Axial Half Wave Research Articles

Free Vibration Characteristics Analysis of Metal-Rubber Cylindrical Shells Based on Viscoelastic Theory

The cylindrical shell made of metal rubber has a strong ability to reduce and absorb vibration, which widens its application in the industrial field. Therefore, it is of great significance to study the vibration characteristics of metal-rubber cylindrical shells (MRCSs). However, there is relatively little research on this aspect. Based on this, the dynamic properties of MRCS are investigated in this paper based on viscoelastic theory, the Rayleigh–Ritz method, and the Gram–Schmidt orthogonal polynomials. The correctness of the proposed model was verified by comparison with the literature and experimental verification. The results show that the preloading state and boundary conditions have significant effects on the natural frequency and modal loss factor of MRCS. The effect of the Pasternak elastic foundation on the natural frequency and modal loss factor of MRCS varies with the change of the axial half wave number m. The effect of the Pasternak elastic foundation on higher-order vibrations is similar to that of the artificial spring technique.

Free vibration characteristics of cord-reinforced air spring with winding formation under preload conditions

A precise transfer matrix was employed to analyze the free vibration characteristics of a cord-reinforced air spring with winding formation under preload conditions. The air spring was abstracted as a rotary combined shell structure with variable winding trajectory characteristics. The rotary structure, which consisted of cylindrical and toroidal shells, was analyzed for its shell pre-stress under preload conditions. The shell pre-stress was combined with the theory of thin shells to create the rotary shell vibration control equations under preload effect. The Helmholtz equation was used to analyze the influence of sound pressure inside shells under the fluid–structure coupling boundary conditions. The rotary shell transfer matrix was derived from the variable winding trajectory equation, geometrical equation, and physical equation of shells. Under the continuous conditions of state vector between the cylindrical and toroidal shells, the shells were combined into a rotary combined shell structure to obtain its total free vibration equation. The shell boundary conditions were considered to solve the natural frequency of the air spring. The calculation results were compared with the prototype test and simulation results to verify the effectiveness and accuracy of the theoretical model. On this basis, we explored the influence of axial half wave number, circumferential wave number, geometrical structure, and material characteristics on the free vibration characteristics of the air spring. The research findings would provide important and significant guidance for the structural design of cord-reinforced air springs with winding formation.

Plateau lower-bounds to the imperfection sensitive buckling of composite shells

The buckling of thin FRP laminated shells are sensitive to initial geometric imperfections. A large number of geometric and material variables prohibit the traditional experiment-based lower-bound design method for metallic shells from being extended to composite ones. As an alternative, the so-called reduced stiffness method (RSM) has been applied to the lower bound buckling of FRP laminated shells. It has been shown for shorter shells the method predicted excellent lower bounds to the nonlinear buckling loads. This paper aims to extend the study to longer shells. It is shown for longer composite shells the lower-bound buckling modes generally occur in the long axial wave mode having one axial halfwave but require imperfection amplitudes of impractical largeness. Further studies of the geometric parameters identify the existence of intermediate and significant plateaus to the lower bounds. These plateau lower-bounds demonstrate the importance of the short axial wave modes having more than one axial halfwave, associated with the imperfection amplitudes of practical smallness. Using the plateau values and modes to predict lower bounds is suggested to provide an important alternative for improving shell buckling design.

Free and forced vibration characteristics of bimodular composite laminated circular cylindrical shells

The free and forced vibration characteristics of bimodular cross-ply laminated clamped–clamped circular cylindrical shells are studied using Bert’s and a recently proposed model considering first order shear deformation theory. The free vibration frequencies in positive and negative half cycles are found to be same for all asymmetric modes and for axisymmetric modes with even number of axial half waves, and different for axisymmetric modes with odd number of axial half waves. However, the negative half cycle amplitude is significantly greater than the positive half cycle amplitude due to the local stiffness of bimodular cylindrical shells being different for positive and negative half cycles even though the natural frequencies are same for positive and negative half cycles of asymmetric vibrations. The relative difference of positive and negative half cycle frequencies is very less for single layer orthotropic shells, and it is significant for cross-ply shells for axisymmetric mode of vibration. The vibration characteristics predicted using proposed and Bert’s constitutive models are quite close except for the transverse to the fiber direction normal stress.

