Docstoc

EMCH Lab Report Template

Document Sample
EMCH Lab Report Template Powered By Docstoc
					 Padnos School of Engineering
     Grand Valley State University




FEEDBACK CONTROL FOR ANTI-SWAY
        COMPENSATION

 EGR 345 Dynamic Systems Modeling and Control




                   12/04/03




                   Team #3

                   Rick Berta
                  Chris Buiter
                   Dan Cole
                  Matt Reimink
                  Bill Rininger




                  Fall 2003
                                        Executive Summary


        The goal of the project was to build a proof-of-concept prototype for an anti-sway system for a

gantry crane. The cart has to travel a maximum distance of twenty inches in two-inch increments along a

2x4. The cart has to travel the desired distance in the least amount of time and have no sway when the

desired position was reached. The design constraints were to have a cart with a mass of 0.2 Kg or less,

and the cost of the cart to be under $150.00. The cart design was first simulated in Scilab, so the design

parameters could be easily changed to minimize the settling time. The cart was built using 0.25 in.

Plexiglas for the sides and the arms of the cart, and the drive wheels were constructed out of aluminum.

An encoder was used to measure the distance the cart traveled, and a potentiometer was used to measured

the angle the hanging mass was swinging. A C program was written to control the motion of the cart. The

encoder feedback was used to determine the position of the cart, and the potentiometer feedback was

used to compensate for the swaying of the mass once the cart had reached its desired position. In the first

test the cart moved from 0-18 inches and compensated for sway in 13 seconds. The cart for the first test

had a mass of 0.5778 Kg, and a cost of $140.00. These values were entered into the equation for the

score, and the score was found to be 24482.62. After the first test, the mass of the cart was greatly

reduced and the program was adjusted. For the second test the cart traveled from 0 to 20 inches and

compensated for sway in 5.5 seconds, a 58% decrease in time. The mass of the cart was reduced to

0.4288 Kg and the cost of the cart for the second tested was $135.21 for a score of 2096.18. This was an

improvement of 22,386.44 points in the score from the first test to the second test. When compared to

the other groups this score was actually the minimum with other groups’ scores ranging from 2925.4 to

infinity, meaning there was no successful run. The cart traveled 20 inches had a settling time of 3.4

seconds for the final test, with a mass of 0.409 Kg. The cart was voted best build quality and finished 3 rd

for the overall, with a score of 748.
                                    Table of Contents

Executive Summary _____________________________________________ 2

Table of Contents _______________________________________________ 3

List of Figures and Tables ________________________________________ 4

1.   Design Description _________________________________________ 5
 1.1 Design Constraints ________________________________________________________________5
 1.2 Bill of Materials __________________________________________________________________5
 1.3 Weight Inventory _________________________________________________________________7
 1.4 Project Budget ___________________________________________________________________8
 1.5 Control Scheme _________________________________________________________________10


2. Test Results ________________________________________________ 16
 2.1 Simulation ______________________________________________________________________16
 2.2 Preliminary Testing_______________________________________________________________17
 2.3 Formal Testing __________________________________________________________________17


3. Conclusions ________________________________________________ 20

Appendices ___________________________________________________ 21
 Appendix A: Drawings _______________________________________________________________22
 Appendix B: Calculations _____________________________________________________________23
 Appendix C: Controller C program _____________________________________________________27
 Appendix C: Controller C program _____________________________________________________28
 Appendix D: Scilab Simulation Program _________________________________________________29
 Appendix E: Receipts and Cost Evidence _________________________________________________30
                                            List of Figures and Tables

Table 1: Bill of materials ................................................................................................................ 7
Table 2: Mass table ......................................................................................................................... 8
Table 3: Project budget ................................................................................................................... 9
Table 4: Test scores ...................................................................................................................... 18


Figure 1: Anti-sway cart.................................................................................................................. 6
Figure 2: Assembly drawing ........................................................................................................... 6
Figure 3: Control system architecture ........................................................................................... 10
Figure 4: Control system block diagram ....................................................................................... 11
Figure 5: Circuit schematic ........................................................................................................... 12
Figure 6: State diagram of the program ........................................................................................ 13
Figure 7: FBD of hanging mass (provided by Dr. Hugh Jack) ..................................................... 13
Figure 8: Scilab simulation results ................................................................................................ 16
Figure 9: Arm hole diagram .......................................................................................................... 24




