Cocconcelli Marco

Fonte Cosimo

Grosso Pasquale

Mottola Giovanni

Rubini Riccardo

Strozzi Matteo

English

The present works were conceived to be exhibited during the 2022 European Researchers’ Night (ERN 2022), at the University of Modena and Reggio Emilia. The idea is to illustrate the key concepts of mechanical vibration through the use of 3D models and virtual simulation analysis. The paper is directed to high school students planning to enroll in a mechanical engineering bachelor’s degree, in order to approach or consolidate some fundamental concepts of mechanical vibration. Topics not easy to explain, such as the natural frequencies of a body, could be presented more effectively using physical models. Mathematical formalism will be kept to a minimum, as it is beyond the scope of this paper

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A European Researchers’ Night project onmechanical vibrations for high school studentsMarco Cocconcelli, Cosimo Fonte, Pasquale Grosso, Giovanni Mottola, MatteoStrozzi, and Riccardo RubiniDepartment of Sciences and Methods for Engineering,University of Modena and Reggio Emilia, 42122 - Reggio Emilia, Italy,marco.cocconcelli@unimore.itAbstract The present works were conceived to be exhibited during the2022 European Researchers’ Night (ERN 2022), at the University ofModena and Reggio Emilia. The idea is to illustrate the key concepts ofmechanical vibration through the use of 3D models and virtual simulationanalysis. The paper is directed to high school students planning to enrollin a mechanical engineering bachelor’s degree, in order to approach orconsolidate some fundamental concepts of mechanical vibration. Topicsnot easy to explain, such as the natural frequencies of a body, could bepresented more effectively using physical models. Mathematical formalismwill be kept to a minimum, as it is beyond the scope of this paper.Keywords: Mechanical vibration, Natural frequencies, Modal analysis1 IntroductionAny motion that repeats itself after an interval of time is called vibration oroscillation [1]. Two main types of vibrations are generally defined: forced and free.A forced vibration happens when the system is subjected to an external (usuallyperiodic) force. On the other hand, if a system, after an initial disturbance,vibrates without external variable forces, one has a free vibration: this introducesthe concept of natural frequency. The study of natural frequencies is extremelyimportant because when bodies are excited at their natural frequency, a resonanceoccurs, which leads to high deformations of the structure [2]. Each naturalfrequency of a body is associated with a shape, a way in which the body vibrates,called natural mode. In the next sessions the natural modes and frequencies forthree different structures will be identified. The first (Sec. 2) is a simple caseof cantilever beams with attached point masses, whose vibrations are manuallyexcited. The second structure (Sec. 3) is used to represent the modes of a three-storey building, excited with a harmonic force generated by a mechanical shaker,to show how these vary using a mass damper. In the last example (Sec. 4), a plateclamped at its center is adopted as object on which perform an experimentalmodal analysis, identifying some natural frequencies and showing mode shapesthrough software interface. All the examples were first explained in the simplestway, then the experiments were performed in order to visualize and convey theconcepts in a more engaging and stimulating way.2 Marco Cocconcelli et al.2 Vertical beams with point massesThe first model developed is a small cart with three vertical beams, each carryinga concentrated mass along its length (Fig. 1). This concept was inspired bysimilar models [3], in which a mechanical system is designed to display severalresonances at low frequencies. This way, the vibrations can be manually excited,and it is easy to distinguish the behavior at different frequencies. Our design hasthe following features, which we used as guidelines in this project.1. Configurability : the design can be easily modified during a presentation tochange the dynamic behavior of the structure. For simplicity, we considera simplified model of a slender cantilever beam (with distributed mass Md)vibrating with small displacements and carrying a point mass Mc at a givenposition a along its length L. The first natural frequency can be foundanalytically, but it requires solving a complex trascendental equation [4, 5].This equation depends on L, a and Mc, on the cross-section properties ofWashersCaster wheelsBeamsLateral screws(a)3D-printed holders3D-printed base Lateralmotion(b)Figure 1. (a): the CAD concept we developed. (b): the final built model of the cart.A European Researchers’ Night 3the beam (the area S and second moment of area I) and on the intrinsicproperties of its material (namely, its stiffness modulus E and density ρ).Thus, the design allows to vary all these parameters independently:– the lengths a and L can be changed by unlocking lateral screws that fixMc along the beam and the beam on the cart, respectively;– the mass Mc can be changed by adding or removing washers on the3D-printed holders that are placed on each beam;– the beam properties can be changed by using beams of different materials(in our tests, we used