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					         Grand Valley State University
        The Padnos School of Engineering

  EGR 345 Dynamic Systems Modeling and Control

                  Andrew Edler

                 October 12, 1999

                  Lab Partner
                 Brian Malkowski

                   Fall 1999
                          Table of Contents

Table of Contents _______________________________________________ 2

List of Figures and Tables ________________________________________ 3

Executive Summary _____________________________________________ 4

1. Introduction __________________________________________________ 5

2. Theory ______________________________________________________ 6

3. Apparatus ____________________________________________________ 8

4. Procedure ____________________________________________________ 9

5. Results _____________________________________________________ 10

6. Analysis and Interpretation ____________________________________ 10

7. Conclusions and Recommendations_____________________________ 11

Appendices ___________________________________________________ 12
                                          List of Figures and Tables

Table 1 - Frequency .......................................................................................................... 10

Table 2 - Percent Discrepancy .......................................................................................... 10

Figure 1 - System Diagram ................................................................................................. 6

Figure 2 - LabVIEW Diagram ............................................................................................ 9

                                  Executive Summary
    The objective of the experiment was to study torsional oscillation using LabVIEW
and data collection. The main goal was to study the effects of applying a radial
displacement to torsional systems consisting of varying materials. In particular, the focus
of the experiment was to gather information on the frequency of the systems’ responses
once they had been displaced.
    The systems consisted of a torsional spring (shaft) which was fixed at one end and
suspended in a vertical position. A mass was fixed to the other end of the shaft. By
applying a radial displacement to the mass on the end of the shaft and releasing it, the
system began to oscillate. The spring constant was determined by measuring the amount
of torque required to displace the system a given number of radians. Then, a
potentiometer was calibrated and used to measure the position of the system. The
potentiometer was connected to a data acquisition card in a PC and the data was collected
using a software package called LabVIEW. The position was plotted and by knowing the
time interval between data points on the plot, the frequency was determined. As a source
of comparison, the frequency was also determined by measuring the amount of time for
the system to complete ten cycles once it had been displaced. Two systems were used:
one used an aluminum shaft and the other used a PVC shaft.
    The natural frequency of a torsional system was calculated and then compared to the
measured results. As expected, the use of the computer proved to be more accurate than
measuring the time for ten cycles when compared to the theoretical natural frequency.
    In conclusion, the results attained for the system using an aluminum shaft agree with
theory better than those obtained with the PVC shaft do. An improvement for this
experiment would be to use the same mass on the end of each shaft in order to eliminate
any discrepancies over the mass’s inertia.

                                     1. Introduction
    This experiment was performed to gain a more complete knowledge of, and better
understanding of the relationship between material stiffness, inertia and radial deflection.
The study of these variables and how they interact with one another made it possible to
understand how all three effect the natural frequency of oscillating systems.
    The calculations of the systems’ natural frequencies were done so under the
assumption that the systems were undamped. Since all unforced systems will eventually
come to rest, they have some internal damping. However, the internal damping for the
systems used in this experiment are very small and can be ignored. In addition, the
calculations of the polar moments of inertia of the masses on the end of the torsional
shafts were calculated under the assumption that the density of each material was uniform

                                   2. Torsional Springs
    A large symmetric rotating mass has a rotational inertia J, and a twisting rod has a
torsional spring coefficient K. Refer to Figure 1. The basic torsional relationships are as

                                  Figure 1 – System Diagram

                           T  J  J  dt  
                                        

                          T  K  K (   0 )

                               J 1G

                                J G                                                  (1)
                           T  1    J   
                                  L        dt 

   The previous equations describe the equality of the torque generated by the rotating
mass and the opposing torque of the torsional shaft.

       The natural frequency of an oscillating system can be found by solving the differential
 equation of motion (Equation (1)). This is done in the following derivation of a second
 order homogeneous differential equation where the assumed solution is of the form
  h  e At :

      J G                               J 1G
                            2                             2
                d                                d
 T  1    J                              J     0 or
         L       dt                       L       dt 

                                  J 1G                        J 1G
   h  e At        0  JA 2                 Ai
                                    L                         LJ

                 J 1G              JG                    J 1G
  h  cos(          t )  i sin( 1 t ) where                  rad
                 LJ                LJ                    LJ

                                           1   J 1G                                   (2)
                                    f              Hz
                                          2   LJ

 J 1  first moment of area of the shaft
 G  shear modulus of the shaft
 J  polar moment of inertia of the rotating mass
 L  Length of the shaft
 T  torque
   angular acceleration
   angular displacement
       Knowing the material properties of the shaft is very important as its shear modulus
 can vary greatly with different materials. Also, accurate measurement of the of the shaft
 dimensions and the dimensions of the rotating mass are important in order to lessen the
 error in calculating the first moment of area of the shaft and the polar moment of inertia
 of the rotating mass is very important.