                                                                 4
                                     1. Design Description
1.1 Design Constraints
The goal of the project was to build a proof-of-concept prototype for an anti-sway system for a
gantry crane. The cart was driven by an electric motor, and rode across the top of a 2x4 piece of
lumber. A 1 Kg mass was hung 40 centimeters below the top of the 2x4. A C program was
written to drive the cart to the desired position through a feedback control and also compensate
for the sway of the mass. The objectives and constraints for the design can be seen below:
               The cost of the cart should be less than $150.
               The target mass of the cart is 0.2 Kg.
               The setting time of the cart should be minimized.
               The cart will travel to a desired position and should be in a range of +/- 0.5 inch,
                of the desired position.
               The total movement of the cart will be a range from 2 to 20 inches in 2-inch
                increments.
               The cart apparatus must be easily mounted and removed to not damage the
                experiment setup.
               The cart will carry a 1 Kg mass mounted on a carriage bolt that has a diameter of
                0.375 inch.
               The mass must be able to freely rotate.
These constraints and objectives were all taken into account when designing the cart, electrical
system and software for the anti-sway prototype design.


1.2 Bill of Materials
When designing the mechanical system of the cart, the main objectives were to minimize weight,
make the cart strong, and make the cart aesthetically pleasing. Plexiglas was used for the hanging
arms and for the sides of the cart. Plexiglas was chosen because it is light, strong, inexpensive,
and has high aesthetic value. The rendered assembly model (minus gears, belts and pulleys for
clarity) of the cart can be seen in Figure 1.



                                                5
                                      Figure 1: Anti-sway cart

The drive wheels of the cart were made of 0.5-inch diameter aluminum rod, and were knurled to
give the cart some traction on the 2x4. The knurled wheels did not give enough traction on the
2x4, so O-rings were added. Side wheels were added to the cart to help stabilize the cart. These
wheels were made of 0.5-inch diameter nylon rod, and were milled down to a thickness of 0.25
inches. The side wheels were mounted to the cart with a 6-32 x 1.25 inch socket cap screws,
which was tapped into the Plexiglas body of the cart. The support rods, composed of 6-32
threaded rods of length of 3 inches, keep the cart together and also make the cart adjustable.
Figure 2, shows the exploded assembly view of the cart.




                                     Figure 2: Assembly drawing


                                             6
A detailed assembly drawing of the cart can be seen in Appendix A. The bill of materials of all
the parts used in the cart can be seen below in Table 1.

                                       Table 1: Bill of materials

                                             Bill of Materials
                                               Item                         Quantity
                        Gearhead motor                                         1
                        Potentiometer                                          1
                        Encoder                                                1
                        68HC11 Axiom board                                     1
                        Power supply                                           1
                        Plastic spur gears 48p                                 4
                        Drive gear (MXL timing pulley (2.5" dia. x 0.25"))     1
                        Drive gear (MXL timing pulley ( 0.5" dia. x 0.25"))    1
                        Drive belt (MXL timing belt (length 10.4" x 0.25"))    1
                                                                                 2
                        Arms - Plexiglas 0.1875"                            34 in
                                                                                 2
                        Cart side (LH) non motor Plexiglas 0.25"            15 in
                                                                                 2
                        Cart side (RH) motor Plexiglas 0.25"                18 in
                        Pivot rod (alum. rod dia. 0.25" x 3.5")               1
                        Alum. drive wheels (dia. 0.5" x 3.5")                 2
                        Side wheels (Nylon rod dia. 0.5" x 0.25")             4
                        Support rod 6-32 Threaded rod (length 3")             2
                        6-32 nut                                              8
                        O-rings ID 0.375 OD 0.625"                            4
                        6-32 x 1.25" socket head cap screw                    2
                        6-32 x 0.75" socket head cap screw                    2
                        4-40 set screw                                        6
                        10 K resistors                                        2
                        47 uF capacitors                                      2
                        Assortment of wire                                    -



1.3 Weight Inventory
A breakdown of the mass of every part on the cart is shown Table 2. The table shows that
progression of all the masses throughout the different stages of the project. The masses of each
part were weighed on a scale. In the design phases of the cart weight was not a huge issue, but as
testing occurred the mass of the cart had to be reduced to lower the overall score. Next, material
was removed from the drive gear, the sides of the cart and the side wheels. The last column of
the table shows the projected mass values before the final test.