Analysis of Fiberglass Winding Angle on Natural Frequency of Cylindrical Shell with Symmetric Boundary Conditions

In order to obtain approximate solution of natural frequencies for the free vibration of anisotropic circular cylindrical shells made of GFRP (glass fiber-reinforced plastic) with symmetric boundary conditions, Loves theory and energy method are used. Computation results show that the fundamental natural frequency comes from different vibration modes while the winding angle varies, the effect of number of axial half waves is stronger than number of circumferential waves on natural frequency of free vibration of anisotropic circular cylindrical shell.

Analysis of Fiberglass Winding Angle on Natural Frequency of Free Vibration of Cylindrical Shell with Asymmetric Boundary Conditions

In order to obtain approximate solution of natural frequencies for the free vibration of anisotropic circular cylindrical shells made of GFRP (glass fiber-reinforced plastic) with asymmetric boundary conditions, Love’s theory and energy method are used. Computation results show that the fundamental natural frequency comes from different vibration modes while the winding angle varies, the effect of number of axial half waves is stronger than number of circumferential waves on natural frequency of free vibration of anisotropic circular cylindrical shell. The effect of shell’s geometrical parameters is also investigated on natural frequencies.

Temperature effects on wave propagation in nanoplates

In this paper, the thermal effects on the ultrasonic wave propagation characteristics of a nanoplate are studied based on the nonlocal continuum theory. The nonlocal governing equations are derived for the nanoplate under thermal environment. The axial stress caused by the thermal effects is considered. The wave propagation analysis is carried out using spectral analysis. The influences of the nonlocal small scale coefficient, the room or low temperature, the high temperature and the axial half wave numbers on the wave dispersion properties of nanoplate are also discussed. Numerical results show that the small scale effects and the thermal effects are significant for larger half wavenumbers. The results are qualitatively different from those obtained based on the local plate theory and thus, are important for the development of graphene-based nanodevices such as strain sensor, mass and pressure sensors, atomic dust detectors, and enhancer of surface image resolution.

Nonlinear vibrations of rotating thin circular cylindrical shell

Nonlinear vibrations of thin circular cylindrical shells are investigated in this paper. Based on Love thin shell theory, the governing partial differential equations of motion for the rotating circular cylindrical shell are formulated using Hamilton principle. Taking into account the clamped-free boundary conditions, the partial differential system is truncated by using the Galerkin method. Sequentially, the effects of temperature, geometric parameters, circumferential wave number, axial half wave number and rotating speed on the nature frequency of the rotating circular cylindrical shell are studied. The dynamic responses of the rotating circular cylindrical shell are also investigated in time domain and frequency domain. Then, the effects of nonlinearity, excitation and damping on frequency responses of steady solution are investigated.

Pasternak foundation effect on the axial and torsional waves propagation in embedded DWCNTs using nonlocal elasticity cylindrical shell theory

The axial and torsional wave propagation in a double-walled carbon nanotube (DWCNT) embedded on elastic foundations are investigated using nonlocal continuum shell theory. The effects of the surrounding elastic medium are considered using the spring constant of the Winkler-type and the shear constant of the Pasternak-type. The van der Waals (vdW) forces between the inner and the outer nanotubes are taken into account. The dynamic response of the carbon nanotube is formulated on the basis of nonlocal elasticity shell theory. The cut-off frequencies are obtained and it has been concluded that the cut-off frequencies are independent of small scale coefficient and shear modulus of the elastic medium. It has been found that the phase velocity sharply decreases by increasing the axial half wave number and approaches a constant value. It has also been concluded that the maximum phase velocity predicted by nonlocal theory is located between 5 and 10 nanometers while for local theories the phase velocity sharply decreases in this interval and approaches a constant value. Results show that the effect of Pasternak-type on phase velocity is significant but the effect of Winkler-type is not really considerable.

