Mechanics Statics

Tuning fork experiments at the undergraduate level usually only demonstrate a tuning forks linear resonance. In this paper, we introduce an experiment that can be used to measure the nonlinear tuning curve of a regular tuning fork. Using double-grating Doppler interferometry, we achieve measurement accuracy within ten microns. With this experiment setup, we observe typical nonlinear behaviors of the tuning fork such as the softening tuning curve and jump phenomena. Our experiment is inexpensive and easy to operate. It provides an integrated experiment for intermediate-level students and a basis for senior research projects.

For narrow band vibration control applications, the alternating stop band characteristics of periodic structures have been widely used. The objective of the present work is to extend this idea for broadband excitations. We seek to synthesize a torsional periodic structure having the largest fraction of the frequencies falling in the attenuation bands of the structure. The unit cell of such a periodic structure is comprised of two distinct regions having different inertial and stiffness properties. Guidelines are derived for suitable selection of inertial and stiffness properties of the two regions in the unit cell, such that the maximal frequency region corresponds to attenuation bands of the periodic structure. It is found that higher the impedance mismatch between the neighbouring regions of the unit cell leads to maximal attenuating frequencies. It is found that more than 98% of the frequencies are blocked in the extreme case. For torsional excitations, it is shown that large, fi...

We present design, fabrication, and characterization of a mechanically coupled MEMS H resonator capable of performing simultaneous mechanical amplification and filtering in air. The device comprises of two doubly clamped polyimide microbeams joined through the middle by a coupling beam of the same size. The resonator is fabricated via a multi-layer surface micromachining process. A special fabrication process and device design is employed to enable operation in air and to achieve mechanical amplification of the output response. Moreover, mixed-frequency excitation is used to demonstrate a tunable wide band filter for low frequency applications. It is demonstrated that through the multi-source harmonic excitation and the operation in air, an improved bandpass filter with flat response and minimal ripples can be achieved.

The static response of an electrostatic micro-catilever beam has been obtained by using Galerkin's method. To make the system bi-stable, a controller has been added and the static response profile is presented using a multi-mode model for the beam. The number of mode shapes leading to convergence has been studied. The softening effect of adding more mode shapes has been investigated along with the effect of changing the system parameters on the static response. Decreasing the controller gain has been found to widen the voltage range of the bi-stability region and increasing the sensor amplification factor is shown to push the upper equilibrium point away from pull-in. Properly choosing these parameters can adjust the range of voltage for bi-stability. By doing a linearization about the stable fixed points, we also found the two natural frequencies for each stable equilibrium point. Finally, we have found the dynamic response of the bistable system using one-and three-mode-model...

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The paper reviews and analyzes features of MEMS accelerometers. As a result, their advantages and disadvantages are revealed. Examples of miniature devices and their demand are given. The key parameters that must be taken into account when choosing an accelerometer are formulated. There are three categories by which accelerometers can be distinguished.