                                               7
                                         Table 2: Mass table

                                         Mass Table (in Kg)
                                   Test 1                   Test 2                  Final Values
 *changes shown in bold*      November 11, 2003       November 19, 2003          November 25, 2003
        Description        Mass Qty. Total Mass Mass Qty. Total Mass          Mass Qty. Total Mass
Gearhead motor             0.1840 1       0.1840  0.1840 1          0.184     0.1840 1       0.1690
Encoder                    0.0110 1       0.0110  0.0110 1          0.011     0.0110 1       0.0110
Potentiometer              0.0060 1       0.0060  0.0060 1          0.006     0.0060 1       0.0060
Alum. drive wheels         0.0166 2       0.0332  0.0166 2         0.0332     0.0096 2       0.0192
Side wheels and axles      0.0160 4       0.0640  0.0043 4         0.0172     0.0033 4       0.0132
Cart side (LH) non motor   0.0350 1       0.0350  0.0281 1         0.0281     0.0281 1       0.0281
Cart side (RH) motor       0.0405 1       0.0405  0.0333 1         0.0333     0.0333 1       0.0333
Arms                       0.0749 2       0.1498  0.0327 2         0.0653     0.0327 2       0.0653
Gears                      0.0024 4       0.0096  0.0024 4         0.0096     0.0024 4       0.0096
Drive gears and belt       0.0177 1       0.0177  0.0148 1         0.0148     0.0148 1       0.0148
Pivot rod                  0.0090 1       0.0090  0.0090 1          0.009     0.0090 1       0.0090
Support rod                0.0090 2       0.0180  0.0087 2         0.0173     0.0056 2       0.0112
        Total Mass                        0.5778                   0.4288                    0.4047




1.4 Project Budget
Table 3 is the complete budget for the cart. The goal for the budget was to limit spending to
$150.00. The prices for electrical components (e.g. capacitors, resistors, encoders,
potentiometers, etc.) are based on bulk unit prices. This is reasonable because if the carts were to
be mass-produced, components would be ordered in bulk.




                                             8
                                        Table 3: Project budget

                                         Material Budget
               Item                Quantity       Supplier              Part Number  Price ($)
Gearhead motor                        1            GVSU                  40791205     10.00
Potentiometer                         1           Digi-key              CT2204-ND      2.52
Encoder                               1           Digi-key              CT3001-ND      2.40
68HC11 Axiom board                    1            GVSU                               89.00
Power supply                          1            GVSU                                0.00
10K Resistor                          2           Digi-key           MFR-25FRF-1K00    0.04
47uF Capacitor                        2           Digi-key              P5528-ND       0.66
L293D chip                            1           Digi-key            296-9518-5-ND    1.28
Plastic spur gears 48p                4     Stock Drive Products      A1M2-TA48048     8.68
MXL timing pulley (dia.=2.5")         1     Stock Drive Products     A6Z16-100DF2508   7.34
MXL timing pulley (dia. = 0.5")       1     Stock Drive Products     A6T16-020DF2508   3.86
MXL timing belt (length = 10.4")      1        McMaster-Carr          24-827887 K82    2.71
                                          2
Plexiglas 3/16"                      33 in            GVSU                               0.55
                                          2
Plexiglas 1/4"                       34 in            GVSU                               0.90
Aluminum rod dia. 0.25" x 3.5"         1              GVSU                               0.50
Aluminum rod dia. 0.5" x 3.5"          2              GVSU                               1.50
O-rings ID 0.375 OD 0.625"             4           McMaster-Carr          9396K63        0.27
Misc. nuts and bolts etc.              -              GVSU                               3.00
                                                                                Total = 135.21


The bill of materials of the cart can be seen on the assembly drawing of the cart in Appendix A.
The miscellaneous nuts and bolts used on the cart were taken from the machine shop in the
Keller Engineering building at Grand Valley. The estimated cost was $3.00 based on values
from McMaster-Carr. The Plexiglas price of $1.45 was calculated by finding the area of
Plexiglas used in the cart. A total of 34 square inches of 0.1875-inch thick Plexiglas was used for
the two arms, and 33 square inches of 0.25-inch thick Plexiglas was used for the side pieces. This
area also includes excess material. The Plexiglas purchased by Grand Valley is purchased in 4’x
8’ sheets for a total area of 4608 in2. The total cost for a 0.25-inch thick sheet of Plexiglas is
$125.00, and 0.1875-inch thick sheet of Plexiglas is $75.00. The total cost for the plastic was
$0.90 for the cart and $0.55 for the arms. All of the other price comparisons are shown in
Appendix E.




