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Mechanisms Assignment 1
Introduction
The aim of this assignment is to investigate the mobility of three different mechanisms, mainly the Walschaerts gear valve, Catalina flying boat float mechanism and wheel retraction system. This is done through the use of Kutzbach mobility equation and simplified kinematic diagrams.
Question 1
Assumptions relating to train kinetic diagrams
The front and behind wheels are powered by the drive wheel and will be discarded to simplify Figure 4. The forward and backward 4 bar mechanism isn’t shown in Figure xx as it doesn’t affect the mobility of the train and thus, are not drawn.
Figure 1 - Chosen train mechanism (Walschaerts valve gear)
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Note that the gear changing mechanism (reach rod and associated 4 bar linkage) has not been included here as the function of the mechanism in one particular direction does not require it to be used. For a representation of the mechanism in its reverse gear configuration, see Figure 7.
Based on Figure 4, there are 2 prismatic joints (Labelled 1 &2) and 11 revolute joints (Labelled 3 to 13).
1 Eccentric Crank 6 Lifting Link 11 Combination Lever
2 Eccentric Rod 7 Expansion Link 12 Valve Stem
3 Reach Rod 8 Radius Rod 13 Valve Spindle
4 Reverse Arm & Shaft 9 Crosshead Arm(Drop Link) 14 Connecting Rod
5 Lifting Arm 10 Union Link 15 Steam Port
Figure 4 - Simplified kinematic diagram of train mechanism
1 2 3 6 7 8 9 5 4 10 1 3 4 6 5 10 9 7 8 11 13 12 2
Figure 3 - Link naming terminology for describing motion on a similar Walschaert gear. This kinetic diagram includes some of the gear changing mechanism links and is captured from a mechanism simulation (Dockstader, 2009).
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3D analysis
Given the planar operation of the train mechanism, the joints and links have the same positions and numbering as shown in Figure 4.
For the 3D mechanism,
( ) ( ) ( )
Based on the number of inputs required to move the mechanism, the true mobility of this mechanism should be 1. The 3D analysis gives a mobility of -11, indicating that the system is over constrained.
This is because of a special geometric condition where the links of the joints are parallel to each other and are joined together to achieve a common movement. The Kutzbach criterion does not account for the perpendicular joints and the parallel links; it assumes that the joints are at different locations and angular orientation in the mechanism.
2D analysis
For the 2D mechanism,
( ) ( ) ( )
It is assumed to be a planar mechanism due to the parallel joint axes. This means that all of the joints must be on the plane for it to move, else it will lock. The Kutzbach criterion of the 2D system is 1, which means it is properly constrained and requires only one input motion to be driven, namely the force provided by the main steam cylinder.
How it Works
1. Shifting of the Train’s Gears
Cells 1 to 6 of Figure 5 shows the piston in a forward configuration. When the reach rod is pulled from the train cab, a simple 4 bar mechanism will shift a link to the reverse configuration as shown in Figure 6. This switching transforms the kinematic diagram of our chosen train mechanism into the version seen in Figure 7.
2. Mechanism Driving the Wheels & Controlling the Valve
With reference to the forward configuration motion in Figure 5, the valve spindle moves and injects steam into the right side of the piston cylinder. The steam will push the piston towards the left, pulling the crosshead arm linked to the combination lever and the union link. Once the piston reaches its maximum displacement to the left, the valve spindle will move to open the left steam port and steam will start pushing the piston to the right. In the 3rd and 4th sections in Figure 5, the eccentric rod will be moved towards the right which will pull the expansion link and the radius rod to hold the steam port open for a few moments to allow enough steam to push the piston to the right. Once the piston is moving to the right, it will now push the crosshead arm towards the right at the end, the left steam port will be held open by the radius bar to allow steam to push the piston back to the left side. While all this is happening, the connecting rod and eccentric rod that is linked to the eccentric crank moves accordingly to drive the wheel. The process will then reiterate.
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Figure 5 - Main piston driving power to the wheel. These cell images were obtained by taking screen captures of an animation (Fritz, 2007). These images were annotated to describe the forces on the pistons at different stages of drive.
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Question 2
3D analysis
By observing the float mechanism, a schematic 3D drawing is created. By recognising that all revolute joints have parallel axes and that many of the linkages operate in parallel, the 3D image is represented in the 2D plane with the multiple joints recognised in the 3D calculation.
