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									       Grand Valley State University
      The Padnos School of Engineering




  MECHANICAL COMPONENTS
EGR 345 Dynamic Systems Modeling and Control




               Heather Boeve

                October 7, 1999

                Lab Partners
                 Becky Engel




                  Fall 1999
                              Table of Contents

Table of Contents _______________________________________________ 2

List of Figures and Tables ________________________________________ 3

Executive Summary _____________________________________________ 4

1. Introduction __________________________________________________ 5

2. Simple Translational Systems ___________________________________ 5
 2.1    A Mass-Spring System ________________________________________________________ 6
 2.2    The Spring Constant __________________________________________________________ 7
 2.3    A Mass-Damper System _______________________________________________________ 7
 2.4    The Damping Coefficient ______________________________________________________ 9
 2.5    A Mass-Spring-Damper System _________________________________________________ 9


3. Apparatus ___________________________________________________ 10

4. Procedure ___________________________________________________ 11

5. Results _____________________________________________________ 12

6. Analysis and Interpretation ____________________________________ 14

7. Conclusions and Recommendations_____________________________ 16
                                     List of Figures and Tables

Table 1 - List of Apparatus ............................................................................................... 10
Table 2 – Experimental Data For the Mass-Spring System .............................................. 12
Table 3 – Experimental Data For the Mass-Damper System ............................................ 13
Table 4 – Comparison of Data For the Mass-Spring System............................................ 14
Table 5 – Comparison of Data For the Mass-Damper System ......................................... 14


Figure 1. A mass-spring system. ........................................................................................ 6
Figure 2. A mass-damper system. ...................................................................................... 8
Figure 3. A mass-spring-damper system. ......................................................................... 10
Figure 4. Displacement vs. Time for Mass-Spring-Damper System. .............................. 13
Figure 5. Comparison of Theoretical and Experimental Motion of the Mass-Spring-
    Damper System. ......................................................................................................... 15




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                               Executive Summary
    The objective of the experiment explained within this report was to analyze simple
translational systems consisting of mechanical components such as masses, springs and
dampers. The procedure consisted of three separate systems. In each of these systems,
experimental values for displacement versus force were collected. The weight of the mass
was used as the force in the systems and the resulting displacement of the mass was
measured as a function of this force. The first system analyzed was a spring and mass
only. The second consisted of a damper and masses. The final system combined the
spring and damper with the masses. Using the experimental values that were obtained,
the spring constant and damping coefficient for the system components were calculated.
When compared with theoretical values for displacement versus force in the systems, all
of the error percentages were below 14%. Though there was some discrepancy between
the theoretical and experimental values obtained, this experiment supported the theory
behind analyzing these types of simple translational systems. The values for displacement
in this experiment were taken with linear measuring devices by hand. More accurate
results could be obtained by utilizing more precise and accurate measuring devices such
as computer-controlled sensors and timers. Also, collecting more data points may have
increased the accuracy of this experiment.




                                             4
                                    1. Introduction
    This experiment was conducted in order to analyze simple translational systems
consisting of masses, springs and dampers by comparing experimental results to values
obtained theoretically, thus supporting the theory behind these systems. Systems of
springs, dampers, and the combination thereof are present in many of the everyday
objects and mechanisms that people use. They provide the useful purpose of opposing or
retarding motion in things like automobile shock systems or door closers. These systems
can be analyzed quite simply with the use of equations such as d’Alembert’s Law,
Hooke’s Law, the definition of the damping coefficient, and common calculus concepts
such as the first and second order derivatives.
    The theory section will explain how all of these equations and concepts fit together to
theoretically analyze systems of springs and dampers. In the procedure, the experiment is
conducted on the translational systems and data is collected to be used in these equations.
The experimental results are compared to the theoretical results in order show that the
translational systems in the real world behave according to the theoretical descriptions of
motion explained in the theory section.



                         2. Simple Translational Systems

    Many mechanical systems can be analyzed by utilizing the basic concept of
Newton’s First Law, as shown below in equation (1). Equation (1) states that the forces
acting on a body is equal to the mass of the body multiplied by its acceleration.


