Matteo Mottola – firstname.lastname@example.org
The angular speed of an operating engine is stable if, for that angular speed, two conditions are verified: equilibrium between the engine torque and the resistant torque, negative derivative respect to angular speed of the sum of both engine and resistant torque.
However, if the engine torque isn’t controlled with an external feedback coming from the angular speed, these conditions must be satisfied by the engine torque characteristics. In fact, to maintain a resistant torque at a certain speed, the engine characteristics must satisfy, for that speed, the two stability conditions.
This problem was first encountered by James Watt in the eighteenth century: he redesigned the centrifugal governor of Christiaan Huygens in order to provide a negative feedback control to the steam admission valve respect to the angular speed.
The governor rotates with the engine as it’s connected to the main shaft with a transmission. The centrifugal force acting in the “flyballs” tends to move them upwards and away from the rotating axis, against the gravity. A four-bar-chain connected to the translating hub of the governor moves the steam admission valve, so the raise of the engine angular speed causes a reduction of the aperture of the valve itself.
The main objective is the modelling of a system made of an engine and a governor in order to comprehend the dynamics and to test the stability of the whole system.
The modelling objective is the creation of a CAD model of a governor, a near-linear linkage and a valve, taking inspiration from historical drawings. The only kinematic analysis to perform will look at the relation between the governor’s hub translation and the rotation of the valve.
The behaviour of the only governor is tested with a dynamic simulation in order to find the relation between the rod angular speed and the flyball “opening”, with the aim to find the eigvalues and the vibration frequencies.
An example turbine is modelled with a rotating cylinder and a characteristic linear torque equation. The closed-loop control is created modulating the engine torque with the carburetor valve opening.
The influence of the governor in the system will be tested with two dynamic simulation, with and without the feedback. A constant resistant moment is interrupted for a few seconds: the aim is to demonstrate that, after the resistant perturbation, the turbine without closed-loop control loses the equilibrium while the stabilized one returns to the initial speed. The stability properties will be ensured by an eigvalue analysis.
The final objective is the performing of a dynamical simulation in order to obtain and discuss the torque characteristics of the new stabilized system.
The modelling problem
In literature are present different configurations of the Watt’s regulator. Various images of drawings and real regulators have been analysed in order to take inspiration.
The previous picture represents a steam engine governor located in a whisky distillery in Glasgow . The CAD model takes inspiration from a fac-simile drawing  founded in an historical archive that allows to well analize the components and the joints involved:
Neglecting the valve four-bar linkage, for a modelling purpose, may be distinguished eight functional bodies:
- Rotating rod (axis A—A’)
- Translating hub (R)
- Left flyball (C—B)
- Right flyball (C’—B’)
- Left link (M—N)
- Right link (M’—N’)
- Hub fork (V)
- Fork link (W)
The carburetor is modelled according to the introduction drawing. The created CAD model is shown in the following renderings:
The ADAMS model is completed with two “stoppers” that act as contact bodies in order to limit the hub travel to a total of 40mm. In the following figure are shown the red stoppers and the regulator in the not-rotating equilibrium position, with the carburetor valve full open:
The chosen material for all the bodies is steel. In the real model may be founded rotational joints in both flyballs and flyball links in order to mantain loads simmetry and structural integrity in the rotating structure. However, the multibody model constraints are changed in order to prevent redundancy. The following tables explicate the adopted constraint configuration:
|N°||Joint type||Body i||Body j|
|5||Spherical||Link SX||Wing SX|
|6||In Line||Link DX||Wing DX|
|N°||Joint type||Body i||Body j|
The real model is explicitly undamped, however every real constraint is affected by friction and lubricated. The flyballs are also affected by air friction: these conditions result in a damped -even vibrating- system. However, the impossibility to find information about the damping value imposes to find a plausible value empirically. Assuming to aim to an under-damped system, the main hypothesis of this project is to place a damper between rod and hub with a damping coefficient of 20 Ns/m. This value comes from the imposition to the vibration decay of the model to be similar to the behaviour of working governors observed in a few videos founded in the web.
