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Analogous electrical networks were previously derived from the Euler–Bernoulli and Kirchhoff–Love theories to represent beams and plates, respectively, for use in multimodal structural vibration damping. However, these networks do not account for shear deformations or rotary inertia, which can result in suboptimal vibration damping performance when used on moderately thick beams and plates. In this paper, we investigate the incorporation of shear deformation and rotary inertia using Timoshenko–Ehrenfest beam theory and Mindlin–Reissner plate theory to develop improved electrical networks that can more accurately represent thick beams and plates. Our findings suggest that the inclusion of shear deformation and rotary inertia can significantly improve the frequency coherence of the electrical networks and multimodal vibration damping for thicker structures. The electrical analogs presented here are of use for various applications, especially to conveniently design complex circuit topologies in fields spanning from vibration attenuation to energy harvesting. [ABSTRACT FROM AUTHOR]

In this paper, the dynamics of a taut horizontal string with a constant velocity moving mass including its rotary inertia is modelled. The equation of motion is solved using Galerkin’s approach, employing appropriate comparison functions. A discontinuity or jump in the trajectory of the mass has been established when the mass is about to leave the string. The consideration of rotary inertia in the model does affect the spatial location of the jump in the trajectory of the moving mass. [ABSTRACT FROM AUTHOR]

Objective: The purpose of this work is to investigate the size-dependent free vibrations of carbon nanotube-reinforced nanowires/nanobeams with movable ends. For this purpose, two movable end conditions are considered for carbon nanotube-reinforced nanowires/nanobeams with different carbon nanotube distributions. The size effect is addressed with the modified couple stress theory, which includes a material length scale parameter. Methods: In this study, a solution approach based on the Fourier sine series and Stokes' transformation is used. With the help of this approach, both rigid and movable end conditions can be investigated. Firstly, equation sets consisting of infinite series and movable end parameters are derived. Then, eigenvalue problems are obtained for the free vibration of carbon nanotube-reinforced nanowires/nanobeams. The eigenvalues of these problems give the frequencies of the carbon nanotube-reinforced nanowires/nanobeams with movable ends. Conclusions: The eigenvalue problems obtained in this study can be used to examine both rigid and movable end conditions. The accuracy of the problems obtained for solution is proven by various comparison studies. Then, a series of analyses are carried out for nanowire/nanobeam reinforced with carbon nanotubes, including both the size effect and the rotary inertia effect. When the frequencies of nanowire/nanobeams reinforced with carbon nanotubes are examined, it is understood that the material length scale parameter, carbon nanotube distribution, carbon nanotube volume fraction, rotational and lateral spring parameters can cause significant changes in free vibration. [ABSTRACT FROM AUTHOR]

A hierarchy of asymptotic models governing long-wave low-frequency in-plane motion of a fluid-loaded elastic layer is established. In contrast to a layer with traction-free faces, modelled by Neumann boundary conditions, a fluid-loaded one assumes more involved conditions along the interfaces, dictating a special asymptotic scaling. The latter corresponds to a fluid-borne bending wave, controlled by elastic stiffness of the layer and fluid inertia. In this case, the transverse inertia of the layer and fluid compressibility do not appear at zero-order approximation. The first-order approximation is associated with a Kirchhoff plate, immersed into incompressible fluid. In the studied free vibration setup, the fluid compressibility has to be taken into account only at third order, along with elastic rotary inertia. Transverse shear deformation enters the second-order approximation along with a few other corrections. The conventional impenetrability condition has to be also refined at second order. Dispersion relations corresponding to the developed asymptotic models are compared with the polynomial expansions of the full dispersion relation, obtained from the plane-strain problem of linear elasticity. [ABSTRACT FROM AUTHOR]

