On the flexural vibration of cylinders under axial loads:Numerical and experimental study

The flexural vibration of a homogeneous isotropic linearly elastic cylinder of any aspect ratio is analysed in this paper. Natural frequencies of a cylinder under uniformly distributed axial loads acting on its bases are calculated numerically by the Ritz method with terms of power series in the coordinate directions as approximating functions. The effect of axial loads on the flexural vibration cannot be described by applying infinitesimal strain theory, therefore, geometrically nonlinear strain–displacement relations with second-order terms are considered here. The natural frequencies of free–free, clamped–clamped, and sliding–sliding cylinders subjected to axial loads are calculated using the proposed three-dimensional Ritz approach and are compared with those obtained with the finite element method and the Bernoulli–Euler theory. Different experiments with cylinders axially compressed by a hydraulic press are carried out and the experimental results for the lowest flexural frequency are compared with the numerical results. An approach based on the Ritz formulation is proposed for the flexural vibration of a cylinder between the platens of the press with constraints varying with the intensity of the compression. The results show that for low compressions the cylinder behaves similarly to a sliding–sliding cylinder, whereas for high compressions the cylinder vibrates as a clamped–clamped one.

Reducing MRI gradient coil vibration with rib stiffeners

This work investigates the effect of rib stiffeners on the free and forced vibration of a gradient coil in a Magnetic Resonance Imaging (MRI) scanner. Several reinforcement schemes are studied in this paper. One scheme utilizes the existing holes in the gradient coil structure (typically reserved for magnetic shims) to produce the reinforcement. Non-ferrous, non-magnetic carbon fibre rib stiffeners are employed to fill these holes in several ways to strengthen a gradient coil. Another scheme replaces the inner half of the gradient coil material with a grid of interconnected axial and circumferential rib stiffeners. It is found that the structural stiffness of the gradient coil increases substantially when the coil is reinforced by carbon fibre rib stiffeners. The reinforcement affects the noise and vibration response of the gradient coil structure in the following ways. It increases the frequency range of forced response of the gradient coil at low frequencies due to the increased resonant frequency of the fundamental mode of the coil. Secondly, it reduces the forced response amplitude of the coil structure (which is governed by the structural stiffness of the coil). Thirdly, it reduces the number of natural modes in the low and medium frequency range and therefore lessens the chance of the coil structure being excited resonantly by magnetic resonance signal acquisition sequences. It is shown that gradient coils modelled by solid finite element models have higher stiffness along the coil’s circumference and lower stiffness in the axial direction than those using shell finite element models.

Infrastructure sustainability : differential axial shortening of concrete structures

High density development has been seen as a contribution to sustainable development. However, a number of engineering issues play a crucial role in the sustainable construction of high rise buildings. Non linear deformation of concrete has an adverse impact on high-rise buildings with complex geometries, due to differential axial shortening. These adverse effects are caused by time dependent behaviour resulting in volume change known as ‘shrinkage’, ‘creep’ and ‘elastic’ deformation. These three phenomena govern the behaviour and performance of all concrete elements, during and after construction. Reinforcement content, variable concrete modulus, volume to surface area ratio of the elements, environmental conditions, and construction quality and sequence influence on the performance of concrete elements and differential axial shortening will occur in all structural systems. Its detrimental effects escalate with increasing height and non vertical load paths resulting from geometric complexity. The magnitude of these effects has a significant impact on building envelopes, building services, secondary systems, and lifetime serviceability and performance. Analytical and test procedures available to quantify the magnitude of these effects are limited to a very few parameters and are not adequately rigorous to capture the complexity of true time dependent material response. With this in mind, a research project has been undertaken to develop an accurate numerical procedure to quantify the differential axial shortening of structural elements. The procedure has been successfully applied to quantify the differential axial shortening of a high rise building, and the important capabilities available in the procedure have been discussed. A new practical concept, based on the variation of vibration characteristic of structure during and after construction and used to quantify the axial shortening and assess the performance of structure, is presented.

