One-dimensional Beam Equation Research Articles

Exploring player’s feelings on clarinet mouthpiece geometry using artificial blowing machines and airflow simulations

The mouthpiece geometry of single-reed instruments has been known to have a great impact on sound quality and playability. Although many researchers have investigated the effects of the geometry using artificial blowing machines or user case studies, few attention has been paid to the essential cause of physical changes in the instrument. In this study, we propose a method combining the artificial blowing machine and numerical flow simulations to explore the player’s feelings about the airflow and sound generation in the clarinet mouthpiece. With the artificial blowing machine, the mouth pressure, as well as lip force, were systematically changed using a pressure regulator and strain gauge. At particular conditions, the numerical flow simulation was conducted by solving fluid–structure interactions with the Navier–Stokes equations and the one-dimensional beam equation. The playability of the mouthpiece was shown by the changes in oscillation threshold pressures and sound spectral characteristics, whereas the cause of the changes could be described by the airflow pressures inside the mouthpieces. The results suggest that the combination of the artificial blower and the flow simulations can be effectively used for clarifying the cause of changes in players’ feelings.

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.

A fully coupled fluid–structure–acoustic interaction simulation on reed-type artificial vocal fold

The sound generation mechanisms of vocal folds’ self-induced oscillations have been investigated using vocal fold replicas. In this study, a compressible flow simulation coupled with a one-dimensional beam equation was conducted to study a reed-type artificial vocal fold. The vibrating reed was expressed by the volume penalization method in the aeroacoustic fields. The sound of the artificial vocal fold was measured in a flow experiment and compared with the simulation. The results showed that the predicted acoustic characteristics agreed well with those measured in the experiment. The flow fields during reed oscillation indicated that the rapid change in the flow rate downstream from the reed formed a strong source in the high-frequency harmonics.

Positive Solutions of a Discontinuous One-Dimensional Beam Equation

We provide sufficient conditions for the existence of one positive solution for a fourth--order beam equation with a discontinuous nonlinear term. Also a multiplicity result is established. They are based on a recent generalization of the Krasnosel’skiĭ fixed point theorem in cones.

Approximate Analytical Solution of One-Dimensional Beam Equations by Using Time-Fractional Reduced Differential Transform Method

In this paper, a recent and reliable method, named the fractional reduced differential transform method (FRDTM) is employed to solve one-dimensional time-fractional Beam equation subject to the appropriate initial conditions. This method provides the solutions very accurately and efficiently in convergent series form with easily computable coefficients. The efficacy and accuracy of this method are verified by means of three illustrative examples which indicate that the present method is very effective, simple, and easy to implement. Finally, it is observed that the FRDTM is the prevailing and convergent method for the solutions of linear and nonlinear fractional-order partial differential equations.

Optimal Actuator Placement and Static Load Compensation for Euler-Bernoulli Beams with Spatially Distributed Inputs

Adaptive structures in civil engineering are based on active structural elements. In this paper, an active beam element is considered with potentially multiple fluidic inputs spatially distributed over the beam’s length. It is modeled by a one-dimensional beam equation. The inputs are realized as torque inputs, where the torque is proportional to the pressure of the fluid. A method to calculate the optimal input values analytically depending on a given static load case is presented in the first part of the paper. In the second part, by generalizing the static load using a sufficient number of discrete loads evenly distributed along the beam, optimal actuator positions can be calculated for a given number of actuators. The cost function is based on extending the idea of the Gramian compensability matrix to systems governed by spatially distributed ordinary differential equations. Since the number of actuators can be a design variable, this yields a Pareto front supporting the user in selecting an appropriate number. A fluidic actuator serves as example to illustrate the potential of the proposed placement algorithm.

On the nonlinear extension-twist coupling of beams

Abstract In many applications, beams carry large axial forces, such as centrifugal forces, for instance, and the resulting axial stresses affect their torsional behavior. To capture this important coupling effect within the framework of beam models, a nonlinear theory must be developed that takes into account higher-order strain deformations. The proposed development starts with a three-dimensional elasticity model of beam-like structures. While strain components are assumed to remain small, the strain energy expression is expanded to retain strain components up to cubic order. This expansion of the three-dimensional model leads to material and geometric stiffness matrices. For linear problems, the three-dimensional warping fields induced by each of the six unit sectional strain components can be obtained. These warping fields are used to reduce the three-dimensional elasticity model to one-dimensional beam equations. Byproducts of this reduction process include the sectional stiffness matrix and six nonlinear stiffness matrices representing the geometric stiffness induced by unit sectional strains. It is shown that the nonlinear extension-twist coupling is captured by these geometric stiffness matrices. Comparison of the predictions of the proposed approach with analytical and experimental results demonstrate their effectiveness and accuracy.