The next step is turbine modelling: a steel cylinder 250 mm long with a 250 mm radius is constrained to the ground with a rotational joint. A coupler relates turbine’s and governor’s rotations with a speed ratio of 50. The resulting Groubler equation counts n°DOF=2 with no redundancy in the constraint equations. Two single component torques, representing engine and resistant actions, are applied to the turbine’s rotational joint. The following picture displays the entire modelled system:
This results in obtaining a linear behaviour with 200 Nm at 1000 rpm as shown in the following graph:
This characteristics are relative to a full-throttle configuration, with the steam/fuel admission valve completely opened: the last modelling objective is the creation of a feedback control of the turbine’s torque provided by the carburetor valve rotation. This is realized by the following equation that represents the engine torque that is applied to turbine’s rotational joint:Where the function γ(θ) is the carburetor equation, which values are supposed to be γ=1 with a full-throttle configuration, so the engine torque is equal to the maximum capability of the turbine at the rotating speed, and γ=0 with a closed-valve configuration so the engine doesn’t provide any torque. The simplest way to synthetize the carburetor equation is with a linear depdendency in the valve angle θ.
Before proceeding, a kinematic analysis is performed in order to obtain a measure of the valve rotation range values that have to been included in the carburetor equation. A motion locks the turbine’s -and so the governor’s- rotation, while another motion imposes a translation in the governor’s hub, consenting to obtain the movement of carburetor’s valve. The following graph provides the obtained results:
The four-bar chain that transfers the hub motion to the valve rotation is well dimensioned as provides a near-linear relation between the two motions. The valve acts between θ=0 rad (full open) and θ=1.476 rad (full closed) so the following carburetor equation is chosen:
With the carburetor equation, the closed-loop control is done and the dynamic system is completely defined. The following block scheme, finally resumes the work done:
Simulations and analysis of results
Before proceeding with a complete system analysis, the only governor behaviour is discussed. While the governor is rotating, centrifugal forces act against gravitational forces determining the opening of flyballs and the raising of the translating hub. As the rotational speed determines the centrifugal forces, it has a fundamental role because a speed change causes the rising or lowering of the mechanism and so a change in the output carburetor signal. Instead of this, fixed the angular velocity, a steady-state translational equilibrium of the hub may be reached and, hand in hand, a vibration around the equilibrium position may identified.
The first dynamic simulation aims to reach a relation between the governor’s angular velocity ω and the equilibrium position of the hub. A motion imposes a quasi-static increase in rod’s rotational speed with an acceleration of 0.01 rad/s²: this constraints the rotation of the system reducing the degrees of freedom at the only opening of the flyballs. The quasi-static increase makes the inertias and the damping forces negligible, allowing to obtain a curve representative of the steady-state equilibrium:
The results suggest that the mechanism works in a speed range between 8.4 rad/s and 10.1 rad/s with a near linear behaviour that is interrupted by the cut-off induced by bottom and top contacts. The three cases are represented in the following picture with the governor, respectively, in contact with lower stopper, in working equilibrium and in contact with upper stopper:
- Left(8 rad/s): contact with lower stopper, full-opened valve
- Middle(9.5 rad/s): working equilibrium, governor free to vibrate, partialized valve
- Right(11 rad/s): contact upper stopper, full-closed valve
The stability analysis of equilibrium positions is performed with a series of dynamical simulations in which a consant rotational speed is imposed. Every simulation lasts for 15s that is enough to reach the equilibrium position. The tested speed range is 8.5rad/s – 11 rad/s with 0.125 rad/s speed steps, with a total of 13 simulations. At the equilibrium position, the system is then linearized and the eigvalues are extracted.
The linearization is, however, influenced by the problem of selecting a correct reference frame with respect to which to compute the linearization: a rotating frame is suggested as the governor is rotating, but the non-rotating four-bar linkage and valve sub-system may affect a correct solution. A frequency analysis is so performed computing the hub’s translation Fast Fourier Transform: this puts in evidence a peak corresponding to the vibration frequency computed linearizing respect to a fixed frame, suggesting to linearize with respect to the ground frame.
The obtained eigvalues are summarized in the following real-imaginary plane and expressed with adams units of measure (Hz), with the velocity increasing from right to left:
As expected, the negative real part suggests that the vibration modes are stable throughout the examined speed range. The vibration frequency is well defined as is near constant between 0.7Hz and 0.8Hz. The damping ratio increases with angular velocity because the speed ratio between the hub translation (which body the damper is connected to) and the flyballs opening also increases, enphatizing damper’s work.
These informations will be useful to identify the vibration mode of the only governor when studying the stability of the two-dof system.
