Dynamics SystemModeling- and ControlsEGR 345 Lab #4Mechanical ComponentsAndrew EdlerOctober, 5 1999Purpose:Determine the coefficient of spring and damping coefficient of a simple spring-damper system.Theory:The spring coefficient refers to the amount a spring will deflect given a force applied to itand can be found by hanging various weights from the spring and measured the spring’s deflection. With the relationship between the spring constant and the deflection being linear, the amount of deflection in the spring will be proportional to the amount of weightplaced on it. The damping coefficient refers to the amount of resistive force that will be applied given the velocity at which the spring is being deflected and can be found calculating the velocity at which the system moves with a given force applied to it.Equipment:1 Door Closer (Black)1 Various Weights1 Force Transducer1 Yard Stick1 Stop Watch1 Various Clamps and Fixtures1 Marking PenProcedure:Determine Spring Coefficient:Method 1: To determine the spring coefficient, various weights were applied to the spring. For each weight the displacement was measured. With the force and the displacement known, the spring coefficient can be calculated.1. Use the force transducer to measure the preload on the spring.2. Hang various weights from the spring and measure the deflection.3. Calculate the spring coefficient by dividing the force by displacement.Method 2: To determine the dampening coefficient, a force indicator was attached to the system. The time for the spring and damper to travel over a three inch length was recorded to calculate the velocity (assuming zero acceleration). The average force over the three inch interval was taken by measuring the force at the midpoint of the interval. The dampening coefficient can be calculated from the force and velocity.Determing Damping Coefficient:1. Mark a 3 inch interval on the door closer rod.2. Measure the spring’s force at the midpoint of the interval using the force transducer.3. Measure the time it takes for the spring to retract over the 3 inch interval.4. Calculate the velocity by dividing the interval distance by the travel time.5. Calculate the damping coefficient by dividing the force at the interval midpoint by thevelocity calculated in step 4.Results: The results that were obtained in this experiment are linked here:Mathcad results http://claymore.engineer.gvsu.edu/~malkowsb/EGR/egr345/lab4.mcdWorking Model simulation http://claymore.engineer.gvsu.edu/~malkowsb/EGR/egr345/lab4.wmBy following the procedure above, the spring coefficient was found to be 4.386 lbf/in. Assuming that the system had a constant velocity over the 3 inch interval used to determine the damping coefficient, the coefficient was found to be 42 (lbf*s)/in using the above procedure. Also, a plot of force vs. displacement was created and the spring coefficient and spring preload was determined using MathCAD functions.MathCAD calculations:Calculation of the spring constant:x16.89 inx27.76 inx39.17 inx1x2x1x10.87 inx2x3x2x21.41 inF118 lbfF223 lbfF328 lbfF1F2F1F15 lbfF2F3F2F25 lbfxavex1x22FaveF1F22KsFavexaveKs4.386 lbf in1Calculation of the damping constant:x10 inx23 int16 st26 st36 stavet1t2t33tave6 sFmid21 lbf(Force at the midpoint of tave)KdFmidtavex2x1Kd42lbf sinGraph of force vs. displacement and the least squares approximation of the line slope:i 0 2xi0.8662.283Fi182328F F lbfx x inFilbfxiin0 2 4020Force vs. displacementintercept x F( ) 18.49 lbfslope x F( ) 4.297lbfinWorking Model results:Due to formatting errors, the working model results could not be included with this file. They can be viewed at: http://claymore.engineer.gvsu.edu/ ~malkowsb/EGR/egr345/lab4.wmConclusionDiscussion of Results:By plotting the force vs. displacement, MathCad’s, slope function was used to determine the spring constant. The slope of the plot was found to be 4.297 lbf/in which was very close to the measured value of 4.386 lbf/in. MathCad’s intercept function was used to determine the preload on the spring and was found to be 18.49 lbf. Again, this was fairlyquite close to the measured value of 13 lbf. A possible reason for the discrepancy between the calculated preload value and the measured value could be due tofriction between the damper’s piston and cylinder. There seemed to be a good deal of static friction that had to be overcome before the piston would move. The damping coefficient was found to be 42 (lbf*s)/in.Percent discrepancy between measured and calculated values:Ksmeasured4.386lbfinKscalculated4.297lbfinPreloadmeasured13 lbfPreloadcalculated18.49 lbf%discrepancyKsKsmeasuredKscalculatedKscalculated100%discrepancyKs 2.071%discrepancyPreloadPreloadmeasuredPreloadcalculatedPreloadcalculated100%discrepancyPreload 29.692T en sio n of S pring 61 6.000 4 8.0008 .0001 6.0002 4.000 (se c)
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