MAE 412

Machines and Mechanisms II

Four-Bar Catapult

Dr. V. Krovi

12/13/02

Group E Craig Jackson Matt Johnson Jon Knechtges Jeff Jaskowiak Tony Kania Ani Ketkar Tae Hun Kim Damon Knapp

TABLE OF CONTENTS

Introduction……………………………………………………………………..

Idea Generation…………………………………………………………………

Analysis………………………………………………………………………...

Physical Construction………………………………………………………….

Testing………………………………………………………………………….

Conclusions…………………………………………………………………….

Appendix A (competition results)……………………………………………..

INTRODUCTION

When given the task of developing a catapult system, our group began by clearly

defining the problem. The three main objectives for our project were to achieve the maximum

throwing distance, produce reliable and repeatable results and to accurately model and analyze

the mechanism using various methods. In addition to these objectives, certain constraints were

placed on the project; such as a two by two foot size restriction, the use of a pre-determined

motor, the inability to use any stored energy (such as pressurized gas), and the use of a four-

bar (minimum) mechanism. Our fundamental strategy for completing this project was to break

up the tasks into smaller groups. This would allow each group to focus on one particular task,

and then bring them all together in the end.

IDEA GENERATION

Once our group had a clear definition of the problem at hand, we held a meeting to

discuss possible ideas. Each member of the group was asked to come up with one or two

solutions to the problem. All ideas were considered and the positive and negative aspects of

each were noted. The many solutions were narrowed down to six mechanisms, which were then

sketched out and thought about further. Being able to clearly visual each of these mechanisms

made it much easier to exclude some due to complexity of design and construction.

Our group met again to discuss the problem and our specific goals further. In addition to

the goals given to us, we set a couple goals of our own. One of these goals was to throw the

ball with repeatability at least ten feet. Also, we wanted the mechanism to be easily set up,

taking only a few moments between consecutive throws.

Fig. 1

One of our ideas, figure 1, consisted of the squash ball being hit off of a stand. The

advantage of this system is that it incorporates the elasticity of the ball, much like a racket hitting

a tennis ball. However, when the squash ball was examined more closely, it was decided that

the ball would probably absorb more energy than it would return.

Fig. 2

Figure 2 shows another potential mechanism to launch a squash ball. This crank-rocker

mechanism was dismissed in part because there are too many overlapping parts. A mechanism

of this complexity could present problems with the spring and pulley systems. Also, the

launching arm would not be as efficient as other mechanisms due to its limited motion.

Fig. 3

This general idea is what we concluded would be our final design. At first, we had

designed the mechanism with r4 > r2, which gave us a Grashof case I mechanism (S+L < P+

Q). However, this was not an optimal design considering the purpose of our mechanism.

We finally decided to use a Grashof case two four-bar mechanism, similar to the one in

figure 3. This mechanism was chosen because there are no limiting conditions, hence free

movement of the input and follower link. This configuration also appeared to be the most stable

because of its wrap-around base. It also offered excellent conversion of velocity into to the

launching arm. From this sketch, our group refined the design. The motor would turn a pulley,

which would pull the mechanism down until the angle ?1 was approximately zero. This would

build up potential energy in the compression spring attached to link r2 (input). The line would

then be released, throwing the mechanism back up, and launching the ball when link r4

(follower) hit a stopper set at 45 degrees. This would give the ball its maximum velocity at the

angle appropriate for maximum distance.

Once the general design was determined, our group arbitrarily chose lengths of the links

to fit the size restriction placed on project. The group played with the lengths of both the input

and follower links to find a pattern. They began with an input link length of 22.5’’ and a

follower length of 15’’. These arbitrary lengths, as well as several other variations of lengths,

were then analyzed using velocity and acceleration methods to find the best combination. In

each case, the combination of input > follower gave ? 4 > ? 2 and a4 > a2. This told us that the

combination was valid, and to produce a design where input length > follower length.

The group found through analysis that the best combination of link lengths, with the only

constraint being the frame in which to work, was 18.0’’ input length and 12.0’’ output length.

This information was then handed down to the construction team. They calculated the input

velocity from the spring force velocity and acceleration analysis. These forces, they figured,

would play a role in the performance of the mechanism. Using the data thy had collected, they

built a few different combinations of link lengths near the lengths they had calculated previously.

