Basic System ComponentsEGR 345 Lab 4a – Mechanical ComponentsLee C. GroenewegOctober 14, 1998Purpose:The purpose of this lab is to reinforce the student’s learning in the classroom of dynamic systems that are in the world.Theory:For all rigid bodies, forces can be summed. In a static system (no motion), these forces, and moments are equal to zero. If there is motion (dynamic system) d’Alembert’s equations must be used. These equations are:Using these equations and appropriate free body diagrams (FBD) for the system, aset of second order differential equations can be written and solved. The values solved for in a translational system are linear position and linear velocity.Equipment:1 – spring with unknown spring constant (to be determined)1 – spring-damper cylinder with unknown spring and damping coefficients (to be determined)1 – engine hoist (to suspend the spring and damper from to take measurements)Various massesMaFJMProcedures:In the first procedure, the spring constant was to be determined. Thus was done bysuspending three different masses and measuring the respective elongations of thespring. To verify that a correct spring constant was obtained, an additional mass was used to verify the results. The results for the spring constant are found in Table 1.Mass Displacement Spring Constant2.0 kg 0.037 m 258.07 N/m3.0 kg 0.075 m 258.07 N/m3.5 kg 0.094 m 222.88 N/m4.0 kg 0.116 m 248.27 N/mTable 1. Experimental results of spring constant calculations.Figure 1 shows the system used to determine the spring constant and frequency ofthe oscillation in Procedure 1 and 2.Figure 1. System used to determine spring constant and frequency for the spring.In the second procedure, the frequency of oscillation for the spring was to be determined. The effect of release height was also to be analyzed. This frequency was determined by hanging a mass from the spring and measuring the time for a fixed number of oscillations. In our analysis, we measured the time for a mass to cycle 25 oscillations. The experimental and theoretical results for this procedure are found in Table 2.Mass Theoretical Result Experimental Results3 kg1.444 Hz1.262 Hz1.286 Hz1.303 Hz4 kg1.25 Hz1.166 Hz1.142 Hz1.149 HzTable 2. Theoretical and experimental frequency values for the spring.In the third procedure, we were to determine the spring constant of the spring-damper cylinder. To determine the spring constant of the cylinder we suspended various masses from the cylinder and measured the respective elongations, similarto the procedure of Procedure 1. The results for the spring constant of the cylinderare found in Table 3.Mass Spring constant5.5 kg 544.814 N/m6.0 kg 843.372 N/m9.44 kg 788.535 N/mTable 3. Results for the spring constant of the spring-damper cylinder.Figure 2 shows the system used to determine the spring constant and damping coefficients of the cylinder in Procedures 3 and 4.Figure 2. System used to determine spring constant and damping coefficient for the cylinder.In the fourth procedure, we were to determine the damping coefficient for the cylinder. This required finding the velocity that the mass was displaced at. This was determined by measuring the time for equal displacements of the cylinder. The force of the spring must be cancelled out in order to determine this value correctly. If however, the velocity measurements are taken over short distances, the force from the spring will be negligible. The results for the damping coefficient are found in Table 4.Mass Damping coefficient9.44 kg 509.946 N*s/mTable 4. Results for the damping coefficient.In the fifth procedure, we were to plot the position of the cylinder with an attached mass as a function of time. In order to be able to make a plot we had to collect data for time and position. This was done by marking equal distance reference points on the whiteboard, pulling downward on the mass-cylinder system and recording the times as the system passed these points. The values the time and position and time are found in Table 5. The position-time graph is labeled as Graph 1.Time (seconds)0.000 0.189 0.647 1.318Position (meters)0.000 0.03 0.06 0.09Table 5. Values for time and position in Procedure 5.Graph 1. Position-Time plot for the cylinder.Summary: The results from Procedure 1 show the first and second values for the spring constant are equal. There is however some slight variation in the next two results. This variation may be due to non-uniformity of the spring, or that the spring had been slightly deformed from either the previous two measurements, or from the analysis group before us. The results from the Procedure 2 are close to the theoretical results with a small margin of difference. This may be due to human error in our analysis technique. This error could have been introduced when the timing began (starter observed the time to begin late) or when the timing ended (starter stopped the observation time late). The results of the Procedure 3 show that the spring constant for the cylinder was not uniform. This may have been due to the viscous friction inside the cylinder (resulting from the fluid inside the cylinder).In Procedure 4, the time used to determine the damping coefficient was calculatedfrom taking successive trials of dropping the mass and recording the times. These times were then averaged to determine the time used for this part of the experiment. Results for the fourth procedure reflect the use of this averaged time. In Procedure 5 the times used for the time-position plot reflects averaged times. In this procedure, we ran successive trials for each position to determine an average time for the displacement. This was to ensure that the data was reasonableand not reflecting erroneous results. This lab required the student to use the knowledge gained in the classroom on how to solve dynamic systems. This lab also required the use of engineering DistancedTimed0 0.5 1 1.500.050.1teamwork to arrive at a plan for analysis. The purpose of this lab was effectively completed. The data and calculations are attached as Appendix
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