plastic bars composed of POM, PVC and Nylon).The cart has three slots for placing the beams, two with a diameter of 6 mmand one with a diameter of 8 mm, so the effect of changing the cross-sectioncan also be evaluated (by comparing the behavior of the three beams).2. Transportability : the model can be disassembled and reassembled easily, forease of use in classroom activities and at public events such as ERN.3. Effectiveness : the design useas cheap elements found in hardware stores, withsome 3D printed parts realized with a standard FDM printer.The cart is manually excited, by moving it back and forth on a flat surface;the caster wheels on its base allow for quick and frictionless motion. It can thenbe seen how one of the beams (the one whose natural frequency is closer tothe excitation frequency) vibrates with higher amplitudes: the demonstration isparticularly effective if the movable masses are placed such that the beams havenatural frequencies that are well apart from each other. A metronome can beused for keeping the rhythm; the experiment can also be designed as a teachinggame, where the students have to find the right frequencies by trial and error.We also developed an Excel worksheet to compute the first natural frequencyof the beams: besides being useful in the design phase, this worksheet can be aninteractive teaching tool, to compare the expected frequencies with the real ones.3 Vibrating building with tuned mass damperThe second structure is a model of a three-storey building vibrating under a lateralload; the CAD concept is shown in Fig. 2. While other similar demonstrationtools exist [6, 7], this model was designed as a configurable and inexpensivetool, like the one presented in Sec. 2: therefore, the same design guidelines werefollowed in this case, too. Thus, the floors can be moved along the length of thefour columns (composed of 6 mm POM beams); their masses can also be tunedby adding or removing nuts that can be clamped on the available holes.This design uses lateral wires for bracing the structure: this way, the shearmotion (along the x axis of the coordinate frame, which is aligned with the axisof a shaker) corresponds to a relatively low stiffness, while other motions (suchas bending or torsion of the whole structure, axial displacements along z or shearmotion along y) have much higher stiffness. Since the shaker operates at lowfrequencies in this case, only the natural modes corresponding to shear alongthe x axis are of interest. We also use stiffening rods, to reinforce the floors(composed of 3D printed plastic and thus relatively flexible); this way, they can4 Marco Cocconcelli et al.Cable eyelets Bracing cablesStiffening rodsConnection joint𝑥𝑥 𝑦𝑦𝑧𝑧Figure 2. CAD model of the building; dimensions in mm. The shaker is also shown.be considered as rigid masses. In any case, it was also verified that the modelcan be used without these wires and rods, to tune its stiffness as desired.The model is connected to the shaker through a rotary joint on the first floor:this was required to avoid misalignment of the shaker connection. In this way,one can also lower the shaker with respect to the model and verify the effect ofhaving both horizontal (shear) and axial excitation forces. For simplicity, theshaker axis was kept horizontal in the tests, by raising the shaker base.The final built model is in Fig. 3. The frequencies of this building may befound through analytical methods [8, 9], by considering the structural complianceto be concentrated in the columns. This, however, proved to be complex dueto the uncertainties in the structural parameters, such as the properties of thematerial used for the columns. Instead, an experimental modal analysis wasperformed, by exciting the structure with an impulse (which is produced by afunction generator) and measuring the vibrational response. Two piezoelectricaccelerometers (with sensitivities of 10 mV/g) were placed on the model: one closeto the shaker connection point, measuring the input excitation, and the otherone on the top floor, which is expected to have the largest oscillations. Three-axial accelerometers were used, in which the channels of interest are the axes ofmeasurement aligned with the x axis; the accelerometers were connected to a NIcDAQ Ethernet chassis sampling at 51200 Hz. By computing the FRF betweenA European Researchers’ Night 5Tuned mass damperShakerBuildingstoreysEthernet chassisFunction generatorOutput accelerometerFigure 3. Photo of the vibrating building clamped on the test table for modal analysis.the input and the output channels, the natural frequencies of the structure werefound at 1, 6.5 and 24 Hz. By exciting the system at these frequencies withsinusoidal forces (again by controlling the function generator), the vibrationmodes can be clearly shown, since the displacement of each floor visibly changes.This model was also designed to be used with a tuned mass damper, whichcan be rapidly connected and disconnected to show the different behavior ineach case. This is composed of a pendulum suspended on the third floor: while asimple point mass attached to a rope could be applied [10], it would be difficult toconstrain its motion along the x axis. Therefore, in our design, we used a threadedbar suspended at its ends by four cables (which pass through cable eyelets