Punktmechanik, Elektrodynamik und Relativitätstheorie sind die Modellstrukturen, die ich nachfolgend kurz diskutiere. Dabei versuche ich die Gesetze so einfach wie möglich und so komplex wie notwendig zu formulieren. Die Mechanik des Massenpunktes, die in vielen Physikbüchern in den ersten Kapiteln abgehandelt wird, liefert bis heute die Blaupause für eine physikalische Theorie. Das elektromagnetische Feld ist ein raumfüllendes, kontinuierlich verteiltes System, das mit raumzeitlich veränderlichen Feldstärken beschrieben wird, wobei die Ladungs- und Stromdichten als Quellendichten anzusehen sind. Weil die Lichtgeschwindigkeit als universelle Konstante in den Feldgleichungen steckt, hat Einstein erkannt, dass diese Grösse gemäss dem Trägheitsprinzip unabhängig von der Geschwindigkeit des Bezugssystems sein muss. Raum und Zeit verschmelzen damit zu einer Einheit, Energie ist mit der trägen Masse gleichzusetzen und die so verallgemeinerte Energie ist zusammen mit dem Impuls zu einem Vierervektor zu erweitern. Die grösste Herausforderung für Einstein war der Entwurf einer Feldtheorie für die Gravitation. Statt einer skalaren Ladung wie in der Elektrodynamik ist der Energie-Impuls-Vektor für die konkrete Form des Gravitationsfeldes verantwortlich. Folglich geht nicht nur eine vierdimensionale, sondern eine viermalvierdimensionale Dichte in die Feldgleichung ein. Diese Grösse, der Energie-Impuls-Tensor, dient uns nachfolgend als Basis für eine Neudarstellung der Mechanik, die statt von einer Ansammlung von Massenpunkten vom Kontinuum ausgeht. Dass mit der Allgemeinen Relativitätstheorie die Gewichtskraft abgeschafft und mit Hilfe des Äquivalenzprinzips zur Trägheits- oder Scheinkraft erklärt wird, sollte man schon in einem einführenden Physikkurs berücksichtigen. Den an unseren Schulen vorherrschende Reformstau löst man nicht mit lustigen Animationen, sondern mit Modellbildung und Simulation. Ein paar Beispiele sollen zeigen, dass eine Modellstruktur über die ikonische Darstellung eines Systemdiagramms (flow chart) genauso gut zu verstehen ist wie als System von Differentialgleichungen. Nur erhöht sich mit dieser Methode die Zahl der möglichen Experten um den Faktor zehn, wenn nicht gar hundert.

In the Newtonian Mechanics, any force exerted by body A on B accelerates B, while the acceleration of B creates an equal and opposite force accelerating A back. We can accelerate one body only at the expense of the opposite acceleration of another body. Therefore, we can only exchange acceleration for acceleration, because force only creates acceleration (F = ma), and acceleration only creates force (a = F/m). With other words, we can equal mathematically, and respectively exchange physical derivatives of the displacement of two bodies only if they are the same (of the same power). But in Classical Mechanics there are formulas that relate force as a function of the product of two velocities (Fc = mv) instead of the function of acceleration (F=ma). If we substitute the two force expressions into Newton's Third Law, it turns out that we mathematically equate acceleration as function of the product of two velocities. That is, we equate derivatives of the displacement of the two bodies of different degrees (acceleration = function (speed times speed). We define this dependence as the step derivative equation of inertial motion. If this dependence is not a product of mathematical formalism, but is real physical, inertial, it means that we can exchange real acceleration of one body for real speed of the other. The disproportion in the equations of the step derivatives of inertial motion affects the Laws of Conservation of Angular Momentum and Momentum. The development has been confirmed experimentally.

The thesis proposes a argument that a mass point moving in a horizontal circle on the Earth's surface can generate a radial centrifugal force of the Earth's center of mass, thus realizing an unconventional anti-gravity path. And through the theoretical analysis of the difference in the distribution of gravitational acceleration at different latitudes caused by the rotation of the earth, and the centrifugal force of a mass point moving in a horizontal circle on the earth's surface is analyzed, the validity of the argument is demonstrated. At the same time, the paper puts forward two possible basic principles, and makes theoretical analysis and demonstration. Conceive of the concrete experimental verification of principle two is proposed. Four technical schemes for antigravity ascent are proposed and basic analysis is made. On the premise of not fully considering the practical technical ability, it is theoretically proved that three of them have realistic operability and development potential. In theory, this paper opens up a different way of anti-gravity for human beings, which will bring new ideas and prospects for human beings to get rid of the bondage of gravity.

This experiment looks at how a mechanical system with an aluminium disk and springs responds to outside forces and damping. It helps us understand resonance, which is important in engineering and physics. By using Rotary Motion Sensors, we track the positions of the disk as it moves. We aim to find the resonant frequency and learn about key factors like spring strength and the disk's ability to rotate. We also study how damping, like friction and magnet effects, affects the system. This experiment helps us see how damping can make resonance wider and affect how well the oscillations happen.

