V1-Final-260805638-converted

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M C G ILL U NIVERSITY D EP ARTMENT of M ECHANICAL E NGINEERING V1 F INAL R EPO r T MECH 362 - M ECHANICAL L ABORA TORIES Free and Forced Vibrations B ERNIER , Emeric 260805638 March 22, 2021

1 1 Final Report Discussion Questions 1.1 Experiment 1 1. Show your results using Table 2 See Excel spreadsheet. 2. Use Rayleigh’s theory to calculate the effective mass and then find the theoret- ical oscillation frequency. Compare with the measured value from the experiment. According to the improved Rayleigh’s theory, the effective mass corresponds to the sum of the exciter mass and the corrected mass that allows for the mass of the simply supported beam of 17 m beam . 17 17 m effective = m exciter + 35 m beam = (4 . 2 kg ) + 35 (1 . 65 kg ) = 5 . 00 kg Then, assuming that the central mass (exciter) acts as a point load, the theoretical oscillation frequency is given by: 1 . 6 E I beam 1 . 6(2 . 0 × 10 11 P a )(2 . 083 × 10 − 9 m 4 ) m eff I 3 3 The measured value of oscillation frequency from the experiment is 15.43 Hz for an effective mass of 5.00 kg, which corresponds to a 0.44% relative error. The highest percent relative error is 0.91% for an effective mass of 5.2 kg (Note: the percent relative errors for all the effective masses are tabulated in the Excel sheet). Therefore, the experimental data matches closely the theory. 3. Plot natural frequency ( ω n ) vs added masses and comment on your results. Figure 1: Graph of the system’s natural frequency as a function of the added mass 3 f = 2 = 2 (5 . 00 kg )(0 . 375 m ) 3 = 15 . 497

From Figure 1, one can see that as mass is added to the system, the natural frequency decreases, which is expected, as theory suggests that the two are inversely proportional. Also, the percent relative error between the experimental and theoretical data was already discussed in the previous question. Graphically, we can see the same trend as the experimental data matches closely the theoretical data, making the experiment accurate. 4. As shown in the Dunkerley’s theory, plot a chart of 1/ f 2 (measured natural fre- quency) as a vertical axis against total mass. Extend the line of the chart to cut the vertical axis and find the theoretical frequency for just the beam. Discuss your results. Figure 2: Graph of 1/ f 2 as a function of the added mass From the extended line in Figure 2, the intercept on the vertical axis occurs at 1/ f 2 = 0.0008, which corresponds to a natural frequency of 35.36 Hz. The theoretical frequency for just the beam is twice of the oscillation frequency (natural frequency) of the system when subject to free vibration. This large difference will be discussed in the expected sources of error. 5. Compare the theoretical frequency that you found with the value from the theory. Remember that you are finding the frequency of two cantilevers (375 mm) and not the entire beam. From Dunkerley’s method the theoretical frequency for just the beam itself is given by fol- lowing equation: . EI beam m bea m I 3 3 . (2 × 10 11 P a )(2 . 083 × 10 − 9 m 4 ) Hence the percent relative error between the frequency extracted from Dunkerley’s theory plot and the theory is 8.4%. The error being relatively low, we can have confidence in the accuracy of the experimental data. (1 . 65 kg )(0 . 375 m ) 3 = 38 . 33 f = ==

