# Linear Differential Concentration And Damping Factors

Introduction
The main objective of this lab is to observe the behavior of a metal block while experiencing uniaxial translation and oscillation in respects to time to calculate the acceleration, spring constant, and damping factor. The experiment was split into two different labs. Part one, the drop test, to take the block to a fixed height and drop it. This would then allow us to calculate the friction force, along with the acceleration of the metal block. Part two, the bounce test, is attaching the metal block to the spring, pushing the block down and releasing it. We would use these data to calculate the damping factor and the spring constant. To accomplish the different experiments we needed the use of a Linear Variable Differential Transformer
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Part two of the lab allowed us to find the spring coefficient and the damping coefficient. To find the spring coefficient we use the equation, k= ((2πf)^2 m)/2 Eq.10
Were f is the frequency and m is the mass. This is showing the spring coefficient which represents the relationship between how hard the spring is able to push the spring down [2]. As the spring constant gets larger the more force needed to push the spring down. The last equation we used was to calculate the damping factor, which was, c=2√km ∙ ζ Eq. 11
Where zeta (ζ) is equal to, ζ= ln⁡〖(x(t_1))/(x(t_2))〗/√(4π^2+ ln⁡〖(x(t_1))/(x(t_2))〗 ) Eq.
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The m value or 0.0334 showed us the sensitivity of the LVDT. We then took the voltages that were given at different displacements by the LVDT during the calibration. Each of these were converted into position by reformatting equation 1 to solve for position. The ‘Normalizing’ was done by saying that when the block was pushed up against the fixed plate that it was equal to zero. After calibration was completed and the calculations were completed we moved on to the drop test. Each of the three test were executed by lifting the block to the top of the system and letting go. We were able to graph the oscillation as seen in Figure 4
Figure 4: Drop Test 1 Displacement vs. Time
Figure 5: Drop Test 1 Free Fall
Figure 5, is showing the curve of the free fall state on the block in drop test one. This is showing that as time increases position increase exponentially. We are able to see that this means velocity is increasing which also means that acceleration is increasing as well. We were then able to find the finite differential velocities, which were computed for the by the use of equation 2. We also found the finite differential acceleration using equation 2 found in the theory section. The finite differential acceleration results are shown in Figure 6, Figure 6: Computed Values of Acceleration of