Stability Analysis of Columns with Imperfection

In this paper, geometrically exact fully intrinsic equations and variational asymptotic beam sections are used to study the buckling and postbuckling of a column under self-weight. Initial curvature effects both generalized one-dimensional beam equations and the cross-sectional stiffness properties. The effects of initial curvature have been studied on buckling and postbuckling in this paper. This problem is a valuable test case for the utility of the fully intrinsic equations and variational asymptotic beam sections.

On the maximum and antimaximum principles for the beam equation

We consider a one-dimensional beam equation with a strictly positive forcing term. Under different assumptions on the restoring force we prove the existence of positive and negative solutions, respectively. Our method is based on non-monotone but convergent iterations and extends related results known in the literature so far.

Positive and negative solutions of one-dimensional beam equation

Abstract In this paper, we show that the usual limitations on the coefficient c = c ( x ) in the linear problem u ( 4 ) + c ( x ) u = h ( x ) with Navier boundary conditions and nonnegative right hand side h are not necessary to get the existence of positive or negative solutions whenever c ( x ) is a nonconstant function.

Global attractors for nonlinear beam equations

This paper is concerned with the dynamics of nonlinear one-dimensional beam equations. We consider nonlinear beam equations with viscosity or with a lower-order damping term instead of the viscosity, and we establish the existence of global attractors for both systems.

The effect of taper on section constants for in-plane deformation of an isotropic strip

The variational-asymptotic method is used to obtain an asymptotically-exact expression for the strain energy of a tapered strip-beam. The strip is assumed to be sufficiently thin to warrant the use of twodimensional elasticity. The taper is represented by a nondimensional constant of the same order as the ratio of the maximum cross-sectional width to the wavelength of the deformation along the beam, and thus its cube is negligible compared to unity. The resulting asymptotically-exact section constants, being functions of the taper parameter, are then used to find section constants for a generalized Timoshenko beam theory. These generalized Timoshenko section constants are then used in the associated one-dimensional beam equations to obtain the solution for the deformation of a linearly tapered beam subject to pure axial, pure bending, and transverse shear forces. These beam solutions are then compared with plane stress elasticity solutions, developed for extension, bending, and flexure of a linearly tapered isotropic strip. The agreement is excellent, and the results show that correction of the section constants using the taper parameter is necessary in order for beam theory to yield accurate results for a tapered beam.

Local Controllability of a One-Dimensional Beam Equation

We prove that the beam equation with clamped ends is locally controllable in a $H^{5+\epsilon} \times H^{3+\epsilon} ((0,1),\mathbb{R})$-neighborhood of a particular trajectory of the free system, with $\epsilon>0$ and with control functions in $H^{1}_{0}((0,T),\mathbb{R})$. Ball,. Marsden, and Slemrod already proved that this equation is not controllable in $H^{2}_{0} \times L^{2}((0,1),\mathbb{R})$ with control functions in $L^{r}_{\text{loc}}(\mathbb{R},\mathbb{R})$, $r>1$. This article justifies that their negative result is due to a choice of functional spaces which does not allow controllability. Our proof uses moment theory and the Nash-Moser theorem.

Stabilization of a hybrid system fo noise reduction

We consider a simple model arising in the control of noise. We assume that the two-dimensional cavity Ω =(0,1)x(0,1) is occupied by an elastic, inviscid, compressible fluid. The potential Φ of the velocity field satisfies the linear wave equation. The boundary of Ω is divided in two parts Γ0 and Γ1. The first one, Γ0 is flexible and occupied by a Bernoulli-Euler beam. On Γ0 the continuity of the normal velocities of the fluid and the beam is imposed. The subset Γ1 of the boundary is assumed to be rigid and therefore, the normal velocity of the fluid vanishes. A dissipative term is assumed to act in the one-dimensional beam equation. We prove the existence and the uniqueness of solutions. Each trajectory is proved to converge to an equilibrium as t→∞. On the other hand we show that the convergence rate of the energy is not exponential. The proof of this r esult uses a perturbation argument allowing to modify the boundary conditions so that separation of variables applies.

Simple formula for dynamic stress intensity factor of pre-cracked Charpy specimen

It has been reported that consideration of inertia effect is important to estimate the stress intensity factor for impact loading. In the present paper a simple expression for the dynamic stress intensity factor of a pre-cracked Charpy specimen is derived by making use of an approximate solution for one-dimensional beam equations. The time variation of the dynamic stress intensity factor obtained retaining only the fundamental mode of bending vibration is in good agreement with the two-dimensional finite element solutions for step-loading and also for final-peak-sawtooth-pulse loading.

Note on a Nontrivial Simple Example of Higher-Order One-Dimensional Beam Theory

Abstract : One-dimensional beam equations are derived for a rectangular-cross section beam consisting of shear webs and corner stringers with one cross-sectional axis of symmetry. It is shown that a systematic statement of this problem involves one more equation than elementary theory, with this additional equation involving the concept of cross-sectional warping and bi-moment. As an application of the general results of the paper, new conclusions are deduced concerning the location of the center of shear of the beam. (Author)