The behaviour of the whole turbine-governor system is now discussed. Thanks to a series of two dynamic simulations, will be observed the difference between a not-stabilized and a governor-stabilized turbine. These tests consist in verifying the system’s speed stability after a temporary quick variation of the resistant action: a stable system should return to initial equilibrium configuration while, after the perturbation, the non-stabilized one may lose the instable initial equilibrium.
The first simulation involves a non-stabilized turbine: the governor translational degree of freedon is therefore locked with a motion that constraints hub’s translations. The single-DOF system should mantain an equilibrium with respect to a constant resistant torque with a magnitude of 250Nm, assuming 4000rpm as working rotational speed. Turbine’s characteristics show that, at 4000rpm, the turbine supplies 500Nm of torque, so the carburetor valve opening is set (with hub’s motion) to a fixed value of 50%: in this way the system supplies the requested 250Nm engine torque. A couple of STEP functions simulate a temporary resistant torque drop to a value of 50Nm within 10s and 12.5s. As turbine isn’t able to reach, starting from ω=0rpm, the equilibrium speed of 4000rpm by itself, it is carried to the target speed with a motion that will be deactivated after 5s.
The following graph displays the obtained results:
This graph displays turbine’s rotational speed (black), engine torque (red) and resistant torque (blue).
- t=[0,5]s: Turbine is brought to target speed with a motion located in it’s rotational joint. As it reaches target speed (4000rpm) it also reaches the target torque (250Nm).
- t=[5,10]s: The motion is deactivated and the unstable equilibrium, between engine and resistant actions, allows the system to mantain a constant speed.
- t=[10,12.5]s: The STEP functions prodives a drop in the resistant torque causing a positive torque resultant in the system wich thus accelerates, causing a drift from the equilibrium speed. As turbine’s speed rises, hand in hand rises also the engine torque that it provides.
- t=[12.5,30]s: After the resistant action is returned to the original value, the engine doesn’t return to the initial equilibrium as gained around 400rpm by the resistant drop. There’s so a positive difference between engine and resistant actions that causes a positive acceleration. As speed rising is accompanied by an engine torque rise, ther’s no possibility to obtain another equilibrium: the system diverges.
The impossibility of a new equilibrium is confirmed by a linearization at simulation’s end with witch two non-vibrating eigvalues of the single-DOF system are obtained wich one is a positive real part:
The second simulation involves the turbine-governor stabilized two-DOF system as the governor’s degree of freedom dynamics completely determine carburetor’s valve opening. The simulation conditions are the same: the system is initially brought by a motion to 4000rpm and a constant 250Nm resistant torque is applied. As in the previous simulation, a couple of STEP functions perturbate for a while the initial equilibrium.
The following graph displays the obtained results while a speeded-up videoclip shows governor’s motions:
The four curves in the graph are turbine’s rotational speed (black), engine torque (red), resistant torque (blue) and engine’s regulation (green). This last parameter is the output of the carburetor equation ϒ(θ) and can display the turbine torque regulation introduced by goveror’s motions.
- t=[0,5]s: The system is brought to the target speed of 4000rpm with a motion located in turbine’s rotational joint. The governor speed isn’t enough to lift the flyballs as they start to work at a governor speed greater than 8.4rad/s (while it’s now rotating at ω≈8.38 rad/s). So, differently from the single-DOF simulation where the valve is locked to half regulation, now the valve is full-opened and the engine provides torque with it’s maximum capability.
- t=[5,15]s: The motion is deactivated. However, differently from the single-DOF simulation, there’s no equilibrium as the engine torque is much higher than the resistant one: the system so accelerates and the speed grows. As seen in the “Translating hub equilibrium positions” graph, starting from 8.4rad/s (around 4010rpm in the turbine), a speed rise causes a hub rise and so causes, through the carburetor equation, a regulation that results in a torque drop. While the speed is rising, the torque is dropping and the acceleration decreases until an equilibrium is reached, around 4448rpm. The resulting governor motion may be seen in the first five seconds of the video: as torque equilibrium is related to rotational speed equilibrium and to governor’s translational equilibrium, the first conclusion is that, in the governor’s working speed range (8.4rad/s-10.1rad/s), governor’s equilibrium is related to speed equilibrium.
- t=[15,17.5]s: The resistant torque drop causes a positive difference between resistant and engine actions. As this causes a speed rise, the same situation of the previous time interval appears, with a hub rise, a regulation drop and so a torque drop. The system is now moving towards a different equilibrium, determined by the magnitude of the new resistant action.