                                         3. Apparatus

       The items shown in Table 1 are the various apparati used in the experiment. More
than one item was used where two different serial numbers are listed in the same box.

                                    Table 1 - List of Apparatus

                    Item             Manufacturer           Model   Serial Number

            Digital Trainer         Cadet              --           GVSU 12196
                                                                    GVSU 217126
            Digital Multimeter      Fluke              --           GVSU 23166
                                                                    GVSU 23117
            Computer                Dell               --           GVSU 125
                                                                    GVSU 124
            Stop Watch              --                 --           GVSU 6-94

            Hot Glue Gun            --                 --           GVSU 8841

            Protractor              --                 --           --

            Calipers                Mitutoyo           --           GVSU 2428301

            Scale                   Ohaus              --           --

            1K Potentiometer        Bourns             53C1         --
            10K Potentiometer       Bourns             3540S        --
            Table Clamp             --                 --           --

            Steel Block (Used as    --                 --           --
            the rotating mass)
            Aluminum Rod            --                 --           --
            (Used as the
            torsional spring)
            19.09” x .249” dia.
            Wood Block (Used        --                 --           --
            as the rotating mass)
            PVC Tubing (Used        --                 --           --
            as the torsional
            shaft) O.D.=.85”

    Figure 1 illustrates the setup of the system. The potentiometer was glued to the
bottom of the rotating mass with the hot glue gun. The input and ground of the
potentiometer were connected to the +5V and ground of the digital trainer. The output of
the potentiometer was connected to a data acquisition card on the back of the Dell

                                     4. Procedure
   Two different pieces of apparatus were set up. One apparatus consisted of a PVC
pipe as the torsional spring and a wooden board as the rotating mass. The PVC pipe was
centered on the board and C-clamps were attached on each end of the board for added
mass. The second apparatus consisted of an aluminum rod as a torsional spring and a
block of carbon steel as the rotating mass.
   As mentioned previously, the apparatus design was set up as shown Figure 1 –
System Diagram from above. A potentiometer was mounted to the bottom of the rotating
mass at its center of rotation using a hot glue gun. A constant voltage was then applied to
the potentiometer. As the mass rotated the wiper of the potentiometer changed the
resistance, which changed the voltage. A LabVIEW program for the data acquisition was
designed to record the change in voltage over time. From this data we can estimate the
natural frequency. The diagram for the LabVIEW program is shown below in Figure 2.

                                 Figure 2 – LabVIEW Diagram

   All measurements were taken with great care and with the highest precision the
available equipment would allow. It is advised to use the highest accuracy available, as
this will lower the experimental error.

                                          5. Results
    The raw data was taken from LabVIEW and analyzed to determine the frequency at
which the system oscillated. Raw data taken from LabVIEW for the system with the
aluminum shaft can be seen in Appendix B. Sample calculations for the first moment of
area of the shafts, polar moments of inertia for the rotating masses, theoretical frequency,
and the frequency measured by using the stopwatch can be found in Appendix A.
    Table 1 – Frequency shows the natural frequency of each system as measured with
the stopwatch and LabVIEW. The table also shows the theoretical frequency for
comparison. Table 2 – Percent Discrepancy shows the percent discrepancy of the values
measured with the stopwatch and by LabVIEW with the theoretical values.