In this paper, we address the general case of a light clock in uniform translational motion parallel to itself and perpendicular to its uniform velocity v, as well as the case of the light clock in relativistic hyperbolic motion. Neither case has been previously addressed in the specialized literature, which typically restricts itself to canonical orientations where the light clock moves parallel to either the vertical or horizontal axis with uniform velocity, without acceleration. Therefore, it becomes interesting to study the more general case where the clock has an arbitrary orientation and/or is accelerated. Our paper is divided into two main sections. The first section deals with the light clock moving with constant velocity, oriented at an arbitrary angle with respect to the x-axis. We prove that the moving clock exhibits a standard time dilation, identical to that of a light clock moving in a canonical orientation. The second section deals with the light clock moving with constant acceleration, i.e., in hyperbolic motion. For the light clock in hyperbolic motion, we derive the period as measured from the perspective of an inertial frame and draw parallels with the case of uniform motion, outlining a term that is similar (but not identical) to the γ factor of uniform motion. We also point out that this factor depends not only on acceleration but also on the height of the light clock. This dependency on the dimension of the light clock distinguishes the accelerated case from the case of uniform motion. The first three sections deal with the theoretical aspects of light (optical) clocks, while the fourth section addresses the experimental implementations of optical clocks.

First of all, the author solemnly declares that he recognizes the energy conservation law (understanding, however, that it will not help). Next we shall show that this law does not contradict the possibility of building a perpetual motion machine that uses the force of gravity. This, of course, reduces the kinetic energy of the Earth, but the author ignores this problem (in the same way as it is ignored by the designers of hydropower plants). Among the projects of perpetual motion the so-called unbalanced wheels are rather common. As described in [3] "the first design of unbalanced wheels was described by Marquis Worchester. From the description, it follows that it was a wheel with two rims-one within the other. To the rims weights are attached by means of strings so that when

The rolling friction phenomenon is encountered in a wide range of applications and when two different materials are involved, quantitative characterization is necessary. The parameter to be determined is the coefficient of rolling friction, for whose estimation a methodology is proposed, based on the damped oscillation of a conical pendulum. The pure rolling contact between a sphere and a plane is obtained when a steel ball is the bob of the pendulum, which rolls on an inclined plate made from a second material from the contacting pair. The mathematical model of the motion of a conical pendulum constructed from a revolution body supported on an inclined plane in the presence of the rolling friction is developed. The dynamic equations of the rigid body with fixed point are applied and the differential equation of motion of the pendulum is obtained together with the expressions of the reaction forces in the contact point. For different pairs of materials, tests are performed on a labo...

The frequency-dependent mechanical loss occurring at the oscillation of a single dislocation under an applied cyclic stress with frequencies in the MHz-GHz range is investigated by a discrete dislocation dynamics simulation and by analytical treatment. With the help of an analytical solution, an influence parameter for the occurrence of inertial effects is derived.

This paper presents the design optimization of a magnetic field energy harvesting platform for self-powered ac power line monitoring devices in smart grid applications. The designed platform can be used with any type of vibrationto-electricity energy conversion schemes including piezoelectric, electromagnetic, and electrostatic energy harvesters. The platform includes a cantilever beam with a miniaturized permanent magnet (PM) at the tip that vibrates due to the interaction with the magnetic field of the ac line. The proposed method analytically determines the optimum geometric orientations of the platform as well as the optimum dimensions of PM to maximize the applied magnetic force to the beam. The validity of the design procedure and its assumptions are verified numerically and experimentally using a piezoelectric energy harvesting platform. The numerical and experimental test results confirm the calculated optimum orientation angles and PM dimensions that maximize the magnetic force. The analysis shows that for a PM with vibration direction in a plane normal to the power line, the maximum force can be achieved at the multiples of +45 degrees with reference to the vibration direction. The optimum orientation angle also depends on the angles between the magnetization vector and the direction of the vibration.

Objectives: This study aimed to evaluate the subjective acceptability and physiological response of an inflatable vest as a wearable device to stimulate calming effects. Methods: A Visual Analogue Scale (VAS) was used as a subjective measurement and an electroencephalogram (EEG) as an objective/physiological measurement. Results: The results showed that the subjective scores of 30 male participants rated the prototype as an 'excellent' device, with an average score of 81. An increase in the theta (4-8 Hz) frequency band was the most noticeable change in EEG, which represented the calming effect behavior experienced in 64% of all participants. Conclusion: The self-regulated inflatable vest could be beneficial for therapeutic purposes by inducing a calming effect.

