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Mechanical Vibration

Add to Wishlist. By: D. Product Description Product Details Focusing on applications rather than rigorous proofs, this volume is suitable for upper-level undergraduates and graduate students concerned with vibration problems. In addition, it serves as a practical handbook for performing vibration calculations. An introductory chapter on fundamental concepts is succeeded by explorations of frequency response of linear systems and general response properties, matrix analysis, natural frequencies and mode shapes, singular and defective matrices, and numerical methods for modal analysis.

Additional topics include response functions and their applications, discrete response calculations, systems with symmetric matrices, continuous systems, and parametric and nonlinear effects. The text is supplemented by extensive appendices and answers to selected problems. This volume functions as a companion to the author's introductory volume on random vibrations see below.

Alternatively, the system of equations 3 can be obtained using the Euler-Lagrange formulation [ 2 ]. For such, the kinetic and potential energy of the system has to be obtained. Equations 7 and 8 present the kinetic and potential energy. As the system has dampers, an equation for the dissipation energy has to be given, which is presented in equation 9 , considering viscous damping. Given the kinetic, potential and dissipation energy equations, the Lagrangian of the system can be obtained using equation Furthermore, the EOM for each mass can be obtained applying the Euler-Lagrange equation shown in equation 11 and The system of equations 11 gives the EOM for each mass shown in equation 3 when the equations 7 , 8 and 9 , for the kinetic, potential and dissipation energy, respectively, are used and the differentiations are done.

As we are interested only in the obtention of the structure's natural frequencies of vibration and to simplify the calculations, the damping of the system will not be considered in the model. Therefore, the system's EOM reduces to,. Substituting equation 13 in 12 , the following eigenvalue and eigenvector problem is obtained,.

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Since we are interested in the nontrivial solutions, the following characteristic equation is obtained. The stiffness [ K ] , obtained using equation 1 , and the mass [ M ] matrices for the studied structure are given by the following. Applying these matrices in equation 16 , which is a characteristic equation for an eigenvalue and eigenvector problem; the first, second and third natural frequencies are obtained analytically, which are 4.

The experimental setup consists of an Arduino board model Mega Rev. The Arduino is an open-source hardware- and software-based electronic prototyping platform of easy use that can be used as a signal acquisition, to control motors and a great number of other things, by means of written programs in an easy-to-use computer language The Arduino board has analog and digital pins as well as some digital communication interfaces, being the SPI e I2C the most used ones [ 4 ].

The linkage of the MPU chip with the Arduino is done by means of the serial pin via I2C protocol [ 3 ], [ 4 ], [ 24 ] and [ 25 ]. The linkage of the MPU with the Arduino can be seen in the appendix. The output signal from the accelerometer is analogical in the form of voltage proportional to the acceleration that the sensor was subjected.

In this case, as the signal is digitalized by the Arduino, its resolution is 10 bits [ 26 — 27 ]. The three story shear-building studied is made by four polypropylene plates with dimensions of x The complete structure possesses dimensions of x mm. Figure 2. The structure is excited by an unbalanced motor mounted in the center of the top story of the building. The motor was unbalanced placing a mass of 6.

The voltage vs frequency graph was obtained experimentally and it's shown in Figure 4. For the data acquisition of the shear-building, it was mounted in an inertial bench, which is an experimental bench mounted on top of springs with the purpose of isolate the system from others form of excitation. The problem of not using an inertial bench to perform the measurements is that frequencies from unknown sources can appear.

The signal acquisition was performed with a sample rate of 1 kHz and points, which were configured in the sketch and uploaded to the Arduino. The measurements were done for the free-vibration of the structure and when it was excited by the unbalanced motor in the steady state with the frequencies of excitation near the natural frequencies of the structure.

Also, measurements were performed considering the transient state of the motor, were the frequency of excitation was varied from 0 to 24 Hz. In this last measurement more points were used, , but the sample rate was maintained. In addition, the accelerometers were mounted in the structure's walls over an aluminum support which was fixed on the structure using bee wax, as shown in Figure 2. The acceleration of the structure was measured in five different scenarios: a free-vibration; a 4.

The excitation frequency was controlled using a variable-voltage power source and defined using the voltage vs frequency graph shown in Figure 4. It was performed ten measurements for each case described. The implementation of the PSD was done using the Python 3. Also, others computers languages and computers programs can be used such as Matlab [ 33 ], Octave [ 34 ], Labview [ 35 ] and Fortran [ 36 ].

Figure 5.

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Figure 6. In Figure 6. This result is expected since the system is in its steady state. Furthermore, Figures 7. It's worth noting that the higher peak appeared in the frequency domain graphs, just like Figure 6. Figures 9 — 12 show the signals obtained using the MPU which are presented in the same order and has similar features as the signals obtained with the ADXL Figure 13 show the signals obtained by measuring the acceleration of the structure when it was subjected to a variable frequency of excitation, where Figure The frequency was varied from 0 to 24 Hz.

From the figures one can identify the resonance peaks, which are the instants were the frequency of excitation got near the natural frequencies of the structure. In addition, Table 1 presents the comparison between the values of the natural frequency given by the two accelerometers used and the analytical value. In this paper presents an expansion of previous works initiated in [ 3 ], which aims at measuring mechanical vibrations using the Arduino microcontroller and low-cost sensors for educational purposes.

This work treated in the evaluation of the Arduino and low-cost MEMS-based accelerometers for analysis and characterization of mechanical systems with multi degrees of freedom. The sensors used showed suitable for the vibration analysis in mechanical systems with multi-DOF in the time and frequency domain.

Also, the results obtained, which are shown in Table 1 , showed that the measurements are in agreement with the analytical values and almost no variation is seen comparing the values given by the two accelerometers, as one can also note from the table. The methodology proposed is adequate to undergraduate and graduate courses of physics and engineering, especially in disciplines such as dynamic of rigid bodies, mechanical vibrations, instrumentation, signal processing, dynamic of structures and so on.


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It's worth repeating the low-cost and simplicity characteristic of the experimental setup presented in this paper, which makes the mechanical vibration analysis more accessible for students. It is important to note that both sensors are suitable for the proposed application. In the literature one may found that the level of precision of MEMS-based sensors has raised significantly [ 37 — 45 ], which allows its use in teaching, research and industrial applications, which require a higher precision in their results.

The same can be spoken about the Arduino microcontrollers present in the market, which are being used in a wide range of applications from teaching and research to industrial. Rao and F. Balachandran and E. Magrab, Vibrations Cengage Learning, Toronto, , 2 nd ed. Varanis, A. Silva, P. Brunetto e R. Varanis A. Silva, A. Mereles, C. Oliveira e J. Boonsawat, J. Ekchamanonta, K. Bumrungkhet and S. Gillet F. Geoffroy, K.

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Zeramdini, A. Nguyen, Y. Rekik and Y. Piguet International Journal of Engineering Education 19 , Stelzer Slovak University of Technology, Bratislava, , p.


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