                                               9
1.5 Control Scheme
The electrical system of the cart is composed of the electric gearhead motor, potentiometer, and
encoder. The wires from these three devices are wired in a wiring harness and go to the 68HC11
Axiom board. The system architecture for the electrical diagram can be seen in Figure 3.




                                Figure 3: Control system architecture


To drive the cart a 12 VDC gearhead motor was used. A 2.5-inch timing pulley was mounted on
the shaft of the motor, and a 0.5-inch timing pulley was mounted on the front aluminum drive
wheel. A 10.4-inch timing belt wrapped around the two timing pulleys to drive the cart. The
potentiometer and the encoder were mounted to the side of the cart. Gears were then mounted on
the shafts of the potentiometer and the encoder. The gear on the shaft of the encoder was attached
to a gear on the drive wheel. As the drive wheel rotated, the encoder read the position of the cart
and updated the position variable. The gear on the shaft of the potentiometer was attached to
another gear on the pivot bar for the mass. As the pivot bar moved forward and backwards, the
gear on the pivot bar turned, which in return turned the gear on the potentiometer. The voltage
generated by the turning potentiometer was used in the anti-sway program.




                                             10
Figure 4 shows the system block diagram.




                               Figure 4: Control system block diagram



    Where,
               C pe  position error
               K pp  porportional gain for position control
               C p  position count from encoder
                p  rotational position of the motor
               C pc  output commandfor position control
               C c  combined motor controloutput
               Vs  effective voltage to the motor
                L  angle of the load from vertical
               VL  voltage proprtional to angle of load
               C L  integer value for load angle
               C LC  output commandfor sway control

The program receives a desired input from the user, and then after compensating for deadband,
proportional and integral gain, the program drives the cart to the target position. The control
loop then continuously reads the voltage from the encoder to adjust position and the
potentiometer compensate for sway.




                                            11
The circuit schematic used to control the cart and anti-sway movement can be seen in Figure 5.




                                    Figure 5: Circuit schematic

The software program designed for the cart is attached in Appendix C. The initial idea for the
program was to read the voltage coming in from the potentiometer and then based on its value,
set an error variable for the amount of sway. Four different variables were used: high_neg_sway,
low_neg_sway, high_pos_sway, and low_pos_sway. These variables not only indicated the
approximate size of the sway but also the direction the mass was swaying. Based on which
variable was set, the speed of the cart was adjusted accordingly. This method ultimately was not
used because it proved to be too time consuming for the 68HC11 to complete the calculations.


The updated program used the potentiometer reading to adjust the pulse width modulation
(PWM) of the cart. The “current” potentiometer reading was continuously updated and
compared to the initial potentiometer value. This “error” value was then multiplied by the gain
value for the potentiometer, which then updated the desired position accordingly. Once the sway


                                           12
of the mass was eliminated, the program drove back to the original desired position when it was
required. The state diagram of the compensating function can be seen in Figure 6.




                                  Figure 6: State diagram of the program

Figure 7, shows the free body diagram of the cart and the hanging mass. The equations show the
forces on the cart and hanging mass systems in the x-direction




                     Figure 7: FBD of hanging mass (provided by Dr. Hugh Jack)

M c  mass of cart
F p  mass of payload
F p  force in suspension arms
Fw  force from wheels
 L  angle of payload from vertical
l  length of suspension arm
rw  radius of cart wheel
                F  M    p
                                                 x
                               g sin  L  M p lL  cos L M p  0



                                                13
                               
                              lL   g sin  L   cos L
                                                   x                                                                            (1)


                   F  F                             
                                     M p g cos L   L M p l  sin  L M p  0
                                                             2
                                p                                x

                                                   2
                              Fp  M p g cos L   L M p l  sin  L M p  0
                                                              x

                    F  M         c
                                          Fp sin  L  Fw  0
                                        x

                                                        2                                    
                               M c   M p g cos L   L M p l  sin  L M p sin  L  Fw  0
                                    x                               x                                                           (2)