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1.2 Experiment 2 1. Compare the natural frequency and damping ratio for all four conditions. The natural frequency, damping ratio and magnification factor for all three conditions (un- damped, fully open and fully shut) is tabulated below. Figure 3: Comparison of the natural frequency, the damping ratio and the magnification factor for all three conditions From Figure 3, one can observe that the natural frequency of the system is equal for the fully open and fully shut conditions, while it is at a maximum when the system is undamped. However, the damped natural frequency should decrease when the damping ratio increases. Therefore, an experimental error may have caused the natural frequencies to be equal for the fully open and fully shut cases. Also, the damping ratio is negligible for the undamped case, and it is 4.5 times greater for the fully shut condition when compared to the fully open condition. The reason for the damping ratio not being equal to zero for the undamped case might come from external forces applied onto the system (i.e. air flow, vibration of the structure, etc.). 2. Calculate the magnification factor for a speed ratio of 1.0 for all four sets of results to compare. Show your calculations. The magnification factor defined as a dynamic increase in amplitude compared to the ampli- tude caused by a statically applied force is described by the relationship between the damping ratio and the forced and natural frequency ratio ω/ω n . 1 β = √ (1 − ω 2 /ω 2 ) 2 + (2 ξω/ω ) 2 n n For a speed ratio of 1.0, the equation may be simplified to: 1 1 β = (1 − (1)) 2 + (2 ξ (1)) 2 = 2 ξ For the undamped condition, the magnification factor is the following: 1 1 β = 2 ξ = 2(0 . 003) ≈ 167 The magnification factor was calculated for a speed ratio of 1.0 for all three conditions and the results are tabulated in Figure 3. It can be observed that as the damping ratio increases, the magnification factor decreases, which is normal since the two parameters are inversely proportional. √

3. Produce charts of Amplitude (vertical axis) against speed ratio for the four cases. Discuss your results in details. A graph of the amplitude as a function of the speed ratio for the four cases is illustrated below. Figure 4: Graph of the oscillation amplitude for the three damping conditions for experiment 2 From Figure 4, one can clearly see the effect of damping on the oscillation amplitude. The resonant frequency occurs at a speed ratio of 1, when the oscillation frequency equals the natural frequency, for the three cases. As expected, damping does not have any effect on the resonant frequency. The shape of the curve for various damping is affected in amplitude, but not in width. Indeed, as the damping increases, the amplitude of the oscillatory motion decreases. Since the width of all the curves is similar, the amplitude of the system grows and decays faster around the resonant frequency for larger damping. 4. Produce charts of Phase Lag (vertical axis) against speed ratio for the four cases experimentally and theoretically. Discuss your results in details. A graph of the phase lag as a function of the speed ratio for the four cases is illustrated below.

Figure 5: Graph of the phase lag for the three damping conditions for experiment 2 As observed in Figure 5, the frequency ratio and any damping may affect the phase lag. In an ideal system with no damping ( ξ = 0), there should be no lag until resonance, where the lag goes instantaneously to 180° past resonance. Given that the damping ratio is not exactly zero, the change from 0° to 180° happens more slowly. Then, as damping increases, the change from 0° to 180° becomes slower and slower. Thus, the slope for the fully open case is less steep, and the one for the fully shut even less.

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1.3 Experiment 3 1. Show your results using Tables 8 & 9. See Excel spreadsheet. 2. Find theoretically the natural frequencies for the 2DOF system and compare them with the experimental results. The theoretical natural frequencies for the 2DOF system are given by the quadratic equation. wher e ω 2 = − b ± √ b 2 − 4 ac 2 a 17 33 a = 2 m 1 m 2 = 2( m exciter + 35 m beam + m absorber )( m mass + 140 m absorber beam ) = 2(4 . 2 kg + = 1 . 55 17 (1 . 65 kg ) + 0 . 438 kg ) (0 . 125 kg + 35 3 3 14 0 (0 . 0744 kg )) = 2(5 . 44)(0 . 14) b = − [2 m 1 k absorber + 2 m 2 ( k beam + 2 k absorber )] = − [2(5 . 44)(1 . 09 × 10 3 ) + 2(0 . 14)(4 . 74 × 10 4 + 2(1 . 09 × 10 3 ))] = − 25 , 991 . 94 c = 2 k beam k absorber = 2(4 . 74 × 10 4 )(1 . 09 × 10 3 ) = 1 . 03 × 10 8 Therefore, the natural frequencies are simply: ω 2 = 6 . 483 × 10 3 ⇒ ω = 80 . 52 ⇒ f = ω 1 = 12 . 81 Hz ω 2 = 1 . 028 × 10 4 ⇒ ω = 101 . 39 ⇒ f = ω 2 = 16 . 14 Hz The experimental natural frequencies for the 2DOF system are 12.76 Hz and 16.35 Hz. There- fore, the percent relative error for the first and second natural frequencies are 1.31% and 0.43% respectively. 3. Plot a chart of the oscillation amplitude against frequency to compare with the 2DOF chart in the theory. Discuss your results in details. Given that a vibration absorber is added to the simply supported beam, a two degree of free- dom (2DOF) system is created. This system has two natural frequencies. The amplitude and frequency chart for a two degree of freedom system is illustrated below using the experimental data. n 1 2 2 2 1 1 2 2