                                     F  ma                                           (1)
where,
F: Force (N)
m: mass (kg)
a: acceleration (m/s2)

Note: Acceleration, a, can also be expressed as the second derivative of a displacement.


                                              5
D’Alembert’s Law is an extension of equation (1) that can is used to sum all of the forces
acting on an object and determine the object’s motion. This equation is given in equation
(2) below.

                                      F  ma                                           ( 2)

   All three of the translational systems analyzed in this experiment utilize these basic
concepts.

2.1 A Mass-Spring System
    A system consisting of a mass, M, and a spring with spring constant Ks will oscillate
when a force is applied to it. The force exerted by the spring on the mass can be
calculated by using Hooke’s law, which is stated in equation (3).
                                     Fs  K s y                                         ( 3)

where,
Fs: Force Exerted by Spring (N)
Ks: Spring Constant (N/m)
y: Displacement of Spring (m)


Figure 1 below illustrates the translational system of a mass and a spring. Utilizing
d’Alembart’s Law and equations (1) and (3), equation (4) can be formulated which
describes the motion of the mass in this system.




                                mg            Fs              y




                                Figure 1. A mass-spring system.




                                              6
                                              d
                                                       2                                 ( 4)
                           Fy   Fs  mg  m dt  y
                                               
Where,
Fs: Force exerted by Spring (N)
m: Mass (kg)
g: Acceleration due to Gravity (9.8 m/s2)
y: Displacement of the Mass (m)


2.2 The Spring Constant
    This experiment will involve calculating a value for the spring constant based on
measurements in the lab. If two different known forces are applied to the spring, two
different displacements will result. By applying the differences in these force values, a
value for the spring constant can be calculated. Equation (3) has been modified below in
equations (5) through (6) in order to get a useful formula for calculating an experimental
value for the spring constant.
                                    F  K s y                                         ( 4)

                                            F                                          ( 5)
                                     Ks 
                                            y

                                          F2  F1                                       ( 6)
                                   Ks 
                                          y 2  y1

Where,
F1: First Force Applied to the Spring (N)
F2: Second Force Applied to the Spring (N)
y1: Displacement of Spring Caused by F1 (m)
y2: Displacement of Spring Caused by F2 (m)


2.3 A Mass-Damper System
     A system consisting of a mass, M, and a damper with damping coefficient Kd will
tend to return to rest after the initial force supplied by the weight of the mass is applied


                                                 7
because the damper dissipates energy. The force exerted by the damper on the mass can
be calculated by using equation (7).
                                          d                                       ( 7)
                                 Fd  K d   y
                                           dt 
where,
Fd: Force Exerted by Damper (N)
Ks: Damping Coefficient (Ns/m)
y: Displacement of Spring (m)


Figure 2 below illustrates the translational system of a mass and a damper. Utilizing
d’Alembart’s Law and equations (1) and (7), equation (8) can be formulated which
describes the motion of the mass in this system.




                                mg                              y
                                                   Fd




                              Figure 2. A mass-damper system.

                                             d
                                                        2                           ( 8)
                          Fy   Fd  mg  m dt  y
                                              
where,
Fd: Force exerted by Damper (N)
m: Mass (kg)
g: Acceleration due to Gravity (9.8 m/s2)
y: Displacement of the Mass (m)




                                            8
2.4 The Damping Coefficient
    This experiment will involve calculating a value for the damping coefficient based
on measurements in the lab. The definition of the first derivative can be used to estimate
the value for the first derivative required in order to solve for the damping coefficient as
shown in equation (7). The weight of the mass will act as the known damping force, Fd,
and the displacement of the mass on the damper will be measure initially and after a
know amount of time has passed. This formula for calculating the damping coefficient is
derived in equations (9) and (10).
                                     y (t  T )  y (t )                             ( 9)
                           Fd  K d                       
                                             T           
                                           Fd T                                       (10)
                              Kd 
                                     y (t  T )  y (t )
Where,
Fd: Force exerted by Damper (N), in this case it is the weight of the mass (N)
y(t): Initial Displacement of the Mass (m)
y(t + Displacement of the Mass after seconds (m)
Time between Displacement values (s)


2.5 A Mass-Spring-Damper System
    A spring and damper are often used combined together in a system, usually by
placing the spring inside of the cylinder of the damper. This type of system can be
analyzed as if the spring and damper mechanisms were assembled in parallel, as shown in
Figure 3. Using d’Alembart’s Law and equations (1), (3) and (7), an equation that
describes the motion of the mass in the system of Figure 3 can be formed as shown in
equation (11).