An efficient method has been developed to analyze the natural frequencies of a Timoshenko-Ehrenfest beam with variable cross-sections undergoing free flexural vibration. The method uses a single element to discretize the beam and combines polynomial and trigonometric shape functions to represent the displacement fields, with the trigonometric shape functions representing the internal degrees of freedom (DOF) of the beam. By incorporating trigonometric shape functions as enriching functions alongside polynomials, the potential issues related to ill-conditioning associated with higher-order polynomials are avoided. The differential equations of motion for the free vibration are derived using Lagrange's equation. The generalized eigenvalue problem is then formulated. Non-dimensional natural frequencies are computed considering different parameter variations such as taper ratios, shear deformation, rotary inertia parameters, as well as end conditions. Incorporating rotary inertia and shear deformation in the analysis of Timoshenko-Ehrenfest beam lowers its natural frequencies compared to simpler beam theories like Euler–Bernoulli. It is observed that as taper ratios increase, the beam's mass decreases toward the tip, causing higher frequency parameter values for the fundamental mode. Increasing rotary inertia parameter leads to a decreasing trend in frequency parameters especially in higher modes. The combined effect of rotary inertia and tapering significantly reduces frequency parameter values for all modes, with the rotary inertia counteracting the tapering influence. The proposed Fourier-p element model accurately predicts Timoshenko-Ehrenfest beam frequencies with high precision compared to other existing models. [ABSTRACT FROM AUTHOR]

Photovoltaic power generation and photothermal power generation are two forms of solar power generation. The development cost of photovoltaic is relatively lower than photothermal, but a certain proportion of energy storage and distributed synchronous condensers are needed to ensure the safe and stable operation of the system. As a result the cost of photovoltaic power generation rises. Photothermal comes with heat storage and output friendliness, so currently there is still a great dispute about the technical route of photovoltaic and photothermal. This paper takes Haixi area of Qinghai province as an example. Firstly, based on the characteristics of solar irradiation resources and output in the same time and space, the power guarantee capabilities of photovoltaic+energy storage and photothermal is compared. Then this paper compares photovoltaic+energy storage+condenser with photothermal from the perspective of power grid security and stability support ability. Finally, this paper compares the economics of solar thermal and photovoltaic based on the forecast of their cost declining trend. Photothermal has both thermal storage and conventional generator functions, providing short circuit capacity, reactive power compensation and moment of inertia support for the system. The results show the economy of photothermal is basically equivalent to, or even better than photovoltaic+energy storage+condenser after the cost reduction in the case of technical equivalence.

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Critical velocities of a two-layer composite tube subjected to a uniform internal pressure moving at a constant velocity are analytically derived by using a first-order shear deformation shell theory incorporating the transverse shear, rotary inertia and material anisotropy. The composite tube consists of two perfectly bonded axisymmetric circular cylindrical layers of dissimilar materials, which can be orthotropic, transversely isotropic, cubic or isotropic. Closed-form expressions for four critical velocities are first derived for the general case by including the effects of transverse shear, rotary inertia, material orthotropy and radial stress. The formulas for composite tubes without the transverse shear, rotary inertia or radial stress effect and with simpler anisotropy are then obtained as special cases. In addition, it is shown that the model for a single-layer, homogeneous tube is included in the current model as a special case. To illustrate the newly derived closed-form formulas, a composite tube with an isotropic inner layer and an orthotropic outer layer is analyzed as an example. The numerical values of the lowest critical velocity of the two-layer composite tube predicted by the new formulas compare well with existing data. [ABSTRACT FROM AUTHOR]

In this study, the transverse vibration of free–free slender beams with two unequal end masses attached were studied. The effects of the rotary inertia of the end masses on the free vibration of the beam were investigated. An exact frequency equation and the boundary conditions were obtained by using the Euler–Bernoulli beam theory and Hamilton's principle. Natural frequencies and mode shapes of the beams in transverse vibrations were calculated for various combinations of physical and geometrical parameters, such as mass ratios, the distances between the attachment point and the center of the masses, etc. The effects of an increase in the rotational inertia of the end masses and different mass ratios on the natural frequencies and mode shapes of the beam are presented. It is shown that the increase in the rotational inertia of the end masses had a greater effect at low frequencies of the beam. In addition, experimental tests were performed to validate the obtained analytical results. A good agreement was obtained between the analytical and experimental results. The main scope of this study was to reveal the effects of the rotary inertia of the end masses on the dynamic behavior of the beam. Thus, the aim is to contribute to the understanding of the properties of the end mass and the effect of rotary inertia on the dynamics of end-mass-attached structures. Furthermore, the results obtained from this research are helpful for designing end-mass-attached structures, such as micromechanical sensors, energy harvesters, and Stockbridge-type dynamic absorbers. [ABSTRACT FROM AUTHOR]

