Modelling the Leminscoidal Linkage of the First Kind by Watt (S24)

ES3323 B16

Project 1

Emily Chretien, Constantine “Tino” Christelis, and Matthew Garcia

29 November 2016

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Table of Contents Table of Contents ................................................................................................................................... 2

Table of Figures ..................................................................................................................................... 2

Introduction ........................................................................................................................................... 3

Background ........................................................................................................................................... 4

Straight line mechanisms ..................................................................................................................... 4

Watt & Reuleaux, Applications ............................................................................................................ 4

Design Synthesis & Calculations, Dimensions, DOF............................................................................ 4

Modelling Strategy ................................................................................................................................ 8

Parts ................................................................................................................................................... 8

Fasteners ............................................................................................................................................. 8

Sub-Assemblies .................................................................................................................................... 8

Assembly ............................................................................................................................................. 9

Mechanism ........................................................................................................................................ 11

Discussion ............................................................................................................................................ 14

Conclusion ........................................................................................................................................... 16

References ............................................................................................................................................ 17

Modeling Strategy: Grounded Link........................................................................................................ 18

Modeling Strategy: Crank ...................................................................................................................... 20

Modeling Strategy: Frame ..................................................................................................................... 24

Table of Figures

Figure 1: Mechanism Overview ................................................................................................ 5

Figure 2: Result of 2-Position Synthesis; Circles indicate ground joints. ........................................ 6

Figure 3: Crank-handle sub-assembly overview. ........................................................................ 9

Figure 4: Assembly Relations. ................................................................................................. 10

Figure 5: Isometric view of assembly and model tree. ............................................................... 11

Figure 6: Snapshot of model; Crank Up starting condition. ........................................................ 12

Figure 7: DOF and Redundancies; Mechanism Tree. .................................................................. 13

Figure 8: Angle of slider guide relative to vertical with full rotation of crank. .............................. 14

Figure 9: X-Position change of coupler point with full rotation of crank. ..................................... 15

Figure 10: Y-Position change of coupler point with full rotation of crank. ................................... 15

Figure 11: Trace line, in blue. ................................................................................................. 16

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Introduction The purpose of this project was to model and analyze a small straight-line Reuleaux

kinematic mechanism using Creo Parametric software. Our team had to select one of the models

on display in Cornell University’s The Kinematic Models for Design Digital Library

(KMODDL) [7]. We had to study this mechanism, determine the link dimension ratios, and

evaluate the 2D and 3D Kutzbach equations to verify that the mechanism has one degree of

freedom. Using this information, we needed to develop a robust modeling strategy for the

structure, creating all necessary parts and subassemblies to form the final assembly in Creo

Parametric, using constraints and relations to accurately represent the design intent of the

mechanism. Once we had the completed model, we had to use the Creo Parametric Mechanism

Application to insert servo motors and perform position and kinematic analyses. From these

analyses, we could measure the degrees of freedom and redundancies in the model, as well as

trace the path of the slider rod and compare it to the ideal straight-line motion the mechanism is

intended to simulate.

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Background

Straight line mechanisms

Straight line mechanism have the purpose of geometrically forming a straight line,

without use of a pre-existing straight line as a guide (like a straight edge). Simply put, these

mechanisms convert rotary motion into pure, straight line motion. Many variations of “straight

line mechanism” have been devised, all with varying amount of accuracy and complexity.

Watt & Reuleaux, Applications

James Watt is the inventor of the straight line mechanism modeled in the project. He is

most noted for his work in steam engine technology. Through his years of work he eventually

developed the term of horsepower and the unit of “Watt” was named after him. The Watt’s

linkage was developed as a constituent component to his engine designs. It approximated straight

line motion which was used for the motion of the cylinder rod and pump in his engine design.