Influence of axial deformation on modal strain energy

Axial loads of load bearing elements impact on the vibration characteristics. Several methods have been developed to quantify axial loads and hence axial deformations of individual structural members using their natural frequencies. Nevertheless, these methods cannot be applied to individual members in structural framing systems as the natural frequency is a global parameter for the entire framing system. This paper proposes an innovative method which uses modal strain energy phenomenon to quantify axial deformations of load bearing elements of structural framing systems. The procedure is illustrated through examples and results confirm that the proposed method has an ability to quantify the axial deformations of individual elements of structural framing systems

The feasibility of vibration analysis as a technique to detect osseointegration of transfemoral implants

One of the main causes of above knee or transfemoral amputation (TFA) in the developed world is trauma to the limb. The number of people undergoing TFA due to limb trauma, particularly due to war injuries, has been increasing. Typically the trauma amputee population, including war-related amputees, are otherwise healthy, active and desire to return to employment and their usual lifestyle. Consequently there is a growing need to restore long-term mobility and limb function to this population. Traditionally transfemoral amputees are provided with an artificial or prosthetic leg that consists of a fabricated socket, knee joint mechanism and a prosthetic foot. Amputees have reported several problems related to the socket of their prosthetic limb. These include pain in the residual limb, poor socket fit, discomfort and poor mobility. Removing the socket from the prosthetic limb could eliminate or reduce these problems. A solution to this is the direct attachment of the prosthesis to the residual bone (femur) inside the residual limb. This technique has been used on a small population of transfemoral amputees since 1990. A threaded titanium implant is screwed in to the shaft of the femur and a second component connects between the implant and the prosthesis. A period of time is required to allow the implant to become fully attached to the bone, called osseointegration (OI), and be able to withstand applied load; then the prosthesis can be attached. The advantages of transfemoral osseointegration (TFOI) over conventional prosthetic sockets include better hip mobility, sitting comfort and prosthetic retention and fewer skin problems on the residual limb. However, due to the length of time required for OI to progress and to complete the rehabilitation exercises, it can take up to twelve months after implant insertion for an amputee to be able to load bear and to walk unaided. The long rehabilitation time is a significant disadvantage of TFOI and may be impeding the wider adoption of the technique. There is a need for a non-invasive method of assessing the degree of osseointegration between the bone and the implant. If such a method was capable of determining the progression of TFOI and assessing when the implant was able to withstand physiological load it could reduce the overall rehabilitation time. Vibration analysis has been suggested as a potential technique: it is a non destructive method of assessing the dynamic properties of a structure. Changes in the physical properties of a structure can be identified from changes in its dynamic properties. Consequently vibration analysis, both experimental and computational, has been used to assess bone fracture healing, prosthetic hip loosening and dental implant OI with varying degrees of success. More recently experimental vibration analysis has been used in TFOI. However further work is needed to assess the potential of the technique and fully characterise the femur-implant system. The overall aim of this study was to develop physical and computational models of the TFOI femur-implant system and use these models to investigate the feasibility of vibration analysis to detect the process of OI. Femur-implant physical models were developed and manufactured using synthetic materials to represent four key stages of OI development (identified from a physiological model), simulated using different interface conditions between the implant and femur. Experimental vibration analysis (modal analysis) was then conducted using the physical models. The femur-implant models, representing stage one to stage four of OI development, were excited and the modal parameters obtained over the range 0-5kHz. The results indicated the technique had limited capability in distinguishing between different interface conditions. The fundamental bending mode did not alter with interfacial changes. However higher modes were able to track chronological changes in interface condition by the change in natural frequency, although no one modal parameter could uniquely distinguish between each interface condition. The importance of the model boundary condition (how the model is constrained) was the key finding; variations in the boundary condition altered the modal parameters obtained. Therefore the boundary conditions need to be held constant between tests in order for the detected modal parameter changes to be attributed to interface condition changes. A three dimensional Finite Element (FE) model of the femur-implant model was then developed and used to explore the sensitivity of the modal parameters to more subtle interfacial and boundary condition changes. The FE model was created using the synthetic femur geometry and an approximation of the implant geometry. The natural frequencies of the FE model were found to match the experimental frequencies within 20% and the FE and experimental mode shapes were similar. Therefore the FE model was shown to successfully capture the dynamic response of the physical system. As was found with the experimental modal analysis, the fundamental bending mode of the FE model did not alter due to changes in interface elastic modulus. Axial and torsional modes were identified by the FE model that were not detected experimentally; the torsional mode exhibited the largest frequency change due to interfacial changes (103% between the lower and upper limits of the interface modulus range). Therefore the FE model provided additional information on the dynamic response of the system and was complementary to the experimental model. The small changes in natural frequency over a large range of interface region elastic moduli indicated the method may only be able to distinguish between early and late OI progression. The boundary conditions applied to the FE model influenced the modal parameters to a far greater extent than the interface condition variations. Therefore the FE model, as well as the experimental modal analysis, indicated that the boundary conditions need to be held constant between tests in order for the detected changes in modal parameters to be attributed to interface condition changes alone. The results of this study suggest that in a clinical setting it is unlikely that the in vivo boundary conditions of the amputated femur could be adequately controlled or replicated over time and consequently it is unlikely that any longitudinal change in frequency detected by the modal analysis technique could be attributed exclusively to changes at the femur-implant interface. Therefore further development of the modal analysis technique would require significant consideration of the clinical boundary conditions and investigation of modes other than the bending modes.

