Geneva Mechanism and Powersaw Machine

Prontnicki, PeterRoque, Mario

VanRoden, JackZicker, Albert

March 21, 2014

I pledge my honor that I have abided by the Stevens Honor System.

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CONTENTS ME-345

Contents

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1 Geneva Mechanism

1.1 Introduction

The Geneva Mechanism is a simple example of an intermittent motion device. Intermittent motionis a delayed motion of the output link while there is a continuos motion of the input link. Whendealing with the Geneva Mechanism our Geneva Wheel is considered the output link whichremains stationary until a pin connected to the crank (input link) enters the slot of the genevawheel causing the wheel to rotate counterclockwise. This movement ends when the pin leavesthe slot which then means that the wheel is again stationary while the crank continues to rotate.The crank also has an angular segment exactly the same size as the space between slots on thegeneva wheel that keeps everything in place. This movement repeats dependent on the angularvelocity of the crank itself. Although our interpretation of the Geneva Mechanism has four slots,a minimum of three slots in necessary for it to be considered a Geneva Mechanism. Some pointsthat are important when creating a Geneva Mechanism is the detail that is necessary for it towork properly. The interaction between the wheel and the crank has to be as smooth as possible.Although small knowing that one must make sure that when the pin goes through the slot of thewheel, it is tangent from the center of the wheel. The need for close to perfection is importantbecause of the interaction between the wheel and the crank.

1.2 Setup and Analysis

When given the situation where there is constant clockwise angular velocity of the crank at 2 rad/sor Wheel A and we have to determine the counterclockwise angular velocity of the geneva wheelwhen θ = 20◦.

tanβ =O1P sin θ

O1O2 −O1P cos θ(1)

Since we know the lengths of O1 and O2 and the angle of θ we were able to solve for β which isjust the angle relative to the geneva wheel.

tanβ =1/√

2 sin θ

1− 1/√

2 cos θ→ β = 35.8◦ (2)

By solving for β we are setting ourselves up to solve for the angular velocity. Taking the derivativeof the first equation will give us our relationship between angular velocity and the lengths andangles of the mechanism itself.

β sec2 β =(1− 1/

√2 cos θ)(1/

√2θ)− (1/

√2 sin θ)(1/

√2θ sin θ)

1− 1/√

2 cos θ(3)

Solving for β will give us our angular velocity for the wheel. We just have to plug in θ = 20◦ andθ = −2 rad/s.

ω2 = β = −1.923rad/s (4)

The angular velocity is negative because it is counterclockwise and we also decided that this resultalso makes sense. Since our angles was not zero we were going to get something close to but nottwo. This idea matches our results; the deflection of 20◦ gives us 1.923 rad/s. From this conclusion

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1.3 SolidWorks Analysis ME-345

we can also make certain assumptions that in theory would be proved by our SolidWorks analysis.Since there is a linear relationship between the angular velocity of the crank to the angular velocityof the wheel, then we can assume that as the velocity of the crank increases so will the velocity ofthe wheel. Also if we increase the angular velocity of the crank there will be in the frequency.

1.3 SolidWorks Analysis

The analysis of our SolidWorks design further enforces our data and assumptions.

Figure 1: geneva mechanism

Our velocity graphs:

Figure 2: Velocity v. Time at 20 rpm(2.1 rad/s)

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1.3 SolidWorks Analysis ME-345

Figure 3: Velocity v. Time at 40 rpm(4.2 rad/s)

Figure 4: Velocity v. Time at 100 rpm(10.5 rad/s)

The graphs show an increase in the amplitude which correlates to the velocity. There is also anincrease in frequency because angular velocity and frequency are directly related by the equation,ω = 2πf , which is why at 100 rpm there are is an enormous increase in peaks and troughs.

In addition to velocity we analyzed the acceleration of the geneva mechanism.

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1.3 SolidWorks Analysis ME-345

Figure 5: Acceleration v. Time at 20 rpm(2.1 rad/s)

Figure 6: Acceleration v. Time at 40 rpm(4.2 rad/s)

Figure 7: Acceleration v. Time at 100 rpm(10.5 rad/s)

When looking at the graphs there was confusion because if velocity is constant, then initiallyyou would assume that the acceleration would be zero because the acceleration is just the timederivative of velocity. When the crank makes contact with the slot there is a torque or contactforce from the crank that is proportional to a centripetal force on the geneva wheel. Which is alsowhy we believe that the crank has to have a certain weight to assure that it can create a forcestrong enough to move the wheel. So that is why if we look at the graphs of acceleration there isonly acceleration when the crank and wheel make contact.

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2 Conclusion for the Geneva Mechanism

The Geneva Mechanism was a clear example of intermittent motion and our results and relationshipsbetween angular frequency, angular velocity, and angular acceleration agree with basic laws ofphysics, reassuring that we made no grave error. Again the more exact the design, the clearer theresults. As far as recommendations for analyzing a Geneva Mechanism further, one could add morepins on the crank and more slots on the wheel. Physically one could assume that this will changethe frequency, but this would also change the design of the mechanism because the crank wouldhave to be completely redesigned thus causing other issues. As seen before the crank needs to forcethe wheel to move. By changing these factors there are many possibilities to creating variations ofa Geneva Mechanism.