They physically tested those after construction, and found that after frictional forces and forces

in the springs are considered, the optimum link lengths and input and output angles are as

follows:

r1 = 4” ?4 = 90°

r2 = 18” ?1 = 90

r3 = 12”

r4 = 12”

Only ?2 and ?3 needed to be found for complete construction. The methods for finding

those angles are demonstrated in the analysis section.

Analysis: (note: blank pages occur where images did not scan correctly into MS Word. We apologize for the inconvenience) In order to accurately assess how to construct our mechanism and how it would perform, we

needed to execute various forms of graphical, computer, and hand analysis. We used these methods to

optimize our device in terms of position, velocity, acceleration, and torque associated with forces in the

system.

The graphical method for position analysis is shown on the next two pages. With given values for

r1, r2, r3, r4, ? 1, and ? 4, we used the method of perpendicular bisectors to find ?2 and ? 3. ? 1 and ? 4 were both

chosen to be 90° for simplicity, and so we needed to find ?2 and ? 3 to complete our position configuration.

We performed the graphical process for both the “cocked” and “uncocked” positions. These positions

corresponded to the throwing position (maximum kinetic energy) and the release position (maximum

potential energy), respectively.

Graphical velocity analysis told us at what velocity a certain point on our device would be

traveling, given our configuration. Point A corresponds to the joint between links 3 and 4. This is where,

we determined, our maximum velocity would exist for our mechanism. The velocity of the point of interest

was calculated using VB/A = VB - VA , where VA is the velocity in question.

Next, we performed hand analysis using Method III in order to find ?2 and ? 3. Since graphical

analysis is more prone to human error, we found that we had achieved more accurate results for our

unknown angles through this method. Here, we kept our known values the same: r1 = 4’’, r2 = 18.5’’, r3 = 12’’,

r4 = 10.5’’, ?1 = 90° and ? 4 = 90°. With these values we were able to find our unknown values of ? 2 and ? 3 in

order to set-up our final configuration. The method is displayed on the next two pages:

? 3 = 202.25° and ? 2 = 59.75°. These values are slightly different from what we had found through graphical

methods, but are more accurate and are what we used in our final configuration.

With the position analysis complete we then moved on to the velocity analysis. We had already

established that all values found in the position analysis were our optimum values, so we can conclude then

that all velocity numbers will also be our optimum values. Velocity analysis was once again completed

using Method III, differentiating the position equations one time. Our velocities of interest existed in the

angles of links 3 and 4. This is the area where the ball was placed, and hence the point of interest.

Velocity results are displayed on the next page:

As was confirmed by the velocity analysis, maximum output angular velocity was possessed by ?4,

which told us that the ball should be held for launch atop link 4.

To further demonstrate this point, it is important to look at the acceleration of the output links as

well as velocity. If the link and its corresponding angle has maximum output velocity as well as maximum

acceleration, the link will be rotating the fastest.

Acceleration analysis was completed by differentiating the velocity equations used in the previous

calculation. Our results are as follows on the two subsequent pages:

? 4 not only possesses the maximum angular velocity, but also the maximum angular acceleration.

The maximum response will be achieved on link 4, and the ball will travel the furthest if it is

placed on this link.

A complete force analysis of our mechanism is displayed on the next three pages. These

calculations were achieved using inverse force analysis. This method shows results for the amount of force

produced at the joints and amount of torque created throughout the system.

In order to confirm our previous findings, we implemented a Matlab program. This program was

set-up to produce position, velocity, and acceleration results (like the ones produced using hand methods)

Matlab: The purpose of this design is to throw the ball to a maximum distance and hit a given target.

The final design that we decided is using four bars. We decide this four bar shape to make it possible to

analysis from what we learned from the class and the shape can make the ball launch to desired direction to

45 degree to get the maximum distance. After we decided the basic shape of the four bars, we try to focus on

finding the best link length that can make one interested point maximum acceleration because the force of

our design will be created by spring with spring constant value K, and also the momentum to the racquet

ball will be generated with contact time delta T with the bar so that we focus on the finding the best link

length that make the acceleration maximum on the out put link.

Position analysis:

theta3=f(a,b,c,d,theta1,theta2) theta4=f(a,b,c,d,theta1,theta2) Velocity analysis: W3=f(r1,r2,r3,r4,theta1,theta2,theta3,theta4,w2) W4=g(r1,r2,r3,r4,theta1,theta2,theta3,theta4,w2) Acceleration analysis: Alpa3=f(a,b,c,d,theta2,theta3,theta4,W2,W3,W4,alpa2 Alpa4=f(a,b,c,d,theta2,theta3,theta4,W2,W3,W4,alpa2) Because our system is initially has two known angle theta1=90, theta2=90

The mat lab code given here is rotated 90 degree to the left in order to use the analysis example in the book.

