Multi-beam Structure Research Articles

Nonlinear dynamical modeling and response analysis of complex structures based on assumed mode weighting

The assumed mode method (AMM) is widely used for dynamical modeling but faces increasing challenges in developing low-dimensional and high-precision models of complex structures for investigations of nonlinear dynamical responses and active vibration control. This is mainly because the current complex structure has an increasing number of components, each of which needs to be discretized by multiple assumed modes in dynamical modeling, leading to increasing degrees of freedom for the system. To address this issue, this paper proposes a dimension-reduced method based on the assumed mode weighting method, referred to as Mode Weighting Method (MWM). The modeling process of a multi-beam structure is fully presented as an example to show the strategy of the MWM. During this process, the assumed mode (AM) model (the dynamic model derived in terms of a set of assumed modes) is developed first where the joints of the structure between its components are considered as artificial springs and each component is discretized by its vibration modes without constraints. The components of the eigenvector of the characteristic equation derived from the AM model are taken as the weighting coefficients to get the approximate analytical global modes of the structure by multiplying the corresponding assumed modes. Consequently, a nonlinear dynamic model with a few degree of freedom is obtained by discretizing the structure with the weighted modes. Moreover, a technique for simply obtaining the weighted mode (WM) models by directly processing the AM model through a transformation matrix is presented. To validate the proposed method, the simulations on the natural characteristics and the dynamical responses for both linear and nonlinear models, are performed respectively based on the WM and the finite element models, where both results are well matched each other. This work provides a new way of developing advanced dynamical modeling methods for structural dynamics design, optimization, and active vibration control.

Topological high-harmonic spectroscopy

Linearly polarized vector beams are structured lasers whose topology is characterized by a well-defined Poincaré index, which is a topological invariant during high-order harmonic generation. As such, harmonics are produced as extreme-ultraviolet vector beams that inherit the topology of the driver. This holds for isotropic targets such as noble gases, but analogous behaviour in crystalline solids is still open to discussion. Here, we demonstrate that this conservation rule breaks in crystalline solids, in virtue of their anisotropic non-linear susceptibility. We identify the topological properties of the harmonic field as unique probes, sensitive to both the microscopic and macroscopic features of the target’s complex non-linear response. Our simulations, performed in single-layer graphene, show that the harmonic field is split into a multi-beam structure whose topology encodes information about laser-driven electronic dynamics. Our work promotes the topological analysis of the high-order harmonic field as a spectroscopic tool to reveal the nonlinearities in the coupling of light and target symmetries.

The Influence of Bottom Sediment on the Propagation of Caustic Beams in Oceanic Waveguides

Numerical modeling with the use of mode theory was used to investigate the regularities of the spatial (depth and horizontal distance) distribution of the acoustic field intensity formed by multiple interaction of a caustic beam with a layered bottom in a shallow oceanic waveguide with an underwater sound channel open to the bottom. It has been established that at the values of the sound velocity at the upper boundary in the sedimentary layer, less than the sound velocity at the bottom in the water layer, the formation of a multi-beam structure of the acoustic field is possible. It was found that, starting from certain distances, newly formed beams can play the main role in the spatial distribution of the acoustic field intensity.

3D-spatial vibration global modes of a flexible arm-supported ring antenna and its dynamic analysis

Due to the complexity and high degrees of freedom of the structure, dynamical modelling of ring truss antenna (RTA) is a challenging job. The three-dimensional (3D) vibration of each component of RTA is spatial and coupled to each other, and thus assumed modes cannot depict its vibration accurately and dynamical models obtained by FEM generally have high dimensions, by which it is very hard to either conduct dynamical analysis or design its vibration controller. To overcome these obstacles, a dynamical modeling method named the 3D-Global Mode Method is proposed to obtain the low-dimensional analytical dynamical model of a flexible arm-supported ring antenna (ASRA) that vibrates spatially. The natural characteristics of a RTA is investigated by simplifying it as a flexible arm-supported closed multi-beam structure. Natural frequencies and the corresponding analytical 3D global modes are obtained by solving the eigen-equation formed by the spatial matching conditions and the boundary conditions. Orthogonality relations of the 3D analytical global modes are established and then used to transform the partial differential equations of spatial motion into a set of reduced-order ordinary differential equations. Taking the simulation results obtained from the FEM as a reference, the validity of the proposed method is verified by comparing the natural frequencies and 3D global mode shapes with those obtained from FEM. The mode localization phenomenon of ASRA is found and the influence of the parameters of RTA on mode characteristics is investigated. Through the proposed model, dynamical responses of the ASRA excited by orbital maneuvering acceleration are simulated numerically. This study paves an effective way to obtain the analytical global modes of spatially vibrating ASRAs and other similar spatial flexible combined structures, and to establish the low-degree-of-freedom dynamical model.