Stiffness and damping coefficients of 3-axial groove water lubricated bearing using perturbation technique

A Computational fluid dynamics (CFD) approach is used to model fluid flow in a journal bearing with three equi-spaced axial grooves and supplied with water from one end. Water is subjected to both velocity (Couette) & pressure induced (Poiseuille) flow. The working fluid passing through the bearing clearance generates driving force components that may increase the unstable vibration of the rotor. It is important to know the accurate rotor dynamic force component for predicting the instability of rotor bearing systems. In this paper a study has been made to obtain the stiffness and damping coefficients of 3 axial groove bearing using Perturbation technique.

Interactive axial shortening of columns and walls in high rise buildings

Concrete is commonly used as a primary construction material for tall building construction. Load bearing components such as columns and walls in concrete buildings are subjected to instantaneous and long term axial shortening caused by the time dependent effects of "shrinkage", "creep" and "elastic" deformations. Reinforcing steel content, variable concrete modulus, volume to surface area ratio of the elements and environmental conditions govern axial shortening. The impact of differential axial shortening among columns and core shear walls escalate with increasing building height. Differential axial shortening of gravity loaded elements in geometrically complex and irregular buildings result in permanent distortion and deflection of the structural frame which have a significant impact on building envelopes, building services, secondary systems and the life time serviceability and performance of a building. Existing numerical methods commonly used in design to quantify axial shortening are mainly based on elastic analytical techniques and therefore unable to capture the complexity of non-linear time dependent effect. Ambient measurements of axial shortening using vibrating wire, external mechanical strain, and electronic strain gauges are methods that are available to verify pre-estimated values from the design stage. Installing these gauges permanently embedded in or on the surface of concrete components for continuous measurements during and after construction with adequate protection is uneconomical, inconvenient and unreliable. Therefore such methods are rarely if ever used in actual practice of building construction. This research project has developed a rigorous numerical procedure that encompasses linear and non-linear time dependent phenomena for prediction of axial shortening of reinforced concrete structural components at design stage. This procedure takes into consideration (i) construction sequence, (ii) time varying values of Young's Modulus of reinforced concrete and (iii) creep and shrinkage models that account for variability resulting from environmental effects. The capabilities of the procedure are illustrated through examples. In order to update previous predictions of axial shortening during the construction and service stages of the building, this research has also developed a vibration based procedure using ambient measurements. This procedure takes into consideration the changes in vibration characteristic of structure during and after construction. The application of this procedure is illustrated through numerical examples which also highlight the features. The vibration based procedure can also be used as a tool to assess structural health/performance of key structural components in the building during construction and service life.

Microcontinuum approach to the pulsatile flow in tubes with and without longitudinal vibration

A fully developed pulsatile flow in a circular rigid tube is analysed by a microcontinuum approach. Solutions for radial variation of axial velocity and cell rotational velocity across the tube are obtained using the momentum integral method. Simplified forms of the solutions are presented for the relevant physiological data. Marked deviations in the results are observed when compared to a Newtonian fluid model. It is interesting to see that there is sufficient reduction in the mass flow rate, phase lag and friction due to the micropolar character of the fluid.

On the axial vibrations of rotating bars

One of the important developments in rotary wing aeroelasticity in the recent past has been the growing awareness and acceptance of the fact that the problem is inherently non-linear and that correct treatment of aeroelastic problems requires the development of a consistent mathematical model [l]. This has led to a number of studies devoted to the derivation of a consistent set of “second order” non-linear equations, for example, those of Hodges and Dowel1 [2], of Rosen and Friedmann [3], and of Kvaternik, White and Kaza [4], each of which differs from the others on the question of the inclusion of certain terms in the equations of motion. The final form of the equations depends first upon the ordering scheme used for characterizing the displacements and upon the consistency with which this is applied in omitting terms of lower order. The ideal way of achieving this would be to derive the equations of motion with all the terms first included regardless of their relative orders of magnitude and then to apply the ordering scheme.

Free vibration of a stochastic beam-column using stochastic fem

The free vibrational characteristics of a beam-column, which is having randomly varying Young's modulus and mass density and subjected to randomly distributed axial loading is analysed. The material property fluctuations and axial loadings are considered to constitute independent one-dimensional, uni-variate, homogeneous real, spatially distributed stochastic fields. Hamilton's principle is used to formulate the problem using stochastic FEM. Vibration frequencies and mode shapes are analysed for their statistical descriptions. A numerical example is shown.

