Proper Orthogonal Decomposition for Nonlinear Radiative Heat Transfer Problems

Analysing large scale, nonlinear, multiphysical, dynamical structures, by using mathematical modelling and simulation, e.g. Finite Element Modelling (FEM), can be computationally very expensive, especially if the number of degrees-of-freedom is high. This paper develops modal reduction techniques for such nonlinear multiphysical systems. The paper focuses on Proper Orthogonal Decomposition (POD), a multivariate statistical method that obtains a compact representation of a data set by reducing a large number of interdependent variables to a much smaller number of uncorrelated variables. A fully coupled, thermomechanical model consisting of a multilayered, cantilever beam is described and analysed. This linear benchmark is then extended by adding nonlinear radiative heat exchanges between the beam and an enclosing box. The radiative view factors, present in the equations governing the heat fluxes between beam and box elements, are obtained with a ray-tracing method. A reduction procedure is proposed for this fully coupled nonlinear, multiphysical, thermomechanical system. Two alternative approaches to the reduction are investigated, a monolithic approach incorporating a scaling factor to the equations, and a partitioned approach that treats the individual physical modes separately. The paper builds on previous work presented previously by the authors. The results are given for the RMS error between either approach and the original, full solution.

Bifurcation of Nonlinear Normal Modes by Means of Synge’s Stability

We study bifurcation of fundamental nonlinear normal modes (FNNMs) in 2-degree-of-freedom coupled oscillators by utilizing geometric mechanics approach based on Synges concept, which dictates orbital stability rather than Lyapunovs classical asymptotic stability. Use of harmonic balance method provides reasonably accurate approximation for NNMs over wide range of energy; and Floquet theory incorporated into Synges stability analysis predicts the respective bifurcation points as well as their types. Constructing NNMs in the frequency-energy domain, we seek applications to study of efficient targeted energy transfers.

Vertically-Aligned Springless Energy Harvester

This paper analyzes a new configuration of a recently proposed “springless” vibration energy harvester. In this study, the harvester is positioned so that its oscillations are aligned vertically acting against gravity. The MPG response is investigated experimentally. Test results show that the VEH behaves as a softening nonlinear oscillator even for small excitations. A mathematical model of the underlying impact oscillator is also derived and its parameters are estimated.

The Use of the Fixed Points Method for Optimal Damping of Harmonically Forced Cantilever Beams

The use of viscous dampers for vibration attenuation in harmonically forced cantilever beams is studied. The system considered is a cantilever beam with a point harmonic force applied at a given location and a viscous damper attached to it from one end, and grounded from the other. An assumed mode model of the system is derived using the first two transverse modes of the beam. For any given positions of the point force and damper, the optimal damping constant which minimizes the maximum of the frequency response function at the tip of the beam is determined analytically. It is shown that the objective function passes through a number of points independent of the damping constant. These inevitable points are used in the determination of the maximum allowable value of the objective function. As the locations of the point force and damper are varied separately from the fixed end of the beam to its tip, a two dimensional region plot is generated illustrating the different regions where each of these points is the highest. The optimal damping constant is determined analytically by forcing the frequency response function to pass horizontally through the highest fixed point which is referred to as the active peak. Four different damping ratios are determined and depending on the positions of the force and damper, the two dimensional map is consulted in the selection of the correct optimal damping ratio. The solution obtained is unique except when the active peak is the static fixed point. In this case, the solution is made unique by modifying the objective function to further enhance the solution at high frequencies.

Multi-Modal Vibration Energy Harvesting Using a Trapezoidal Plate

Vibration-based energy harvesters are usually designed to exhibit natural frequencies that match those of the excitation for maximum power output. This has spurred interest into the design of devices that respond to variable frequency sources. In this work, an electromagnetic energy harvester in the form of a base excited trapezoidal plate is proposed. The plate geometry is designed to achieve two closely spaced vibration modes in order to harvest energy across a broader bandwidth. The ensuing bending and twisting vibrations are utilized in this capacity by placing a magnet on the plate tip that moves past a stationary coil. A dynamic model is presented to predict the system performance and is verified experimentally.

A Highly Accurate Beam Finite Element for Curved and Twisted Helicopter Blades

Blade optimization is more than ever a crucial activity for helicopter manufacturers, always looking for performance improvements, noise reduction and vibratory comfort increase. Latest studies have led to design new blade concepts including a double swept plan shape, an evolutionary and increased twist angle at the tip and a new layout for internal components like roving spars. Such blades exhibit a highly coupled behavior between torsion, longitudinal and bending motions that should be accurately modeled for predictive numerical tools. In this research a highly accurate beam finite element is formulated in the rotating frame to improve the static deformation calculation under aerodynamic and centrifugal loads and thus enhance dynamic and stability analysis usually performed for a helicopter development. Numerical and experimental investigations are performed to demonstrate the model reliability both for academic beams with extreme shape and for actual blade design.