A low computational cost optimization procedure for axial load carrying capacity of the multi-beam structure

The paper presents an improved algorithm to compute an axial load carrying capacity of the specific multi-beam structure in order to failure avoidance. The former approach based on the finite element method is reduced to determine a relation between the axial load and axial displacement for a single beam, while the time-consuming FE procedure applied on several beams is replaced with a numerical optimization process. The developed approach provides faster and simpler computation of the minimum axial load carrying capacity of the structure considering the beam length tolerances, where the results are in a good agreement with the complex FE analysis. Supporting theoretical and experimental analyses were performed to confirm the correctness of the mentioned method where the load carrying capacity of the multi-beam structure made up of six steel circular beams was analyzed.

Three-to-One Internal Resonance of L-Shaped Multi-Beam Structure with Nonlinear Joints

In this paper, a reduced-order analytical model for an L-shaped multi-beam structure with nonlinear joints is presented to investigate the nonlinear responses of the system with three-to-one internal resonances conditions. Firstly, the global mode shapes are used to obtain an explicit set of nonlinear ordinary differential equations of motion for the system. Then, the first two natural frequencies of the system are calculated to determine the specific tip mass that results in three-to-one internal resonance. Subsequently, an approximation of the analytical solution of the dynamic model with two-degree-of-freedom is derived by using the multi-scale method. The accuracy of the approximation solution is verified by comparing it with the numerical solution obtained from the original motion equations. Based on the nonlinear dynamical model obtained by this paper, the frequency response curves are given to investigate the nonlinear dynamic characteristic of the L-shaped multi-beam structure with nonlinear joints. The results show that the nonlinear stiffness of the joints has a great influence on the nonlinear response of the system with three-to-one internal resonance conditions.

Multi-beam vortex generation induced by the non-linear optical anisotropy of graphene

We analyse the high harmonic emission from single-layer graphene driven by infrared vector beams. We demonstrate that graphene’s anisotropy offers a privileged scenario to explore non-trivial light spin-orbit couplings, which substantially extends the possibilities for the generation of high-harmonic structured beams currently studied in atomic and molecular targets. In our case, graphene’s crystal symmetry introduces a spin-dependent diffraction pattern that, coupled with the fundamental conservation of the driver’s topological phase, leads to the splitting of the harmonic field in a multi-beam structure, composed of spatially diverging vortices. Our work demonstrates that anisotropic targets are extraordinary tools to sculpt complex structured short-wavelength beams.

Rotational multi-beam energy harvester with up-conversion mechanism in an extremely low frequency scenario

We propose a novel frequency-up converting device in an extremely low frequency rotating scenario based on a multi-beam piezoelectric energy harvester and multiple impacts. The rotating multi-beam structure consists of a piezoelectric sheet added to a top beam, a large seismic mass added to a bottom beam and an elastic stop which can be eventually replaced by a buzzer. At half of the rotating cycle, the bottom beam impacts the top beam by means of a spring generating energy by a frequency up-conversion mechanism. At the other half of the cycle, the up and bottom beam separate from each other and the bottom beam impacts an elastic stop or a buzzer which are used to simultaneously recover the kinetic energy of the collision, generate energy (buzzer) and prevent possible damage to the structure. An analytical model that governs the electromechanical equations and the impact dynamics is proposed to provide a means to predict the generated output energy. The model is based on the Euler-Bernoulli structural theory for the beams and a piezoelectric nonlinear constitutive model. Two different energy generation possibilities are studied, depending on the selection of the elastic stop or the buzzer. When the elastic spring acts as stop, the collision between the bottom beam and the stop is more elastic. Replacing the spring stop with a piezo buzzer adds to the system an additional mechanism of energy conversion. By making several experimental tests, the analytical model is validated by computing the cumulated energy. The results demonstrate that the buzzer element significantly improves the harvested power when it replaces the elastic stop. To estimate the usable power of the device, rectifier circuits coupled to the piezo elements are built and test. The results show that in the case of the buzzer used as a stop, a rectified power that overcomes 6.4 times in average other previous designs is harvested in the range of 0.8–2.2 Hz over an electrical load of 10 kΩ. The amount of collected power is suitable for autonomous sensing of wind turbines of 30 kW with rotational speeds of between 50 and 150 rpm.

