Linear Beam Equation Research Articles

Regularity results in 2D fluid–structure interaction

We study the interaction of an incompressible fluid in two dimensions with an elastic structure yielding the moving boundary of the physical domain. The displacement of the structure is described by a linear viscoelastic beam equation. Our main result is the existence of a unique global strong solution. Previously, only the ideal case of a flat reference geometry was considered such that the structure can only move in vertical direction. We allow for a general geometric set-up, where the structure can even occupy the complete boundary. Our main tool—being of independent interest—is a maximal regularity estimate for the steady Stokes system in domains with minimal boundary regularity. In particular, we can control the velocity field in W2,2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W^{2,2}$$\\end{document} in terms of a forcing in L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document} provided the boundary belongs roughly to W3/2,2.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W^{3/2,2}.$$\\end{document} This is applied to the momentum equation in the moving domain (for a fixed time) with the material derivative as right-hand side. Since the moving boundary belongs a priori only to the class W2,2,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W^{2,2},$$\\end{document} known results do not apply here as they require a C2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^2$$\\end{document}-boundary.

Convergence of thin vibrating rods to a linear beam equation

We show that solutions for a specifically scaled nonlinear wave equation of nonlinear elasticity converge to solutions of a linear Euler–Bernoulli beam system. We construct an approximation of the solution, using a suitable asymptotic expansion ansatz based upon solutions to the one-dimensional beam equation. Following this, we derive the existence of appropriately scaled initial data and can bound the difference between the analytical solution and the approximating sequence.

Classification of asymptotic profiles for the Cauchy problem of damped beam equation with two variable coefficients: Effective damping case

We consider the Cauchy problem for the linear beam equation with two variable coefficients:utt+b(t)ut+uxxxx−a(t)uxx=0,(t,x)∈R+×R. The purpose of this study is to classify the behavior of solution with respect to (α,β) when the coefficients a and b behave as a(t)∼(1+t)α and b(t)∼(1+t)β. In this article we shall give the asymptotic profile formula in effective damping cases, by using the method of scaling variables developed by Gallay and Raugel [9].

On the solution linear and nonlinear fractional beam equation

On the solution linear and nonlinear fractional beam equation

KAM theory for the reversible perturbations of 2D linear beam equations

In the present paper, we prove an infinite dimensional reversible Kolmogorov-Arnold-Moser (KAM) theorem. As an application, we study the existence of KAM tori for a class of two dimensional (2D) non-Hamiltonian completely resonant beam equations with derivative nonlinearities. The Birkhoff normal form theory is also used since there are no external parameters in the equations.

Existence of local strong solutions to fluid–beam and fluid–rod interaction systems

We study an unsteady nonlinear fluid–structure interaction problem. We consider a Newtonian incompressible two-dimensional flow described by the Navier–Stokes equations set in an unknown domain depending on the displacement of a structure, which itself satisfies a linear wave equation or a linear beam equation. The fluid and the structure systems are coupled via interface conditions prescribing the continuity of the velocities at the fluid–structure interface and the action-reaction principle. Considering three different structure models, we prove existence of a unique local-in-time strong solution, for which there is no gap between the regularity of the initial data and the regularity of the solution enabling to obtain a blow up alternative. In the case of a damped beam this is an alternative proof (and a generalization to non zero initial displacement) of the result that can be found in [20]. In the case of the wave equation or a beam equation with inertia of rotation, this is, to our knowledge the first result of existence of strong solutions for which no viscosity is added. The key points consist in studying the coupled system without decoupling the fluid from the structure and to use the fluid dissipation to control, in appropriate function spaces, the structure velocity.

Implicit analytic solutions for the linear stochastic partial differential beam equation with fractional derivative terms

Analytic solutions in implicit-form are derived for a linear stochastic partial differential equation (SPDE) with fractional derivative terms, which can model the dynamics of a stochastically excited Euler–Bernoulli beam resting on a viscoelastic foundation. Specifically, the original initial–boundary value problem of the SPDE is reduced to an initial value problem of a second-order stochastic differential equation in an appropriate Hilbert space. Next, addressing the abstract Cauchy problem, employing cosine and sine families of operators, and representing the fractional derivative term in a suitable form, a variation of parameters treatment yields the solution in implicit-form. The limiting purely viscous and purely elastic modeling cases are also studied within the same framework. The herein proposed technique and derived implicit-form solutions can be construed as an extension of available results in the literature to account for fractional derivative terms. This generalization is of significant importance given the vast utilization of fractional calculus modeling in engineering mechanics, and in viscoelastic material behavior in particular. In this regard, the herein proposed analytical treatment also supplements existing more numerically oriented solution schemes available in the engineering mechanics literature.

