skip to main content


The NSF Public Access Repository (NSF-PAR) system and access will be unavailable from 11:00PM ET on Friday, December 15 until 2:00 AM ET on Saturday, December 16 due to maintenance. We apologize for the inconvenience.

This content will become publicly available on January 26, 2024

Title: Asymptotic distribution of the eigenvalues of the bending‐torsion vibration model with fully nondissipative boundary feedback

Asymptotic and spectral results on the initial boundary‐value problem for the coupled bending‐torsion vibration model (which is important in such areas of engineering sciences as bridge and tall building designs, aerospace and oil pipes modeling, etc.) are presented. The model is given by a system of two hyperbolic partial differential equations equipped with a three‐parameter family of non‐self‐adjoint (linear feedback type) boundary conditions modeling the actions of self‐straining actuators. The system is rewritten in the form of the first‐order evolution equation in a Hilbert space of a four‐component Cauchy data. It is shown that the dynamics generator is a matrix differential operator with compact resolvent, whose discrete spectrum splits asymptotically into two disjoint subsets called the α‐branch and the β‐branch, respectively. Precise spectral asymptotics for the eigenvalues from each branch as the number of an eigenvalue tends to ∞ have been derived. It is also shown that the leading asymptotical term of the α‐branch eigenvalue depends only on thetorsioncontrol parameter, while of the β‐branch eigenvalue depends on twobendingcontrol parameters.

more » « less
Author(s) / Creator(s):
Publisher / Repository:
Date Published:
Journal Name:
Studies in Applied Mathematics
Page Range / eLocation ID:
p. 996-1025
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. Mathematical analysis of the well known model of a piezoelectric energy harvester is presented. The harvester is designed as a cantilever Timoshenko beam with piezoelectric layers attached to its top and bottom faces. Thin, perfectly conductive electrodes are covering the top and bottom faces of the piezoelectric layers. These electrodes are connected to a resistive load. The model is governed by a system of three partial differential equations. The first two of them are the equations of the Timoshenko beam model and the third one represents Kirchhoff’s law for the electric circuit. All equations are coupled due to the piezoelectric effect. We represent the system as a single operator evolution equation in the Hilbert state space of the system. The dynamics generator of this evolution equation is a non-selfadjoint matrix differential operator with compact resolvent. The paper has two main results. Both results are explicit asymptotic formulas for eigenvalues of this operator, i.e., the modal analysis for the electrically loaded system is performed. The first set of the asymptotic formulas has remainder terms of the order O ( 1 n ) , where n is the number of an eigenvalue. These formulas are derived for the model with variable physical parameters. The second set of the asymptotic formulas has remainder terms of the order O ( 1 n 2 ) , and is derived for a less general model with constant parameters. This second set of formulas contains extra term depending on the electromechanical parameters of the model. It is shown that the spectrum asymptotically splits into two disjoint subsets, which we call the α -branch eigenvalues and the θ -branch eigenvalues. These eigenvalues being multiplied by “i” produce the set of the vibrational modes of the system. The α -branch vibrational modes are asymptotically located on certain vertical line in the left half of the complex plane and the θ -branch is asymptotically close to the imaginary axis. By having such spectral and asymptotic results, one can derive the asymptotic representation for the mode shapes and for voltage output. Asymptotics of vibrational modes and mode shapes is instrumental in the analysis of control problems for the harvester. 
    more » « less
  2. null (Ed.)
    The distribution of natural frequencies of the Euler–Bernoulli beam resting on elastic foundation and subject to an axial force in the presence of several damping mechanisms is investigated. The damping mechanisms are: ( i ) an external or viscous damping with damping coefficient ( − a 0 ( x )), ( ii ) a damping proportional to the bending rate with the damping coefficient a 1 ( x ). The beam is clamped at the left end and equipped with a four-parameter (α, β, κ 1 , κ 2 ) linear boundary feedback law at the right end. The 2 × 2 boundary feedback matrix relates the control input (a vector of velocity and its spacial derivative at the right end) to the output (a vector of shear and moment at the right end). The initial boundary value problem describing the dynamics of the beam has been reduced to the first order in time evolution equation in the state Hilbert space of the system. The dynamics generator has a purely discrete spectrum (the vibrational modes). Explicit asymptotic formula for the eigenvalues as the number of an eigenvalue tends to infinity have been obtained. It is shown that the boundary control parameters and the distributed damping play different roles in the asymptotical formulas for the eigenvalues of the dynamics generator. Namely, the damping coefficient a 1 and the boundary controls κ 1 and κ 2 enter the leading asymptotical term explicitly, while damping coefficient a 0 appears in the lower order terms. 
    more » « less
  3. Classical statistical mechanics has long relied on assumptions such as the equipartition theorem to understand the behavior of the complicated systems of many particles. The successes of this approach are well known, but there are also many well-known issues with classical theories. For some of these, the introduction of quantum mechanics is necessary, e.g., the ultraviolet catastrophe. However, more recently, the validity of assumptions such as the equipartition of energy in classical systems was called into question. For instance, a detailed analysis of a simplified model for blackbody radiation was apparently able to deduce the Stefan–Boltzmann law using purely classical statistical mechanics. This novel approach involved a careful analysis of a “metastable” state which greatly delays the approach to equilibrium. In this paper, we perform a broad analysis of such a metastable state in the classical Fermi–Pasta–Ulam–Tsingou (FPUT) models. We treat both the α-FPUT and β-FPUT models, exploring both quantitative and qualitative behavior. After introducing the models, we validate our methodology by reproducing the well-known FPUT recurrences in both models and confirming earlier results on how the strength of the recurrences depends on a single system parameter. We establish that the metastable state in the FPUT models can be defined by using a single degree-of-freedom measure—the spectral entropy (η)—and show that this measure has the power to quantify the distance from equipartition. For the α-FPUT model, a comparison to the integrable Toda lattice allows us to define rather clearly the lifetime of the metastable state for the standard initial conditions. We next devise a method to measure the lifetime of the metastable state tm in the α-FPUT model that reduces the sensitivity to the exact initial conditions. Our procedure involves averaging over random initial phases in the plane of initial conditions, the P1-Q1 plane. Applying this procedure gives us a power-law scaling for tm, with the important result that the power laws for different system sizes collapse down to the same exponent as Eα2→0. We examine the energy spectrum E(k) over time in the α-FPUT model and again compare the results to those of the Toda model. This analysis tentatively supports a method for an irreversible energy dissipation process suggested by Onorato et al.: four-wave and six-wave resonances as described by the “wave turbulence” theory. We next apply a similar approach to the β-FPUT model. Here, we explore in particular the different behavior for the two different signs of β. Finally, we describe a procedure for calculating tm in the β-FPUT model, a very different task than for the α-FPUT model, because the β-FPUT model is not a truncation of an integrable nonlinear model. 
    more » « less
  4. null (Ed.)
    Abstract Painful herniated discs are treated surgically by removing extruded nucleus pulposus (NP) material (nucleotomy). NP removal through enzymatic digestion is also commonly performed to initiate degenerative changes to study potential biological repair strategies. Experimental and computational studies have shown a decrease in disc stiffness with nucleotomy under single loading modalities, such as compression-only or bending-only loading. However, studies that apply more physiologically relevant loading conditions, such as compression in combination with bending or torsion, have shown contradicting results. We used a previously validated bone–disc–bone finite element model (Control) to create a Nucleotomy model to evaluate the effect of dual loading conditions (compression with torsion or bending) on intradiscal deformations. While disc joint stiffness decreased with nucleotomy under single loading conditions, as commonly reported in the literature, dual loading resulted in an increase in bending stiffness. More specifically, dual loading resulted in a 40% increase in bending stiffness under flexion and extension and a 25% increase in stiffness under lateral bending. The increase in bending stiffness was due to an increase and shift in compressive stress, where peak stresses migrated from the NP–annulus interface to the outer annulus. In contrast, the decrease in torsional stiffness was due to greater fiber reorientation during compression. In general, large radial strains were observed with nucleotomy, suggesting an increased risk for delamination or degenerative remodeling. In conclusion, the effect of nucleotomy on disc mechanics depends on the type and complexity of applied loads. 
    more » « less
  5. Abstract

