Effects Of Rotary Inertia Research Articles

Transient Vibroacoustic Analysis of Functionally Graded Plates

Abstract This paper presents the vibroacoustic response of pure functionally graded (FG) plates under transient loading of mechanical nature. The functionally graded plate is modeled using the conventional first-order shear deformation theory (FSDT) to incorporate the effects of transverse shear and rotary inertia. The mid-surface variables are determined using the finite element method. Transient structural response is determined using Newmark Beta time marching scheme and the acoustic pressure in the free field is obtained using the time-domain Rayleigh integral. The effective material properties of the FG plate and the transient response of both the structural and acoustic fields have been computed in matlab. The influence of the volume fraction index, thickness ratio, and boundary conditions of pure FG plate on its transient vibroacoustic response is investigated by a detailed parametric study.

Effect of reinforced cutouts on the buckling and vibration performance of hybrid fiber metal laminates

The cutouts are commonly found in fiber metal laminates to facilitate to pass electrical wires, water/fuel lines and sometime to facilitate easy access and inspection. Sometimes, such cutouts need to be reinforced by providing stiffener around the cutouts to avoid pre-mature failures. This article addresses the effects of reinforced cutouts and non-linearly varying edge loads on the vibration and buckling performance of inter laminar hybrid fiber metal laminates (HFMLs) by utilizing finite element approach. Toward this a 9-noded heterosis plate element has been used to discretize both plate and stiffener by taking into account the effect of shear deformation and rotary inertia. The stiffener considered in this study is spread in nature. The displacement compatibility by the plate and stiffener is maintained by using transformation matrix. The effect of eccentricity of the spread stiffener is also incorporated within the transformation matrix. Since the stress distribution within the plate is highly non-uniform in nature, the dynamic approach has been used to solve the buckling problems. The present study consists of aluminum metal face sheets bonded with four layered symmetric hybrid cross-ply laminates, in which six different hybrid configurations have been considered. The performance of each hybrid configuration along with different sized and positioned reinforced cutouts is well investigated under various nonlinear loading conditions.

Effect of compression and tension types of concentrated edge loads on buckling and vibration behavior of interlaminar hybrid fibre metal laminates

The present study deals with the numerical investigation on the buckling and vibration behaviour of interlayer hybrid fibre metal laminates (HFMLs) subjected to different types of concentrated edge compression and tension loads. The term interlayer hybrid represents carbon and glass fibre laminates are sandwiched in between aluminum face sheets. Totally six different hybrid configurations are taken into consideration separately for cross-ply and angle-ply layup schemes. To serve this purpose a finite element code using MATLAB is developed to examine the effects of hybrid configuration, aspect ratio, fiber orientation, type of loading and boundary conditions have been considered to study the buckling responses of HFMLs. The plate is modelled by using 9-noded heterosis elements by considering the effect of shear deformation and rotary inertia. It is revealed from the investigation that the hybrid configurations and loading conditions significantly affect the buckling and vibration characteristics of HFMLs.

Enhancing bandwidth of metamaterial plate with linear and nonlinear passive absorbers

In this work, passive vibration absorbers are used to obtain a metamaterial vibrating plate. For this purpose, a set of linear and nonlinear passive absorbers are used. The plate is modeled based on Mindlin’s theory in which the effects of rotary inertia and shear deformation are considered. The metamaterial plate is considered to be both linear and nonlinear, with a set of linear, nonlinear, and viscous absorbers (Maxwell and Maxwell–Voigt). By optimizing the absorber parameters, a noticeable reduction in the vibration amplitude of the plate is obtained, without imparting a significant mass to the original plate. The effects of mass, stiffness, damping of the absorber as well as the number of absorbers are also investigated so that the sensitivity of the behavior of the system with changing the parameters is minimized. The effect of absorbers at lower frequencies is the main focus of this work. There is more than 77% reduction in the vibration amplitude of the plate, while the total added mass is lower than 10%, which is much less than the values given in similar works. Also, the nonlinear effects of the plate have been greatly reduced and no new resonances with sufficient bandwidth have emerged, and linearization of the nonlinear plate with absorber is achieved. Finally, the performances of the used absorbers are compared.