- t=[17.5,40]s: Upon the return of the initial resistant action, the resistant torque excess reverses the situation, causing the deceleration of the system. The speed drop causes the hub lowering, the carburetor valve opening and so a rise in the turbine torque regulation. This effect fades when the engine torque reaches the resistant one:once again the system reaches the equilibrium speed of 4448rpm that has been estabilished before the resistant torque perturbation.
The system stability is discussed linearizing with respect to the final (t=40s) system configuration. The extracted eigvalues are:All the eigvalues are characterized by a negative real part: the system is stable and governor’s stabilizing effect is verified. The first two eigvalues represent the stable rotating behaviour of the system. The third vibrating eigvalue, with the respective eigvector visualization and with a comparison to the analysis performed to the only governor, is attributable to governor’s opening mode.
It’s noticeable that isn’t possible to determine a requested rotational speed as happens in the case of a manual positioning of the carburetor valve. The only way to change the speed is thus changing the speed ratio between turbine and governor.
Indeed, taken in consideration a resistant torque, the equilibrium speed is determined by regulator’s dynamics. Considering the whole engine as an integrated system of turbine and regulator with a closed-loop control, may be obtained new torque characteristics: this is the topic discussed in the last simulation.
In order to obtain the system torque characteristics, a motion applied to rod’s rotational joint imposes a constant acceleration of 0.01rad/s² starting from ω=0rad/s. This quasi-static speed rise makes the inertias and the damping negligible, allowing to assume that istant by istant the regulator is in an equilibrium position: the result is a good approximation of the steady-state torque characteristics. The simulation lasts for 1025s, the time necessary to arrive at a regulator speed of 10.25rad/s, that is the speed at wich the valve becomes completely closed.
The results are resumed in a graph that expresses the engine torque (red) and the hub translation (blue) as function of the angular velocity:
Above 4010rpm, the original linear turbine characteristic curve is interrupted by the closed loop introduced by the regulator. In the regulator working range (≈4000÷4800rpm) the system behaviour has a negative derivative respect to velocity, and it’s suitable to obtain an engine stable working with a wide range of resistant actions, providing that the peak torque isn’t reached and the total shaft torque derivative, respect to the rotational speed, it’s still negative.
The governor acts as a new subsystem characterized by a second translational equilibrium: the torque equilibrium become so consequence of the governor’s. However the first conclusion is that, with the governor, the operating speed can’t be chosen as it depends on its dynamics. Considering the entire engine system as a relation between its resistant torque input and the speed output, new characteristics may be extracted, as the previous graph demonstrates. The extracted one is a didactic example, but different turbine characteristics and different governor shapes could provide different curves, wich all have a right branch with a negative derivative. This property carries to a second conclusion: an engine with torque characteristics that aren’t suitable to maintain a load at a stable and constant speed, may be stabilized with the introduction of a proper Watt’s governor.
The third conclusion is that the governor may act also as a speed limiter: modifying turbine’s characteristics and causing a torque drop at a certain speed, it allows to reduce the operating speed of a system. The following picture represent, indeed, what would have happened if a quadratic speed-dependent load had been applied to the turbine: differently from some linear speed-dependent loads (with low derivative), an equilibrium would be reached, but at a high speed.However it’s possible that could be necessary to reduce the operating speed as the achievement of high speeds may cause mechanical failure in the turbine or in the load. In the steam age Watt’s governor was one of the most efficient way to achieve this result. In the following picture is shown how the governor could limit the operating speed with the previous quadratic load:
The fourth conclusion concerns a warning: Watt’s governor doesn’t act as an engine stabilizer. It acts as a torque characteristics changer that facilitates the stabilization. As seen in the last example, the turbine characteristics that before were thought as “not predisposed to the equilibrium” are instead stable with another kind of load. Hand in hand, there are loads that could not reach the equilibrium once applied to the governor-turbine system, wich for the simulation performed was stable: it’s enough to find a load that implies a positive total torque derivative respect to the speed that also a governor stabilized turbine would never find a stable equilibrium.
- R. Routledge: “Discoveries & Inventions of the Nineteenth Century”, 1900
- Klaus-Werner Friedrich, “Centrifugal governor of 1880 steam engine”, Locke’s whiskey distillery, Glasgow
- Mary Evans Picture Library: “Pictures from historical archive”, maryevans.com