Mutual Information Maximizing Wideband Multi-User (wMU) mmWave Massive MIMO

To enable next-generation mmWave cellular, it is vital to design high-performance precoding schemes for wideband multi-user (wMU) mmWave massive MIMO (mMIMO). As existing approaches are mostly ad-hoc, thereby lacking performance guarantee, we will tailor an enhanced transceiver design explicitly for wMU mmWave mMIMO, with the goal of maximizing mutual information (MI). The proposed scheme follows the prevalent hybrid block diagonalization (HBD-)based framework that is well-known for balancing the transmitter-end processing flexibility and the user-end detection complexity. In this paper, we for the first time prove that HBD is optimal in the sense of MI. In terms of the transceiver design, we start by decoupling the hybrid processing into a two-stage analog and digital processing, and then derive the MI bounds associated with HBD. By optimizing the tight MI bound, excellent HBD transceivers are devised for both the multi-aperture structure (MAS) and the multi-beam structure (MBS). The proposed HBD technique does not rely on substantial computational complexity, striking channel sparsity, or high-resolution analog beamformers, and can achieve a superb MI performance even with inferior hardware configurations.

Vibration analysis and distributed piezoelectric energy harvester design for the L-shaped beam

The vibration characteristics of a cantilever L-shaped beam perpendicular to the horizontal plane are investigated in this paper. The dynamical model of the L-shaped multi-beam structure is established by using the Euler-Bernoulli beam theory, where vibration equations of each substructure, matching conditions of the connection and boundary conditions are included. Based on the modal orthogonality, the discrete dynamical model with finite degrees of freedom is obtained. Then the distributed piezoelectric energy harvester is proposed to collect the vibration energy of the L-shaped beam. Different the most traditional energy harvester that the piezoelectric material covers the surface of the structure, the distributed energy harvester is placed on the selected points. The genetic algorithm is adopted to obtain the optimal position that can provide the maximum output power. The validation and accuracy of the present model is verified by comparison with the finite element model. Numerical results show the discrete model can perform the dynamic characteristics of the structure well with only two modes. By the comparison of different configurations, it concludes that the optimal position of the distributed piezoelectric energy harvester is close to the fixed end of the whole cantilever structure.

Nonlinear dynamics of a foldable multibeam structure with one to two internal resonances

Nonlinear dynamics of a foldable multibeam structure subjected to a base excitation with one to two internal resonances are investigated in this paper. The nonlinear partial differential governing equations of the multibeam structure is derived by Hamilton's principle. To identify the existing one to two internal resonance for a foldable multibeam structure, the first two natural frequencies of the foldable multibeam structure versus the folding angle are calculated and validated by the finite element method. The characteristic of the one to two internal resonances are illustrated by the dynamic response of the multibeam structure. The modulation equations for the steady state motion of the foldable multibeam structure under this internal resonance condition are obtained by the multiple scales method. The approximation solution derived from the modulation equation is compared with the numerical solution obtained from the original motion equation. A comprehensive study of the nonlinear dynamics of the foldable multibeam structure is performed by considering two cases of harmonic excitation. The saturation and jumping phenomena are observed for the foldable multibeam structure.

Thermal Performance of Micro Hotplates with Novel Shapes Based on Single-Layer SiO₂ Suspended Film.