An Accurate Spatial Discretization and Substructural Method With Application to Moving Elevator Cable-Car Systems: Part I—Methodology

A spatial discretization and substructure method is developed to calculate the dynamic responses of one-dimensional systems, which consist of length-variant distributed-parameter components such as strings, rods, and beams, and lumped-parameter components such as point masses and rigid bodies. The dependent variable, such as the displacement, of a distributed-parameter component is decomposed into boundary-induced terms and internal terms. The boundary-induced terms are interpolated from the boundary motions, and the internal terms are approximated by an expansion of trial functions that satisfy the corresponding homogeneous boundary conditions. All the matching conditions at the interfaces of the components are satisfied, and the expansions of the dependent variables of the distributed-parameter components absolutely and uniformly converge. The spatial derivatives of the dependent variables, which are related to the internal forces/moments, such as the axial forces, bending moments, and shear forces, can be accurately calculated. Assembling the component equations and the geometric matching conditions that arise from the continuity relations leads to a system of differential algebraic equations (DAEs). When some matching conditions are linear algebraic equations, some generalized coordinates can be represented by others so that the number of the generalized coordinates can be reduced. The methodology is applied to moving elevator cable-car systems in Part II of this work.

The Underwater Effects of High Power, High Frequency Acoustic Noise on MEMS Gyroscopes

Unlike their macroscale counterparts, MEMS gyroscopes use a vibrating proof mass rather than a rotational mass to sense changes in angular rate. They are also smaller and less expensive than traditional gyroscopes. For this reason, MEMS gyroscopes are being used in many new applications, some of which include operation in harsh environments. There has been much research on the negative effects of the performance of MEMS gyroscopes in environments that experience mechanical shock, high frequency vibration, and high frequency acoustic noise in air. However, MEMS gyroscopes are beginning to be used in underwater applications such as autonomous underwater vehicles, digital compasses, and torpedo guidance systems. The results of this experiment demonstrate that MEMS gyroscopes submerged in water are susceptible to high power, high frequency acoustic noise at and near the resonant frequency of the proof mass. These effects are demonstrated using the ADXRS300 MEMS gyroscope.

Comparing the Performance of a Nonlinear Energy Harvester in Mono- and Bi-Stable Potentials

The quest to develop broadband vibratory energy harvesters (VEHs) has recently motivated researchers to explore introducing nonlinearities into the harvester’s design. Some research efforts have demonstrated that this new class of nonlinear harvesters can outperform their traditional linear (resonant) counterparts; some others however concluded that nonlinearities can diminish the harvester’s transduction. Through this effort, we compare the performance of a nonlinear VEH operating in mono- and bi-stable potentials. With that objective, we consider an axially-loaded clamped-clamped piezoelectric beam which functions as an energy harvester in the mono-stable (pre-buckling) and bistable (post-buckling) configurations. For the purpose of fair performance comparison, the oscillation frequency around the stable equilibria of the harvester is tuned to equal values in both configurations. The harvester is then subjected to harmonic base excitations of different magnitudes and a slowly-varying frequency which spans a wide range around the tuned oscillation frequency. The output voltage measured across an arbitrarily chosen electric load is used as a relative performance measure. Both numerical and experimental results demonstrate that the shape of the potential function plays an essential role in conjunction with the magnitude of the base excitation to determine whether the bi-stable harvester can outperform the mono-stable one and for what range of frequencies.

Study of Start-Up Vibration Response for Oil Whip and Dry Whip Through Hilbert Huang Transform

Oil whip induces self-excited vibration in fluid-handling machines, and what is worse, it can cause self-excited reverse precessional full annular rub, known as “dry whip” which is a secondary phenomenon resulting from a primary cause and may lead to a catastrophic failure of machines; that is, “coincidence of oil whip and dry whip” that occurs repeatedly with constant frequency and amplitude in small clearance cases of fluid-handling machines. Early detection of rub malfunction is essential to avoid damage. Hilbert Huang Transform, which included an empirical mode decomposition and Hilbert spectral analysis, is applied. Hilbert Huang Transform is a great method for analyzing non-linear and non-stationary signals, such as rotor startup signals. Hilbert Huang Transform clearly indicates instability at its initiation stage, and energy concentration changes by different stages. Malfunctions like rub can not observe by Hilbert spectrum. Hilbert spectrum combining full spectrum is developed, as know full Hilbert spectrum, to interpret the rub. The coincidence of oil whip and dry whip is observed definitely through FHS. The advantage of the full Hilbert spectrum is to offer a faster, more efficient method to diagnose fluid-induced instability.