                                                9
                                                                    y



                           Fs         Fd




                                       mg
                           Figure 3. A mass-spring-damper system.


                                               d
                                                           2                              ( 8)
                       Fy   Fd  Fs  mg  m dt  y
                                                
where,
Fd: Force exerted by Damper (N)
Fs: Force exerted by Spring (N)
m: Mass (kg)
g: Acceleration due to Gravity (9.8 m/s2)
y: Displacement of the Mass (m)


                                     3. Apparatus
    Most of the equipment used in this experiment are common elements used in
everyday life. Table 1 below lists these items and descriptions individually.

                                  Table 1 - List of Apparatus

         Item         Manufacturer           Model    Serial Number          Range   Resolution

 Silver Damper        NA                --            --                --           --

 Spring from Silver   NA                --            --                --           --




                                               10
          Item        Manufacturer         Model   Serial Number        Range   Resolution

 Damper

 Surface Clamp       NA               --           --              --           --

 Metal Rod           NA               --           --              --           --

 Individual Masses   NA               --           --              --           --

 Linear Measuring    NA               --           --              0 – 0.05 m   0.001 m
 Device

 Stop Watch          NA               --           --              --           0.01 s

 Computer with       NA               --           --              --           --
 Mathcad and

 Working Model

 Software




                                     4. Procedure
    The surface clamp and metal rod were used to secure the mass-spring system
as shown in Figure 1. Three different masses were place on the top of the spring and the
displacement was measured for each mass.
    The spring was then replaced with the damper and three different masses were placed
on top of the damper as shown in Figure 2. The velocity was calculated as a function of
time by measuring the time required to displace the damper a fixed distance.
    The spring was then placed inside the damper and secured onto the surface
clamp as shown in Figure 3. The amount of spring compression as a result of adjusting
the damper to its neutral position was measured. The velocity was calculated as a
function of time by measuring the time required to displace the damper for three equal,



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consecutive fixed distances with a single mass. The response of the system was
calculated using Mathcad.
       Throughout the experiment, each lab partner verified the displacement values in
order to reduce the risk of human error. Successively larger forces should have produced
larger distance displacement/smaller time differences and so this property of the systems
was taken into consideration when evaluating whether or not the collected data seemed
reasonable.


                                        5. Results
    The force and displacement data for the mass-spring system are tabulated below in
Table 2. These values were used in equation (6) in order to evaluate the experimental
value of the spring constant. The individual spring constants were averaged and the
spring constant was found to be 1064 kg/s2.

                     Table 2 – Experimental Data For the Mass-Spring System

               Force (N)                Displacement (m)        Spring Constant (kg/ s2)

                   26.69                     0.023                       1160

                   29.76                     0.029                       1026

                   40.23                     0.040                       1006



    For the mass-damper system, the first derivative of the displacement, or velocity, had
to be measured. Since the distance of displacement was fixed, time for the system to
move through this displacement was the only experimental data collected. The force and
time data for this system are tabulated below in Table 3. These values were used in
equation (10) in order to evaluate the experimental value of the damping coefficient. The
individual damping coefficients were averaged and the damping coefficient was found to
be 505.139 kg/s.




                                              12
                   Table 3 – Experimental Data For the Mass-Damper System

               Force (N)                    Time (s)               Damping Coefficient

                                                                          (kg/ s)

                 4.63                        10.15                        469.8

                 9.53                         5.91                        563.3

                 7.57                         6.37                        482.3



    For the mass-spring-damper system, experimental values of position in the y-
direction (m) versus time (s) were collected. These values are plotted below in the graph
of Figure 4. Note that the system was released at a position of –0.06 m.