Dynamic Analysis of Functionally Graded Sandwich Plates under Multiple Moving Loads by Ritz Method with Gram–Schmidt Polynomials

This paper investigates the dynamic behavior of functionally graded sandwich plates under multiple moving loads. The first-order shear deformation theory of plates is adopted with the effects of shear deformation and rotary inertia included. By using Lagrange’s equations, the equations of motion for the dynamic behavior of the plate are derived. Then they are solved by the Ritz and Newmark time integration methods for the free and forced vibrations of the plates with different boundary conditions. To guarantee that all terms in the admissible functions can cope with the essential boundary conditions, the Gram–Schmidt procedure is used to generate the shape functions for the Ritz method. The influences of several factors on the dynamic response of the plates, such as layer thickness ratio, boundary condition, velocity, excitation frequency, phase angle, etc., are examined and discussed in detail. The numerical study indicates that the dynamic deflection has initial fluctuated growth in the low range of moving load velocity before reaching the peak at the critical velocity, which is followed by the considerable decrease in magnitude. Besides, the gaps or distances between the moving loads also play an important role in predicting the dynamic deflections of the plate when subjected to more than one moving loads.

Dynamics and stability analysis of a flexible liquid-filled rotor in a constant thermal environment

In this study, the dynamics and stability of a flexible rotor containing liquid in a constant thermal environment are investigated. According to thermoelastic theory, the thermal axial force exerted on the rotor is calculated by using the analytical method. A spinning Rayleigh beam is used as a simplified model of the rotor. Applying the Hamilton principle, the governing equation of motion for the flexible liquid-filled rotor system is derived. Using the obtained model, the stability prediction model and the critical spinning speed for the rotor system are formulated. To demonstrate the validity of the developed model, the present analysis is compared with the results reported in the previous study, and good agreement is observed from the comparison results. Finally, numerical results based on the obtained model are performed for a better understanding of the parameters including filling parameters, mode number, rotatory inertia and thermal effect on the stability, and critical spinning speed of the rotor system.

Hydroelastic Modeling of Wet Deck Slamming on Multihull Vessels

Slamming against the wet deck of a multihull vessel in head sea waves is studied analytically and numerically. The theoretical slamming model is a two-dimensional, asymptotic method valid for small local angles between the undisturbed water surface and the wet deck in the impact region. The disturbance of the water surface as well as the local hydroelastic effects in the slamming area are accounted for. The elastic deflections of the wet deck are expressed in terms of "dry" normal modes. The structural formu­lation accounts for the shear deformations and the rotatory inertia effects in the wet deck. The findings show that the slamming loads on the wet deck and the resulting elastic stresses in the wet deck are strongly influenced by the elasticity of the wet deck structure.

Free and Forced Wave Vibration Analysis of a Timoshenko Beam/Frame Carrying a Two Degrees-of-Freedom Spring-Mass System

Abstract In this paper, free and forced vibrations of a transversely vibrating Timoshenko beam/frame carrying a discrete two-degrees-of-freedom spring-mass system are analyzed using the wave vibration approach, in which vibrations are described as waves that propagate along uniform structural elements and are reflected and transmitted at structural discontinuities. From the wave vibration standpoint, external excitations applied to a structure have the effect of injecting vibration waves to the structure. In the combined beam/frame and two-degrees-of-freedom spring-mass system, the vibrating discrete spring-mass system injects waves into the distributed beam/frame through the spring forces at the two spring attached points. Assembling the propagation, reflection, transmission, and external force injected wave relations in the beam/frame provides an analytical solution to vibrations of the combined system. In this study, the effects of rotary inertia and shear deformation on bending vibrations are taken into account, which is important when the combined structure involves a short beam element or when higher frequency modes are of interest. Numerical examples are given, with comparisons to available results based on classical vibration theories. The wave vibration approach is seen to provide a systematic and concise solution to both free and forced vibration problems in hybrid distributed and discrete systems.