In this paper, two kinds of suspended micro hotplate with novel shapes of multibeam structure and reticular structure are designed. These designs have a reliable mechanical strength, so they can be designed and fabricated on single-layer SiO2 suspended film through a simplified process. Single-layer suspended film helps to reduce power consumption. Based on the new film shapes, different resistance heaters with various widths and thicknesses are designed. Then, the temperature uniformity and power consumption of different micro hotplates are compared to study the effect of these variables and obtain the one with the optimal thermal performance. We report the simulations of temperature uniformity and give the corresponding infrared images in measurement. The experimental temperature differences are larger than those of the simulation. Experimental results show that the lowest power consumption and the minimum temperature difference are 43 mW and 50 °C, respectively, when the highest temperature on the suspended platform (240 × 240 μm2) is 450 °C. Compared to the traditional four-beam micro hotplate, temperature non-uniformity is reduced by about 30–50%.

Dynamics of a multi-beam structure connected with nonlinear joints: modelling and simulation

A dynamic modelling approach for a multi-beam structure connected with nonlinear joints is presented and subsequently used to obtain its reduced-order analytical model. Firstly, by applying the matching and boundary conditions, the natural frequencies and global mode shapes of the system are derived. According to the nonlinear transmitted torque formulations of the joints, the joints nonlinearities are introduced into the system by using the equilibrium conditions between the beams and the joints. Then, the global mode shapes and their orthogonality relations are used to obtain an explicit set of reduced-order nonlinear ordinary differential equations(ODEs) of motion for the flexible structure with nonlinear joints. A flexible structure composed of 4 beams and nonlinear joints is given as an example to illustrate the application of the modelling approach proposed. And a comparison between the natural frequencies obtained by the proposed approach and those from the finite element method is given to verify the validity of the derived model. Through the nonlinear ODEs obtained by the Galerkin truncation of the original dynamic model, a study on the variation in dynamic responses of the system with different numbers of modes is performed. The results are used to determine the number of modes taken in nonlinear vibration analysis for certain range of the vibration amplitude of the external excitation. The dynamic responses of the system with various joint parameters are worked out, which show that the transmission characteristics of the joints can strongly affect the dynamic behavior of the whole structure.

Optimization of EH Multi-beam Structures for Freight Car Vibration

Autonomous sensors (AS) became a suitable alternative to the constant monitoring of mechanical systems, preventing unexpected failure of the components and resulting in significant savings in maintenance costs and time. Regarding railroad transportation, these autonomous sensors can provide useful information of the freight cars’ conditions, preventing damage and even derailments. However, many freight cars equipped with pneumatic control valves do not have an electric power source to keep the AS running. Energy harvesting (EH) appears to be a viable alternative to powering the AS by converting the vibration energy naturally present in the freight cars into electricity by means of piezoelectrics. This study searched for and found an optimized configuration of a piezoelectric multi-beam structure adequate for freight cars’ vibrational frequencies, resulting in an output power of 13 mW, which is enough to power a conventional vertical accelerometer.

Global bifurcations and chaotic motions of a flexible multi-beam structure

Global bifurcations and multi-pulse chaotic motions of flexible multi-beam structures derived from an L-shaped beam resting on a vibrating base are investigated considering one to two internal resonance and principal resonance. Base on the exact modal functions and the orthogonality conditions of global modes, the PDEs of the structure including both nonlinear coupling and nonlinear inertia are discretized into a set of coupled autoparametric ODEs by using Galerkin’s technique. The method of multiple scales is applied to yield a set of autonomous equations of the first order approximations to the response of the dynamical system. A generalized Melnikov method is used to study global dynamics for the “resonance case”. The present analysis indicates multi-pulse chaotic motions result from the existence of Šilnikov’s type of homoclinic orbits and the critical parameter surface under which the system may exhibit chaos in the sense of Smale horseshoes are obtained. The global results are finally interpreted in terms of the physical motion of such flexible multi-beam structure and the dynamical mechanism on chaotic pattern conversion between the localized mode and the coupled mode are revealed.

