Dynamics Of Flexible Plates Research Articles

Mathematical modeling of nonlinear thermodynamics of nanoplates

The theory of nonlinear statics and dynamics of flexible plates, taking into account the modified couple stress theory and temperature field, is developed. The theory is based on the classical Kirchhoff model for an isotropic elastic body. Geometric nonlinearity is taken into account according to the von Kármán model. Variational differential equations are yielded by the Hamilton principle, and partial differential equation (PDE's) related to displacements of the middle surface and deflection are derived. The hypotheses of the modified couple stress theory implied an increase of the order of the system of partial differential equations due to the fact that moments of higher order appeared. The resolved PDE's are reduced to the Cauchy problem by the method of finite differences of the second order accuracy, which is solved by methods of the Runge-Kutta type (from the fourth to the eighth order of accuracy) and the Newmark method. No restrictions are imposed on the temperature field, and it determined from the solution of the three-dimensional heat equation using the finite element method (FEM). The convergence of the proposed algorithms is investigated depending on the number of finite elements and the time step. Static problems are solved by the dynamic approach. Nonlinear dynamics is analyzed based on (Fourier spectrum, wavelets of different types and phase portraits). The analysis of Lyapunov exponents obtained by different methods (Kantz, Wolf, Rosenstein, Sano-Sawada and the authors method) is carried out, which validated occurrence of chaotic vibrations. Hyperchaotic vibrations have been detected and studied. It is also illustrated how increase of the temperature field influences localization of the chaotic zones for size-dependent plates.

Dynamics of flexible plates and flow under impulsive oscillation

The induced flow and the dynamics of flexible plates under heaving characterized by impulsive acceleration patterns Π(t) were experimentally inspected for various Cauchy numbers Ca and fractions τa of the time under acceleration with respect to the period of oscillations T. Simultaneous measurements of the plate motion along their span and the surrounding flow were obtained with particle tracking velocimetry (PTV) and particle image velocimetry (PIV). The Π(t) function was imposed at the root of a low-aspect-ratio plate vertically submerged in a quiescent water tank. Results show that Ca and τa significantly modulated the plate dynamics as well as the induced flow and coherent motions. We developed a simple model to estimate the velocity difference Δu between the root and tip of the plates; it is based on the damped harmonic oscillator and exhibits good performance across the fifteen scenarios inspected (three Ca by five τa). In general, the plate dynamics and flow exhibited three distinctive patterns characterized by flapping-like motions, translation-like motions and another intermediate state. The type of motion was induced by the relative instant at the reverse direction within the forced Π(t)-heaving within one of three characteristic stages; two of them depending on the sign of Δu and the other when Δu→0, i.e., in the limit of pure translation.

On the Dynamics of Flexible Plates under Rotational Motions

The reconfiguration of low-aspect-ratio flexible plates, required power and induced flow under pure rotation were experimentally inspected for various plate stiffness and angular velocities ω. Particle tracking velocimetry (PTV) and particle image velocimetry (PIV) were used to characterize the plate deformation along their span as well as the flow and turbulence statistics in the vicinity of the structures. Results show the characteristic role of stiffness and ω in modulating the structure reconfiguration, power required and induced flow. The inspected configurations allowed inspecting various plate deformations ranging from minor to extreme bending over 90∘ between the tangents of the two tips. Regardless of the case, the plates did not undergo noticeable deformation in the last ∼30% of the span. Location of the maximum deformation along the plate followed a trend s m ∝ l o g ( C a ) , where C a is the Cauchy number, which indicated that s m is roughly fixed at sufficiently large C a. The angle (α) between the plate in the vicinity of the tip and the tangential vector of the motions exhibited two distinctive, nearly-linear trends as a function of C a , within C a ∈ ( 0 , 15 ) and C a ∈ ( 20 , 70 ), with a matching within these C a at C a > 70, α ≈ 45 ∘. Induced flow revealed a local maximum of the turbulence levels at around 60% of the span of the plate; however, the largest turbulence enhancement occurred near the tip. Flexibility of the plate strongly modulated the spatial distribution of small-scale vortical structures; they were located along the plate wake in the stiffer plate and relatively concentrated near the tip in the low-stiffness plate. Due to relatively large deformation, rotational and wake effects, a simple formulation for predicting the mean reconfiguration showed offset; however, a bulk, constant factor on ω accounted for the offset between predictions and measurements at deformation reaching ∼ 60 ∘ between the tips.

Vibration Control of Plates Using Electromagnetic Compressional Damping Treatment

A new class of structural damping treatments is introduced. This class is the Electromagnetic Compressional Damping Treatment (ECDT) which relies in its operation on a viscoelastic damping layer sandwiched between an electromagnet and a permanent magnet. Interaction between the magnets generates magnetic forces that enhance the compressional damping mechanism of the viscoelastic layer. With proper tuning of the magnetic forces, in response to the structural vibration, undesirable resonances and catastrophic failures can be avoided. The fundamentals and the underlying phenomena associated with the ECDT are investigated theoretically and experimentally. A finite element model is developed to describe the interaction between the dynamics of flexible plates, the visco-elastic damping layer and the electro-magnetic actuators. The validity of the developed finite element model is checked experimentally using aluminum plates treated with single and multi-ECDT patches. The plate/ECDT system is subjected to sinusoidal excitations and its multi-mode response is monitored when the electro-magnetic actuator is activated or not. Several control strategies are considered to activate the electro-magnetic actuator including simple PD controllers. The performance of the uncontrolled and controlled system is determined at various operating conditions. Comparisons with conventional Passive Constrained Layer Damping (PCLD) treatments emphasize the potential of the ECDT treatment as an effective means for controlling structural vibrations.