The propagation of surface gravity waves in the presence of finite floating elastic plate over varying sea bottom profile is investigated using the Timoshenko-Mindlin plate theory. The continuity of velocity and pressure at the interfaces along with the continuity of deflection, slope, bending moment and shear force is employed with the support conditions at the plate edges. The numerical computation is performed to obtain the hydroelastic behaviour of the floating elastic plate due to abrupt change in bottom topography. The validation of the present analytical model is performed with the known results available in the literatures. Further, a detailed comparison of the numerical results is presented for different step bottom topography on the hydroelastic characteristics of a floating elastic platform. The present study will provide an insight into the effect of the ocean bottom profile on wave propagation due to the presence of a large floating elastic plate. [ABSTRACT FROM AUTHOR]

The attenuation of the incident wave interacting with a very large floating structure (VLFS) in the presence of vertical barriers is analysed considering small amplitude wave theory. The VLFS is considered to be articulated and is modelled based on Timoshenko-Mindlin plate theory. The eigenfunction expansion method along with the orthogonal mode-coupling relation is employed for the case of finite water depth. The numerical study is performed to analyse the wave reflection, transmission and dissipation characteristics due to the articulated floating plate for the case of bottom standing and surface piercing vertical porous barriers. The hydroelastic behaviour in terms of deflection and strain for an articulated floating thick elastic plate in the presence of porous barriers is analysed. The study reveals that the magnitude of wave attenuation is enhanced due to the presence of vertical porous barriers and also provides an understanding in mitigating the structural response. [ABSTRACT FROM AUTHOR]

In this study, the transverse vibration of free–free slender beams with two unequal end masses attached were studied. The effects of the rotary inertia of the end masses on the free vibration of the beam were investigated. An exact frequency equation and the boundary conditions were obtained by using the Euler–Bernoulli beam theory and Hamilton’s principle. Natural frequencies and mode shapes of the beams in transverse vibrations were calculated for various combinations of physical and geometrical parameters, such as mass ratios, the distances between the attachment point and the center of the masses, etc. The effects of an increase in the rotational inertia of the end masses and different mass ratios on the natural frequencies and mode shapes of the beam are presented. It is shown that the increase in the rotational inertia of the end masses had a greater effect at low frequencies of the beam. In addition, experimental tests were performed to validate the obtained analytical results. A good agreement was obtained between the analytical and experimental results. The main scope of this study was to reveal the effects of the rotary inertia of the end masses on the dynamic behavior of the beam. Thus, the aim is to contribute to the understanding of the properties of the end mass and the effect of rotary inertia on the dynamics of end-mass-attached structures. Furthermore, the results obtained from this research are helpful for designing end-mass-attached structures, such as micromechanical sensors, energy harvesters, and Stockbridge-type dynamic absorbers.

Flywheel resistance training devices can be classified by their shaft shape. The objective of this study was to analyze whether using two flywheel resistance training devices shaft shapes can influence force and velocity production, regardless of the inertia used. Thirty-nine (n = 39) healthy active men participated in this study. They were randomized to perform 3 sets of 7 repetitions at maximal concentric voluntary execution, followed by a break in the last third of the eccentric phase in the half squat exercise. A progressive rotational inertial setting of 0.11, 0.22, and 0.33 kg·m2 was used. Force- and velocity-time profiles were captured using two force plates and a synchronized linear encoder. Statistical parametric mapping was used to compare biomechanical output between the flywheel devices. The level of significance was set at p < 0.05. Force application was significantly higher in the horizontal cylinder-shaped device for the three moments of inertia used in the eccentric phase (p < 0.001). In the concentric phase, force application was significantly higher in the horizontal cylinder-shaped device in 0.11 (p < 0.001) and 0.22 kg·m2 (p < 0.001). The resultant speed was higher in the vertical cone-shaped device in the concentric phase and the eccentric phase for the three moments of inertia (p < 0.001). In conclusion, the flywheel shaft type determines the mechanical output of the half squat exercise, regardless of the moment of inertia used. While a horizontal cylinder-shaped device is more suitable to achieve higher forces, especially in the eccentric phase, a vertical cone-shaped device can be used to achieve higher speeds during the execution of the exercise. [ABSTRACT FROM AUTHOR]