After reading this chapter the student will be able to 1. Differentiate between various structural forms such as beams, plane truss, space truss, plane frame, space frame, arches, cables, plates and shells. 2. State and use conditions of static equilibrium. 3. Calculate the degree of static and kinematic indeterminacy of a given structure such as beams, truss and frames. 4. Differentiate between stable and unstable structure. 5. Define flexibility and stiffness coefficients. 6. Write force-displacement relations for simple structure.

The paper deals with a thermo-mechanical problem for a slender microbeam subjected to an electric actuation; the purpose of the study is to improve the knowledge of loss mechanisms in micro-electro-mechanical devices and to predict accurately the dynamical behaviour. The thermoelastic damping in microbeam resonators is strictly correlated to the mechanical behaviour, and the thermoelastic response of the structure changes significantly near critical frequencies. To fully understand the thermoelastic coupling effects, we add the description of the thermal phenomena to the mechanical problem obtaining a system of two coupled PDEs. The proposed governing equations, by making use of a unified model, are able to describe the response by using the classical thermoelastic formulation and two distinct generalized theories, namely, the Lord-Shulman and the Green-Lindsay models. The study is carried out by means of a spectral approximation method and numerical simulations. The results show the influence of the relaxation times and the presence of dissipation peaks in the different formulations.

The frequency response curves of a non-uniform beam undergoing nonlinear oscillations are determined analytically by the multiple time scale method, which provides approximate, but accurate results. The axial inertia in neglected, and so the equations of motion are statically condensed on the transversal displacement only. The nonlinearity due to the stretching of the axis of the beam is considered. The effects of variable cross-section, of variable material properties and of the distributed axial loading are taken into account in the formulation. They have been illustrated by means of two examples and are also compared with existing results. The main result of this work is that the effects of any type of non-uniformity can be detected by simple formulas. Keywords Non-uniform beam • Nonlinear curvature • Nonlinear oscillations • Multiple time scale method • Frequency response curves Communicated by Francesco dell'Isola and Giuseppe Piccardo.

In vielen Kampfsportarten verhilft ein geschicktes Umsetzen physikalischer Kräfte zum Sieg. Judoka versuchen den Partner zu Fall zu bringen, indem sie mit seiner Bewegung arbeiten. Liegt er auf dem Boden, dann klemmen sie gerne seinen Arm so ein, dass er durch den für ihn ungünstigen Hebel nicht mehr freikommt. Boxen hat das Ziel, den Gegner k.o. zu schlagen. Manche Kopftreffer beschleunigen das Gehirn so erheblich, dass sie Schäden verursachen. Karatekas können tatsächlich ohne Tricks verblüffend dicke Holzbretter und Betonstücke mit der bloßen Hand durchschlagen.

We investigate the rich potential of the multi-modal motions of electrostatically actuated asymmetric arch microbeams to design higher sensitivity and signal-to-noise ratio (SNR) inertial gas sensors. The sensors are made of fixed–fixed microbeams with an actuation electrode extending over one-half of the beam span in order to maximize the actuation of asymmetry. A nonlinear dynamic reduced-order model of the sensor is first developed and validated. It is then deployed to investigate the design of sensors that exploit the spatially complex and dynamically rich motions that arise due to veering and modal hybridization between the first symmetric and the first anti-symmetric modes of the beam. Specifically, we compare among the performance of four sensors implemented on a common platform using four detection mechanisms: classical frequency shift, conventional bifurcation, modal ratio, and differential capacitance. We find that frequency shift and conventional bifurcation sensors have ...

A piecewise-linear model with a single degree of freedom is derived from first principles for a driven vertical cantilever beam with a localized mass and symmetric stops. The resulting piecewise-linear dynamical system is smoothed by a switching function (nonlinear homotopy). For the chosen smoothing function it is shown that the smoothing can induce bifurcations in certain parameter regimes. These induced bifurcations disappear when the transition of the switching is sufficiently and increasingly localized as the impact becomes harder. The bifurcation structure of the impact oscillator response is investigated via the one-and two-parameter continuation of periodic orbits in the driving frequency and/or forcing amplitude. The results are in good agreement with experimental measurements.