This mechanism and its principles have been used in some automotive applications in modern

day as a replacement to a pan hard bar. This prevents side to side motion between the axle and

the chassis of the car. In a traditional suspension setup a pan hard rod centers the axle according

to one side of the vehicle. With the watt’s linkage setup the center of the axle is mounted to the

point where straight line motion occurs. The two links connected to the vertical bar then keep the

center of the axle in line with both sides of the vehicle. [4, 5]

Franz Reuleaux has developed many varieties of mechanical models to demonstrate

various mechanisms and their kinematics. He developed several ways to describe and classify

mechanisms already in existence, which lead to the invention of many more of his own. Most

notably these included the four-bar linkage, which can be seen in our chosen model, and a crank,

also in our model. He abstracted the idea that the four-bar linkage as simple in its design, could

be formed from combinations of many different joints thus producing many different iterations

of a simple four-bar linkage. [6, 7]

Design Synthesis & Calculations, Dimensions, DOF

The principles of the workings of the mechanism are generally straightforward. The

mechanism can be seen below, and consists of two parts: a 4-bar (red), and a crank-driven dyad

(blue).

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Figure 1: Mechanism Overview

The red portion of the assembly forms a double rocker four-bar, with which a coupler

point centered on the coupler moves an approximate straight line. As the linkage goes through its

motion, the coupler point diverges from its straight motion and curves off once the coupler point

passes above or below the ground joints [3]. In fact, a full coupler curve of the midpoint reveals

the full motion is that of a figure eight [2]. This results in only ~29 degrees of rocker motion; a

value which will come into play later with the dyad design. The Watt Straight-Line Linkage

requires the following link relations:

L1 = 4 (ground)

L2 = 2

L3 = 1 (coupler)

L4 = 2

We can evaluate the Grashof condition of the mechanism using the following equation [1]:

S + L ≤ P + Q

In the above equation, S is the shortest link (coupler, which is 1 unit), L is the longest

link (ground, which is 4 units), and P and Q are the other link lengths (the two rockers, which are

2 units). With the left side of the equation being greater than the right, the mechanism is

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determined to be non-Grashof. This indicates that no link in the assembly can make a full

rotation, and thus cannot be driven via a direct rotary input as we would like.

This is where the blue portion of the assembly, the dyad, comes into play. With a dyad,

we can achieve controlled rocker motion (motion of the 4-bar) with a constant rotary input

motion as initially desired. Because we know the rockers length and the angle it must be rotated,

we can use standard two-position linkage synthesis to calculate the dyad coupler and crank

dimensions. [1]

Figure 2: Result of 2-Position Synthesis; Circles indicate ground joints.

Because of some nice geometrical coincidences (the crank ground joint being collinear

with the rocker end points), the calculation is simplified. The crank comes out to a length of 0.5

units (or half of the 4-bar coupler length), and the dyad coupler comes out to be the length of the

vertical distance between the upper right ground joint of the 4-bar and the ground joint of the

crank. With all of the link lengths and positions established, we have a design ready to be

realized in a CAD environment.

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Because the entire assembly should be controlled via a single crank alone, the assembly

should only have one DOF (degree of freedom). We can use the Kutzbach mobility equation to

confirm this. [1]

(2d) Mobility = 3(L - 1) - 2J1 - J2

(3d) Mobility = 6(L - 1) - 5J1 - 4J2- 3J3- 2J4- J5

Using the above equation for 2d mobility, the assembly can be proven to indeed have

only one degree of freedom. We have 6 links in total including ground, 7 full joints (1-DOF), and

no half joints, which provides:

(2d) Mobility = 3(6 - 1) - 2(7) - 0 = 1

It should be noted that when calculating for mobility in 3d, we will have to assume some

joints to be cylinder (2-DOF) or ball (3-DOF) joints rather than pin joints, in order to eliminate

redundancies.

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Modelling Strategy

Parts

See attached modelling strategies.

Fasteners

Spacers, nuts, and bolts were all used in our solid model. We wanted to use common,

standard fastener sizes for manufacturability and ease of assembly/repair. We chose to use ¼-20

threaded fasteners, as this is a very common fastener size that fits the scale of our model. Thus

we modeled all of the fastener holes in the assembly as 0.25 inches in diameter. We downloaded

the ¼-20 nuts and bolts as STEP files from the McMaster-Carr website, selecting several

different bolt models so we would have a variety of lengths (0.50, 0.75, and 1.0 inches) to choose

from when assembling the model. We selected hex-driven, button-head bolts so that they would

have a low-profile that would not interfere with the mechanism motion. For simplicity, we

modeled our own spacers. Our spacer has an inner diameter of 0.25 inches and is 0.20 inches

thick. This is double the thickness of each link, to provide the appropriate clearance needed

between moving components for bolt heads.

