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Title Introduction to Mechanical System Modeling

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Title Introduction to Mechanical System Modeling

    Introduction to Mechanical System Modeling

    EGR 345: Lab 4a

    Authors: Heather Boeve and Becky Engel

    Date: October 7, 1999

1. Purpose

    This experiment analyzed a simple translational system consisting of a mass, spring, and damper.

2. Theory

    Newton’s equation of linear motion defines the motion/acceleration of a rigid body, as shown below in equation form:

    (FMa

A system consisting of a mass, M, and a spring with spring constant K will oscillate s

    when a force is applied to it. The motion of the mass-spring system will follow Hooke’s

    law, shown below:

    sFKx

    sFKx

    F sKx

    21FFsKx2x1

    The natural frequency of the mass-spring system is found by completing one time period of 2:

    2KsT/MKs/Mf

     1fKs/M2

    If the spring is preloaded, the force applied must overcome the preload force in order for the system to oscillate due to the fact that the spring is already under tension or compression.

    If the spring is replaced by a damper, the damper dissipates energy and causes the system to return to rest. The resistance force of the damper is proportional to the velocity of the system, as shown below by using the definition of velocity. Note that as the velocity increases, the damping force also increases. Likewise, when the velocity is zero, the force is zero.

    d?vx?dt~?

    ?d??FKdvKdx?dt~?

    x(t;T)x(t)?FKd?T~?

    FTdKx(t;T)x(t)

The equation governing the response of the spring-mass system under an applied force is

    shown below:

    Fig. 1: Mass-spring system

     Mg Fs y

    2d?yFFsFgMy?dt~?

    2d?s Ky;MgMy?dt~?

    2d?My;KsyMg?dt~?

The equation governing the response of the mass-damper system under an applied force

    is shown below:

    Fig. 2: Mass-damper system

     Mg Fd y

    2d?FFd;MgMyy?dt~?

    2?dd???Ky;MgMy d??dtdt~?~?

    2?dd???My;KdyMg??dtdt~?~?

The equation governing the response of the spring-mass system under an applied force is

    shown below:

     y

    Fig. 3: Mass-spring-

     damper system

     Fs Fd

     Mg

    2d?FFdFs;MgMyy?dt~?

    2?dd???KyKy;MgMyds??dtdt~?~?

    2?dd???My;Kdy;KsyMg??dtdt~?~?

Each of the three system examples results in non-homogeneous differential equation that

    can be solved using numerical techniques and/or explicit solutions.

3. Equipment

    3.1 The following list of equipment was used to conduct this experiment:

    1 silver damper (spring included)

    1 surface clamp

    1 metal rod

    Individual weights

    1 ruler or tape measure

    Computer with Mathcad and Working Model Software

    4. Procedure

    4.1 The surface clamp and metal rod were used to secure the spring-mass system as shown in Fig. 1. Three different masses were place on the top of the spring and the displacement was measured for each mass. The spring constant, K, s

    was calculated (see Mathcad Attachment #1) and compared with the theoretical values obtained using Working Model example (see Attachment #2). The natural frequency of the system was also calculated in Mathcad based on the masses and experimental spring constant.

    4.2 The spring was then replaced with the damper and three different masses were placed on top of the damper as shown in Fig. 2. The velocity was calculated as a function of time by measuring the time required to displace the damper a fixed distance. The damping coefficient, K, was calculated d

    (see Mathcad Attachment #1) and compared with the theoretical values obtained using Working Model (see Attachment #3).

    4.3 The spring was then placed inside the damper and secured onto the surface clamp as shown in Fig. 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 fixed distances. The response of the system was calculated using Mathcad (see Mathcad Attachment #1) and compared with the Working Model example (see Attachment #4).

    5. Results

    5.1 Spring-Mass System: 2A spring constant of 1064 kg/sec was calculated by averaging the spring

    coefficients, K, in Mathcad (see Attachment #1). The force and s

    displacement data are tabulated below and compared with the displacement values from Working Model for the experimental spring constant value:

    Force (N) Displacement Working % error

    (m) Model (displacement

    Theoretical vs. Working

    Displacement Model)

    (m)

    26.69 0.023 0.025 8%

    29.76 0.029 0.028 4%

    40.23 0.040 0.038 5%

    The natural frequency of the system for the three masses is shown below:

    Mass (kg) Frequency (1/s)

    2.72 3.148

    3.03 2.983

    4.1 2.564

    5.2 Mass-Damper System

    A damping coefficient of 505.139 kg/sec was calculated by averaging the damping coefficients, B, in Mathcad (see Attachment #1). The force and time data are tabulated below and compared with the time-displacement values from Working Model for the experimental damping coefficient:

    Force (N) Experimental Working % error

    Time (s) Model Time (Experimental

    (s) vs. WM time)

    4.63 10.15 10.3 1.5%

    9.53 5.91 5.2 13.7%

    7.57 6.37 6.6 3.5%

    5.3 Mass-Spring-Damper System

    The system response was analyzed in Mathcad using the Runge-Kutta method, rkadapt (see Attachment #1). The rkadapt function returns a matrix in which: (1) the first column contains the points at which the solution is evaluated and (2) the remaining columns contain the corresponding values of the solution and its first n-1 derivatives. Unlike the rkfixed function that integrates in equal size steps to reach a solution, rkadapt examines how fast a solution is changing and adapts its step-size accordingly. Rkadapt uses non-uniform step sizes internally when it solves the differential equation, but will return the solution at equally spaced points. The dots on the graph represent the experimental time-displacement points that were taken. The system response is shown below:

    0.2.2

    0.1

    ?!1S

    yt()

    0

    0.10.10123450?!05St

6. Discussion

    The results of the spring-mass 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 have 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 damper-mass 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-damper-spring system that was created in Mathcad (see Attachment #1) 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 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 response of this system in the Working Model example (see Attachment #4) has the same tendency as the Mathcad-generated response, leveling off near zero at about 3 seconds; however, the damping is much greater in the range of t = 0 to 1 seconds in the Working Model example. The largest source of error in this part of the experiment is again most likely due to human error in the time measurements.

7. Conclusion

    The experimental results obtained in this experiment for spring-mass, mass-damper, and spring-damper-mass systems support the theory behind analyzing these types of simple translational systems. 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. More accurate results could be obtained for this experiment by utilizing more precise measuring devices such as computer-controlled sensors and timers.

8. Attachments

     8.1 Mathcad Document

     8.2 Spring-Mass System in Working Model (see Web Page)

     8.3 Damper-Mass System in Working Model (see Web Page)

     8.4 Spring-Damper-Mass System in Working Model (see Web Page)

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