Group 4 , Lab 7

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California State University, Northridge *

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317

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Mechanical Engineering

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Feb 20, 2024

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Fall 2023 California State University, Northridge Experiment 7 Vibration of a Rigid Beam with a Dashpot Damper Mechanics Lab AM 317 Written By:

Abstract: Vibrations are the oscillatory motion or periodic movement of a mechanical system about its equilibrium position. There are various types of vibrations. Free vibrations occur when a system oscillates without any external forces or excitation. Forced vibrations occur when an external force is applied to the system. Resonance occurs when a forced frequency matches the system’s natural frequency. Finally, damped vibrations occur when a damping mechanism, such as a viscous damper, is introduced into a system. The damper acts to reduce the amplitude of vibrations and control the system’s response. Dampers are devices designed to reduce the amplitude of vibrations and dissipate energy in mechanical systems. These devices are critical for controlling vibrations in engineering applications. There are many different types of dampers. Friction dampers use friction between surfaces to dissipate energy. When a system with a friction damper vibrates, the friction forces absorb and dissipate energy and reduce the amplitude of motion. Pneumatic dampers use gases to absorb energy. When a system with a pneumatic damper vibrates, the compression and expansion of gases within the damper reduce the amplitude of motion. Finally, Viscous dampers use a fluid to resist motion. When a system with a viscous damper vibrates, the fluid in the damper experiences shear and reduces the amplitude of motion. For the most part we saw what we expected to see in this lab. For the tables that we made from the data collected, we saw a sharp increase of magnification factor around 6.8 HZ for both the fully opened damper and the fully closed damper. This sharp increase in magnification factor indicates that the vibrations matched the beam’s resonance frequency.

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Introduction: Vibrations represent the periodic or oscillatory movement of a mechanical system around its equilibrium position. Dampers are used to help reduce the vibrations that occur in various systems. There are various types of dampers that an engineer can use in order to reduce the vibrations in a system. Dampers that most people know about are in one’s car. The dampers in one’s car act as shock absorbers to give the driver and passengers a smoother ride. Other places that have a big affect on people is dampers in buildings and bridges. Dampers in buildings and bridges serve to reduce the vibrations caused by wind in order to reduce the risk of structural damage. Of the different applications and different types of dampers, the one that we will be testing is a viscous damper that uses a fluid to dampen vibrations, most commonly found in automotive applications. For this lab, we made use of newer equipment. The testing machine is called the TQ TM1016 Free and Forced Vibrations Apparatus. This machine let us set the acceleration and displacements of the motor, as well as the motor’s speed, and recorded the data onto one of the lab computers. We also made use of the accompanying software, TecQuipment VDAS Hardware and Software in order to record what the testing machine was reading out. Lastly, we used a tape measure to get measurements of the equipment that we were testing. There are a few things that we expect to see in this lab, one of which is a bell-shaped graph. As the frequency gets closer and closer to the material’s natural frequency, the more the beam will vibrate.

Theory: Because there are so many different variables in this lab, there are also many different equations to consider when performing this lab. The first of which is the mass inertia of the system. The four different mass inertia terms that we need are the mass inertia of the beam, exciter, spring unit, and dashpot. For the following equations, m is the mass, and L is the length. MoI, beam: I b = 1 3 m b L b 2 MoI, exciter: I e = m e L e 2 MoI, spring unit: I s = ( 1 3 m s + m fixing ) L s 2 MoI, dashpot: I d = m d L d 2 The moment of inertia of the system is all the moments of inertia added together. MoI, system: I A = I b + I e + I s + I d The next equation that we need is the angular natural frequency. The angular natural frequency is calculated by multiplying the spring constant k, in this case the spring constant is given as 3800, times L s , the length from the pivot to the center of the spring fixing squared, over the I A ., the moment of inertia of the system, all under the square root. ω n = √ k L s 2 I A