Gradient blow-up for a fourth-order quasilinear Boussinesq-type equation

The Cauchy problem for a fourth-order Boussinesq-type quasilinear wave equation (QWE-4) of the form \begin{document}$u_{tt} = -(|u|^n u)_{xxxx}\;\;\; \mbox{in}\;\;\; \mathbb{R} × \mathbb{R}_+, \;\;\;\mbox{with a fixed exponent} \, \, \, n>0, $ \end{document} and bounded smooth initial data, is considered. Self-similar single-point gradient blow-up solutions are studied. It is shown that such singular solutions exist and satisfy the case of the so-called self-similarity of the second type. Together with an essential and, often, key use of numerical methods to describe possible types of gradient blow-up, a homotopy approach is applied that traces out the behaviour of such singularity patterns as \begin{document}$n \to 0^+$\end{document} , when the classic linear beam equation occurs \begin{document}$u_{tt} = -u_{xxxx}, $ \end{document} with simple, better-known and understandable evolution properties.

Asymptotic profile of solution for the Cauchy problem of beam equation with variable coefficient

We consider the Cauchy problem for the linear beam equation: u t t + u t + u x x x x − a ( t ) u x x = 0 , ( t , x ) ∈ R + × R , where a ( t ) ∼ ( 1 + t ) α . The purpose of this study is to clarify the behavior of solution depending on the rate α . Here we shall give the asymptotic behavior of the solution in the case α > − 1 ∕ 2 , by using the method of scaling variables developed by Gallay and Raugel (1998).

Reducibility of beam equations in higher-dimensional spaces

In this paper, we reduce a linear d-dimensional beam equation with an x-periodic and t-quasi-periodic potential for most values of the frequency vector via the KAM theorem. We focus on the measure estimates of small divisor conditions and the estimation on the coordinate transformation.

KAM for the nonlinear beam equation

In this paper we prove a KAM theorem for small-amplitude solutions of the non linear beam equation on the d-dimensional torus $$u_{tt}+\Delta^2 u+m u + \partial_u G(x,u)=0, \quad t \in {\mathbb{R}}, x \in {\mathbb{T}^d}, \quad (*)$$ where \({G(x,u)=u^4+ O(u^5)}\). Namely, we show that, for generic m, many of the small amplitude invariant finite dimensional tori of the linear equation \({(*)_{G=0}}\), written as the system $$u_t=-v,\quad v_t=\Delta^2 u+mu,$$ persist as invariant tori of the nonlinear equation \({(*)}\), re-written similarly. The persisted tori are filled in with time-quasiperiodic solutions of \({(*)}\). If \({d\ge2}\), then not all the persisted tori are linearly stable, and we construct explicit examples of partially hyperbolic invariant tori. The unstable invariant tori, situated in the vicinity of the origin, create around them some local instabilities, in agreement with the popular belief in the nonlinear physics that small-amplitude solutions of space-multidimensional Hamiltonian PDEs behave in a chaotic way.

Approximation of the controls for the linear beam equation

This article deals with the approximation of the boundary controls of a 1-D linear equation modeling the transversal vibrations of a hinged beam using a finite-difference space semi-discrete scheme. Due to the high frequency numerical spurious oscillations, the semi-discrete model is not uniformly controllable with respect to the mesh size and the convergence of the approximate controls corresponding to initial data in the finite energy space cannot be guaranteed. In this paper we analyze how do the initial data to be controlled and their discretization affect the result of the approximation process. We prove that the convergence of the scheme is ensured if the continuous initial data are sufficiently regular or if the highest frequencies of their discretization have been filtered out. In both cases, the minimal weighted \(L^2\)-norm discrete controls are shown to be convergent to the corresponding continuous one when the mesh size tends to zero.

Existence of Global Strong Solutions to a Beam–Fluid Interaction System

We study an unsteady nonlinear fluid–structure interaction problem which is a simplified model to describe blood flow through viscoelastic arteries. We consider a Newtonian incompressible two-dimensional flow described by the Navier–Stokes equations set in an unknown domain depending on the displacement of a structure, which itself satisfies a linear viscoelastic beam equation. The fluid and the structure are fully coupled via interface conditions prescribing the continuity of the velocities at the fluid–structure interface and the action–reaction principle. We prove that strong solutions to this problem are global-in-time. We obtain, in particular that contact between the viscoelastic wall and the bottom of the fluid cavity does not occur in finite time. To our knowledge, this is the first occurrence of a no-contact result, and of the existence of strong solutions globally in time, in the frame of interactions between a viscous fluid and a deformable structure.

Analytical model for instantaneous lift and shape deformation of an insect-scale flapping wing in hover

In the analysis of flexible flapping wings of insects, the aerodynamic outcome depends on the combined structural dynamics and unsteady fluid physics. Because the wing shape and hence the resulting effective angle of attack are a priori unknown, predicting aerodynamic performance is challenging. Here, we show that a coupled aerodynamics/structural dynamics model can be established for hovering, based on a linear beam equation with the Morison equation to account for both added mass and aerodynamic damping effects. Lift strongly depends on the instantaneous angle of attack, resulting from passive pitch associated with wing deformation. We show that both instantaneous wing deformation and lift can be predicted in a much simplified framework. Moreover, our analysis suggests that resulting wing kinematics can be explained by the interplay between acceleration-related and aerodynamic damping forces. Interestingly, while both forces combine to create a high angle of attack resulting in high lift around the midstroke, they offset each other for phase control at the end of the stroke.