Analogous piezoelectric networks have been shown to be effective for multimodal vibration attenuation in structures, including beams, plates, and rings. Previous studies for rings have only accounted for in-plane transverse vibration attenuation and disregarded the out-of-plane vibration modes. Furthermore, these previous numerical models and experiments have only been studied on thin rings, which ignore the effects of shear deformation and rotary inertia. As a result, these networks are not suitable for attenuating vibrations in thick rings. This study enhances on the previous electrical networks, considering both shear deformation and rotary inertia for both in-plane and out-of-plane vibrations. A new passive network topology is developed for the out-of-plane dynamics of thick rings, and the existing passive analogous network of in-plane vibration of thin rings is enhanced by considering the effects of shear deformation and rotary inertia. Combined, these new networks are capable of multimodal vibration damping of a thick ring in three-dimensions, encompassing primarily six types of vibration modes: the inextensional bending modes, the extensional modes, the thickness-shear modes, the coupled twist-bending modes, the torsional modes, and the transverse thickness-shear modes. By using piezoelectric elements to couple two separate analogous passive electrical networks derived from both the in-plane and out-of-plane governing equations of a ring and optimizing the internal resistance in each unit cell, it becomes possible to replicate the dynamics and effectively attenuate different types of vibration modes. This study serves as a theoretical foundation for implementations of passive vibration attenuation in ring structures. [Display omitted] • Passive electrical networks are designed to mimic ring dynamics for in-plane and out-of-plane displacements. • Adding shear stiffness and rotary inertia analogous effects in ring electrical networks enhance frequency coherence. • Electrical networks coupled to a piezoelectric ring induce significant broadband damping for different types of modes. [ABSTRACT FROM AUTHOR]

Free vibration analysis of a twisted, double-tapered blade mounted on a rotating disk undergoing overall motion is presented here. The Lagrangian approach is adapted to study the modal characteristics of the blade modeled as a rotating Rayleigh beam. The expressions for the kinetic energy and potential energy of the cantilever blade are derived using hybrid deformation variables. The continuous deformation variables in these equations are discretized using a series of basis functions that satisfy all boundary conditions of the cantilever beam. The equations governing the coupled stretch–bending–torsion motion of the rotating blade are derived using Lagrange's approach. The equations are then transformed into a non-dimensional form which are then solved for the eigenvalue problem for the modal characteristics of the blade. The results of the present model are verified with the results available in the literature. The variation of the natural frequencies with the rotating speed, taper ratio and pre-twist angle is presented. The tuned angular speed of the blade at which the angular frequency matches with any of the natural frequency of the blade resulting in the resonance is investigated. The Campbell diagram is plotted for the specific problem to identify the resonance where the natural frequency matches with the harmonics of the rotating speed. [ABSTRACT FROM AUTHOR]

In this paper, the natural vibration characteristics of elastically supported functionally graded material plate are investigated using the dynamic stiffness method (DSM). Power-law functionally graded (P-FG) plate, the material properties of which vary smoothly along the thickness direction following the power-law function, that has been used for the analysis. Classical plate theory and Hamilton's principle are used for deriving the governing differential equation of motion and associated edge conditions for P-FG plate supported by elastic foundation. During the formulation of dynamic stiffness (DS) matrix, the concepts of rotary inertia and neutral surface are implemented. Wittrick–Williams (W-W) algorithm is used as a solving technique for the DS matrix to compute eigenvalues. The results thus obtained by DSM for the isotropic, P-FG plate, and the P-FG plate with elastic foundation compare well with published results that are based on different analytical and numerical methods. The comparisons indicate that this approach is very accurate. Furthermore, results are provided for elastically supported P-FG plate under four different considerations in order to see the differences in frequencies with the inclusion or exclusion of neutral surface and/or rotary inertia. It is noticed that the inclusion of rotary inertia and neutral surface influences the eigenvalues of P-FG plate, and that cannot be discounted. The study also examines the influence of plate geometry, material gradient index, edge conditions, and elastic foundation modulus on the natural frequency of P-FG plate. [ABSTRACT FROM AUTHOR]

This paper analyzes the free vibration frequencies of a beam on a Winkler–Pasternak foundation via the original Timoshenko–Ehrenfest theory, a truncated version of the Timoshenko–Ehrenfest equation, and a new model based on slope inertia. We give a detailed comparison between the three models in the context of six different sets of boundary conditions. In particular, we analyze the most common combinations of boundary conditions deriving from three typical end constraints, namely the simply supported end, clamped end, and free end. An interesting intermingling phenomenon is presented for a simply-supported (S-S) beam together with proof of the 'non-existence' of zero frequencies for free-free (F-F) and simply supported-free (S-F) beams on a Winkler–Pasternak foundation. [ABSTRACT FROM AUTHOR]