Liegt der Kraftvektor in der x, y-Ebene, d. h., sind z und F z gleich null, so folgt (Bild 2 c) M DM z D.F y x F x y/e z I M DjM jDM z DF y x F x y DF r sin' DF h: B1.2 Zusammensetzen und Zerlegen von Kräften mit gemeinsamem Angriffspunkt B1.2.1 Ebene Kräftegruppe Zusammensetzen von Kräften zu einer Resultierenden. Kräfte werden geometrisch (vektoriell) addiert, und zwar

In the present study an oscillator system formed by a seesaw connected to a simple pendulum coupled to a mobile platform with a certain slope, is analyzed. The observed properties of the system when faced with a possible displacement of the mobile are affected in terms of energy loss, which can be reflected in the angle, the variation of apparent weight and the height. The possible variations which can be introduced modify these parameters, so that the system tries to compensate them in order to preserve its symmetry. The working of this system can be described by means of a transformation matrix where the nature of the movement is represented. Such tool enables us to observe the variations that the system may experience as rotation and translation functions. Therefore, the matrix offers a clear visualization of the values expressed and thus of the need to either provide or extract energy to or from the system.

In this study, the properties of an oscillating system composed of a pendulum connected to a seesaw and placed on a moving platform with a certain slope are analyzed. Using complex numbers to collect the information contained in the system proves to be crucial in order to observe the properties described by both cross and dot products. The representation of physical quantities in complex numbers reveals that for certain angles, precisely those where the oscillation translates into a displacement, the properties of the system are expressed in the real plane.

The nonlinear transversal oscillations of a cantilever beam system at primary, superharmonic, and subharmonic resonance cases are investigated within this work. Time-delayed position-velocity controller is proposed to suppress the considered system nonlinear vibrations. The multiple scales homotopy approach is employed to analyze the controlled nonlinear model. The amplitude-phase modulating equations that govern the system dynamics at the different resonance cases are extracted. The stability charts of the loop-delay are obtained. The influence of the different controller parameters on the system vibration behaviors is explored. The acquired analytical results revealed that the loop-delay has a great influence on the controller efficiency. Accordingly, the optimal values of the loop-delay are reported and utilized to enhance the applied controller performance. Finally, numerical validations of the accomplished analytical results are performed, which illustrated an excellent agreement with the obtained analytical ones.

Atomic fountain clocks launch laser-cooled balls of atoms upwards to some height h in the local gravitational field where they experience both second-order Doppler shifts and gravitational frequency shifts. It is shown in this paper that the net shift, relative to a reference at the launch point, is gh/3c 2. We derive the next most significant correction to this expression, and show that the value of g should be corrected for Coriolis and centripetal effects.

Atomic fountain clocks launch laser-cooled balls of atoms upwards to some height h in the local gravitational field where they experience both second-order Doppler shifts and gravitational frequency shifts. It is shown in this paper that the net shift, relative to a reference at the launch point, is gh/3c 2. We derive the next most significant correction to this expression, and show that the value of g should be corrected for Coriolis and centripetal effects.

Purpose A close form solution was developed in this paper to find the nonlinear tuning parameters of symmetric/asymmetric rotor-Squeeze film damper system. Methods Initially, close form solution was developed to find the optimum tuning criteria using linear models. Later, it has been extended to nonlinear unbalanced rotor damper system using circular centre orbit condition. Analytical modeling of Squeeze film damper forces is carried out considering viscous, inertial and temporal contributions under laminar and turbulent conditions. Modified Reynolds equation with short damper approximation is used to derive the SFD forces for 2π-film. The solution of the system of equations helped to predict optimum tuning parameters, such as cross-over frequency and maximum possible amplitudes. Contributions of various governing parameters are discussed. Conclusion Mass ratio of damper-to-rotor mass and nonlinear radial/tangential damper forces play an important role in finding the tuning parameters of symmetric system. Additional shaft parameter, f p , plays an important role in getting the optimum tuning parameters of asymmetric system as compared to symmetric system. Keywords Flexible rotor • Squeeze film dampers • Nonlinear fluid forces • Tuned mass damper List of symbols M s Mass of rotor and disc system, Kg, Ms = M for asymmetric system, Ms = M/2 for symmetric system M s M/2 for symmetric system M Mass of disc, kg M d Mass of damper, kg K1 Shaft stiffness of top half-length of shaft, N/m K2 Shaft stiffness of bottom half-length of shaft, N/m K3 Stiffness of retainer spring, N/m K s Stiffness of the shaft, N/m K s = K1 + K2 for asymmetric system K s = (K1 + K2

