Resonance Frequency Of Beam Research Articles

External and internal resonances of thin-walled curved beams under three-directional moving harmonic loads

This paper deals with the internal and external resonances of simply supported curved beams subjected to three-directional harmonic moving loads. Firstly, based on the differential transformation methods and the Glerkin methods, the analytical solutions are newly proposed for vertical, torsional, radial and axial response of thin-walled curved beams under three-directional harmonic moving loads. Then, in the proposed analytical solutions, this study attains the conditions for the external resonance for in-plane and out-of-plane vibrations by letting the denominator of the proposed analytical solutions equal to zero. Besides, the semi-analytical solution for the internal resonance of curved beams is first proposed in this paper by the coincidence of both two mentioned-above resonant frequencies of beams. The internal resonance will occur when the beam is designed with the critical length for the given cross-section. The derived condition for the internal resonance can offer a basis for structural design to avoid the resonant appearance. All the analytical solutions are in good agreement with the results addressed in existing literature and numerical ones obtained from the finite element method (FEM). The results reveal that the corresponding motion of curved beams diverges, reaching a maximum value during the duration of the harmonic moving loads, when in the external resonance. Moreover, under the internal resonance condition, the curved beams are in repetitive transition between the out-of-plane mode and the in-plane mode without any external energy input. Additionally, in this scenario, the excitation frequency can coincide with the two resonance frequencies of beams resulting in simultaneous occurrences of out-of-plane and in-plane resonances on curved beams. The aim of this study highlights the potential resonant appearances on beams and provide reference in the design phase to avoid the internal resonance appearance and ensure the safety of structures.

Simulation design based on piezoelectric photoelectric composite energy harvester

A simulation design based on piezoelectric-photoelectric composite energy harvesters is proposed to solve the limitation that different energy harvesters can only work under certain conditions. The design of composite energy harvesting can capture sunlight and produce electrical energy and can also harvest vibrational energy in the environment to generate electricity from it. Cantilever beams with 6 different structures are simulated. The resonance frequency of the T-shaped cantilever beam is only 40 Hz, which can be used to collect vibration energy in the environment. The effect of different thicknesses of perovskite solar cells on the output performance was simulated. When the perovskite layer was 500 nm thick, the open circuit voltage could reach 1.08 V. The piezoelectric module and the photoelectric module are integrated into the energy collection circuit, which can collect vibration energy and light energy at the same time. The maximum output power may be obtained as 290 μW when the load resistance is 35 KΩ, and the output voltage at this time is 3.18 V. The structure broadens the application environment of the energy harvester and has the function of improving the output.

Cantilevered adaptive fiber-optics collimator based on piezoelectric bimorph actuators.

A novel, to the best of our knowledge, cantilever construction design of an adaptive fiber-optics collimator (AFOC) based on piezoelectric bimorph actuators for tip/tilt control is introduced. With this new cantilever structure, an AFOC with a diameter of only 6 mm was developed, and the output laser beam deviation angle and resonance frequency of the device were measured. The experimental results show that this new AFOC can provide more than 1 mrad deflection angle at a 20 V driving voltage, and the first resonance frequency is about 500 Hz. Further, in order to verify whether the cantilever structure can be used in a high-power fiber collimator, a high-power X-Y positioner with an 8 mm diameter fiber end cap was developed. The experimental results show that the high-power X-Y positioner can output more than 2 kW laser power and provide about 330 µm displacement of the fiber end cap in the X direction and about 770 µm in the Y direction at a 150 V driving voltage.

Study on excitation force characteristics in a coupled shaker-structure system considering structure modes coupling

The interaction between an elastic structure and electrodynamic shakers commonly exists in Ground Flutter Simulation Tests (GFST) with multi-point excitations, causing a considerable discrepancy between the practical excitation forces and desired ones. To investigate the excitation force characteristics on a cantilever beam excited by a voltage-sourced electrodynamic shaker, the coupled shaker-beam system is modeled to derive the excitation force formula using Hamilton’s principle and Galerkin’s approach. Simulation results using the multi-mode beam model coupled with the shaker model are in good agreement with experimental results, verifying that the proposed multi-mode method can accurately predict the excitation force. Furthermore, parametric studies show that the influence of system parameters on the excitation force is related to the shaker's operating mode. Unlike in current mode of shaker, when the beam resonant frequency approaches the suspension frequency of shaker armature, the variation of excitation force amplitude in voltage mode is no longer minimal. Meanwhile, if the exciting point in the GFST is located far away from the modal node, it is essential to compensate the force because the accuracy of tests can be reduced dramatically. The coupled shaker-beam model proposed in this paper can provide the basis for compensation measures.