                                K2        K F r
                   w  w 
                                     VS    w w
                                     
                                 JR        JR  J

                                          K       J       K2 
                                 Fw  VS                    r R
                                               w    w 
                                         r R                 
                                          w        rw     w 

given x   w rw
                               K   J              K2 
                              
                      Fw  VS          2   x  2 
                                      x          
                                                    r R
                               rw R   rw          w 
plugging in Equation (2)
              x                   
         M c   M p g cos L   L 2 M p l   sin  L M p sin  L  V S 
                                                x                            K   J
                                                                                    x
                                                                              r R    2
                                                                                                    
                                                                                                      x K   0
                                                                                                    
                                                                                                        
                                                                                                           2 
                                                                                                          r 2 R 
                                                                              w   rw                   w 

                                                                                           K        K2 
          M c  2  sin  L 2 M p   M p g cos L sin  L   L 2 M p l sin  L  VS 
                    J                                                                                     
          x
                                                                                           r R   x 2   0
                  rw                                                                       w        rw R 

           M c rw 2  J  sin  L 2 M p rw 2   
                                                    M g cos sin     2 M l sin   V  K   x  K 
                                                                                               
                                                                                                          2
        
        x                                                                  
                                                                                           S    
                                                                                            r R
                                                                                                             
                         rw 2                        p       L     L       L p       L              r 2 R 
                                                                                           w       w    
                                        M p grw 2                                   M p l sin  L rw 2         
          cos L sin  L 
        x                                                              2
                                                                                                                 
                               M r 2  J  sin  2 M r 2               L                                     
                                                                              M c rw  J  sin  L  M p rw
                                                                                     2                  2     2
                               c w               L    p w                                                       
                                                                                                  K2                    
                              VS                                                 x                                      
                                               Krw
                                                                                                                     
                                                                                    
                                  R M r 2  J  sin  2 M r 2                      R M r 2  J  sin  2 M r 2       
                                     c w              L    p w                          c w              L    p w        




State Equations
               xv
                

                                                            14
                                       K2                                              M p grw
                                                                                                   2
                                                                                                              
                      v                                       cos sin                               
                    v
                           
                          R M r 2  J  sin  2 M r 2
                              c w              L    p w       
                                                               
                                                                        L     L
                                                                                 M r 2  J  sin  2 M r 2 
                                                                                 c w                L   p w 



                              M p l sin  L rw
                                                2
                                                                        Krw                        
                    L 
                       2                                 V                                        
                        M r  J  sin   M r
                        c w
                             2
                                              L
                                                 2    2 
                                                   p w 
                                                            S
                                                               R M r  J  sin  2 M r 2
                                                                   
                                                                   c w
                                                                       2
                                                                                  L    p w          
                                                                                                     

                    L   L
                           g           v cos L
                                        
                    L 
                             sin  L 
                            l              l

The above state equations were verified using the motor parameters found in Appendix B,

               V
K  2.706
              rad
R  19.8
M C  0.4288Kg
J  0.007Kgm2
 L  10  1.59rad
M P  1Kg
l  0.040m



xv


v  v(51.148))  0.05986



             rad
 L  6.283
               s

 L  v0.48  245 .205
     




                                                   15
                                         2. Test Results
2.1 Simulation
Before any fabrication took place the design was tested using Scilab. The Scilab program used to
simulate the anti-sway cart can be seen in Appendix D. Figure 8, shows the graph of the
simulation.




                                   Figure 8: Scilab simulation results

From the Scilab simulation it was determined that the settling time for the mass was similar to
the actual results. The second test resulted in a settling time of 5.5 seconds. From the Figure 8 it
can be seen that the settling time is around 3 to 3.5 seconds. In the final test the cart had a settling
time of 3.4 seconds which matches the simulation from Scilab. Also, as the cart moves forward
the sway moves backward and the cart tries to compensate for the sway throughout the
movement of the cart. The Scilab program was changed to include the gear ratio of the cart,
which accurately simulated the sway of the hanging mass.



                                              16
2.2 Preliminary Testing
The first objective in preliminary testing was to move the cart to the desired position, and make
sure the cart was within the +/-0.5 inch tolerance. During the first test the cart did not travel
smoothly, it had a stutter step every 2 inches, which had a big impact on the settling time of 13
seconds. At first it was assumed that the anti-sway program was compensating too early, but after
that portion of the code was commented out, the motor still had its stutter. The stutter step was
eliminated by clipping the c_wanted values that were set in the deadband. If the c_wanted value
was larger than 255, the c_adjusted value would overflow causing the stutter in the motor.