The thermal effect on buckling analysis of a DWCNT embedded on the Pasternak foundation

In this article, the thermal effect on the buckling analysis of a double-walled carbon nanotube (DWCNT) embedded in an elastic medium subjected to a uniform external pressure is investigated. Based on the nonlocal continuum cylindrical shell theory, the following effects are studied: small scale effect, van der Waals (vdW) forces between inner and outer tubes, surrounding elastic medium, and influence of temperature change in high-temperature environment. The interaction between matrix and the outer tube is modeled as a Pasternak foundation. The obtained results of numerical simulation indicate that for any specific circumferential wave number ( n), the nonlocal critical buckling pressure ( p crit ) is related directly to the axial half wave number ( m). Furthermore, the effect of temperature change on the critical buckling pressure is negligible, especially for stiff elastic medium; however, this is not the case if the elastic medium is soft. The strength of the DWCNT is directly related to the Winkler and shear moduli; hence, increase in the latter leads to enhanced p crit .

Single Walled Carbon Nanotube for Sensing Small Molecules

We present an analytical model for investigating pressure dependence of the radial mode resonant frequencies of single walled carbon nanotube (SWCNT). Mechanical shell model is used to in the calculation. SWCNTs with 1nm and 10nm radiuses have been analyzed, and it is found that frequency dependence of the radial mode vibration of SWCNT is related to diameter, axial half wave number, m and the circumferential wave number, n. Based on the modeling, a device concept of molecule sensor has been proposed, which is capable of detecting concentrations of small size molecules.

Geometrically nonlinear forced vibrations of the symmetric rectangular honeycomb sandwich panels with completed clamped supported boundaries

Abstract Geometrically nonlinear forced vibrations of symmetric rectangular honeycomb sandwich panels with clamped supported boundaries at the four edges are investigated using the homotopy analysis method (HAM). The honeycomb core of hexagonal cells is modeled as a thick layer of orthotropic material whose parameters of physical and mechanical properties are calculated by the corrected Gibson’s formula. The basic formulation of nonlinear forced vibrations has been developed based on the classical plate theory (CPT) and the nonlinear strain–displacement relation. The equilibrium equations have been obtained using Hamilton’s principle. Effects of axial half-waves, height and height ratio on the nonlinear free vibration response have been investigated for honeycomb sandwich panels.

Nonlinear dynamic response of rotating circular cylindrical shells with precession of vibrating shape—Part I: Numerical solution

Abstract The nonlinear dynamic response of a cantilever rotating circular cylindrical shell subjected to a harmonic excitation about one of the lowest natural frequency, corresponding to mode ( m =1, n =6),where m indicates the number of axial half-waves and n indicates the number of circumferential waves, is investigated by using numerical method in this paper. The factor of precession of vibrating shape ς is obtained, with damping accounted for. The equation of motion is derived by using the Donnell’s nonlinear shallow-shell theory, and is general in the sense that it includes damping, Coriolis force and large-amplitude shell motion effects. The problem is reduced to a system of ordinary differential equations by means of the Galerkin method. Three different mode expansions are studied for finding the proper one which is more contracted and accurate to investigate the principal mode (i.e., m= 1, n= 6) response. From the present investigation, it can be found that for principal mode resonant response, there are two traveling waves with different linear frequencies due to the effect of precession of vibrating shape of rotating circular cylindrical shells; the effects of additional modes n and k (multiples of frequency) on the principal mode resonant response are insignificant compared with an additional mode m , showing that it is better to adopt two neighboring axial modes to study the principal resonant response of the system.