Sub-Assemblies

Each sub-assembly consists of a group of parts that are all stationary relative to one

another, so that each sub-assembly moves as a single body in the final assembly. Each fastener

was fully constrained to one of the adjacent links in a sub-assembly, because the one degree of

rotational motion in each washer does not impact the overall motion of the mechanism. There are

three sub-assemblies in the model: one for the base frame and all “grounded” parts, one for the

ternary and its associated fasteners, and one for the crank and handle.

The crank and handle assembly is simple in its purpose, but required more attention to

detail than many other areas of the project. This was mostly due to the fact that parts in this

assembly had to be constrained such that they would rotate with each other, which meant pins

had to be created to force-fit with the crank/handle and the shaft. The crank-handle sub-assembly

is made of 9 unique parts, all of which are labeled in the figure below.

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Figure 3: Crank-handle sub-assembly overview.

Many of the features in the assembly were driven by relations, so that part dimensions

could be changed and not break the design intent of the other parts. There are also tolerances

taken into account with the hole sizing, with the holes the pins go through being the same

diameter as the pins for a force-fit, and the holes through the crank and handle for the shaft

having a normal tolerance, so that it may be easily slipped into position.

Assembly

The ground sub-assembly (base and frame) was the first component inserted into the

assembly, and was fully constrained to the default assembly datum planes, with the assembly

origin located at the center of the crankshaft hole. This was chosen as the origin because it is the

location from which the entire mechanism will be driven.

Each link was connected to the assembly using joint connections. To allow for small

deviations from perfect straight-line motion, the slider rod was initially constrained using a

cylinder joint to the guide piece, which was connected with a pin joint to the ground frame. The

ability of the guide to pivot allows for the small amount of X-component motion in the sliding

rod without over-constraining the assembly.

Several relations were written in order to ensure that all parts and subassemblies were the

proper dimensions/ratios and fit together correctly. The grounded frame was used as the

reference, so all variables in the main assembly are driven by the dimensions of the grounded

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frame. One relation set the link ratios (driven by the length of the grounded link bar), one

relation set all of the bolt holes to be the same diameter, and other relations made sure that the

lengths of components such as the crankshaft and slider guide were set correctly to align with the

links.

Figure 4: Assembly Relations.

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Figure 5: Isometric view of assembly and model tree.

Mechanism

In order to evaluate the precision of our straight-line motion model, we set up a

mechanism analysis. We placed a servo motor on the crankshaft axis and set it to rotate at a

constant angular velocity of 180 degrees per second, simulating a person turning the crank at a

constant rate. An initial condition was created by taking a snapshot of the mechanism when the

crank handle is in its most upward position. Using this motor and initial condition, a position

analysis was constructed for one full revolution of the motor to analyze the motion of the slider.

We created measures for the X- and Y-components of the position of the slider and for the

angular position of the slider guide.

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Figure 6: Snapshot of model; Crank Up starting condition.

In order to ensure the most accurate position analysis results, the assembly must be

modeled as a working single-driver mechanism, with 1 Degree of Freedom, with 0 redundancies.

We can refer back to the Kutzbach equation to determine the proper joint types to achieve this. In

the 3d model, we have 8 links, including the crank and slider guide.

(3d) Mobility = 6(8 - 1) - 5J1 - 4J2- 3J3- 2J4- J5

To find the “J” values, we started by only considering the initial 4-bar linkage, and used

CREO to conduct a kinematic analysis on this piece. We adjusted the joint types and re-analyzed

until we obtained 1 DOF and 0 redundancies. Then we added in the dyad and crank, and

performed the analysis again, making joint adjustments until we reached 1 DOF and 0

redundancies again. Finally, we added in the slider and slider guide and repeated the process to

find our final configuration. Our final results were 4 pin joints, 3 cylinder joints, and 3 ball

joints. This can be verified using the Kutzbach equation:

Number of Drivers (motors) = 3d Mobility = 6(8 - 1) - 5(4) - 4(3) - 3(3) - 2(0) - (0)

= 42 - 20 - 12 - 9 = 1

It should be noted that when using a CREO kinematic analysis to conduct the DOF and

redundancy calculations, the servo motor must be removed from the analysis. The servo motor

eliminates one degree of freedom, so when the motor is included in the analysis, we find DOF =

0.