Figure 3 - Simplified kinematic diagram of train mechanism in reverse gear
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Linkages: n = 7, (note: linkage 6 and 7 are parallel) Joints with respect to Figure 9 3D equivalent positions:
Revolute (f1) = 9
o Revolute at joint 1
o Two parallel revolute at joint 2 o Revolute at joint 4
o Two parallel revolts at joint 5 o Revolute at joint 6
o Two parallel revolts at joint 7
Prismatic (f1) = 1
o Worm gear at joint 3
This implies f1) = 10 m = 6(n-1) - 5f1 - 4f2 - 3 f3 - 2f4 - f5 m = 6(7-1) – 5(10) – 0 = 36 – 50 = -14
Notes on reinforcement
An important requirement of the two symmetrical float mechanisms on the plane is that the floats connected to linkage 1 must be lowered and remain parallel to each other. If the floats are not parallel to each other and the fuselage, then they will attempt to move in different directions when contact is made with the water. As the floats used to land the plane are connected to link 1 of the mechanism, it is desirable to have this linkage re-enforced. Reinforcement in and out of the plane of Figure 10 at link 1 also resists the torque due to landing resistance on the floats. Links 3, 4 and 6 make hold the float down rigidly during landing. Re-enforcements to this linkage consist of two parallel revolute joints connecting links 1 and 3. These two joints are represented by the single joint 2 in Figure 10 as they are separated spatially in a direction perpendicular to the mechanisms plane of motion.
1 2 3 4 6 5 7
Figure 4 - Creation of float 3D schematic diagram
1 2 4 5 3 6 7 1 2 3 5 4 6
Figure 5 - Simplified kinematic diagram of float mechanism with 3D parallel linkage
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The mobility equation shows this mechanism to be over constrained and statically indeterminate due to a negative mobility.
An over constrained solution to the three dimensional mobility problem is due to mechanically redundant components which serve to provide structural integrity to the mechanism. Whilst having no effect on the mechanical operation of the system, these extra linkages have a geometrical component perpendicular to the mechanism’s plane of motion. This is to provide reactant forces to prevent the linkage buckling and also to prevent linkage 1 from being allowed to rotate when
under strain of practical applications such as landing on water. The mobility equation used above counts individual motion restrictions multiple times.
2D analysis
This system can be modelled as a 2D system accurately given the over constrained 3D mobility and the special geometric conditions such as parallel revolute joint axes and parallel linkages.
Linkages: n = 6 (note Figure 10)
Joints with respect to Figure 10 positions:
Revolute at joints 1, 2, and 4-7 f1 = 6
Prismatic at joint 3 f1 = 1 This implies f1) = 7 m = 3(n-1) - 2f1 - 1f2 m = 3(6-1) – 2(7) – 0 = 15 – 14 = 1
The mobility equation shows this mechanism to have a mobility of 1. This is what is expected from this analysis as one actuator (the worm screw) is used to drive the mechanism.
How it works (with reference to Figure 15)
This mechanism is driven by a worm gear which drives linkage 2 horizontally (with respect to Figure 10). Such a worm gear is used as it prevents this linkage from slipping parallel to its range of motion when under load,a safety precaution taken to ensure there is no failure in the link during landing. If a hydraulic ram was used, a burst fluid lead could cause a disastrous collapse. The gear pulls linkage 4 away from the wing tip, collapsing links 3 and 6 inwards. As these two links retract more, the float is drawn upwards to form the tip of the wing, close to joint 7.
1 2 4 5 3 6 7 1 2 3 5 4 6
Figure 6 - Simplified kinematic diagram of float mechanism in 2D
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Figure 11 - Simplified kinematic diagram of float mechanism retracted
Extended
Retracted
Figure 11 - Simplified kinematic diagram of float mechanism retracting part 1
Figure 11 - Simplified kinematic diagram of float mechanism extended
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Question 3
Figure 16 shows the entire wheel raising mechanism. It should be clear from the picture that there is only one actuator (a hydraulic ram), this would suggest that the entire system has mobility of one which intuitively makes sense when we consider that we only want the wheel to extend out of the planes body and not move relative to the direction of motion of the plane.
Due to the parallel axis of the revolute joints connecting the wishbone arms used for re-enforcement, it is
expected that a 3D analysis will yield an over constrained result.
