Giangreco Luca – firstname.lastname@example.org
One of the most known centrifugal regulator is Watt’s Governor, a mechanical system capable of adjusting the rotation speed of its shaft according to itself.
Watt’s Governor was invented by James Watt and was used to regulate the velocity of steam machines. According to its structure, shown below, the governor is able to correct the rotation speed of the spindle if a change in the mass of the machine or in the flow rate of the fluid occurs. These events lead to a change in the angle of the valve; in case of valve opening, the speed of the mechanism increases and the centrifugal force acting on the masses leads to their lifting. The latter, through a kinematic chain, leads to a new closing of the valve, until the mechanism reaches a new stability. An opposite process takes place in the event of initial closure of the valve.
The main objective was to study the stability of the system and his behavior when he suffers disturbances.
The four bar linkage that connects the sleeve to the throttle valve has been modeled using members of sufficient length to allow the latter a movement angle greater than ninety degrees, a body named Block_sup was used to limit this movement to the only range of interest.
A first analysis was made in order to obtain the relation between the throttle valve angle and the position of the sleeve. Subsequently, other analyses was made to find the range of functional velocity of the single rotating pendulum and the relation between the spindle angular velocity and the position of the sleeve using only one of the two degrees of freedom of the system. Eigenvalues was used to evaluate the stability of the system.
The behaviour of the entire system was than studied imposing a torque to the throttle valve and to the spindle, to understand the effect of the retroactive chain. Also here eigenvalues was used to evaluate the stability of the system.
Finally, three analyses were performed to verify the behavior of the mechanism when it is deviated from its equilibrium point.
The modelling problem
The image in the introduction and the one below have been used as a guide line to create the governor’s bodies and joints.
Here is reported the list of the bodies and joints present in the model:
|Joints||Body i||Body j|
And a picture of the model in its resting position:
Two contact were used to limit the movement of the throttle valve. One between the sleeve and Left_fly_ball and the other one between the throttle valve and Block_sup.
To take in account the effect of the different frictions that occour during the motion in the real mechanism, a damper was placed between the upper part of the spindle and the sleeve. Simulations with constant angulare velocity of the spindle and different value of the damping coefficient was used to estimate a value of the latter which would allow to reach the equilibrium position with a relative small number of oscillation.
At this point an analysis was performed to evaluate the relation between the throttle valve angle and the position along Y of the sleeve:
The angle of 90 degrees correspond to the open valve configuration, instead 0 degrees correspond to the complitely close valve. It can be see fom the figure that the relation is non linear. In particular, for a valve only slightly open, near 0 degree, a relative small variation of the sleeve position cause a relatively high variation of the angle. This imply that when the system is working in high velocity there’s the possibility of unstable equilibium configuration.
Last thing was to create the relation between the torque applied to the valve from the flow rate of the fluid before the valve and the torque applied to the spindle from the flow rate of the fluid after the valve. In a simplified way we can consider that the torque(T) applied from the fluid to the valve or the spindle is equal to the flow rate of the fluid(FR) multiply for a constant(c):
So the difference between the torques is equal to the difference between the flow rates.The relation applied take impose that the flow rate of the fluid after the valve is equal to the flow rate before the valve, multiply for the sin of the throttle angle. This bring to the relation of the torque:
FRafter=FRbefore*sin(Throttle_Angle) → Tspindle=Tvalve*sin(Throttle_angle)
Simulations and analysis of results
The first simulation was done in order to obtain the range of functional velocity of the simple rotating pendulum and to try to obtain the relation between the angular velocity and the equilibrium configuration. To do that an angular accelleration of 0.01 deg/s2 was applied to the spindle. This relatively small accelleration should allow us to obtain a curve of the stady state equilibrium positionof the sleeve, because impose a quasi-static increase in the angular velocity. However, this is true only for the first part of the simulation. As shown in the figure below, over a certain value of the angular velocity, because of the relation between the sleeve position and the throttle valve angle, the throttle valve suddenly close.
This simulation was however useful to get an idea of the minimum rotation speed required by the mechanism to lift the sleeve (≈175 deg/s).
Using the design evaluation tool, a series of simulations were performed at different rotation speeds. To avoid problems due to the sudden change of position, a step function was used to bring the speed to the interested level in 10 seconds and then the value was kept constant for another 50 seconds. At the end of the simulation the eigenvalues relative to the center of mass of the sleeve were extracted.
In the first figure is shown the phase plane plot relative to the throttle valve angle and in the second are shown the eigenvalues for the different velocity. Notice that the last 2 simulation dosen’t reach their equilibrium position, they are unstabile, confermed by their eigenvalues.
Here is a zoom of the phase plane plot of the last part of the simulation for the spindle angular velocity of 260 deg/sec.
Next simulations show that the entire mechanism of the watt governor can work with higher velocity than the one shown before without going in instability.
For this simulation were imposed 3 torques:
- Fluid_torque: a constant torque applied to the throttle valve
- Imposed_Spindle_torque: applied to the spindle and directly proportional to the angle of the throttle valve and to Fluid_Torque
- Momento_resistente: a constant torque of -10 N*m applied to the spindle, with opposite sign respect to Imposed_Spindle_Torque, to simulate the resistant action of a mechanism attacched to the governor (the governor work even without this torque)
As before the design evaluation tools was used to carry out the same simulation with different value of Fluid_torque. The tested valueof fluid torque go from 10 to 100 N*m. The simulations last 50 seconds.
Here we show the last value of the Spindle angular velocity and the eigenvalues evaluated at the end of each simulation, referred as before to the sleeve CM.
As the first image show, the governor is able to reach higher velocity that the one shown before and the entire system has only stable equilibrium configurations, because the eigenvalues of the second image have real part less then zero. Now the system have 4 eigenvalues, 2 relative the vibration of the rotating pendulum and 2 relative the vibration of the throttle valve. The only simulation that bring to an eigenvalue with positive real part was the one with the Fluid_torque equal in modulus to Momento_resistente, so in the case of a stopped mechanism, maybe an error caused by the contact of the Left_fly_ball with the Sleeve.
In the last three simulations the variable Momento_resistente was replace by the variable Momento_resistente_con_alterazione. The latter one begin the simulation with a value of -10 N*m, at 50 seconds a step function bring the value of the variable to 0 and at 70 seconds another step function bring back the value to -10. It can be see in the figure below that the governor is able to reach a new equilibrium position during the perturbation and to came back to its previous equilibrium when the perturbation end.
The three simulations was performed with this parameters, regulated manually for each simulation:
- Fluid_Torque=15 N*m; Mass of left and right fly ball=23.8213524905 Kg
- Fluid_Torque=60 N*m; Mass of left and right fly ball=23.8213524905 Kg
- Fluid_Torque=60 N*m; Mass of left and right fly ball=5 Kg
A lighter mass allow the system to reach the stability in less time, but leads to a higher angular velocity of the spindle.
The first conclusion is that Watt’s Governorn is a mechanism able to reach a new stable equilibrium configuration in the case of a perturbation to the entire system.
As shown in the last two figure, the Spindle angular velocity strictly depend on the mass applied on the system, so a second conclusion could be that with an adeguate regulation of the mass, we could be able to let work the governor in a specific range of velocity.
- Mark Denny 2002 Eur. J. Phys. 23 339
- Daniel Marlow, “The Flyball Governor”,Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544(Mar. 5, 1997)