Figure 6: Graph of the oscillation amplitude as a function of the frequency for the 2DOF system The chart illustrated in Figure 6 is similar to the 2DOF chart presented in the theory [1]. The two notable peaks correspond to the first mode and the second mode of frequency of the system. The first one occurs at 12.75 Hz and has an amplitude of 4.03 mm, while the second one occurs at 14.75 Hz and has an amplitude of 0.82 Hz. Since the first peak corresponds to the resonance of the absorber, and the second to the beam’s resonance, the peak sizes vary. Moreover, the point of ‘antiresonance’ can be observed by looking at the point where the amplitude of the simply supported beam reduces to zero between the two frequencies. The antiresonance frequency can be graphically extracted to be roughly 14.35 Hz.

2 Expected Sources of Errors The expected source of errors in this laboratory are the following: • Loss of energy through friction at the ends. • Since we assume the beam to be simply supported, it neglects movement in the x-direction which might introduce a moment on the beam. • The presence of particles such as dirt or grease may affect the measurement of the natural frequency of the beam. • The system is small, meaning it is not robust. It makes it sensitive to external perturbations: i.e. vibrations (vibrations from other experiments in the lab, air flow, etc.), temperature shifts, noise from nearby apparatus, etc. [2] • In free vibration, the motion of the beam and the recording is initiated by the student, which renders an inaccurate initial displacement and time. 3 Conclusion In summary, this experiment allowed us to use a simply supported beam, an exciter, a damping unit, and a vibration absorber to study the relationship between oscillation amplitude, phase lag, magnification factor and speed ratio, and investigate the effect of added mass on the natural fre- quency of the simply supported beam. Several learning objectives were achieved in the execution of this laboratory through the investigation of the following phenomena: • Undamped response of a system to one-by-one mass addition • Damping effect on the oscillations of a forced vibration in a simply supported beam assembly • Absorption of vibrations of a simply supported beam using an auxiliary oscillating system (a vibration absorber) • Key frequencies of a two degrees of freedom (2DOF) system, and resonance at upper and lower system natural frequencies. In the first experiment, the natural frequency of the simply supported beam is measured by one- by-one mass addition, and the theoretical frequency for just the beam is determined graphically. In the second experiment, the oscillation amplitude and the phase lag were determined as a func- tion of the speed ratio for three different damping conditions: undamped, fully open, fully shut. The magnification factor for the three cases was also determined. In the third experiment, a vi- bration absorber was added to the beam assembly so that it brought the system to the point of anti-resonance, at which the beam no longer vibrates. The two natural frequencies of the 2DOF system were determined and agreed closely with the theory. In order to further investigate the effects of adding masses on the natural frequency of the sim- ply supported beam, and study further the relationship between oscillation amplitude, phase lag, magnification factor and speed ratio, it would be interesting to perform the experiment on a larger system with larger mass increments.

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References [1]S. Amar, “V1 lab manual: Free and forced vibrations.” McGill myCourses, 2020. [2]The University of North Carolina at Chapel Hill, “Measurements and error analysis.” https: //www.webassign.net/question_assets/unccolphysmechl1/measurements/manual.ht ml , 2011. [Online; accessed 18-March-2021].