After we finish position analysis, we could fine the two unknown values, angle theta 3 and theta4. Now we

did assume a velocity of input link to find the two unknown velocity W3 and W4. After we finish the

velocity analysis, we assumed an acceleration of input link to find the two unknown acceleration alpa3 and

alpa4. We could assume the input link velocity and Acceleration to find a best link combination because

whatever the velocity and acceleration of input link is, we can get the best link length combination of all four

links. After we decide a positive link length, we tried to do force analysis to find minimum required torque

because we don’t want to burn our motor up.

Matlab code: In this matlab code, link length and angle theta assigned with different name.

put is match with the real value. In order to make it simple, we used the same configuration as the textbook

and made our system rotated 90 degree to left.

Link length: a=r4 b=r3 c=r2 d=r1 Angle: Theta1=theta4 theta2=theta4 theta3=theta3 theta4=theta2 Example values:

r1=4, r2=18.5 r3=12 r3=12 r4=10.5 Initial fixed angles: Theta1=0 theta4=180 (after rotation) Random value: W2=10 (a random W2 and angle Alpha 2 to optimize the link length we need a just constant w2) Alpha2=10 Position analysis a=r4 (90degree top) b=r3 c=r2 (bottom link) d=r1 (ground link) r1=4; r2=18.5; r3=12; r4=10.5; function[alpa3_1]=optlinklength(r1,r2,r3,r4) r4=a; % defining new value r3=b; r2=c; r1=d; Theta2=180*pi/180; %90+90=180 degrees Ax=a*cos(theta2); Ay=a*sin(theta2); Dx=d; Dy=0; K1= (-(Dx^2+Dy^2) + Ax^2+Ay^2 - b^2 + c^2)/(2*(Ax-Dx)); K2= -(2*(Ay-Dy))/(2*(Ax-Dx)); K3=K1-Dx; P=K2^2+1; Q=2*K2*K3-2*Dy; R=K3^2+Dy^2-c^2; BY1= (-Q + sqrt(Q*Q-4*P*R))/(2*P); BY2= (-Q - sqrt(Q*Q-4*P*R))/(2*P); BX1= K1 + K2*BY1; BX2= K1 + K2*BY2; BY1= (-Q + sqrt(Q*Q-4*P*R))/(2*P); BY2= (-Q - sqrt(Q*Q-4*P*R))/(2*P); BX1= K1 + K2*BY1; BX2= K1 + K2*BY2; theta3_1=atan2((BY1-Ay),(BX1-Ax))*180/pi; theta4_1=atan2((BY1-Dy),(BX1-Dx))*180/pi; theta3_2=atan2((BY2-Ay),(BX2-Ax))*180/pi; theta4_2=atan2((BY2-Dy),(BX2-Dx))*180/pi; disp('theta3_1='); disp(theta3_1);