After eliminating the stutter the next goal was to compensate for the sway of the hanging mass
using the potentiometer, and decrease settling of the cart. From the second test it was evident
that the cart had to start adjusting for the sway before it got to the desired position. During the
second test the cart would move to the desired position then start compensating for sway, which
was taking time. Also, during testing the voltage supplied to the cart was varied to increase the
speed of the cart, and to make sure the voltage would not damage the circuit.


Testing the program without compensating software made it evident that the total settling time
for the cart is much faster with the software. Without compensation software, the cart had to
travel at a much slower speed in order to keep the mass from swaying. With compensation
software, the cart was able to travel much faster to the desired position and then compensate for
the sway in less time than without the software.


2.3 Formal Testing
    On November 12, 2003 the first test took place. The cart traveled 18 inches and stopped
within the +/-0.5 inch range in 13 seconds. The mass of the cart at the time of the first test was
0.5712 Kg.
    Table 4 shows the theoretical score after the first test. The scores in Table 4 were calculated
using Equation (4).




                                              17
                                               2
                                         t 
                                 score   s  4 200 10 10 2 0.2
                                                    C                M
                                                           B    T

                                         d                                                         (4)


                                           Table 4: Test scores

                                           Scoring Equation
                          Item               Variables  1st Test   2nd Test Final Test
           The time to settle (s)                ts         13        5.5       3.4
           Total cost of part ($)                C         140      135.21    135.21
           Distance moved in test (m)            d       0.4572      0.508     0.508
           Build quality score (0 to 1)          B         0.1        0.1       0.1
           Theory quality score (0 to 1)         T         0.1        0.1       0.1
           Mass of apparatus (kg)                M       0.5712     0.4288     0.409
                                             Score     24482.6217 2096.1817     748


After the first test took place, there were a number of issues that needed to be changed. The
weight of the car had to be reduced. This was done by milling material off of the sides of the cart.
The arms were made out of 0.1875-inch Plexiglas instead of 0.25-inch and a slot was milled out
of the arms to help reduce the weight. Also, the side support wheels were changed from a
thickness of 1 inch to a thickness of 0.25 inch. Finally, the program was updated to delay the
anti-sway program till the end of motion


After the second test, the main goal was to decrease the settling time of the cart, and reduce the
mass of the cart. This was done by drilling 0.125 inch holes through the drive axle and the pivot
rod. The support rods were trimmed down which saved 7 grams. The motion profile of the cart
was removed to decrease the size of the program and also to decrease the time it took the cart to
travel the desired length. The motion profile was originally included to smooth the travel of the
cart and reduce sway. However, the motion profile did not reduce the sway, and it also slowed
the time of the cart.


The final column in Table 4 relates to the final values from the final test. The cart finished 3rd
overall with a score of 748. The cart was voted best build quality. Table 5 shows the final score
of each of the 12 groups.

                                                18
               Table 5: Final scores

                       Score
Team 1 Team 2 Team 3     Team 4        Team 5   Team 6
 1533   2245   748        900           969      939
Team 7 Team 8 Team 9 Team 10 Team 11 Team 12
 575    1186   478     781    18562   8004




                   19
                                3. Conclusions
   Plexiglas was an optimal material to use because of its high strength to weight ratio

    and because it is easy to machine.

   It was determined that the weight of the cart needed to be reduced, and that the speed

    of the cart had to be increased to continue to be competitive.

   From the second test it was concluded that the sway compensation should take place

    before the cart reaches the final position, which will save time at the end of the run.

   The sway compensation worked properly in the test. The only thing that could have

    been improved was how fast the cart traveled to the final position.




                                     20
Appendices




 21
Appendix A: Drawings




      22
                                       Appendix B: Calculations


Stress Strain Calculations:

Wp = weight of hanging mass in pounds

         Wp  2.210lb
Wa = weigh of the arm in pounds
         Wa  0.072lb
Wc = total mass of cart in pounds
         Wc  0.946lb
w = width of the arm in inches
d = diameter of the hole in inches
t = thickness of the material in inches
  19000 psi ultimate tensile strength of Plexiglas

Arms:
The lower hole on the arms was determined to be an important hole because it is what holds the
carriage bolt, which holds the 1 Kg mass. The bottom hole had to support 2.21 lbs.