Establishing an accurate dynamic model for gear transmission is challenging because of the complex dynamic mechanical behavior of the gear transmission, particularly under difficult working conditions such as heavy loads and frequent starts and stops. This feature significantly increases the difficulty level of the dynamic analysis, control, and real-time fault diagnosis of such a gear transmission, given that sufficiently accurate dynamic models are indispensable for them. Thus, this study proposes a piecewise nonlinear model for gear transmissions by considering rotary inertia, backlash, and friction in control applications under difficult working conditions. The model parameters of gear transmissions can be calculated by using analytical calculation formulas. The experiments are conducted for the gear transmission system of a heavy-duty manipulator in start-stop and continuous rotation conditions. The experimental results show that the proposed piecewise nonlinear model is in good agreement with the practical dynamic gear transmission behavior, which is feasible for the control applications of gear transmissions under difficult working conditions.

The paper theoretically and experimentally analyzes free vibration characteristics of statically loaded moving boundary type curved beam considering rotary inertia and shear deformation effects. Effects of rotary inertia and shear deformation are observed for different thickness to span ratios of curved beam. The subject problem is decoupled into two interrelated problems: determining equilibrium configuration under static load and finding the corresponding free vibration frequency. The static problem is analyzed incrementally in body fitted curvilinear frame as it involves geometric nonlinearity due to generalized curvature, large deformation, and moving boundaries. Variational energy principle is employed to derive governing equation. The nonlinear governing equation associated with complicated boundary conditions is solved through iterative geometry updation. Once static problem is solved for current load step, governing equation for dynamic characteristics is derived using Hamilton's principle. The governing equation gets linearized by using the static configuration, which finally yields a linear eigenvalue problem. Experiment is performed in a dedicated setup with two master leafs having different thickness to span ratios. The roller supported specimens are excited with an instrumented hammer and response signals are captured by accelerometers. The excitation and response signals are recorded using HBM-MX840B data acquisition system. Frequency response functions of the curved beam systems under different static loads are obtained from postprocessing of the dynamic signals in MATLAB®. First two natural frequencies of the specimens are noted from the experimental results and the corresponding theoretical results are generated. The specimens are also modeled in ABAQUS® CAE and finite element results are computed. Comparison between the theoretical, experimental, and finite element results validates the present model. The study also provides some meaningful observations on effects of rotary inertia and shear deformation. Based on the observations, more results are generated for different thickness to span ratios and findings are reported suitably. [ABSTRACT FROM AUTHOR]

In this work, expressions for estimating the modal density, speed of the bending wave, critical frequency and coincidence frequency of panels are derived considering orthotropic properties of the face sheets, transverse shear deformation and the rotary inertia. Presence of rotary inertia results in an increase in the modal density and a reduction in the speed of the bending waves. The influence is significant at higher frequencies. The critical and coincidence frequencies increase due to rotary inertia. Results for a typical equipment panel of spacecraft are presented and they show the need for incorporating rotary inertia while determining these parameters. [ABSTRACT FROM AUTHOR]

New traveling wave solutions are presented for motion of an inextensible, unshearable, planar Kirchhoff beam endowed with rotary inertia and a generalized strain energy function for bending which models nonlinear stiffening and softening. It is shown that although sonic waves (i.e., wave traveling at the bar speed in the beam) do not exist for constant bending stiffness, nonlinear bending stiffness supports new classes of solutions that admit nonlinear waves traveling with axial tension or compression. In addition, it is shown that waves traveling at the bar wave speed may be compactons (i.e., solitary waves of finite span). Surprisingly, for a large class of constant and nonlinear bending stiffness, initially circular rods do not support traveling waves. [ABSTRACT FROM AUTHOR]

Thick composite disks are utilized in fast-rotating machines, including turbine disks and flywheels. Dynamic equations of motion for a rotating composite disk have been formulated in a polar coordinate system using Hamilton's principle, and numerical analysis has been performed by finite element interpolation. The natural frequencies of isotropic and laminated composite disks have been obtained when the rotational speed changes. The effects of transverse shear and rotary inertia on the vibration characteristics of rotating disks have also been investigated. [ABSTRACT FROM AUTHOR]