Static and free vibration analysis of doubly-curved functionally graded material shells

Static and free vibration analysis of doubly-curved (cylindrical, spherical, hyperbolic paraboloid, and elliptical paraboloid) functionally graded material (FGM) shells is presented here using various equivalent single-layer shell theories considering the shear deformation and rotary inertia effects. The kinematics of the various shell theories are presented using a generalized shell theory in which the displacement field is independent of the selection of transverse shear strain function which results in a theoretical unification of various equivalent single-layer shell theories. The present generalized formulation yields traction free boundary conditions on the top and bottom surfaces of the shell using constitutive relations. Equations of motion are derived by using Hamilton’s principle which are further solved analytically using the Navier’s technique of assuming unknown variables in the double trigonometric series. Displacements, stresses, and frequencies are presented for functionally graded cylindrical, spherical, hyperbolic paraboloid, and elliptical paraboloid shells. The present results are compared with previously published results wherever possible to verify the accuracy and efficiency of the present generalized shell theory. Also, this study contributes many new numerical results which can be useful for future researchers.

Multi-moving Loads Induced Vibration of FG Sandwich Beams Resting on Pasternak Elastic Foundation

The dynamic behavior of functionally graded (FG) sandwich beams resting on the Pasternak elastic foundation under an arbitrary number of harmonic moving loads is presented by using Timoshenko beam theory, including the significant effects of shear deformation and rotary inertia. The equation of motion governing the dynamic response of the beams is derived from Lagrange’s equations. The Ritz and Newmark methods are implemented to solve the equation of motion for obtaining free and forced vibration results of the beams with different boundary conditions. The influences of several parametric studies such as layer thickness ratio, boundary condition, spring constants, length to height ratio, velocity, excitation frequency, phase angle, etc., on the dynamic response of the beams are examined and discussed in detail. According to the present investigation, it is revealed that with an increase of the velocity of the moving loads, the dynamic deflection initially increases with fluctuations and then drops considerably after reaching the peak value at the critical velocity. Moreover, the distance between the loads is also one of the important parameters that affect the beams’ deflection results under a number of moving loads.

Effect of rotational speed and inertia on the mechanical properties and microstructural evolution during inertia friction welding of 8630M steel

The effects of varying rotational speed and inertia on the mechanical properties and microstructural evolution in 8630M low-alloy steel during inertia friction welding (IFW) were investigated. The results show that increasing rotational speed and decreasing inertia magnitude detrimentally affect the weld’s mechanical properties. Microstructural observations suggest that the temperature in the weld zone (WZ) and thermo-mechanically affected zone (TMAZ) reached Ac3 and in-between Ac1-Ac3, whereas that of the HAZ stayed below Ac1. Microstructure of the weld was dominated by simple shear texture rotated under the influence of the compression at early stages of IFW, implying that the deformation was a combination of shear and compression.

Computational Analysis of Curved Beams - In and Out-of-Plane Vibrations

In the present work are shown results obtained from a computational model for the dynamic analysis of planed curved beams with constant cross-section and curve radius, based on the exact solution of the differential equilibrium equations of a beam element. This model allows the estimation of the forced harmonic motion, both in-plane and out-of-plane of curvature on a free-free support condition. The input data of the model is the geometry section of the beam, material properties, type of motion, consideration of the shear stiffness, and rotatory inertia effects. The output result is possible either in a frequency response of a fixed parameter curved beam or a plot representation of the natural frequencies curves for an opening angle range, from an approximated straight beam to a closed opened ring. Each of those curves is defined as family of modes and performed on present work. The objective of this work is to clarify the general behaviour of curved beams with the mentioned conditions by using an accurate model showing results as a reference for the scientific community. The model was validated by testing and comparing results with three beam specimens previously studied in other publications. The biggest aim is also to present new results and hypothesis, contesting other authors regarding the natural frequencies of curved beams.