disp('theta4_1='); disp(theta4_1); Velocity analysis: w2=f(r1,r2,r3,r4,theta1,theta2,theta3,theta4,w2) w3=g(r1,r2,r3,r4,theta1,theta2,theta3,theta4,w2) w2=10; %random, constant velocity w2=10 w3_1=(a*w2/b)*sin(theta4_1-theta2)/sin(theta3_1-theta4_1); w4_1=(a*w2/c)*sin(theta2-theta3_1)/sin(theta4_1-theta3_1); w3_2=(a*w2/b)*sin(theta4_2-theta2)/sin(theta3_2-theta4_2); w4_2=(a*w2/c)*sin(theta2-theta3_2)/sin(theta4_2-theta3_2); disp('w3_1='); disp(w3_1); %display the value w3_1 and w4_1 disp('w4_1='); disp(w4_1); %w3_2 and w4_2 is closed configuration Acceleration analysis % alpha3=f(a,b,c,d,theta2,theta3,theta4,W2,W3,W4,alpa2) % alpha4=f(a,b,c,d,theta2,theta3,theta4,W2,W3,W4,alpa2) alpha2=0.040077; %random, constant alpha2 F=mass*acceleration we have a constant mass %so we will get a constant acceleration on one of link which connected to spring A=c*sin(theta4_1); B=b*sin(theta3_1); C=a*alpha2*sin(theta2)+a*w2^2*cos(theta2)+b*w3_1^2*cos(theta3_1)-c*w4_1^2*cos(theta4_1); D=c*cos(theta4_1); E=b*cos(theta3_1); F=a*alpa2*cos(theta2)-a*w2^2*sin(theta2)-b*w3_1^2*sin(theta3_1)+c*w4_1^2*sin(theta4_1); alpha3_1=(C*D-A*F)/(A*E-B*D); alpha4_1=(C*E-B*F)/(A*E-B*D); disp('alpha3_1='); disp(alpha3_1); disp('alpha4_1='); disp(alpha4_1); A=c*sin(theta4_2); B=b*sin(theta3_2); C=a*alpa2*sin(theta2)+a*w2^2*cos(theta2)+b*w3_2^2*cos(theta3_2)-c*w4_2^2*cos(theta4_2); D=c*cos(theta4_2); E=b*cos(theta3_2 F=a*alpha2*cos(theta2)-a*w2^2*sin(theta2)- b*w3_2^2*sin(theta3_2)+c*w4_2^2*sin(theta4_2); alpha3_2=(C*D-A*F)/(A*E-B*D); alpha4_2=(C*E-B*F)/(A*E-B*D); end figure (1) %plotting two intersection of two circle to check the result of

hold off %the position analysis with naked eyes axis([-30,30,-30,30]) % range of plot hold on axis equal grid on plot(Ax,Ay,'r*',Dx,Dy,'bx') for i=1:360 plot(Ax+b*cos(i*pi/180),Ay+b*sin(i*pi/180),'r:'); plot(Dx+c*cos(i*pi/180),Dy+c*sin(i*pi/180),'b'); end return; Example of output with value given previously: theta3_1= 88.0239 (so, the real theta 3 =88.0239-90=-1.97) theta2_1=139.5892 (so, the real theta 2=139.5892-90=49.5892) w3_1=8.8787 (w3,w4,alpa3,alpa4 will not match with real value w4_1=0.3491 because we just assumed w2 and alpha3) alpha3_1=2.6608 alpha4_1=-5.7653 In conclusion, we could get the relative acceleration of our interested point “Alpha3” by assuming a

random velocity and acceleration to optimize our link length.

PHYSICAL CONSTRUCTION

This picture shows the original prototype design

The physical construction of the catapult mechanism was limited to using wood for the

frame and links. Wood allowed for quick and easy construction, lightweight and enough strength

to suit our needs. The links were all made from 3/4 “ X 1 1/4 “ pinewood.

Holes were drilled into the coupler link to reduce the mechanisms rotational weight.

Fig. 4

The catapult also utilized two strong tension springs attached on the sides of the

mechanism to keep it balanced, and to avoid any contact with the links. This was an

improvement over our original idea of using only one compression spring attached to link R4.

One spring would have the tendency to buckle when compressed; it would also need to bend

since link R4 is not always parallel with the ground of the spring. Two tension springs solve both

of these problems. Our group decided to use a pulley to allow the motor to pull the mechanism

into a cocked position, and fishing line was used as the cable.

A custom fabricated pulley with a threaded setscrew was created to increase the

amount of line retracted per motor revolution. Also, a custom fabricated spacer was made in

order for the bearing to fit on the motor’s output shaft. These improvements allowed for a

quicker overall launch time.

Fig 5

During the construction process, a few over looked details arose. One such detail was

the release mechanism. In order to launch the ball, the four-bar mechanism would need to

release from the pulley system.

Our solution to this dilemma was found in an already existing piece of hardware. An

archer’s bow release allowed for a simple, quick and very reliable way to release the pulley

from the mechanism.

Fig. 6

Once the release point was determined, a block of wood was placed at the

appropriate height to hit the bow release trigger. A second detail that was not decided upon

until the construction phase was how the ball was to be held on the catapult. The solution for

this problem was found in a metal coffee spoon that had a slightly larger diameter than the

squash ball we had to launch.

The spoon was screwed into the link, and bent to give the ball a launch angle of 45

degrees.

Fig. 7

The electric motor used to stretch the spring and store energy used a six -volt battery.

This was more voltage than the motor specifications called for, but the group felt that the

increase in voltage would improve the performance of the motor, and was within safe limits for

the mechanism’s brief operating time.