Nanoscale real-time detection of quantum vortices at millikelvin temperatures

Since we still lack a theory of classical turbulence, attention has focused on the conceptually simpler turbulence in quantum fluids. Reaching a better understanding of the quantum case may provide additional insight into the classical counterpart. That said, we have hitherto lacked detectors capable of the real-time, non-invasive probing of the wide range of length scales involved in quantum turbulence. Here we demonstrate the real-time detection of quantum vortices by a nanoscale resonant beam in superfluid 4He at 10 mK. Essentially, we trap a single vortex along the length of a nanobeam and observe the transitions as a vortex is either trapped or released, detected through the shift in the beam resonant frequency. By exciting a tuning fork, we control the ambient vortex density and follow its influence on the vortex capture and release rates demonstrating that these devices are capable of probing turbulence on the micron scale.

Modeling and Analysis of Multiple Attached Masses Tuning a Piezoelectric Cantilever Beam Resonant Frequency

Mechanical vibrations have been an important sustainable energy source, and piezoelectric cantilevers operating at the resonant frequency are regarded as one of the effective mechanisms for converting vibration energy to electricity. This paper focuses on model and experimental investigations of multiple attached masses on tuning a piezoelectric cantilever resonant frequency. A discrete model is developed to estimate the resonant frequencies’ change of a cantilever caused by multiple masses’ distribution on it. A mechanism consisted of a piezoelectric cantilever with a 0.3 g and a 0.6 g movable mass along it, respectively, is used to verify the accuracy of the proposed model experimentally. And another mechanism including a piezoelectric cantilever with two 0.3 g attached masses on it is also measured in the designed experiment to verify the discrete model. Meanwhile, the results from the second mechanism were compared with the results from the first one in which the single attached mass is 0.6 g. Two mechanisms have wildly different frequency bandwidths and sensitivities although the total weight of attached masses is the same, 0.6 g. The model and experimental results showed that frequency bandwidth and sensitivity of a piezoelectric cantilever beam can be adjusted effectively by changing the weight, location, and quantity of attached masses.

Multiple bandgap formation in a locally resonant linear metamaterial beam: Theory and experiments

This study presents a strategy of vibration suppression of a beam when multiple resonant frequencies of the structure are excited. The methodology operates on the ability of local resonators attached to the beam to create frequency ranges corresponding to locally resonant bandgaps in which the vibration of the beam is attenuated. This study discusses the mechanism of multiple bandgap formation by first deriving the equations of motion of the metastructure based on Hamilton’s principle and then utilizing a modal analysis approach to obtain analytical expressions for the edge frequencies of the created bandgaps. As a consequence of this assumption, the lattice constant of the resulting metastructure is much smaller than the operating flexural wavelength of the host beam. These edge frequencies are derived by assuming an infinite number of resonators tuned to different resonant frequencies of the beam at which a bandgap is desired to be centered at. Parametric studies on the steady state dynamic response of the beam, however, reveal that only a finite number of resonators is required to create these bandgaps and that their bandwidths largely depend on the ratio of the mass of the resonators to that of the beam. The proposed analytical approach is used to demonstrate bandgap formation at the first and second resonant frequencies of a cantilever beam both numerically using a commercial finite element solver as well as experimentally. Numerical modal analysis results of the metastructure compare well with experimentally measured modal analysis results and the steady-state response of the tip displacement of the structure clearly demonstrates the creation of two bandgaps both numerically as well as experimentally.