Flutter of a beam in supersonic flow: truncated version of Timoshenko–Ehrenfest equation is sufficient

This paper deals with flutter due to gas flow of a uniform and homogeneous beam with shear deformation and rotary inertia effects taken into account. It utilizes the truncated, consistent Timoshenko–Ehrenfest beam model in contrast to the original Timoshenko–Ehrenfest equations. Omission of the fourth order derivative term in the governing differential equation of the original equations of Timoshenko and Ehrenfest is shown to be more consistent than retaining it as is done in the original Timoshenko–Ehrenfest set. The original Timoshenko–Ehrenfest equations are exposed in almost all textbooks published after 1921. However, this paper shows that these original equations are unnecessarily overcomplicated, and simpler set is consistent. This truncation significantly simplifies the analytical and numerical analyses considerably and agrees with the principle of parsimony, i.e. the most acceptable explanation of a phenomenon is the simplest, involving the fewest entities and assumptions. The critical flutter velocities are compared with the results obtained by using the original set of Timoshenko–Ehrenfest equations.

Seismic Wave Propagation in Framed Structures by Joint-Based Wave Refraction Method

This paper studies the dynamic behavior of the structural frames subjected to seismic loadings using a joint-based wave refraction method. The beams of the frame are modeled by the Timoshenko beam theory to consider the rotary inertia and shear deformation effects. The whole frame is considered as an assemblage of the waveguides connected with joints and boundary conditions, including discrete spring–dashpots for modeling the foundation supports that are generally treated as discontinuities. The wave refraction matrices at the discontinuities are derived analytically and the final assembled system of equations are solved numerically. The accuracy of the method is validated using an experimental setup and its performance is assessed for a 15-story concrete moment frame subjected to real ground motions. The soil structure interaction effect is also considered in the simulations using discrete spring–dashpot elements. The results show that the wave refraction method can be effectively used as an alternative means for the seismic analysis of the frames. In comparison with numerical methods such as finite element method, the proposed method is computationally efficient, while its accuracy and cost are independent of the loading frequency and the length of waveguides.

Energy harvesting from nonlinear vibrations of an L-shaped beam using piezoelectric patches

In the present study, the energy harvesting from nonlinear vibrations of an L-shaped beam with the use of piezoelectric patches is evaluated. The beam was assumed to be a Euler–Bernoulli type in which the effects of rotational inertia and shear forces are neglected. Also, the effective mass of the beam is assumed to be at the end of the beam. The axial displacement of the beam is included with the assumption of small excitation amplitude. The effect of the piezoelectric patch on the equations of motion is also considered. First, the nonlinear motion equations of an L-shaped beam are extracted by the use of the Lagrange method, and then, the obtained equations of motion are solved by multiple scales technique. Also, the mode shapes of the system are calculated for the linear undamped system. The dimensions of the L-shaped beam are chosen in such a way that the ratio of the first-to-second natural frequency is 1 to 2, in order to increase the bandwidth of the energy harvesting. Finally, the results of the analytical solutions are compared and validated with the experimental results. Good agreement was observed between the theoretical values and the measured data.

Wave characteristics of nonsymmetric cross-ply laminated composite beams

The free vibration characteristics of a nonsymmetric cross-ply laminated composite beam coupled in bending and longitudinal deformation is studied using a wave approach. The effects of shear deformation and rotary inertia are included in the analysis. Exact analytical expressions are derived for the natural frequencies, mode shapes, and the power flow of the propagating waves. The derived expressions are validated using the results from past literature and provide a benchmark for numerical models. The advantages of the wave approach over conventional free vibration analysis methods are highlighted. Specifically, the wave approach is used to derive a simplified expression for the mode count function of the composite beam. Additionally, the wave approach is also used to investigate the power flow and cross-conversion of the propagating wavetypes across various classical boundary conditions. The influence of the number of cross-ply layers on the natural frequencies and power flow are also investigated. The efficacy of the wave analysis is illustrated through several numerical examples.

Stability and critical spinning speed of a flexible liquid-filled rotor in thermal environment with nonlinear variable-temperature

This paper deals with the stability and critical spinning speed of a liquid-filled rotor in thermal environment with nonlinear variable-temperature. The nonlinear temperature field model of the rotor is developed by using Laplace transform. The thermal axial force exerted on the rotor as a result of the thermal effect is calculated as functions of temperature rise rate and rotor thickness ratio. Spinning Timoshenko beam model is employed to establish the structural dynamics equations of the rotor system. The governing equation of the motion is derived by using the Hamilton principle. The validity of the developed model is confirmed by comparing with the numerical solutions available in the literature. The numerical results based on the obtained analytical solutions are given for a better understanding of the effects of the shear deformation, rotary inertia, filling parameters and thermal effect on the system stability and critical spinning speed. The results show that the system stability is dependent on the thermal axial force and cavity ratio. Moreover, the results also highlight the role of thermal effect, rotary inertia, cavity ratio and mass ratio on the critical spinning speed.