We present a mechanism that converts in-plane to out-of-plane motions, which is fully compatible with massfabrication processes. By using selective stiffening, we induce coupling between in-plane and out-of-plane responses of the mechanism. The conversion ratio is constant (i.e., linear response), and it can be easily tailored by the number of stiffeners used in an otherwise unchanged design planform. The linearity of the motion conversion and the possibility to tailor it, are demonstrated experimentally using dedicated test devices. As a specific example of application, we use the motion conversion mechanism to achieve a parallel out-of-plane motion of a flat stage, which is driven by in-plane comb-drives.

The moment produced by two equal, opposite, and non collinear forces Mo=F(a+d)-Fa=F*a+F*d-F*a=F*d The magnitude of the couple is independent of the distance (a) which locates the forces with respect to the moment center O. It follows that the moment of a couple has the same value for all moment centers.

The rolling friction phenomenon is encountered in a wide range of applications and when two different materials are involved, quantitative characterization is necessary. The parameter to be determined is the coefficient of rolling friction, for whose estimation a methodology is proposed, based on the damped oscillation of a conical pendulum. The pure rolling contact between a sphere and a plane is obtained when a steel ball is the bob of the pendulum, which rolls on an inclined plate made from a second material from the contacting pair. The mathematical model of the motion of a conical pendulum constructed from a revolution body supported on an inclined plane in the presence of the rolling friction is developed. The dynamic equations of the rigid body with fixed point are applied and the differential equation of motion of the pendulum is obtained together with the expressions of the reaction forces in the contact point. For different pairs of materials, tests are performed on a labo...

The double-cone system is known as one of the mechanical miracles of Physics. Our task for today is to systematically derive the equation of motion of the ’Double-Cone’ system using Lagrangian formalism with some approximations and [then] experimentally verify our theoretical predictions based on that [to be derived] model.

We consider the sticky collision between two sphere of equal rest mass, moving with equal but opposite speeds in a horizontal plane. The collision takes place on the pan of a high sensitivity balance. The conditions of mechanical and thermal equilibrium are studied showing that heat developed in the rest frame of the system has a corresponding inertial mass.

Free vibration of an axially loaded silicon microbeam in MEMS under electrostatic actuation is studied on the basis of a sinusoidal shear deformation theory. Based on the modified couple stress theory (MCST) and von Kármán geometric nonlinearity, a nonlinear finite element formulation is derived and employed to construct the discrete nonlinear governing equations for the microbeam. The deflection of the microbeam at a given DC voltage is firstly calculated using the Newton Raphson based iterative procedure, and used to evaluate the tangent stiffness matrix. The natural frequencies of the microbeam corresponding the DC voltage are then computed. The influence of the applied voltage, the axial force as well as the material length scale parameter on the frequencies of the microbeam is studied in detail and highlighted. The dependence of the pull-in voltage upon the material length scale and the axial force is also examined and discussed.

The present paper concerns the study of geometrically non-linear forced vibrations of beams resting on two different types of springs: rotational and translational. Assuming that the motion is harmonic, the displacement is extended as a series of spatial functions determined by solving the linear problem. Hamilton’s principle and spectral analysis are used to reduce the problem to a non-linear algebraic system solved using a previously developed approximate method. The effects of the nature of the added springs and their location on the non-linear behaviour of the beam are examined. A multimode approach is used in the forced case to obtain results over a wide range of vibration amplitudes. This leads to examining the non-linear forced dynamic response for different positions of each spring and different levels of excitations. Following a parametric study, the non-linear forced mode shapes and their associated bending moments are presented for different levels of excitations and for ...