Analysis of Dynamic Behavior of V-Shaped Atomic Force Microscope Cantilever in Contact and Tapping Modes by Considering Various Immersion Surroundings

In atomic force microscope (AFM) as a powerful device for scanning the samples, a cantilever is used to sense the variations of its dynamic characteristics due to the tip–sample interaction. Theoretical models that can accurately simulate the surface-coupled dynamics of the cantilever are necessary for quantifiable and qualitative explanation and understanding of measured results. A good understanding of the dynamics of AFM cantilevers vibrating in liquid is needed for the interpretation of scanning images, selection of AFM operating conditions and evaluation of sample’s mechanical properties. In this paper, the amplitude of frequency response functions (FRF) of vertical and rotational movements for a V-shaped atomic force microscope beam immersed in various surroundings has been surveyed for the first time. For rising the precision of theoretical model, we have considered all needful details for the beam and sample. The amplitude of FRF of vertical and rotational movements and resonant frequency of beam have been surveyed by supposing cantilever thickness, breadth and length, the angle between cantilever and sample surface, tip height and normal and lateral tip–sample interaction force. In this study, carbon tetrachloride (CCL4), methanol, acetone, water and air have been considered as surroundings. The results show that an increase in the tip–sample interaction force considerably raises the amount of resonant frequency. By increasing the liquid viscosity, the amplitude of FRF for vertical and rotational movements and resonant frequency reduce. Moreover, the amplitude of FRF of vertical and rotational movements is reduced by raising the rectangular and tapered parts lengths, but increased by raising the rectangular part breadth and cantilever thickness. The resonant frequency goes down by increasing the rectangular and tapered parts lengths and rectangular part breadth, but increases by rising the cantilever thickness. In this paper, we have tried to model the realistic dynamic behavior of V-shaped AFM beam in contact and tapping modes with the most accuracy. Results show good agreement. Based on the experimental results, the present theoretical model can forestall better the realistic behavior of tapping mode than the contact mode for V-shaped AFM cantilever. Based on the complexity of the V-shaped AFM cantilever, this study has been done for the first time in both of the contact and tapping modes.

A Low Power Micro-Electromechanical Resonator-Based Digital to Analog Converter

We present a novel design of a low-power micro-electro-mechanical digital to analog converter (DAC) based on a clamped-clamped micro-beam resonator with multiple split electrodes and capacitive airgaps of various widths. The proposed n-bit DAC device operates at a single drive frequency and can access one of $2^{\mathbf {n}}$ distinct stable states based on the applied digital combination. The digital inputs (binary numbers) modulate the beam resonance frequency using the electrostatic softening effect, where the larger binary numbers result in higher beam resonance frequency. Although the tested device consumes 151 pJ per conversion step with a conversion rate of 0.64 kHz, we show that by moderate downscaling of the device dimensions, femto-joules energy consumption and MHz conversion rates are attainable. [2019-0264]

Modeling and control of resonance effects in steel casting mold oscillators

A methodology for capturing resonance in a continuous-casting mold oscillator software testbed incorporating two coupled hydraulically actuated Timoshenko beams is proposed. The mechanism of the mold motion distortion generation in the mold oscillator is clearly delineated. Nontrivial damped natural frequency and resonance frequency calculations are carried out. Then, for a finite-difference analytical beam model approximation, a discovery of monotonic dependence on spatial approximation accuracy of the beam resonance frequency under the fixed mass and of the mass under the fixed resonance frequency is demonstrated numerically. Based on these findings, a novel beam parameters selection procedure for the precise attainment of the desired resonance and damped natural frequencies by analytical and numerical models under the relevant boundary conditions and runtime constraints is developed. Using this procedure, fitting of the beam model parameters to match the actual resonance frequencies exhibited by thin and thick slab casting mold oscillators at, respectively, the Nucor Decatur and the AK Steel Dearborn steel mills, is demonstrated. The resulting resonance frequency value for the latter is then used to guide the internal-model-principle-based controller design for resonance suppression in the software testbed for the AK Steel caster mold oscillator.

Autonomous Sensors Powered by Energy Harvesting from von Karman Vortices in Airflow.

In this paper an energy harvesting system based on a piezoelectric converter to extract energy from airflow and use it to power battery-less sensors is presented. The converter is embedded as a part of a flexure beam that is put into vibrations by von Karman vortices detached from a bluff body placed upstream. The vortex street has been investigated by Computational Fluid Dynamics (CFD) simulations, aiming at assessing the vortex shedding frequency as a function of the flow velocity. From the simulation results the preferred positioning of the beam behind the bluff body has been derived. In the experimental characterization the electrical output from the converter has been measured for different flow velocities and beam orientations. Highest conversion effectiveness is obtained by an optimal orientation of the beam, to exploit the maximum forcing, and for flow velocities where the repetition frequency of the vortices allows to excite the beam resonant frequency at its first flexural mode. The possibility to power battery-less sensors and make them autonomous has been shown by developing an energy management and signal conditioning electronic circuit plus two sensors for measuring temperature and flow velocity and transmitting their values over a RF signal. A harvested power of about 650 μW with retransmission intervals below 2 min have been obtained for the optimal flow velocity of 4 m/s.