Nonlinear Dynamics of Flexible L-Shaped Beam Based on Exact Modes Truncation

Nonlinear dynamics of flexible multibeam structures modeled as an L-shaped beam are investigated systematically considering the modal interactions. Taking into account nonlinear coupling and nonlinear inertia, Hamilton’s principle is employed to derive the partial differential governing equations of the structure. Exact mode functions are obtained by the coupled linear equations governing the horizontal and vertical beams and the results are verified by the finite element method. Then the exact modes are adopted to truncate the partial differential governing equations into two coupled nonlinear ordinary differential equations by using Galerkin method. The undamped free oscillations are studied in terms of Jacobi elliptic functions and results indicate that the energy exchanges are continual between the two modes. The saturation and jumping phenomena are then observed for the forced damped multibeam structure. Further, a higher-dimensional, Melnikov-type perturbation method is used to explore the physical mechanism leading to chaotic behaviors for such an autoparametric system. Numerical simulations are performed to validate the theoretical predictions.

Global dynamics of an autoparametric beam structure

Global dynamics of an autoparametric beam structure derived from a flexible L-shaped beam subjected to base excitation with one-to-two internal resonance and principal resonance are investigated. Hamilton’s principle is employed to obtain the nonlinear partial differential governing equations of the multi-beam structure. A linear theoretical analysis is implemented to derive the modal functions, and the orthogonality conditions are established. The analytical modal functions obtained are then adopted to truncate the partial differential governing equations into a set of coupled nonlinear ordinary differential equations via the Galerkin’s procedure. The method of multiple scales is applied to yield a set of autonomous equations of the first-order approximations to the response of the dynamical system. The Energy-Phase method is used to study the global bifurcation and multi-pulse chaotic dynamics of such autoparametric system. The present analysis indicates that the chaotic dynamics results from the existence of Silnikov’s type of homoclinic orbits and the parameter set for which the system may exhibit chaotic motions in the sense of Smale horseshoes are predicted analytically. Numerical simulations are performed to validate the theoretical results.

A piezoresistive micro-accelerometer with high frequency response and low transverse effect

With the purpose of measuring vibration signals in high-speed machinery, this paper developed a piezoresistive micro-accelerometer with multi-beam structure by combining four tiny sensing beams with four suspension beams. The eight-beam (EB) structure was designed to improve the trade-off between the sensitivity and the natural frequency of piezoresistive accelerometer. Besides, the piezoresistor configuration in the sensing beams reduces the cross interference from the undesirable direction significantly. The natural frequency of the structure and the stress on the sensing beams are theoretically calculated, and then verified through finite element method (FEM). The proposed sensor is fabricated on the n-type single crystal silicon wafer and packaged for experiment. The results demonstrate that the developed device possesses a suitable characteristic in sensitivity, natural frequency and transverse effect, which allows its usage in the measuring high frequency vibration signals.

Analytical modeling of orthogonal spiral structures

This paper presents the analytical modeling of orthogonal spiral structures (OSS), a promising option for small-scale energy harvesting applications. This unique multi-beam structure is analyzed using a distributed parameter approach with Euler–Bernoulli assumptions. First, an aluminum substrate is evaluated to determine if the proposed design can be used to capture vibration energy in the desired frequency range using a twelve beam OSS. Finite element calculations are used to validate the analytical model. This model is then modified to include the electromechanical effects of a piezoelectric layer added to the aluminum substrate. Lastly, the effects of the beam width and the number of beams is analyzed for a particular surface area of the OSS. Results show that increasing the number of beams causes a reduction in the first natural frequency. From those results, it is possible to conclude that OSS can be used as an alternative to current energy harvesting systems for MEMS applications, allowing the capture of environmental energy in the frequency range of common mechanical systems.

Criterion of detachment for grip-and-release devices with slanted multi-beam structure using Ti-Ni wire

A multi-beam structure using Ti-Ni wire to grip rough surfaces and control detachment was manufactured. The grip-and-release mechanism of the structure was investigated. The experimental results were analyzed with a model assuming beam theory and using a fracture criterion at the adhered interface. The fracture criterion of the adhesion interface is suggested to depend on the shear stress.