The motor wire, and all of the circuit wire, was upgraded to 16 gauge. While not

necessary to the circuit, we felt it would improve the durability of the system, and prevent

possible wire breaks. Additionally, a steel project box housed the on/off switch to provide

stability to the switch, all wire connections were soldered and heat shrink tubing was used

where applicable.

Fig. 8

The bow release was hardwired into the motor’s power circuit, so that the circuit would

be broken as soon as the mechanism was released, turning off the motor when it is no longer

needed. The final step of the construction was painting the base black to give the catapult an

industrial look and make it more aesthetically appealing.

Testing

Once the device was constructed, it was necessary to run several successful tests

before competition.

We first wanted to test the string that was pulling our device down in order to fire. We

originally used old “fabric” fishing line. But over time and trials, the line began to fray and

eventually broke. Our solution was to use 40 pound test synthetic fishing line. The strength of

this line was more than we needed, and the material allowed for little or no wear once wheeled

down by the pulley.

Our release mechanism was next to be tested. We needed the mechanism to release

the string from the link at precisely the point specified. This is the one area where we had little

or no trouble. The release performed perfectly, and even surprised us a little how it performed

in shutting the motor off upon release.

We realized early in the testing phase that where and at what angle the ball released

from the holding cup, would play a big role in how far the ball would travel in the horizontal

direction. Since the link we had placed the ball on was completely vertical, we knew that if we

bent the cup to an angle of 45 degrees, we would get the maximum performance upon release,

since the ball would initially travel at the same angle upon release. We played with this angle a

bit to ensure our conclusions were correct. Any angle slightly less than or slightly greater than

45 degrees resulted in less horizontal distance traveled.

To further guarantee that we were correct in placing the cup on link 4, as was found in

earlier analysis, we placed the cup on link 2 for a few trials, and found that the horizontal

distance traveled was significantly less. We knew we had the correct configuration.

We did perform Solid Edge testing of our device to model its performance before actual

physical testing. However, those results do not appear in this report because of loss of contact

with the group member responsible for this testing.

Our device did perform as expected however, throwing consistently exactly 25 feet.

CONCLUSIONS

Our four-bar catapult performed as expected. While the maximum distance achieved

was not as great as some of the other groups, the distance for each trial was the most consistent

(see appendix A). Unfortunately, this worked to our disadvantage when it was time to hit the

target, since the basket could only be placed at five-foot increments. The best solution for this, if

we were to make improvements, would be an adjustable release height. It would be very easy

to replace the solid block used to hit the release mechanism with a height adjustable block.

Releasing the catapult at different heights would achieve different distances. Overall, we feel that

all of the required objectives were met, but the desired objectives fell a bit short. We were

hoping for greater maximum throw distance. Another possible improvement would be to use

springs with a greater stiffness. The motor had no trouble pulling the mechanism down with its

current configuration, and stronger springs would add a substantial amount of distance to the

throw. On a side note, one of our group goals was to split up the work of the project into sub-

teams. Ideally, this would have led to a very efficient completion of all the tasks involved.

Unfortunately this did not work as expected. Instead of separating hand-method analysis from

Matlab and Dynamic Designer simulations, we feel these areas would have benefited from

working more closely together. It would have also been much easier to share information if the

groups overlapped; having one or two people of each group working in conjuncture with

another group. In the field of Engineering, experience is just as important as knowledge, and the

experience gained by this project will be very useful in future group projects.

Appendix A

Project Competition Results

Farthest Distance Shooting Precision Shooting Group Trial 1 Trial 2 Trial 3 Farthe

st Ranking Number of Successful Shoots

A 38.5 ft 37 ft X 38.5 ft 2 2 B 14 ft 30 ft X 30 ft 7 2 C 24 ft 30 ft 32 ft 32 ft 5 1 D 10 ft 12 ft 14 ft 14 ft 12 0 E 25 ft 25 ft 25 ft 25 ft 9 0 F 17.5 ft 18 ft 20 ft 20 ft 10 0 G 38 ft 37 ft 35.5 ft 38 ft 3 1 H 26 ft 31.5 ft 25 ft 31.5 ft 6 0 I 20 ft 10 ft 15 ft 20 ft 10 0 J 18.5 ft 27.5 ft 38 ft 38 ft 3 2 K 60 ft 57.5 ft - 60 ft 1 2 L 24.5 ft 22.5 ft 27 ft 27 ft 8 2