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Figure 7: DOF and Redundancies; Mechanism Tree.

An interference check was performed on the kinematic analysis. The only interferences

that were returned were from the threads on the pre-fabricated nuts and bolts. This is acceptable

and unavoidable, as the modeled threads cannot be properly aligned in the solid model as they

would in reality.

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Discussion Our model represented straight-line motion quite accurately. To determine the deviation

from true straight-line motion, we set up three measures in our kinematic analysis: the X and Y

components of the position of the end of the slider rod, and the angular position of the pin joint

for the slider guide. The Y-position of the slider rod as a function of time appears to be very

close to parabolic, which is what would be expected for a mechanism being driven by a servo at

a constant velocity. The position ranges from approximately -3.35 inches to -5.0 inches.

Throughout this 2.7-inch range of motion, the X-component of the slider rod position only varies

by 0.0035 inches, which equates to a deviation of 0.0013 horizontal inches per inch of vertical

motion. This is a very small deviation, which shows that our model represents straight-line

motion very well. Additionally, the X-position deviation is symmetric about the central Y-axis,

making the net horizontal deviation nearly zero, which is even closer to the ideal straight-line

motion. The model’s accuracy can be further proved by observing the angular position of the pin

joint as a function of time. The pin joint is only forced to rotate a total of 0.09 degrees during one

period of the mechanism, a very slight divergence from the ideal lack of rotation for perfect

straight-line motion.

Figure 8: Angle of slider guide relative to vertical with full rotation of crank.

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Figure 9: X-Position change of coupler point with full rotation of crank.

Figure 10: Y-Position change of coupler point with full rotation of crank.

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Conclusion The general creation of the assembly was a success, with both simulated and calculated

results showing that we were able to achieve one degree of freedom and zero redundancies. The

crank was able to make a full rotation, with no parts colliding / interfering with each other

throughout the mechanism’s motion (besides screws and nuts with threads that didn’t coincide).

With the symmetrical deviation in X-motion and angular position of the coupler point

through the mechanisms motion being extremely small, it is clear that the model accomplished

the task of approximating a straight line. Unless major features of the design are changed, the

motion cannot become any straighter than it already is. A good visual depiction of the

mechanism’s straight line motion can be observed by looking at a coupler curve, which is created

in Creo via trace lines.

Figure 11: Trace line, in blue.

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References [1] R. Norton, Design of Machinery, 5th ed. New York: McGraw-Hill, 2012.

[2] "File:Watts linkage.gif - Wikimedia Commons", Commons.wikimedia.org, 2016. [Online].

Available: https://commons.wikimedia.org/wiki/File:Watts_linkage.gif. [Accessed: 22- Nov-

2016].

[3] "Watt Straight-Line Linkage Mechanism", YouTube, 2016. [Online]. Available:

https://www.youtube.com/watch?v=3fBZgoXhrgE. [Accessed: 22- Nov- 2016].

[4] "James Watt", HISTORY, 2016. [Online]. Available:

http://www.history.co.uk/biographies/james-watt. [Accessed: 25- Nov- 2016].

[5] "Watt Biography", Egr.msu.edu, 2016. [Online]. Available:

https://www.egr.msu.edu/~lira/supp/steam/wattbio.html. [Accessed: 25- Nov- 2016].

[6] Asme.org, 2016. [Online]. Available: https://www.asme.org/getmedia/f47f8dae-5d5c-4b9e-

abd0-1ff665b17100/232-Reuleaux-Collection-of-Kinematic-Mechanisms-at-Cornell-

University.aspx. [Accessed: 25- Nov- 2016].

[7]"KMODDL - Kinematic Models for Design Digital Library", Kmoddl.library.cornell.edu, 2016.

[Online]. Available: http://kmoddl.library.cornell.edu/. [Accessed: 20- Nov- 2016].

