Some Theoretical and Experimental Extensions Based on the Properties of the Intrinsic Transfer Matrix

The work approaches new theoretical and experimental studies in the elastic characterization of materials, based on the properties of the intrinsic transfer matrix. The term ‘intrinsic transfer matrix’ was firstly introduced by us in order to characterize the system in standing wave case, when the stationary wave is confined inside the sample. An important property of the intrinsic transfer matrix is that at resonance, and in absence of attenuation, the eigenvalues are real. This property underlies a numerical method which permits to find the phase velocity for the longitudinal wave in a sample. This modal approach is a numerical method which takes into account the eigenvalues, which are analytically estimated for simple elastic systems. Such elastic systems are characterized by a simple distribution of eigenmodes, which may be easily highlighted by experiment. The paper generalizes the intrinsic transfer matrix method by including the attenuation and a study of the influence of inhomogeneity. The condition for real eigenvalues in that case shows that the frequencies of eigenmodes are not affected by attenuation. For the influence of inhomogeneity, we consider a case when the sound speed is varying along the layer’s length in the medium of interest, with an accompanying dispersion. The paper also studies the accuracy of the method in estimating the wave velocity and determines an optimal experimental setup in order to reduce the influence of frequency errors.

AIMED CONTROL OF THE FREQUENCY SPECTRUM OF EIGENVIBRATIONS OF ELASTIC PLATES WITH A FINITE NUMBER OF DEGREES OF MASS FREEDOM BY INTRODUCING ADDITIONAL GENERALIZED KINEMATIC DEVICES

As is known, targeted regulation of the frequency spectrum of natural vibrations of elastic systems with a finite number of degrees of mass freedom can be performed by introducing additional generalized constraints and generalized kinematic devices. Each targeted generalized constraint increases, and each generalized kinematic device reduces the value of only one selected natural frequency to a predetermined value, without changing the remaining natural frequencies and all forms of natural vibrations (natural modes). To date, for some elastic systems with a finite number of degrees of freedom of masses, in which the directions of mass movement are parallel and lie in the same plane, special methods have been already developed for creating additional constraints and generalized kinematic devices that change the frequency spectrum of natural vibrations in a targeted manner. In particular, a theory and an algorithm for the creation of targeted generalized constraints and generalized kinematic devices have been developed for rods. It was previously proved that the method of forming a matrix of additional stiffness coefficients, specifying targeted generalized constraint, in the problem of natural vibrations of rods can also be applied to solving a similar problem for elastic systems with a finite number of degrees of freedom, in which the directions of mass movement are parallel, but do not lie in the same plane. In particular, such systems include plates. The distinctive paper shows that the method of forming a matrix for taking into account the action of additional inertial forces, specifying targeted kinematic devices in the problem of natural vibrations of rods can also be applied to solving a similar problem for elastic systems with a finite number of degrees of freedom, in which the directions of mass movement are parallel, but do not lie in the same plane. However, the algorithms for the creation of targeted generalized kinematic devices developed for rods based on the properties of rope polygons cannot be used without significant changes in a similar problem for plates. The method of creation of computational schemes of kinematic devices that precisely change the frequency spectrum of natural vibrations of elastic plates with a finite number of degrees of mass freedom is a separate problem and will be considered in a subsequent paper.

An energy efficiency index for elastic actuators during resonant motion

The energetic advantages of series and parallel elastic actuators have been characterized in the literature considering different elastic systems and different tasks. These characterizations usually determine the energy consumption of a specific system during a specific task and generalize poorly. This paper proposes an energetic characterization of elastic actuators, following an analytical approach, rather than a data-driven one. In particular, this work analyzes the energy consumption of elastic actuators during resonant motion and introduces a novel efficiency index. This index characterizes energy consumption as a function of inherent actuator parameters only, generalizing over the specific tasks. The proposed analysis is validated using simulations and experiments, demonstrating its coherence with analytical results.