onsupports fixed to the floor): since the cables are of the same length, the baralways remains horizontal and no inertia torque is introduced. Thus, its behavioris dynamically equivalent to that of a simple pendulum: if an excitation is appliedat the frequency of the pendulum (which depends only on the cables’ lengths),the energy introduced is in large part absorbed by the pendulum, which swingswith large amplitudes, while the vibration on the rest of the structure is reduced.This was verified experimentally, by measuring the output vibration both withand without the damper: the RMS amplitude of the acceleration is reduced by> 10 %. This effect can be varied by changing the mass of the pendulum (onwhich nuts can be screwed), without changing its natural frequency.6 Marco Cocconcelli et al.4 Plate with center point clamped4.1 SetupThis section presents the setup used to carry out the experimental modal analysisof a square plate with the central point clamped. As shown in Fig. 4, the plate wasconstrained in its central point by interlocking to a massive vertical support bar.Being interested in out of plane modes, a uniaxial accelerometer with sensitivityof 10 mV/g was fixed on the lower face of the plate with vertical acquisitiondirection, while the impulses were supplied by an instrumented hammer witha sensitivity of 2.25 mV/N with a metal tip. Both devices were connected tothe acquisition system case through BNC connectors cables. Siemens equipment,LMS SCADAS Mobile (hardware) and LMS Test.LAB (software) were used foracquisition and analysis respectively.Figure 4. Setup4.2 MeasurementsThe plate (200x200x3 mm) was discretized giving a regular grid of 121 points tobe excited. The measurements were carried out by exciting the structure with 5impacts per point, averaged in coherence and frequency response function.A European Researchers’ Night 74.3 ResultsStarting from the complete set of acquisitions, in the post-processing phase aglobal frequency response function was calculated in order to analyze all the datasimultaneously, and thanks to the Least Square Complex Exponential method,estimate values of natural frequencies and related damping percentage has beenobtained (see Tab. 1). The range covered is up to 1000 Hz, due to limits relatedto impact energy supplied manually, and a good measurement accuracy is alsorequired. The corresponding modal shapes has been finally obtained, Fig. 5shows 2 of the 8 identified modes: for each of them, the maximum amplitudedeformations in opposite phases are shown, nodal lines are marked in red.Table 1. Natural frequencies detected [0-1000 Hz]mode frequency [Hz] damping [%]1 63.259 1.422 190.424 2.053 230.459 0.654 339.916 0.495 542.651 4.506 601.901 0.757 750.183 2.248 997.938 1.10Figure 5. Modal shapes8 Marco Cocconcelli et al.5 ConclusionsThe different projects define a course in mechanical vibration analysis, startingfrom systems with a single degree of freedom (Sec. 2) and moving to models withmany degrees of freedom that better approximate a continuous body (e.g. thethird project). Positive and full of admiration feedback are been received fromhigh school students, who, driven by a strong interest in mechanics, have expressedtheir desire to enroll in a degree course. In particular, they appreciated how theuse of physical models is fully effective in understanding complex phenomena.References1. Rao SS (2017) Mechanical vibration (sixth edition), Pearson, chap 1: Fundamentals ofvibration2. Green D, Unruh WG (2006) The failure of the Tacoma bridge: A physical model. AmJ Phys 74(8):706–716, DOI 10.1119/1.22018543. Hipschman R (2002) Exploratorium Cookbook, vol II, Exploratorium, chap 63 -Resonator. URL https://www.exploratorium.edu/snacks/resonator4. Laura PAA, Pombo JL, Susemihl EA (1974) A note on the vibrations of a clamped-freebeam with a mass at the free end. J Sound Vib 37(2):161–168, DOI 10.1016/S0022-460X(74)80325-15. de Rosa MA, Franciosi C, Maurizi MJ (1996) On the dynamic behaviour of slenderbeams with elastic ends carrying a concentrated mass. Comput Struct 58(6):1145–1159,DOI 10.1016/0045-7949(95)00199-96. Bigdeli Y, Kim D (2016) Damping effects of the passive control devices on structuralvibration control: TMD, TLC and TLCD for varying total masses. KSCE J Civ Eng20(1):301–308, DOI 10.1007/s12205-015-0365-57. Varanis M, Silva A, Mereles A, Pederiva R (2018) MEMS accelerometers for mechanicalvibrations analysis: a comprehensive review with applications. J Braz Soc Mech SciEng 40:527, DOI 10.1007/s40430-018-1445-58. Zalka KA (2001) A simplified method for calculation of the natural frequencies ofwall–frame buildings. Eng Struct 23(12):1544–1555, DOI 10.1016/S0141-0296(01)00053-09. Piccardo G, Tubino F, Luongo A (2019) Equivalent Timoshenko linear beam modelfor the static and dynamic analysis of tower buildings. Appl Math Model 71:77–95,DOI 10.1016/j.apm.2019.02.00510. Cuvalci O, Ertas A (1996) Pendulum as vibration absorber for flexible structures:experiments and theory. J Vib Acoust 118(4):558–566, DOI 10.1115/1.2888335

A European Researchers’ Night project on mechanical vibrations for high school students

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A European Researchers’ Night project on mechanical vibrations for high school students

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