Uniform controllability for the beam equation with vanishing structural damping

This paper is devoted to studying the effects of a vanishing structural damping on the controllability properties of the one dimensional linear beam equation. The vanishing term depends on a small parameter ɛ ∈ (0, 1). We study the boundary controllability properties of this perturbed equation and the behavior of its boundary controls \({v_\varepsilon }\) as ɛ goes to zero. It is shown that for any time T sufficiently large but independent of ɛ and for each initial data in a suitable space there exists a uniformly bounded family of controls \({({v_\varepsilon })_\varepsilon }\) in L2(0, T) acting on the extremity x = π. Any weak limit of this family is a control for the beam equation. This analysis is based on Fourier expansion and explicit construction and evaluation of biorthogonal sequences. This method allows us to measure the magnitude of the control needed for each eigenfrequency and to show their uniform boundedness when the structural damping tends to zero.

Existence results for some higher-order evolution equations with operator coefficients

In this paper we study and obtain the existence of C(r)-pseudo almost automorphic solutions to some higher-order differential equations with operator coefficients whose forcing is C(s)-pseudo almost automorphic. We next use these abstract results to study the existence of C(r)-pseudo almost automorphic solutions to some second-order differential equations in Hilbert space which, among other things, model the so-called linear beam equation.

The formation of shocks and fundamental solution of a fourth-order quasilinear Boussinesq-type equation

As a basic higher-order model, the fourth-order Boussinesq-type quasilinear wave equation (the QWE-4) is considered. Self-similar blow-up solutions are shown to exist that generate as t → T− discontinuous shock waves.The QWE-4 is also shown to admit a smooth (for t > 0) global ‘fundamental solution’ i.e. having a measure as initial data. A ‘homotopic’ limit n → 0 is used to get being the classic fundamental solution of the 1D linear beam equation

A spectral-Tchebychev technique for solving linear and nonlinear beam equations

This paper presents a spectral-Tchebychev technique for solving linear and nonlinear beam problems. The technique uses Tchebychev polynomials as spatial basis functions, and applies Galerkin's method to obtain the spatially discretized equations of motion. Unlike alternative techniques that require different admissible functions for each different set of boundary conditions, the spectral-Tchebychev technique incorporates the boundary conditions into the derivation, and thereby enables the utilization of the solution for any linear boundary conditions without re-derivation. Furthermore, the proposed technique produces symmetric system matrices for self-adjoint problems. In this work, the spectral-Tchebychev solutions for Euler–Bernoulli and Timoshenko beams are derived. The convergence and accuracy characteristics of the spectral-Tchebychev technique is studied by solving eigenvalue problems with different boundary conditions. It is found that the convergence is exponential, and a small number of polynomials is sufficient to obtain machine-precision accuracy. The application of the technique is demonstrated by solving: (1) eigenvalue problems for tapered Timoshenko beams with different boundary conditions, taper ratios, and beam lengths; (2) an Euler–Bernoulli beam problem with spatially and temporally varying forcing, elastic boundary, and damping; (3) large-deflection (nonlinear) Euler–Bernoulli beam problems with different boundary conditions; and (4) a micro-beam problem with nonlinear electrostatic excitation. The results obtained from the spectral-Tchebychev solutions are seen to be in excellent agreement with those presented in the literature.

Numerical and Experimental Studies of Gas Pulsations in the Suction Manifold of a Multicylinder Automotive Compressor

This study predicts gas pulsations in the suction manifold of a multicylinder automotive air-conditioning compressor using a comprehensive simulation model of a reciprocating compressor. On the basis of the first law of thermodynamics and a simplified fourth-order Bernoulli-Euler linear differential beam equation for suction valves, the pressure in a cylinder and resultant pressure pulsation in the suction manifold are predicted. The mass flow rate through the valve is estimated assuming one-dimensional compressible flow through an orifice. All of the equations are then solved together in a sequence to obtain the pressure in the cylinder, valve response, and the mass flow rate. A complicated suction manifold geometry is modeled as a simplified cylindrical annular cavity to study gas pulsations in a multicylinder compressor, but the discharge process has not been considered in this study. Using the calculated mass flow rate, pressure pulsations in a simplified cylindrical annular cavity with an area change to consider “mode splitting” are predicted based on the characteristic cylinder method. It is shown that the simulation code can be a useful tool for predicting gas pulsations in the suction manifold of a multicylinder automotive compressor.

Gas Pulsation Reductions in a Multicylinder Compressor Suction Manifold Using Valve-to-Valve Mass Flow Rate Phase Shifts

This paper investigates the pressure pulsations caused by each mass flow rate through the suction valves and ports of a multicylinder compressor in order to attribute high-pressure pulsation responses to certain valves. By staggering the valve configurations appropriately, it is shown that the level of gas pulsations in the suction manifold of a multicylinder automotive compressor can be reduced. First, the equation for a compression cycle, a Bernoulli-Euler linear differential beam equation for the suction valves, and the piston kinematics are considered in order to calculate the mass flow rates through the compressor suction valves. The pressure pulsations in the suction manifold are then predicted based on the characteristic cylinder method using the calculated mass flow rates. In order to investigate the effects of each mass flow rate, the characteristics and phases of the mass flow rates through the suction valves are changed by modifying the clearance volume.