In the present study, the wave interaction with the very large floating structures (VLFSs) is analyzed considering the small amplitude wave theory. The VLFS is modeled as a 2D floating elastic plate with infinite width based on Timoshenko-Mindlin plate theory. The eigenfunction expansion method along with mode-coupling relation is used to analyze the hydroelastic behavior of VLFSs in finite water depth. The contour plots for the plate covered dispersion relation are presented to illustrate the complexity in the roots of the dispersion relation. The wave scattering behavior in the form of reflection and transmission coefficients are studied in detail. The hydroelastic performance of the elastic plate interacting with the ocean wave is analyzed for deflection, strain, bending moment, and shear force along the elastic plate. Further, the study is extended for shallow water approximation, and the results are compared for both Timoshenko-Mindlin plate theory and Kirchhoff's plate theory. The significance and importance of rotary inertia and shear deformation in analyzing the hydroelastic characteristics of VLFSs are presented. The study will be helpful for scientists and engineers in the design and analysis of the VLFSs. [ABSTRACT FROM AUTHOR]

An improved mathematical model of a piezoelectric vibrating gyroscope is constructed with the effects of electromechanical coupling and rotary inertia taken into account. The gyroscope consists of a cantilever beam with tip mass attached to its free end. The piezoelectric materials are adhered to the four surfaces of the rectangular beam. By using two Euler angles and extended Hamilton’s principle, the electromechanical coupled partial differential equations governing the flexural-flexural motions of the piezoelectric gyroscope are obtained. The accuracy and effectiveness of the comprehensive mathematical model are validated by the existing models in literature. The necessity of taking rotary inertia into account is discussed. The natural frequencies of the rotating piezoelectric beam in both drive and sense directions have been investigated by considering the effects of the tip mass, angular speed, and external impedance. The dynamic behaviors of the piezoelectric gyroscope with different parameters, such as the tip mass, the lengths of piezoelectric materials and beam, and the external impedance, have been studied. In particular, the optimal combination of the external excitation frequency and the external impedance for the best calibration curve of the piezoelectric gyroscope is identified. Finally, the dynamic performance of the piezoelectric gyroscope under varying boundary conditions has been investigated for the potential practical applications. [ABSTRACT FROM AUTHOR]

The hydroelastic response of very large floating structures (VLFS) under the action of ocean waves is analysed considering the small amplitude wave theory. The very large floating structure is modelled as a floating thick elastic plate based on Timoshenko-Mindlin plate theory, and the analysis for the hydroelastic response is performed considering different edge boundary conditions. The numerical study is performed to analyse the wave reflection and transmission characteristics of the floating plate under the influence of different support conditions using eigenfunction expansion method along with the orthogonal mode-coupling relation in the case of finite water depth. Further, the analysis is extended for shallow water depth, and the continuity of energy and mass flux is applied along the edges of the plate to obtain the solution for the problem. The hydroelastic behaviour in terms of reflection and transmission coefficients, plate deflection, strain, bending moment and shear force of the floating thick elastic plate with support conditions is analysed and compared for finite and shallow water depth. The study reveals an interesting aspect in the analysis of thick floating elastic plate with support condition due to the presence of the rotary inertia and transverse shear deformation. The present study will be helpful for the design and analysis of the VLFS in the case of finite and shallow water depth. [ABSTRACT FROM AUTHOR]

Abstract The findings of the experimental and numerical investigations on vibration characteristics of laminated skewed shells with cutout are presented in this paper. The shell laminates have been fabricated in the laboratory by adopting resin infusion technique. Experimental analysis is conducted using Photon plus data acquisition system along with accelerometer and impact hammer to identify the influence of skew angle and cutout on the natural frequencies of laminated shell. The results obtained in the present investigation indicate that cutout and skew angle bear remarkable influencing factor to control the frequencies of laminated skewed shells. Additional results obtained from extensive parametric study on various shells with cutout are presented which may be helpful for future researchers. [ABSTRACT FROM AUTHOR]

We discuss the classic rotary inertia notion and extend it for microstructured beams introducing new microinertia parameters as an additional dynamic response to microstructure changes. Slender structures made of beam- or platelet-lattice metamaterials may exhibit not only large translations and rotations but also general deformations of inner structure. Here we considered a few examples of beam-like structures and derive their inertia properties which include effective mass density, rotary inertia and microinertia. Extended dynamic characteristics related to enhanced kinematics may be crucial for description of origami-like structures or other beam-lattice metamaterials. • Rotary inertia and its extensions for a few microstructured beams was discussed. • Rotary inertia and microinertia maybe dominant for some microstructures. • Rotary and microinertia may essentially depend on deformations. • Correspondence to other generalized models of slender structures and media is given. [ABSTRACT FROM AUTHOR]