                                    Frequency (Hz)
                                    Stopwatch LabVIEW Calculated
                            PVC        0.56       0.625  0.287
                         Aluminum      2.91        2.86  2.963
                                     Table 1 - Frequency

                                    % Discrepancy
                                        Stopwatch LabVIEW
                                 PVC       95.12       117.77
                              Aluminum      1.79        3.48
                                    Table 2 - Discrepancy

                          6. Analysis and Interpretation
   The results from Section 5 tend to agree well with the theory from Section 2 for
system with the aluminum shaft. However, the results for the system with the PVC shaft
show that there is an unacceptable level of error from theory as seen in Table 2.
   The sources of error for the system with the PVC shaft are most likely due to the
calculation of the polar moment of inertia of the rotating mass. The rotating mass for this

system was a wood board with C-clamps clamped at the end for added weight. There was
some difficulty in accurately calculating the polar moment of inertia for a board with C-
clamps on each end. The parallel axis theorem was utilized to calculate the polar
moment, but the difficulty lied in calculating the polar moment around the clamps’ axis
of rotation. Once the polar moment around the clamps’ axis of rotation was determined,
the parallel axis theorem was used to determine the polar moment of the clamps about the
torsional spring’s axis of rotation.
   Other sources of error for both systems include the accuracy to which time was
measured for the stopwatch calculations. Since the time was measured pressing the
start/stop button on the watch after 10 cycles were completed, there has to be some degree
of human error. Also, error in the LabVIEW data could be due to inaccuracy of the
potentiometers used. If the wipers in the potentiometers are damaged there may not be
good correlation between the position of the rotating mass and the output voltage of the

                     7. Conclusions and Recommendations
       The objective of the experiment was simply to study torsional vibration. By
performing this experiment, one learns how different materials, masses, and geometry
will effect the natural frequency of this type of system. In conclusion, the analytical
model with the aluminum shaft agreed to theory fairly well while the analytical model
with the PVC shaft did not. Percent discrepancies for both models can be seen in Table
2. Possible causes of error are discussed in Section 6.

                                                                          Appendix A
PVC apparatus:
Calculation of the firs t moment of area of the torsional shaft :

D PVCout         0.85 in             H PVC 24  in

D PVCin      0.59 in

           D PVCout D PVCin
Ix    
            D PVCout D PVCin
Iy    

J PVC I x I y

                        4             4
J PVC 4.486 10                  in

Calulation of the polar moment of inertia of the rotating mass including the masses
(c lamps) at the ends of the board:

M board      1.64 lb                         L board           24  in     W board      3.5 in

                  M board  L board
I xxboard
                                                                    I xxboard  78.72 lb  in

                     M board  W board
I yyboard
                                                                    I yyboard  1.674 lb  in

J board     I xxboard            I yyboard
                                                                    J board  80.394 lb  in

Now, concider the masses (clamps)

M mass       4.20 lb                         D       11  in

                    ( 3.5 in )               ( 2  in )
                                2                        2
J0    M mass 

                   M mass  D
J mass      J0                                       (Parallel axis theorem)

J total    J board       2  J mass
                                                                J total  1.108 10 lb  in
                                                                                  3        2

               5 lbf
G PVC 5.004 10 
                          J PVCG PVC
              1       
            (2  )             
                          H PVCJ total

 f PVC 0.287 s

 Stop watch calculation:

 Cycles         4

 Time       7.14 s


 f  0.56 s

 Steel apparatus:

 Calculation of the first moment of area torsional shaft:

 D rod       0.249 in                 H rod      19.09 in

                    4
 J rod          D rod

                          4        4
 J rod  3.774 10             in

 Calculation of the polar moment of inertia on the mass:

     M st    15.45 lb                 L st    7  in         W st   4  in

                       M st  L st
                                   2                 2
 J stmass                                     W st

 J stmass  83.688 lb  in

                    6 lbf
 G rod       3.8 10

           1               J rod  G rod
f st               
         (2  )        H rod  J stmass

f st  2.963 s

Stop watch calculation:
Cycles        10

Time      3.44 s


f  2.907 s

                                                        Appendix B
Time (s)                   Voltage (V)
    0.05                    0.01
      0.1                   0.12
    0.15                   0.281
      0.2                  0.352 *1st Peak
    0.25                   0.317
      0.3                  0.195
    0.35                   0.088
      0.4                  0.071
    0.45                   0.134
      0.5                  0.254
    0.55                   0.293 *2nd Peak
      0.6                  0.271
    0.65                   0.161
      0.7                  0.088
    0.75                    0.09
      0.8                  0.183
    0.85                   0.298

                                 Voltage vs. Time
                                     (Aluminum Shaft)

  Voltage (V)

                       0         1               2           3       4
                                             Tim e (s)


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