Abstract: Sustained resonance in a linear oscillator is achievable with a drive whose constant frequency matches the resonant frequency of the oscillator. In oscillators with nonlinear restoring forces, ie, Duffing-type oscillators, resonant frequency changes with ...

During the last decades the size of finite element models, used in industrial and research applications, is constantly growing. The increased of computational power and the accuracy of FEM software are leading an interest in nonlinear behaviour of the structures. However, the solution of large set of nonlinear equations is still computationally expensive. The idea of any Model Order Reduction (MOR) is to reduce the number of degrees of freedom of a given structure, so that it is possible to drastically decrease the number of unknowns of the full model into a subset of equations faster to solve. The aim of this work is to formulate a reduced order basis (ROB) without computing the solution of full model. In this way, all the equations will be projected on ROB able to capture the nonlinearity of the whole system. Therefore, the idea is to project a large system of equations onto a smaller subspace to create a lower dimensional space of reduced unknowns. In this thesis, the "nonlinear static condensation" and the "modal derivatives" have been presented to efficiently formulate a subspace of nonlinear differential equations of a geometrically nonlinear model. Moreover, it has been developed a method to analyse the dynamics of the 3D finite element turbofan jet engine compressor and fan blades. The nonlinear forced response (NFR) of the structure can be fast computed projecting of all the equations on a reduced order basis. The computational procedure has been designed to be adapted to a tip rubbing problems, so that it has been possible to consider the contact and geometric nonlinearities (large deformations) during the developing of the in-house code. The flexibility of the code allows to include both the geometric nonlinearities or the contact. The blade-casing interaction in turbofan jet engines can lead the structure to large deformations and this phenomenon cannot be neglected during the structural analysis.

This research work is concerned with the static and dynamic stability study of an exponentially tapered revolving beam having a circular cross-section exposed to an axial live excitation and a variable temperature grade. The stability is analysed for clamped-clamped, clamped-pinned, and pinned-pinned end arrangements. Hamilton’s principle is used to develop the equation of motion and accompanying end conditions. Then, the non-dimensional form of the equation of motion and the end conditions are found. Galerkin’s process is used to find a number of Hill’s equations from the non-dimensional equations. The parametric instability regions are acquired by means of the Saito-Otomi conditions. The consequences of the variation parameter, revolution speed, temperature grade, and hub radius on the instability regions are examined for both static and dynamic load case and represented by a number of plots. The legitimacy of the results is tested by plotting different graphs between displacement...

This paper presents the design optimization of a magnetic field energy harvesting platform for self-powered ac power line monitoring devices in smart grid applications. The designed platform can be used with any type of vibrationto-electricity energy conversion schemes including piezoelectric, electromagnetic, and electrostatic energy harvesters. The platform includes a cantilever beam with a miniaturized permanent magnet (PM) at the tip that vibrates due to the interaction with the magnetic field of the ac line. The proposed method analytically determines the optimum geometric orientations of the platform as well as the optimum dimensions of PM to maximize the applied magnetic force to the beam. The validity of the design procedure and its assumptions are verified numerically and experimentally using a piezoelectric energy harvesting platform. The numerical and experimental test results confirm the calculated optimum orientation angles and PM dimensions that maximize the magnetic force. The analysis shows that for a PM with vibration direction in a plane normal to the power line, the maximum force can be achieved at the multiples of +45 degrees with reference to the vibration direction. The optimum orientation angle also depends on the angles between the magnetization vector and the direction of the vibration.

In this paper, we calculate the gravitational acceleration by using simple pendulums within the designated World Pendulum Alliance: a network constituted by fourteen institutions in eight countries, each provided by a pendulum that can be accessed remotely via the Internet. As a pedagogical option for remote laboratory experiences, we show how to access them and measure the pendulum's period for N oscillations. We discuss the role of the repetitions and samples to minimize the experimental uncertainty, and by using the averaged value of the period, we compute the gravitational acceleration locally. Also, we present the corrections due to the moment of inertia of the system. Finally, by taking into account the geographical location of each pendulum, we report gravity's dependence on latitude, longitude, and altitude by discussing the implications on the Earth's shape. We hope this tool will be helpful for introductory and university physics courses due to its remote access features and its relation to fundamental concepts.