Fastener models downloaded:

https://www.mcmaster.com/#91879a029/=158lyti

https://www.mcmaster.com/#91255A537

https://www.mcmaster.com/#91255A540

https://www.mcmaster.com/#91255a542/=158m1qx

https://www.mcmaster.com/#standard-rounded-head-screws/=158m5ir

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Emily Chretien

Project 1

Modeling Strategy

Modeling Strategy: Grounded Link

1. Origin Selection: Place origin at the center of the part

2. Midplane Extrusion

a. Sketch on the Top Surface

First, two construction centerlines were created, horizontal and vertical. The four

construction circles were sketched on the horizontal centerline (two on each side of the vertical

centerline). Next, two horizontal lines with their endpoints on the two middle construction circles

were sketched, and then the four angled lines were sketched with endpoints coincident with the

construction circles. Then the six arcs were sketched along the construction circles, with their

endpoints coincident with those of the sketched lines. This created the closed-loop sketch profile.

Horizontal constraints were placed on the two center lines, then these lines were set to be

of equal lengths and symmetric about the horizontal centerline. Each pair of construction circles

was constrained to have equal radii and be symmetric about the vertical centerline. The

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endpoints of the arcs were set to be symmetric about the horizontal centerline. Finally, the four

angled lines were constrained to have equal lengths. This fully constrained the sketch profile to

the design intent.

The entire sketch was selected and modified with the scale locked, to set one of the

dimensions and rescale the entire model. Then dimensions were applied to fully define the

sketch.

b. Midplane Extrusion, depth = 0.1”

3. Hole, diameter = 0.25” (fastener holes for moving links). Center is coincident with the

center of the arc of the extrusion.

4. Hole, diameter = 0.25” (fastener hole for attaching to frame). Center is coincident with the

center of the arc of the extrusion. This was made as a different feature so that the

diameters can be adjusted if the manufacturer wants to use different fasteners for the two

functions.

5. Mirror the two holes across the Right Datum Plane.

This concludes the modeling strategy for the Ground Link part.

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Tino Christelis

Project 1

Modeling Strategy

Modeling Strategy: Crank

1. Origin Selection: Origin placed at center of hole feature for shaft, as we can make use of the

ground planes to construct a useful axis for a hole feature later on.

2. Midplane Extrusion

Sketched on the top place, two arcs of arbitrary diameter / radius were constructed, with

straight tangent lines bridging the two. The distance of their centers is also arbitrary, as this is to

be calculated and to be defined in the top assembly level through a reference dimension based on

the lengths of other links. The arc dimensions will be calculated via relations at the end of the

part creation.

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The sketch is then extruded 0.2”.

3. Axis Construction

Two axis were constructed through the centers of the curved surfaces created through the

previous extrusion. These will be used for through holes for the shaft and dyad connection.

4. Blind Extrusion

A circle of equal diameter to the larger arc from the first step is sketched on the bottom of

the newly-extruded solid, and extruded 0.25”. This will be a “spacer” for the crank, giving

clearance for nuts / bolts on the other end of the crank which might otherwise impede rotation by

colliding with the frame.

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5. Hole Features

Holes are created through the part positioned along the previously defined axis, (from left

to right) one for the shaft and one for the fastener connecting the dyad link. The sizes of these

holes will be defined through references in the crank-handle sub-assembly, so that the part won't

fail if the size of the shaft or fasteners change.

6. Axis Construction

An axis was constructed through the TOP and RIGHT planes.

7. Hole Feature

Using this new axis, a hole is placed and sent to go through the entire part. This will be

used to place a pin, which will lock the crank’s rotation relative to the shaft.

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8. Chamfer

A 0.05” chamfer is added, to minimize the surface area of the bottom face, since it will be

in contact with the frame and minimal friction is optimal.

9. Relations

Relations are added to define the dimensions of the arcs created in the first step. The arcs

dimensions are to be 0.2” larger than the hole diameters. This way, if any sizing of any of the

wholes change, the part dimensions will compensate so that the holes are never larger than the

actual part. In the crank-handle sub-assembly, the diameters of the pin holes and shaft hole are

defined.

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Matthew L. Garcia

Project 1

Modeling Strategy

Modeling Strategy: Frame

1. Extrusion of frame outline.

a. Outline of frame first drawn using spline lines and then offset to produce

thickness of sides.

b. Hollows features were then added.

c. Feet were added

d. Extrusion to provide depth of frame

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2. Using datum points hole features are added for crank mounting point, slider guide

mounting point, and ground bar mounting points.

3. Chamfer Edges

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4. Rounds placed on all interior angles and edges of feet to remove internal 90 degree

angles. This is an important manufacturing consideration.