    For brittle friction and rock deformation, the coefficientαin the general effective stress relationσe = σ − αPpcan be approximated as unity with sufficient accuracy. However, it is uncertain ifαdeviates from unity for semibrittle flow when both brittle and intracrystalline‐plastic deformation is involved. We conducted triaxial and isostatic compression experiments on synthetic salt‐rocks (∼300 ppm water) at room temperature to test the effective stress relation in the semibrittle regime using silicone oil and argon gas as pore fluids. Confining and pore pressures were cycled while their difference (differential pressure) was kept constant, such that changes in the mechanical behavior would indicate deviation ofαfrom unity. Microstructural observations were used to determine the dependence ofαon true area of grain contact from asperity yielding. In triaxial compression experiments, semibrittle flow involves grain boundary cracking and sliding, and intragranular dislocation glide and cracking. Flow strength remains constant for changes in pore fluid pressure of more than two orders of magnitude. In isostatic compression experiments, samples show combined processes of microcracking, grain boundary sliding, dislocation glide, and fluid‐assisted grain boundary migration recrystallization. Volumetric strain depends directly on the differential pressures (i.e.,αequals one). Analysis of grain‐contact area in both experiments indicates thatαis independent of the true area of contact defined by plastic yielding at grain boundaries. The observation ofαeffectively equals one may be explained by operation of pressure‐independent intracrystalline‐plastic mechanisms and transmission of pore pressure at grain boundaries through thin fluid films.

    more » « less