     Lower Hole

             Normal Stress:

                          Wp              2.21
                 n             
                       ( w  d )t (1  0.375)(0.1875)
                  n  18.81 psi
                 18.81 psi  2.2 psi


             Calculate the safety factor:
                        19000
                  n     
                       n 18 .81
                  n  1010
             Bearing Stress:
                       Wp         2.210
                 B         
                       d * t (0.375 )( 0.1875 )
                  B  47 .07 psi

             Calculate the safety factor:



                                                  23
              19000
        n      
             B 47 .07
        n  404.07

Upper Hole

The upper hole of the arms had to support the load of the hanging mass. The holes had a
diameter of 0.25”. One side of the hole had a tapped hole going through the side of the
piece; this was used for holding the arm to the pivot rod. Since one side of the hole had a
hole going to it all of the stress from the hanging mass was concentrated to the other side
of the hole. This can be seen in Figure 9. The hole also acts as a stress rise and therefore a
stress concentration factor was calculated to be sure that the arms would be strong
enough.




                               Figure 9: Arm hole diagram




   Calculate the stress concentration factor:
                           hole radius               hole radius 2           hole radius 3
        K  3  3.13(2 *               )  3.66 (2 *            )  1.53(2 *            )
                                w                         w                       w
        K  2.422
   Normal Stress:



                                        24
                       (Wp  Wa ) * K (2.210  0.072 ) * 2.422
                 n                  
                           ( w  d )t    (1  0.25 )( 0.1875 )
                  n  78 .412 psi
             Calculate the safety factor:
                          19000
                  n     
                       n 78 .412
                  n  242.03
             Bearing Stress:
                       Wp  Wa       2.210  0.072
                 B              
                          d *t      (0.25 )( 0.1875 )
                  B  47 .02 psi

             Calculate the safety factor:
                        19000
                  n      
                       B 47 .02
                  n  404.07


Cart: Holes for Horizontal Axel
        These holes supported the aluminum drive wheels. These holes supported the weight of the cart and also the
hanging mass. These holes on the cart where very important, if these holes were to crack the cart would fail.

             Bearing Stress:
                       Wc  Wp 0.946  2.210
                 B              
                         d *t       (0.25 )( 0.25 )
                  B  25 .25 spi

             Calculate the safety factor:
                        19000
                  n      
                       B 25 .25
                  n  752.6


Motor Parameters of the 12VDC gearhead motor


Resistance of the motor was found by measuring the resistance across the motor with the DMM.
 Rmotor  19 .8

The velocity of the motor was found by counting the number of rotations the motor made in 10
seconds.

                                                   25
   10rot 2π rad         rad
              6.283
   10 sec rot           sec
           rad
  6.283
           sec
The voltage supplied to the motor.
VS  17V

Setting different voltages to the motor and plotting the resulting velocity found the time constant
of the motor.
  0.019sec
The motor coefficient K was then found.
    VS      17VS        V
K               2.706 S
    rad 6.283rad        rad
           V
 K  2.706 S
           rad
After K was found the rotational inertia, J, of the motor was found.
     K2       2.7062
J                       0.007Kgm 2
       1        1 
    R  19.8         
             0.019 
 J  0.007Kgm 2


After all of the motor parameters were found the friction of the motor was then found using
equation (5),
       K       R 
  VS    TF  2                                                            (5)
       R      K 
At steady state the velocity is zero so equation (5) becomes,


        K      2.706 
TF  VS    17       
        R      19.8                                                               (6)
TF  2.32N


The motor parameters can be inserted into the equation (6),

                                            26
                                 K 2 
                                      
                1  RT         JR           1  RT
 (t )    VS    2F e 
                        
                                       
                                            VS    2F                                          (7)
               K K                          K K




Plugging in motor parameter values found above into equation (7) the velocity as a function of time becomes,


 (t )  0.009013 e 52.8314   0.009013




                                                    27
Appendix C: Controller C program




                                   28
Appendix D: Scilab Simulation Program




              29
Appendix E: Receipts and Cost Evidence




               30

				
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
views:7
posted:4/7/2011
language:English
pages:30