AIMED CONTROL OF THE FREQUENCY SPECTRUM OF EIGENVIBRATIONS OF ELASTIC PLATES WITH A FINITE NUMBER OF DEGREES OF FREEDOM OF MASSES BY SUPERIMPOSING ADDITIONAL CONSTRAINTS

As is known, for some elastic systems with a finite number of degrees of freedom of masses, for which thedirections of motion of the masses are parallel and lie in the same plane, methods have been developed for creatingadditional constraints that purposefully change the spectrum of natural frequencies. In particular, theory and algorithm forthe formation of aimed additional constraints have been developed for the rods, the introduction of each of which doesnot change any of the modes of natural vibrations, but only increases the value of only one frequency, without changingthe values of the remaining frequencies. The distinctive paper is devoted to the method of forming a matrix of additionalstiffness coefficients corresponding to such aimed constraint in the problem of natural vibrations of rods. This method canalso be applied to solving a similar problem for elastic systems with a finite number of degrees of freedom, in which thedirections of motion of the masses are parallel, but not lie in the same plane. In particular, such systems include plates.However, the algorithms for the formation of aimed additional constraints, developed for rods and based on the propertiesof rope polygons, cannot be used without significant changes in a similar problem for plates. The method for the formationof design constraint schemes that purposefully change the spectrum of frequencies of natural vibrations of elastic plateswith a finite number of degrees of freedom of masses, will be considered in the next work.

Puckering and wrinkling in a growing composite ring

Pattern formation driven by differential strain in constrained elastic systems is a common motif in many technological and biological systems. Here we introduce a biologically motivated case of elastic patterning that allows us to explore the conditions for the existence of local puckering and global wrinkling patterns: a soft growing composite ring adhered elastically to a constraining rigid ring. We explore how differential growth of the soft ring and the elastic resistance to shear and stretching deformations induced by soft adherence lead to a range of phenomena that include uniform aperture-like modes, localized puckers that are Nambu–Goldstone-like modes and global wrinkles in the system. Our analysis combines computer simulations of a discrete rod model with a nonlinear stability analysis of the differential equations in the continuum limit. We provide phase diagrams and scaling relations that reveal the nature and extent of the deformation patterns. Overall, our study reveals how geometry and mechanics conspire to yield a rich phenomenology that could serve as a guide to the design of programmable localized elastic deformations while being relevant for the mechanical basis of biological morphogenesis.

Vibration analysis of nonlinear and linear elastic systems

Objective. In this study, the task is to establish the theoretical prerequisites for the operability of a regressive-progressive elastic mechanism by comparing the amplitude-frequency characteristics and phase trajectories with a linear elastic system of comparable stiffness in a static equilibrium position.Methods. The article presents a comparative dynamic analysis of vibrations of elastic systems with linear rigidity and regressive-progressive characteristics obtained as a result of the use of elastic elements in the form of high flexibility rods with longitudinal eccentric compression. Such elastic elements in various design variants have been tested and patented as damping elements for use in the construction of vibration dampers for construction structures and vehicle suspensions, and have experimentally shown their effectiveness in damping vibrations.Results. The regressiveprogressive elastic characteristic obtained by the elliptic parameters method and using the ANSIS calculation complex is used in the dynamics equations in an approximated form, which expands the capabilities of the method. It is shown that increasing the energy intensity of a curvilinear system reduces the vibration amplitude.Conclusion. The regressive-progressive change of the stiffness of curvilinear elastic systems can be achieved using an elastic element with eccentric longitudinal compression; the regression plot of elastic properties is achieved due to eccentric compression; the progressive plot – through the use of a guide or other design solutions. The implementation of this characteristic allows using such elastic mechanisms in systems where the accumulation of potential energy occurs with a smaller compression stroke for the same perturbation than for linear systems.

Asymptotic finite-dimensional approximations for a class of extensible elastic systems

<abstract><p>We consider the equation</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ u_{tt}+\delta u_t+A^2u+{\lVert{A^{\theta/2} u}\rVert}^2A^\theta u = g $\end{document} </tex-math></disp-formula></p> <p>where $ A^2 $ is a diagonal, self-adjoint and positive-definite operator and $ \theta \in [0, 1] $ and we study some finite-dimensional approximations of the problem. First, we analyze the dynamics in the case when the forcing term $ g $ is a combination of a finite number of modes. Next, we estimate the error we commit by neglecting the modes larger than a given $ N $. We then prove, for a particular class of forcing terms, a theoretical result allowing to study the distribution of the energy among the modes and, with this background, we refine the results. Some generalizations and applications to the study of the stability of suspension bridges are given.</p></abstract>