This picture also shows several other levers which while not
connected to the wheel mechanism are
important to the
mechanisms movement. The red topped bar is there to give the wheel and initial push outwards to help start the wheel lowering. The flap on top of the wheel with the sign on it is designed to fold over the top of the wheel, closing the carriage cavity during landing. Figure 18 shows the other folding mechanism which is outside the scope of our calculations regarding the wheel mechanism mobility.
3D Analysis
n = 8, F1 = 9, F2 = 3Joints with respect to figure 17 3D Equivilent positions:
Revolute (f1) = 9 o Revolute at joint 1, 2, 3, 4, 5, 6, 7, 8, 9 Cylindrical (f2) = 1 o Cylindrical joint at 11 Universal (f2) = 2 o Universal joints at 10, 12 M = 6 (n -1) – 5 F1 – 4 F2 – 3 F3 – 2 F4 – F5 = 42 – 45 – 12 M = -15
A negative mobility indicates an overly constrained system. Reinforcing over links 1 and 2 on Figure 17 contribute to this over constraint. Since there is only one degree of input being the hydraulic arm, this over constrained condition is confirmed. With special geometric conditions considered we can perform 2D analysis.
Figure 12 - Image of Catalina wheel mechanism
Figure 14 - Bottom folding 4-bar
Figure 13 - 3D representation of wheel mechanism 3 1 2 4 6 7 5 8
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2D Analysis
The simplified kinematic diagram (Figure 19) shows only the important or active members of the wheel mechanism in 2D. Other links can be removed because they do not remove any more degrees of freedom and are clearly included for safety reasons. The two universal (f2) joints at joints 1 and 5 (as
represented in figure 19) may now be represented as revolute joints as their second degree of freedom has been removed using the 2D assumption. The same may be said for the cylindrical at joint 4 which is now represented as a prismatic. n = 6, F1 = 7
Joints with respect to figure 19 2D representation:
Revolute (f1) = 6
o Revolute joints at 6 and 7
o Parallel Revolute with inline axis at joints 2 and 3
o Revolute joints at 1 and 5
Prismatic (f1) = 1
o Prismatic at joint 4 M = 3 ( n – 1 ) – 2 F1
M = 3 ( 6 – 1 ) - 2 (7) = 1
As we can see from the 2D mobility equation, we get a mobility of 1 which is what we would expect from the actual mechanism and our understanding of the desired movement of the wheel. As the mobility equations show this mechanism is clearly over constrained and only works because of the special geometric conditions,
having all the revolute joints axes parallel to each other and perpendicular to the plane of motion. The mechanism also has multiple extra links which are not necessary for it to work
but improve the strength and safety of the wheel. It has double wishbone arms (labelled 1 & 2 in Figure 17) which increase the strength of the wheel in the direction of wheel roll, making it less likely to buckle or bend backwards if it hits something
hard. It also has an arm (labelled 4 & 5 In Figure 17) which locks into place once the wheel is fully extended. This arm is there to prevent damage to the actuator if the wheel is forced upwards such as when landing and also to protect from hydraulic failure.
1 2 3 5 4 6 1 2 4 5 3 6 7
Figure 15 - Simplified kinematic diagram for Catalina wheel
Figure 16 - Locking latch on the landing reinforcement arm, protecting against hydraulic failure
Figure 17 - Universal joint on Figure 13 hydraulic ram
Figure 18 - Showing joint 5 of Figure 13 with parallel revolute joint axes
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Figure 21 - Kinematic diagram for Catalina wheel receded
Figure 21 - Kinematic diagram for Catalina wheel receding
Extended
Retracted
Figure 21 - Kinematic diagram for Catalina wheel with bend bar
How it works
As the hydraulic ram is retracted, the reinforcing landing arm bends down. Links 4 and 5 in Figure 15 move upward until the wheel is in its fully retracted position as shown in this flow chart.
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Bibliography
Dockstader, C. (2009, March 20). Steam engine valve gear on the computer. Retrieved September 28, 2011, from Welcome to billp.org: http://www.billp.org/Dockstader/valgr10a.zip
Fritz, G. (2007, May 20). File:Walschaert gear reversing.gif. Retrieved September 28, 2011, from Wikipedia: http://en.wikipedia.org/wiki/File:Walschaert_gear_reversing.gif
Note that photos of the train mechanism are of museum items in The Railway Museum, Bassendean. Photos of components of the Catalina sea plane were of the plane housed in the Bull Creek Aviation Heritage Museum, Perth.