In this paper, the free vibration of axially-loaded, multi-cracked Timoshenko beams with differing boundary conditions, namely, hinged-hinged, fixed-fixed, fixed-hinged, and fixed-free is studied. The cracked beam system is represented as several beam segments connected by massless rotational springs with sectional flexibility. Each segment is assumed to obey the Timoshenko beam theory. A simple transfer matrix method is used to derive the characteristic equation of the axially-loaded, multi-cracked beam with differing boundary conditions. The characteristic equation and corresponding mode shapes are a function of natural frequency, crack size and location, and physical parameters of the beam. In this paper, the effects of crack depth, number of cracks, position of cracks, axial load, shear deformation and rotary inertia on the dynamic behavior of multi-cracked beams are studied in detail. It is found that there is good agreement between the results obtained in this study and results available in the literature. Additionally, interesting observations overlooked by other researchers are obtained.

The most important issue in the vibration study of an engineering system is dynamics modeling. Axially moving continua is often discussed without the inertia produced by the rotation of the continua section. The main goal of this paper is to discover the effects of rotary inertia on the free vibration characteristics of an axially moving beam in the sub-critical and super-critical regime. Specifically, an integro-partial-differential nonlinear equation is modeled for the transverse vibration of the moving beam based on the generalized Hamilton principle. Then the effects of rotary inertia on the natural frequencies, the critical speed, post-buckling vibration frequencies are presented. Two kinds of boundary conditions are also compared. In super-critical speed range, the straight configuration of the axially moving beam loses its stability. The buckling configurations are derived from the corresponding nonlinear static equilibrium equation. Then the natural frequencies of the post-buckling vibration of the super-critical moving beam are calculated by using local linearization theory. By comparing the critical speed and the vibration frequencies in the sub-critical and super-critical regime, the effects of the inertia moment due to beam section rotation are investigated. Several interesting phenomena are disclosed. For examples, without rotary inertia, the study overestimates the stability of the axially moving beam. Moreover, the relative differences between the super-critical fundamental frequencies of the two theories may increase with an increasing beam length. [ABSTRACT FROM AUTHOR]

Identifying cracks and damages in structures using measured vibrational characteristics has received considerable attention in the past few decades. The possibility of using frequency changes due to the application of a mass appended to the structure has also been considered. In this paper an analytical proof to show that the natural frequencies of a cracked beam with a roving body possessing mass and rotary inertia will generally change abruptly as the body passes over a crack, provided that the crack permits differential flexural rotations, is presented. A novel explicit closed form solution of the governing equation of an Euler-Bernoulli beam with a roving body possessing mass and rotary inertia, in the presence of multiple cracks is also proposed. The presented exact solution is used to conduct a parametric analysis of cracked beams. Numerical results for natural frequencies are provided and a procedure to exploit the occurrence of frequency shifts to detect and locate each crack, without having to perform any additional calculation, is described. [ABSTRACT FROM AUTHOR]

Engineers and designers are often faced with the task of selecting materials that minimizes structural weight whilst meeting the required strength and stiffness. In many cases fibre reinforced composites (FRCs) are the materials of choice since they possess a combination of high strength and low density. Depending on the application, composites are frequently constructed to form long slender beam-like structures or flat thin plate-like structures. Such structures when subjected to random excitation have the potential to excite higher order vibratory modes which can contribute significantly to structure-borne sound. Statistical Energy Analysis (SEA) is a framework for modeling the high frequency vibration of structures. The modal density, which is typically defined as the number of modes per unit Hertz in a frequency band, is a fundamental parameter when applying SEA. This study derives formulas for the modal density of a fibre reinforced composite beam coupled in bending and torsion. The effects of shear deformation and rotary inertia are accounted for in the formulation. The modal density is shown to be insensitive to boundary conditions. Numerical analyses were carried out to investigate the variation of modal density with fibre orientation including and excluding the effects of shear deformation and rotary inertia. It was observed that neglecting such effects leads to underestimating the mode count in a particular frequency band. In each frequency band there exists a fibre orientation for which the modal density is minimized. This angular orientation is shown to be dependent on the shear rigidity as well as the bending, torsional and coupling rigidities. The foregoing observation becomes more pronounced with increasing frequency. The paper also addresses the modal density beyond the wave-mode transition frequency where the beam supports three propagating waves. [ABSTRACT FROM AUTHOR]