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M3 Vibrating Beam Eoin Kearney 07383002 Objective The aim of this experiment was to determine the response of a spring-mass-dashpot system to a disturbance and investigate the phenomena associated with the vibration of a beam. Theory Moments about pivot neglecting mass of beam Mx1L1 + Cx3L3 + Kx3L3 =0 Including mass of the beam (I/ (L1) ^2 + M) x1L1 + Cx3L3 +Kx3L3 =0 To find I the inertia of the beam I = (m (L^2))/3 Where for the above M=mass of weight X1=amplitude of oscillation X3= amplitude of oscillation L1= length from the mass L2= length from the mass L3= length from the mass m= mass of beam C=Dashpot resistance I=moment of the beam about its pivot K=spring constant Taking x1/L1 = x2/L2 = x3/L3 Gives the equations X1+X2 [CL3^2/(ML1^2+I)] + X1[kL3^2/(ML1^2+I)] = 0 And X1+ 2PwnX1 + wn^2 X1
Results L1 = 27.5cm L2 = 45cm L3 = 10cm Beam length = 75cm Mass of beam 1.813kg of was calculated from hookes law. A theoretical frequency was calculated from Wn^2 = KL3^2/ (mL1^2 + I) Our theoretical value for wn was 26.05 rad/s Our average values with the dashpot were wn = 23.92Hz Our average values without the dashpot were wn = 25.14Hz We obtained a value of 2.308 for C the dashpot constant no. n L1 X0 Xn C t0 tn wn T 1 9 0.275 0.33 0.23 6.38 244 469 25.13 2.25 2 9 0.275 0.26 0.18 6.5 250 474 25.25 2.24 3 9 0.275 0.6 0.39 7.62 248 476 24.8 2.28 4 9 0.275 0.27 0.18 7.17 250 473 25.36 2.23 1 9 0.275 0.26 0.055 0.0275 242 478 23.96 2.36 2 8 0.275 0.53 0.14 0.0265 274
Such errors could have occurred from Measuring distances with a large measuring ruler Not taking in to account additional masses hanging on the beam at the spring location Computer errors from not placing pointers correctly The beam bending The dashpot lowering the frequency of vibrations from 25.14Hz to 23.92Hz Looking at the effect of the dashpot we observe that the natural frequency did not change a lot but the dampening ratio is clearly higher without the dashpot this is due to the dashpots strong response to the applied force. When the dashpot is connected the damping increases the oscillations reduced and instead of linear regression normally seen a logarithm decrease in amplitude is observed due to the damping. When distance of the mass along the beam was varied we observed the smaller the length the higher the frequency this is caused by greater deflection in the beam with greater length.
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