3 Powersaw Mechanism

3.1 Introduction

The power saw mechanism is supplied all of its power through the input link (link 2). The velocityof the saw blade is strongly connected to angular velocity of the input link.

d =aω2 sin θ3 − θ2

cos θ3(5)

Above is the mathematical equation that models the velocity of the slider. In this equation d is thevelocity of the slider, a is the length of the input crank, ω is the angular velocity of the input crank,and θ3,4 are the relative angles the other links form. As you can see from the equation above, adirect increase in either the size or speed of the input crank will increase the velocity of the slide.This relationship can also be modeled graphically as seen below.

Figure 8: Input Angular Velocity = 1 rad/s Figure 9: Input Angular Velocity = 100 rad/s

The first figure shows the saw with an input angular velocity of 1 rad/s, while the second figuremodels the same saw at 100 rad/s. Both of them appear to have similar velocity profiles, buta closer look will reveal that the second figure, at a higher angular acceleration yielded a resultthat was nearly 100x greater than that of the first figure. The two mechanisms had the exactsame dimensions. Because they were that same size at every independent input angle both of themechanisms would have the same θ3 and θ4. The only variable in the equation above was theangular velocity, and because of where it lies in the equation, an increase in the angular velocity ofthe input link would lead to an increase in the slider velocity.

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3.2 Setup and Analysis ME-345

3.2 Setup and Analysis

Figure 10: Band Saw Threshold

Above is an example of band saw cutting speed thresholds for various materials. All of these ma-terials are fairly common in many industries, and it would be reasonable to assume that a powersaw, like the one that was modeled, would be used to cut materials similar to these. For safety,and to preserve the life on the cutting blade, and other parts of tool, there are usually maximumspeeds to which different material can be cut. Using the equation mentioned above that related theslider velocity to the input angle velocity we are able to approximate a safe and efficient angularvelocity for the input crank to cut each one of these materials. At every different maximum slidervelocity the θ3,θ4 and length of th crank were always the same values. Knowing the maximumcutting speed, of the maximum speed of the blade/slider, the only unknown was the input crankangular velocity which could easily solved for using the slider-crank equation.

Material Average Cutspeed (fpm) Average Cutspeed(mm/s) Max Crank Velocity (rad/s)

Aluminum 1100 5588 57.755Bakelite 550 2794 28.878

Brass Soft 237.5 1206.5 12.47Brass Hard 112.5 571.5 5.907

Brass Sheets 550 2794 28.878Bronze 112.5 571.5 5.907

Cast Iron 75 381 3.938Monel Metal 75 381 3.938Rubber Hard 200 1016 10.501

Steel Alloy 75 381 3.938Steel High Carbon 75 381 3.938

Steel Sheet 70 355.6 3.675Steel Stainless 62.5 317.5 3.282

Steel Tool 100 508 5.25

3.3 SolidWorks Analysis

Below are our figures for the bandsaw in different positions.

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Figure 11: Band Saw Figure Figure 12: Band Saw Figure 2

Figure 13: Band Saw DisplacementFigure 14: Band Saw Velocity

Figure 15: Band Saw Acceleration

As expected the displacement and the velocity are both sinusoidal which agrees with our initialequation analysis.

4 Comparison

Below are all our figures for the analytical model and analysis of the PowerSaw. The analyticalmodel and the SolidWorks simulation output were both illustrated the linear velocity correctly. Themaximum and minimum points are the same for both systems however they were just the inverseof each other ( the maximum of the SolidWorks output was the minimum of the Linkages outputand vice versa). This is arbitrary because it is only caused by the rotational direction of the powersaw when simulated. Additionally, the SolidWorks output illustrates the power saw over a fivesecond time period (roughly three full rotations), while the Linkages software just illustrates onefull, 360◦ rotation. However, the graphs are identical when only looking at one full rotation in theSolidWorks graph. The main point was to illustrate that the rotation causes the same minimumand maximum vales, and produces the same path when graphed.The same reversed minimum and maximum event occurred for the acceleration graph as well, butagain is only due to the motor rotating in the opposite direction. They are still identical, justinversely. The graphs follow the same path as one another for the most part, so the behavior ofthe acceleration is clearly correct for the model.The displacement graphs were slightly different in displacement value; however this is to be expectedbecause the spots chosen for the graph in SolidWorks was different from that of the one chosen

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in the Linkages program. The graphs do show and identical path though, which proves that themodels were accurate because the pieces portrayed graphically were moving in the same motion.

Figure 16: Simple Analysis Image

Figure 17: Displacement Analysis

Figure 18: Velocity Analysis Figure 19: Acceleration Analysis

5 Conclusion for the PowerSaw

The PowerSaw mechanism we created was a correct representation as proven not only by ourSolidWorks motion analysis and also through basic mathematics. We were able to prove the correctdisplacement, velocity and acceleration through the motion analysis and by knowing that thederivative of the displacement is velocity and derivative of velocity is acceleration.

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