Size Effect Impact on the Mechanical Behavior of an Electrically Actuated Polysilicon Nanobeam based NEMS Resonator

In this paper, the dynamic response of resonating nano-beams is investigated using a strain gradient elasticity theory. A nonlinear model is obtained based on the Galerkin decomposition method to find the dynamic response of the investigated beam around its statically deflected position. The mid-plane stretching, axial residual stress and nonlinear interaction due to the electrostatic force on the deflected beam are included in the proposed nonlinear beam model. Comparing the beam natural frequency using strain gradient theory with experimental data shows an excellent agreement among both approaches. The normalized natural frequency is shown to be increasing nonlinearly with the decrease of the applied DC voltage as well as beam thickness. The results also reveal that increasing the tension axial stress increases the natural frequency; however its influence decreases when decreasing the beam thickness. To investigate the effect of AC actuation voltage on the beam resonant frequency, a Lindstedt-Poincare based perturbation method is utilized and validated by comparison with experimental data. The results show that increasing the AC actuation voltage makes the beam stiffer by increasing its resonant frequency.

Identification and Analysis of Artifacts in Amplitude Modulated Atomic Force Microscopy Array Operation

To increase measurement throughput of atomic force microscopy (AFM), multiple cantilevers can be placed in close proximity to form an array for parallel throughput. In this paper, we have measured the relationship between amplitude and tip-sample separation distance for an array of AFM cantilevers on a shared base actuated at a constant frequency and amplitude. The data show that discontinuous jumps in output amplitude occur within the response of individual beams. This is a phenomenon that does not occur for a standard, single beam system. To gain a better understanding of the coupled array response, a macroscale experiment and mathematical model are used to determine how parameter space alters the measured amplitude. The results demonstrate that a cusp catastrophe bifurcation occurs due to changes in individual beam resonant frequency, as well as significant zero-frequency coupling at the point of jump-to-contact. Both of these phenomena are shown to account for the amplitude jumps observed in the AFM array.

Three-dimensional piezoelectric energy harvester with spring and magnetic coupling

This letter proposes a three-dimensional piezoelectric energy harvester (3D PEH) that enables energy scavenging from vibrations in arbitrary directions. The proposed harvester comprises a spring, a primary beam, an inner beam, and two magnetic masses. The beams capture vibrations along their surface normal directions, and the spring, which acts as a mechanical energy buffer, is used to sense vibrations that occur parallel to the length direction of the inner beam. Magnetic coupling is introduced to perform frequency up-conversion and thus transfer vibration energy from the inner beam to the primary beam. As a benefit of the proposed structure, the 3D PEH can sense and convert all vibration energies occurring within the 3D space. When compared with use of the same beams in a linear configuration, the peak voltage of the proposed configuration is more than 3 V, and the bandwidths of voltages exceeding 5 V are wider by more than 3 Hz. Simultaneously, the primary beam resonant frequency is reduced by more than 6 Hz. Experimental results indicate that the proposed 3D PEH offers the advantages of higher open-circuit output voltages and wider frequency bandwidths at a lower resonant frequency.

Mechanical Properties and Residual Stress of Thin 3C-SiC(100) Films Determined Using MEMS Structures

3C-SiC(100) was grown on Si (100) in a thickness range between 40 and 500 nm by low pressure chemical vapor deposition. The mechanical properties and the residual stress were determined using the length dependence of the resonance frequencies of cantilevers and beams. Taking into account the influence of the cantilever bending and the stress gradients the Young’s modulus of the 3C-SiC(100) was obtained. It decreases with decreasing thickness of the epitaxial layer grown on Si (100).

An analytical model for the determination of resonance frequencies of perforated beams

In this paper, we develop closed expressions for the equivalent bending and shear stiffness of beams with regular square perforations, and apply them to the problem of determining the resonance frequencies of slender, regularly perforated clamped–clamped beams, which are of interest in the development of MEMS resonant devices. We prove that, depending on the perforation size, the Euler–Bernoulli equation or the more complex shear equation needs to be used to obtain accurate values for these frequencies. Extensive finite element method simulations are used to validate the proposed model over the full practical range of possible hole sizes. An experimental verification of the model is also presented.