Effect of rotational inertia on building response to earthquakes via a closed-form solution

In this article, a widely used building model, comprised of uniform coupled flexural and shear beams, herein improved by allowing for the effects of rotational inertia. Closed-form solutions in terms of trigonometric and hyperbolic functions are obtained, allowing for the explicit formulation of period ratios, modal participation factors (MPFs), mode shapes, and mode shape derivatives based solely on displacement response, without coupling with chord rotations, which is the case for the Timoshenko beam model. This makes the model proposed in this study more convenient for assessing building behavior to ground motion, by explicitly highlighting the effect of rotational inertia in their response to earthquakes, making the case for studying its beneficial effects in mitigating response of buildings to ground motion. It is observed that rotational inertia induces mild fundamental period lengthening, while notably reducing period ratios between higher modes and the fundamental one. This can lead to an enhanced response to ground motion showcasing narrow-band characteristics. However, the most severe effect is found to be on the MPF, which is significantly diminished. This leads to important reductions on the overall response, as a consequence of attenuation of the first-mode response, along with severe undercutting of the effects of higher modes. The results demonstrate that without considering the rotational inertia in the assessment of in-plane structural response to horizontal ground motion can lead to conservative results. Moreover, the results showcase the advantages of providing supplemental rotational inertia as a way to improve the seismic behavior of buildings.

On the damping influence on the dynamic analysis of functionally graded beams resting on elastic foundation by Green’s function method

This paper deals with dynamic analysis of functionally graded beams on elastic foundation considering damping effect. It is assumed that the mechanical properties of the material vary along the beam thickness. The fourth-order differential equation is analytically solved by means Green's function method, and exact answer is achieved. Damping effect, which is an intrinsic property of structures with great influence on the forced vibration of mechanical systems is included in the analysis. However, due to difficulty involved, it is usually neglected in the analysis of structures. Furthermore, the effect of rotary inertia is considered in the problem formulations. Green’s function method is a straightforward way to model physical problems subjected to arbitrary loads and affected by various parameters, such as, elastic foundation, damping, geometry and mechanical properties. It could be a unique solution to gain the exact answers via the superposition manner. To demonstrate the accuracy and efficiency of the presented solution, dynamic response of the beam under different harmonic loads is obtained and compared to available results in some cases.

Effects of turbocharger rotational inertia on engine and turbine performance in a turbocharged gasoline direct injection engine under transient and steady conditions

The effects of turbocharger (T/C) rotational inertia on engine and turbine performance under transient and steady engine conditions were analyzed in a 2.0 L 4-cylinder turbocharged-gasoline direct injection (T-GDI) engine. The test T/Cs consisted of heavy and light compressor wheels (C/W) and turbine wheels (T/W). The study was conducted in two research stages. First, transient engine load tests were conducted to evaluate the effect of T/C rotational inertia on transient response of the T/C, combustion performance, and fuel consumption. Seconds, steady engine load tests were conducted to find out if a light inertia T/C can run at higher efficiency under the same exhaust pulsating flow conditions within one engine cycle. In order to evaluate the engine on-board turbine instantaneous performances in the units of crank angle degree (CAD), T/C rotation speed and pressure data were measured in the experiment. The instantaneous exhaust gas mass flow rate and the temperature of upstream and downstream of the turbine were extracted by 1-D simulation. Turbine efficiency and mass flow rate parameters were calculated by combining these data. In the results, there existed positive effects of light inertia T/C on response and specific fuel consumption under transient conditions. It was also found that the light inertia T/C could show higher T/C speed fluctuation under the same exhaust pulsating flow conditions. Consequently, blade speed ratio (BSR) and turbine efficiency of light inertia T/C were partially higher than that of conventional one. However, it was not led to higher engine efficiency.