The approximate factorisation of the stability quartic is valid provided:

::$frac\left\{H^2\right\}\left\{JA\right\}>>frac\left\{Wh\right\}\left\{A\right\}+frac\left\{k\right\}\left\{J\right\}$

The feedback term is assumed similar to the toppling moment term, so robustness to gyro slow down can be ensured by choosing the gyro angular momentum such that:

::$frac\left\{H^2\right\}\left\{JA\right\}=20frac\left\{Wh\right\}\left\{A\right\}$

Or $frac\left\{I\right\}\left\{J\right\}Iomega^2 = 20 Wh$

where $omega$ is the angular velocity of the gyro.

Treating the gyro as a thin disc and ignoring the gimbal frame:

:::$frac\left\{I\right\}\left\{J\right\}=2$

The moment of inertia of a disc is:

::$I=frac\left\{Mr^2\right\}\left\{2\right\}$

where M is its mass and r the radius. An estimate of the required gyro size is therefore given by:

::$M=20frac\left\{Wh\right\}\left\{\left(romega\right)^2\right\} = 20frac\left\{Wh\right\}\left\{U^2\right\}$

where U is the peripheral speed.

Maximum spin rate

Gas turbine engines are designed with peripheral speeds as high as 400m/s [Rogers G.F.C. and Y.R.Mayhew - Engineering Thermodynamics, Work and Heat Transfer. Third Edition. Longman 1972, p433] , and have operated reliably on thousands of aircraft over the past 50 years. Hence, an estimate of the gyro mass for a 10 tonne vehicle, with cg height at 2m, assuming a peripheral speed of half what is used in jet engine design, is a mere 140kg. Brennan's recommendation of 3-5% of the vehicle mass was therefore highly conservative.

References

* Schilovsky P P — The Gyroscope, Its construction and Practical Application, E Spon Publications 1922

* Cousins H — The Stability of Gyroscopic Single Track Vehicles, Engineer Nov 21, Nov 28, Dec. 12 1913

* Graham R — Brennan, His Helicopter and other Inventions, Aeronautical Journal, Feb. 1973

* Mee A — Harmsworth Popular Science, Volume 3, Pages 1680 to 1693, 1912

* Eddy, H.T. - The Mechanical Principles of Brennan's Mono-Rail Car. Journal of the Franklin Institute, Vol CLXIX p467

* Tomlinson, N - Louis Brennan, Inventor Extraordinaire. John Hallewell Publications. 1980. ISBN 0-905540-18-2

* Anon - The Scherl Gyroscopic Monorail Car. Scientific American. January 22, 1910

* Anon - The Brennan Mono-Track Vehicle. The Commercial Motor. 18th November 1909.

* Anon - The Schilowski Gyroscopic Monorail System. The Engineer, Jan 23, 1913.

* Hamill, P.A. et al. - Comments on a Gyro-Stabilised Monorail Proposal. Canada. Control Systems Laboratory. LTR-CS-77. December 1972
* Anon - Monorail Vehicles, Engineering. June 14, 1907. p794.

* Rogers G.F.C. and Y.R.Mayhew - Engineering Thermodynamics, Work and Heat Transfer. Third Edition. Longman 1972, p433

* Bicycle and motorcycle dynamics
* Gyrocar
* Segway HT

* [http://www.monorails.org/tMspages/Gyro-Dynamics.html Monorail Society Special Feature on Swinney's monorail]

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