A resonant sensor composed of quartz double ended tuning fork and silicon substrate for digital acceleration measurement

Presented in this paper is a micro-resonant acceleration sensor based on the frequency shift of quartz double ended tuning fork (DETF). The structure is silicon substrate having a proof mass supported by two parallel flexure hinges as doubly sustained cantilever, with a resonating DETF located between the hinges. The acceleration normal to the chip plane induces an axial stress in the DETF beam and, in turn, a proportional shift in the beam resonant frequency. Substrate is manufactured by single-crystal silicon for stable mechanical properties and batch-fabrication processes. Electrodes on the four surfaces of DETF beam excite anti-phase vibration model, to balance inner stress and torque and imply a high quality factor. The sensor is simply packaged and operates unsealed in atmosphere for measurements. The tested natural frequency is 36.9 kHz and the sensitivity is 21 Hz/g on a nominally ±100 g device, which is in good agreement with analytical calculation and finite element simulation. The output frequency drifting is less than 0.5 Hz (0.0014% of steady output) within 1 h. The nonlinearity is 0.0019%FS and hysteresis is 0.0026%FS. The testing results confirm the feasibility of combining quartz DETF and silicon substrate to achieve a micro-resonant sensor based on simple processing for digital acceleration measurements.

Research of Piezoelectric Cantilever Beam Diaphragm Pump

The design in this paper is a rectangular piezoelectric vibrator driven Cantilever Beam Diaphragm Pump. By analyzing the working principle and characteristics of the piezoelectric cantilever beam pump, we developed the dynamics model and FEA model of the piezoelectric cantilever beam and optimized through tests the main factors that affect the output flow of the piezoelectric cantilever pump. Additionally, we designed the piezoelectric cantilever pump prototype, and by using an impedance analyzer, we measured the resonance frequencies of cantilever beams of different lengths; and we used the laser micrometer to test and measure the beams of different structures under loaded and unloaded condition. The maximum displacement of the output terminal of loaded cantilever beam is 50.5μm. Furthermore, through the experiments we tested the output flow and its corresponding variation under different diameters of transmission pistons and different excitation frequencies. Experimental tests show: when selecting the cantilever beam and the drive piston with optimal performance with the input voltage at 120V, and the frequency at 197Hz, the optimum output flow rate is 89ml/min.

Size-dependent bending and vibration behaviour of piezoelectric nanobeams due to flexoelectricity

Flexoelectricity, representing a spontaneous electric polarization induced by a non-uniform strain field (or strain gradient), is believed to become manifest and be responsible for the size-dependent properties of dielectric materials at the nanoscale. In this paper, the influence of the flexoelectric effect on the static bending and free vibration of a simply supported piezoelectric nanobeam is investigated based on the extended linear piezoelectricity theory and the Timoshenko beam model. The governing equations of the piezoelectric nanobeam with non-homogeneous boundary conditions are obtained from Hamilton's principle. Explicit expressions of the beam deflection and resonant frequency are derived to show the size-dependency of the flexoelectric effect. It is found that the flexoelectricity has a significant effect on the deflection of the bending beam and may reverse the deflection direction under certain loading conditions. Simulation results also indicate that the influence of the flexoelectricity on the vibration behaviour of the piezoelectric nanobeam is more prominent for beams with smaller thickness. Thus, it is suggested that possible frequency tuning of piezoelectric nanobeams by adjusting the applied electrical load should incorporate the flexoelectric effect. The current study can be claimed as helpful for qualitatively characterizing the trend of the flexoelectric effect on the mechanical responses of piezoelectric nanobeams.

The Research of Piezoelectric Impedance Technology Applying to the Dynamic Signal Excitation Beam

The technology of damage detection by the piezoelectric impedance has been researched extensively, and in which the static load is usually used. To further expand the research of range, the dynamic signal excitation beam is used. A piezoelectric coupling beam model was established to deduce formulas of beam admittance. The resonance frequency of piezoelectric coupling beam and the dynamic signal excitation beam were studied in experiment. In the low frequency non-resonant frequency region, the experiment results are consistent with the theory. Research shows resonant frequency of coupling beam is equal to its natural frequency.