Geometrically nonlinear free vibrations of the symmetric rectangular honeycomb sandwich panels with simply supported boundaries

Abstract Geometrically nonlinear free vibrations of symmetric rectangular honeycomb sandwich panels with simply supported boundaries at the four edges are investigated using the homotopy analysis method (HAM). The honeycomb core of hexagonal cells is modeled as a thick layer of orthotropic material whose parameters of physical and mechanical properties are calculated by the corrected Gibson’s formula. The basic formulation of nonlinear free vibrations has been developed based on the third-order shear deformation plate theory and the nonlinear strain–displacement relation. The equilibrium equations have been obtained using Hamilton’s principle. Effects of axial half-waves, height and height ratio on the nonlinear free vibration response have been investigated for honeycomb sandwich panels.

Breakdown of structural models for vibrations of single-wall zigzag carbon nanotubes

Free vibrations of zigzag single-wall carbon nanotubes (SWCNTs) of aspect ratio (length/diameter) ∼6 and with ends traction-free have been studied using molecular mechanics (MM) simulations with the MM3 potential. It is found that the frequencies of inextensional (the Love and the Rayleigh) modes of an (n, 0) SWCNT saturate at the circumferential wave number of either (n−1)/2 or n/2 where n is odd or even. This is explained in terms of its molecular structure. Since the frequencies of the inextensional modes of vibration of a thin cylinder made of an isotropic linear elastic material do not saturate with an increase in the circumferential wave number, a continuum structure cannot represent all modes of vibration of a zigzag SWCNT. This result is independent of the value assigned to the wall thickness of the SWCNT. We have also found values of material and geometric parameters of a shell and a hollow cylinder by equating their frequencies of the inextensional, the radial breathing, the axial and the torsional modes of vibrations to the corresponding ones of a zigzag SWCNT, and by taking their mean diameter and length equal to those of the SWCNT. The frequencies of the extensional modes of oscillations of the two continuum structures for various axial half wave numbers and circumferential wave numbers are found to match well with those of the SWCNT obtained from the MM simulations. However, the frequencies of the inextensional modes of the continuum structures deviate noticeably from those of the SWCNT, and this deviation increases with an increase in the circumferential wave number.

Fundamental modes of tank-liquid systems under horizontal motions

This paper reports results on the fundamental impulsive modes of vibration of cylindrical tank-liquid systems anchored to the foundation under horizontal motion. The analyses are performed using a general purpose finite element (FE) program, and the influence of the hydrostatic pressure and the self-weight on the natural periods and modes is considered. The roof and walls are represented with shell elements and the liquid is modeled using two techniques: the added mass formulation and acoustic finite elements. Tank height to diameter ratios from 0.40 to 0.95 were used, with a liquid level at 90% of the height of the cylinder. The effect of the geometry on the fundamental modes for the tank-liquid systems is studied using eigenvalue and harmonic response analyses. Similar fundamental periods and mode shapes were found from these two approaches. The fundamental modes of tank models with aspect ratios ( H / D ) larger than 0.63 were very similar to the first mode of a cantilever beam. For the shortest tank ( H / D = 0.40 ) , the fundamental mode was a bending mode with a circumferential wave n = 1 and an axial half-wave ( m ) characterized by a bulge formed near the mid-height of the cylinder.