Next, we need the excitation speed, ω 0 . The excitation speed is a simple calculation, which is the motor speed in hertz, times 2π. Next, the β of the system needs to be calculated. It is also a simple calculation of the natural frequency, ω n , over the excitation speed ω 0 . The next equation we need uses parts from all the previous equations. The magnification factor uses the moment of inertia of the whole system, I A , over the product of l, the length from the pivot to the displacement sensor, times the L E , the length from the pivot to the center of the exciter, times the m 0 , the eccentric mass added to the rotating unit of the exciter, times r, the distance from the center of the exciter to the eccentric mass, that fraction, times y adj the y value found during the experimental phase minus the y shift . MF = I A l L E m 0 r y adj

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Procedures: For the most part the procedures of the lab this week were done on the computer that was gathering data from the free and forced vibrations machine. First, we had to make sure that the vibrations machine was on. There is a switch on the surge protector as well as a switch on the machine that when switched on, a green light would appear and the screen on the machine would turn on. We then had to change the acceleration dial and displacement dials to zero. The lights above those dials would turn off when they were in the zero position. Once we were sure that the machine was properly set up, we logged onto the computer with one of our CSUN credentials. We opened “TecQuipment VDAS” and changed the settings to what was given detailed in the lab. We tested to see if the vibration machine was properly outputting data to the computer by exciting the cantilever bar with our hand. In our case, the vibration machine did not output the readings to the computer. We decided to turn everything off, restart the computer and restart the lab procedures. This worked and we were able to proceed with the rest of the lab. The next part of the lab was the data collection portion. We set the VDAS software to the appropriate settings, made sure that the oil damper was open, and pressed the green start button on the machine to start the motor. When looking at the computer we saw a sine graph. The scaling on the computer can be changed in order to see the full graph. To start testing we adjusted the exciter motor dial until it was at 5.25 Hz. We waited for the reading to steady out before we took the displacement amplitude. We adjusted the exciter motor’s dial until we recorded all the readings that we needed for the first portion of the lab. The next part was to

repeat the same steps except with the damper disk closed. When we were done, we closed the VDAS software and exited the computer.

Results: Table 1: Specific Measurements and Calculations Component Rigid Beam Dashpot Exciter Eccentric Mass Displacement Sensor Spring and fixing Spring constant k= 3800 N/m) Table 2: Experimental Records and Calculations for Fully-Open Damping Condition Recommended Motor Speed (Hz) Excitation Speed ω 0 (rad/s) β =ω 0 /ω n Measured y (10 -3 m) Adjusted y* (10 -3 m) MF = ( I A / l L e m 0 r )*y adj 5.25 32.987 0.741 0.488 0.039 561.833 5.50 34.558 0.777 0.510 0.061 878.765 5.75 36.128 0.812 0.568 0.119 1714.311 6.00 37.699 0.847 0.625 0.176 2535.452 6.20 38.956 0.875 0.727 0.278 4004.862 6.40 40.212 0.904 0.864 0.415 5978.480 6.50 40.841 0.918 0.979 0.530 7635.168 6.60 41.469 0.932 1.321 0.872 12562.012 6.70 42.097 0.946 2.086 1.637 23582.584 6.80 42.726 0.960 3.470 3.021 43520.457 6.90 43.354 0.974 2.340 1.891 27241.703 7.00 43.982 0.988 1.390 0.941 13556.024 7.10 44.611 1.003 1.127 0.678 9767.252 7.30 45.867 1.031 0.864 0.415 5978.480 7.50 47.124 1.059 0.727 0.278 4004.862 7.75 48.695 1.094 0.659 0.210 3025.255 8.50 53.407 1.200 0.556 0.107 1541.440 9.00 56.549 1.271 0.533 0.084 1210.102