Optimum design of a helicopter rotor for low vibration using aeroelastic analysis and response surface methods

An aeroelastic analysis based on finite elements in space and time is used to model the helicopter rotor in forward flight. The rotor blade is represented as an elastic cantilever beam undergoing flap and lag bending, elastic torsion and axial deformations. The objective of the improved design is to reduce vibratory loads at the rotor hub that are the main source of helicopter vibration. Constraints are imposed on aeroelastic stability, and move limits are imposed on the blade elastic stiffness design variables. Using the aeroelastic analysis, response surface approximations are constructed for the objective function (vibratory hub loads). It is found that second order polynomial response surfaces constructed using the central composite design of the theory of design of experiments adequately represents the aeroelastic model in the vicinity of the baseline design. Optimization results show a reduction in the objective function of about 30 per cent. A key accomplishment of this paper is the decoupling of the analysis problem and the optimization problems using response surface methods, which should encourage the use of optimization methods by the helicopter industry. (C) 2002 Elsevier Science Ltd. All rights reserved.

Thermal vibration analysis of orthotropic nanoplates based on nonlocal continuum mechanics

This paper presents the thermal vibration analysis of orthotropic nanoplates such as graphene, using the two variable refined plate theory and nonlocal continuum mechanics for small scale effects. The nanoplate is modeled based on two variable refined plate theory and the axial stress caused by the thermal effects is also considered. The two variable refined plate theory takes account of transverse shear effects and parabolic distribution of the transverse shear strains through the thickness of the plate, hence it is unnecessary to use shear correction factors. Nonlocal governing equations of motion for the nanoplate are derived from the principle of virtual displacements. The closed form solution for thermal-vibration frequencies of a simply supported rectangular nanoplate has been obtained by using Navier's method of solution. Numerical results obtained by the present theory are compared with available solutions in the literature and the molecular dynamics results. The influences of the small scale coefficient, the room or low temperature, the high temparature, the half wave number and the aspect ratio of nanoplate on the natural frequencies are considered and discussed in detail. It can be concluded that the present theory, which does not require shear correction factor, is not only simple but also comparable to the first-order and higher order shear deformation theory. The present analysis results can be used for the design of the next generation of nanodevices that make use of the thermal vibration properties of the nanoplates. (C) 2012 Elsevier B.V. All rights reserved.

Thermal vibration analysis of monolayer graphene embedded in elastic medium based on nonlocal continuum mechanics

This paper presents the thermal vibration analysis of single-layer graphene sheet embedded in polymer elastic medium, using the plate theory and nonlocal continuum mechanics for small scale effects. The graphene is modeled based on continuum plate theory and axial stress caused by the thermal effects is also considered. Nonlocal governing equation of motion for this graphene sheet system is derived from the principle of virtual displacements. The closed form solution for thermal-vibration frequencies of a simply supported rectangular nanoplate has been obtained by using the Navier's method of solution. Numerical results obtained by the present theory are compared with available solutions in the literature and the molecular dynamics results. The influences of the small scale coefficient, the room or low temperature, the high temperature, the half wave number and the aspect ratio of nanoplate on the natural frequencies are considered and discussed in detail. The thermal vibration analysis of single- and double-layer graphene sheets are considered for the analysis. The mode shapes of the respective graphene system are also captured in this work. The present analysis results can be used for the design of the next generation of nanodevices that make use of the thermal vibration properties of the graphene.

An approach for assessing the accuracy of accelerometer responses during axial impact tests

The primary objective of the present study is to show that for the most common configuration of an impactor system, the accelerometer cannot exactly reproduce the dynamic response of a specimen subjected to impact loading. An equivalent Lumped Parameter Model (LPM) of the given impactor set-up has been formulated for assessing the accuracy of an accelerometer mounted in a drop-weight impactor set-up for an axially loaded specimen. A specimen under the impact loading is represented by a non-linear spring of varying stiffness, while the accelerometer is assumed to behave in a linear manner due to its high stiffness. Specimens made of steel, aluminium and fibre-reinforced composite (FRC) are used in the present study. Assuming the force-displacement response obtained in an actual impact test to be the true behaviour of the test specimen, a suitable numerical approach has been used to solve the governing non-linear differential equations of a three degrees-of-freedom (DOF) system in a piece-wise linear manner. The numerical solution of the governing differential equations following an explicit time integration scheme yields an excellent reproduction of the mechanical behaviour of the specimen, consequently confirming the accuracy of the numerical approach. However, the spring representing the accelerometer predicts a response that qualitatively matches the assumed force-displacement response of the test specimen with a perceptibly lower magnitude of load.