               Figure 4. Displacement vs. Time for Mass-Spring-Damper System.

                               0.2
                          .2




                               0.1


                 y( t )


                                 0




                   0.06 0.1
                                 0.5    1   1.5        2     2.5           3
                                 0.75              t                    2.69




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                         6. Analysis and Interpretation
    In order to provide for some means of comparison between theoretical and
experimental values, the Working Model 2D program was utilized. Each system was
created in the program and the experimental spring constant and known masses were
applied as inputs into the system. The outputs of displacement for the mass-spring
system, and time for the mass-damper system from Working Model were compared to the
experimental values obtained. Tables 4 and 5 below list these theoretical and
experimental values and the percentage errors between them.

                   Table 4 – Comparison of Data For the Mass-Spring System
                                         Theoretical
            Experimental                                             % Error
                                       Displacement (m)
           Displacement (m)

                 0.023                      0.025                       8%

                 0.029                      0.028                       4%

                 0.040                      0.038                       5%



                  Table 5 – Comparison of Data For the Mass-Damper System

         Experimental Time (s)       Theoretical Time (s)            % Error

                 10.15                      10.3                       1.5%

                 5.91                        5.2                      13.7%

                 6.37                        6.6                       3.5%



    For the mass-spring-damper system, a second order differential equation was
obtained by rearranging equation (8). The solution to this equation is the theoretical
motion of the mass in the combined system. Mathcad was used to solve and plot the
solution as a function of time. The data point collected for this part of the experiment



                                             14
were superimposed on this graph to provide a means of comparing experimental and
theoretical values for the system. This graph is shown below in Figure 5.

                                 0.2
                            .2




                                 0.1
                    1 
                   S

                   y( t )

                                   0




                       0.1 0.1
                                    0   1     2              3   4       5
                                   0               0                   5
                                                  S     t

Figure 5. Comparison of Theoretical and Experimental Motion of the Mass-Spring-Damper System.


    The results of the mass-spring system produced errors ranging from 4 – 8% between
the Mathcad calculations and the Working Model predictions. Possible sources of error
in this part of the experiment include human error when measuring the exact
displacement of the mass, and friction from contact between 1) the walls of the cylinder,
which the spring was encased in and 2) the plunger which rested on top of the spring. We
had limited control over these types of error sources, especially human, and so the
discrepancies between experimental and theoretical values are reasonable.
    The results of the mass-damper system produced errors of 1.5%, 3.5%, and 13.7%
between the Mathcad calculations and the Working Model predictions. Human error
when measuring the time values is the most likely cause for these errors. A quarter of a
second hesitation, the typical response time of human data collection, would produce
large errors.
    The graph of the response of the mass-spring-damper system that was created in
Mathcad is dependent on the spring and damper coefficients calculated in the first two
parts of the experiment. It shows a smooth line that approaches zero from the original
displacement and levels off near zero at about 3 seconds. The dots on this graph are the
actual experimental values obtained for this part of the experiment. Though the dots

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proceed in time with the same tendency towards zero, they are all situated well below the
graph of the smooth line generated by the numerical integration rkadapt function. The
largest source of error in this part of the experiment is again most likely due to human
error in the time measurements.


                  7. Conclusions and Recommendations
   Three types of simple translational systems, mass-spring, mass-damper, and mass-
spring-damper, were evaluated in this experiment. The experimental results obtained for
these systems support the theories that were explore. Though there was some discrepancy
between theoretical and experimental values of displacement and time, the calculated
values for spring and damper coefficients seemed reasonable based on this experience in
the real world with such systems. It is concluded that the analytical models agree with the
observed behavior of the system within 15% with a confidence level of 90% for the
conditions of this experiment. More accurate results for all parts of this experiment could
be obtained by utilizing more precise measuring devices such as computer-controlled
sensors and timers. Taking additional experimental data points may also have led to
more conclusive results.




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