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9.25 58.119 1.306 0.522 0.073 1051.636 Table 3: Damping Ratio and Viscous Damping Coefficient Calculations for Fully-Open, Minimum, Damping Condition. ζ using Eq. (7.10) 0.0155 ζ using Eq. (7.11) 0.0000 Average ζ 0.0078 Viscous Damping Coefficient 17.3991 Table 4: Experimental Records and Calculations for Fully-Shut Damping Condition Recommended Motor Speed (Hz) Excitation Speed ω 0 (rad/s) β =ω 0 /ω n Measured y (10 -3 m) Adjusted y* (10 -3 m) MF = ( I A / l L e m 0 r )*y adj 5.25 32.9867 0.7413 0.4990 0.0500 720.2989 5.50 34.5575 0.7766 0.5220 0.0730 1051.6363 5.75 36.1283 0.8119 0.5790 0.1300 1872.7770 6.00 37.6991 0.8472 0.6130 0.1640 2362.5802 6.20 38.9557 0.8755 0.6930 0.2440 3515.0584 6.40 40.2124 0.9037 0.7730 0.3240 4667.5366 6.50 40.8407 0.9178 0.8190 0.3700 5330.2115 6.60 41.4690 0.9320 0.9100 0.4610 6641.1554 6.70 42.0973 0.9461 1.0130 0.5640 8124.9711 6.80 42.7257 0.9602 1.0360 0.5870 8456.3085 6.90 43.3540 0.9743 1.0130 0.5640 8124.9711 7.00 43.9823 0.9884 0.9330 0.4840 6972.4929 7.10 44.6106 1.0026 0.8990 0.4500 6482.6897 7.30 45.8673 1.0308 0.8070 0.3580 5157.3398 7.50 47.1239 1.0590 0.7160 0.2670 3846.3959 7.75 48.6947 1.0943 0.6700 0.2210 3183.7209 8.50 53.4071 1.2002 0.5680 0.1190 1714.3113 9.00 56.5487 1.2708 0.5560 0.1070 1541.4395 9.25 58.1195 1.3061 0.5450 0.0960 1382.9738 Table 5: Damping Ratio and Viscous Damping Coefficient Calculations for Fully-Open, Minimum, Damping Condition. ζ using Eq. (7.10) 0.05102040 8 ζ using Eq. (7.11) 5.91275E-05 Average ζ 0.02553976 8

Viscous Damping Coefficient 57.1333135 7 Graph 1: Magnification factor versus frequency ratio, fully opened damper 0.600 0.700 0.800 0.900 1.000 1.100 1.200 1.300 1.400 0.000 5000.000 10000.000 15000.000 20000.000 25000.000 30000.000 35000.000 40000.000 45000.000 50000.000 MF versus β Fully Open Frequency Ratio, β Magnification Factor, MF Graph 2: Magnification factor versus frequency ratio, fully closed damper 0.6000 0.7000 0.8000 0.9000 1.0000 1.1000 1.2000 1.3000 1.4000 0.0000 1000.0000 2000.0000 3000.0000 4000.0000 5000.0000 6000.0000 7000.0000 8000.0000 9000.0000 MF versus β Fully Closed Frequency Ratio, β Magnification Factor, MF

Conclusions and Discussion: For the most part, we saw in our graphs and tables the results that we expected to see from the lab. In Table 1, we took measurements of length of different points of the system. The table also came with values for mass given. Using both the length measurements and the given mass values, we were able to calculate the moments of inertia for the beam, exciter, spring unit, and dashpot. Using those values, we were able to calculate the moment of inertia for the whole system. In Table 2 and 4, we recorded our measured values of y. There we also calculated the excitation speed, calculated the β, found the y adjusted value, and calculated the magnitude factor. For the magnitude factor column, we also took note of the highest value that we calculated. For table 2 we calculated the highest magnitude factor to be 43,520.457, which happened when we tested the system at 6.8 Hz. For table 4 we calculated the highest magnitude factor to be 8,456.3085, which also happened when we tested the system at 6.8 Hz. For Table 3 and 5, we calculated the ζ of each of the systems using two different methods as well as calculating the viscous damping coefficient. We found that the viscous damping coefficient was much higher for the fully closed damper, which makes sense because more fluid was affecting the damper. For our graphs, we found that the values we calculated produced a bell curve graph, where our highest value was the highest point on the graph. This is similar to the sample graph that was provided in the lab manual.

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