Non-stationary flexural vibration behaviour of a cylindrical shell

An analytical/numerical method involving a small number of generalized coordinates is presented for the analysis of the non-linear vibration and dynamic stability behaviour of imperfect anisotropic cylindrical shells. The small amplitude vibration mode is characterized by m axial half-waves and ℓ circumferential full waves. The radial excitation is assumed to have the same spatial distribution as the small amplitude vibration mode. For large excitations, a coupled mode response can occur, consisting of the directly excited ‘driven mode’ and the so-called ‘companion mode’. The latter has the same spatial form as the driven mode, but is circumferentially 90° out-of-phase. Donnell-type governing equations are used and classical laminate theory is employed. The assumed modes approximately satisfy ‘simply supported’ boundary conditions. A formulation is used which can account for a possible skewedness of the asymmetric modes. Certain axisymmetric modes which are essential for a satisfactory description of the non-linear behaviour are also included in the assumed deflection function. The effect of axial and torsional inertia, including the inertia effect of a ring or disk at the loaded end of the shell, can be taken into account approximately. Viscous modal damping is included in the analysis. The shell is statically loaded by axial compression, radial pressure, and torsion. A two-mode imperfection model, consisting of an axisymmetric and an asymmetric mode, is used. The static state response is assumed to be affine to the given imperfection. In order to find approximate solutions for the dynamic state equations, Hamilton's principle is applied to derive a set of modal amplitude equations. The dynamic response is obtained via numerical time integration of the set of non-linear ordinary differential equations. The coupled mode non-linear vibration behaviour of an axially loaded isotropic shell is simulated using this approach. Non-stationary vibrations, where the response drifts between single mode and different types of coupled mode solutions, are observed in a small frequency region near resonance.

Buckling of thick orthotropic cylindrical shells under external pressure based on non-planar equilibrium modes

A formulation based on the three dimensional theory of elasticity is employed to study the buckling of an orthotropic cylindrical shell under external pressure. In this paper, a non-zero axial displacement and a full dependence of the buckling modes on the three coordinates is assumed, as opposed to the ring approximation employed in the earlier studies. The results from this elasticity solution are compared with the critical loads predicted by the orthotropic Donnell and Timoshenko non-shallow shell formulations. Two cases of end conditions are considered; one with both ends of the shell fixed, and the other with both ends capped and under the action of the external pressure. Moreover, two cases of orthotropic material are considered with stiffness constants typical of glass/epoxy and graphite/epoxy. For the isotropic material case, the predictions of the simplified (single expression) Donnell and the Flügge and the Danielson and Simmonds theories are also compared. In all cases, the elasticity approach predicts a lower critical load than the shell theories, the percentage reduction being larger with increasing thickness. The degree of non-conservatism depends strongly on the material properties, being smaller for the isotropic case. Furthermore, although it is a commonly accepted notion that the critical point in loading under external pressure occurs for n = 2 and m = 1 (number of circumferential waves and number of axial half-waves, respectively), it was found that this is not the case for the strongly orthotropic graphite/epoxy material and the moderately thick construction; for this case, the value of m at the critical point is greater than 1 (yet, in all cases n = 2).

Stability Loss in Thick Transversely Isotropic Cylindrical Shells Under Axial Compression

The stability of equilibrium of a transversely isotropic thick cylindrical shell under axial compression is investigated. The problem is treated by making appropriate use of the three-dimensional theory of elasticity. The results are compared with the critical loads furnished by classical shell theories. For the isotropic material cases considered, the elasticity approach predicts a lower critical load than the shell theories, the percentage reduction being larger with increasing thickness. However, both the Flu¨gge and Danielson and Simmonds theories predict critical loads much closer to the elasticity value than the Donnell theory. Moreover, the values of n, m (number of circumferential waves and number of axial half-waves, respectively, at the critical point) for both the elasticity, and the Flu¨gge and the Danielson and Simmonds theories, show perfect agreement, unlike the Donnell shell theory.

Design of beneficial geometric imperfections for elastic collapse of thin-walled box columns

This paper deals with the design of beneficial geometric imperfections for elastic collapse of thin-walled box columns of square cross-section under axial compression. From the point of view of high elastic post-collapse stiffness, it is desirable to force the column to collapse with a larger number of axial half-waves than with the preferred wave number which is approximately equal to the aspect ratio. By introducing a beneficial geometric imperfection with such a larger wave number and if the magnitude of such imperfection exceeds the transitional value, it is found that the column will collapse in the beneficial mode in the initial postbuckling (assume to be elastic) finite-deflection regime. Equilibrium paths of typical box columns are plotted and analyzed. The two-mode potential energy is found to fall into the category of a double-cusp catastrophe.