Practical solution of torsional vibration problems, with examples from marine, electrical, aeronautical, and automobile engineering practice,

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On the influence of the mode-shapes of a marine propulsion shafting system on the prediction of torsional stresses

Torsional vibration predictions and measurements of a marine propulsion system, which has both damping and a highly flexible coupling, are presented in this paper. Using the conventional approach to stress prediction in the shafting system, the numerical predictions and the experimental torsional vibration stress curves in some parts of the shafting system are found to be quite different. The free torsional vibration characteristics and forced torsional vibration response of the system are analyzed in detail to investigate this phenomenon. It is found that the second to fourth natural modes of the shafting system have significant local deformation. This results in large torsional resonant responses in different sections of the system corresponding to different engine speeds. The results show that when there is significant local deformation in the shafting system for different modes, then multi-point measurements should be made, rather than the conventional method of using a single measurement at the free end of the shaft, to obtain the full torsional vibration characteristics of the shafting system.

Nonlinear drillstring dynamics analysis

This paper studies the dynamical response of a rotary drilling system with a drag bit, using a lumped parameter model that takes into consideration the axial and torsional vibration modes of the bit. These vibrations are coupled through a bit-rock interaction law. At the bit-rock interface, the cutting process introduces a state-dependent delay, while the frictional process is responsible for discontinuous right-hand sides in the equations governing the motion of the bit. This complex system is characterized by a fast axial dynamics compared to the slow torsional dynamics. A dimensionless formulation exhibits a large parameter in the axial equation, enabling a two-time-scales analysis that uses a combination of averaging methods and a singular perturbation approach. An approximate model of the decoupled axial dynamics permits us to derive a pseudoanalytical expression of the solution of the axial equation. Its averaged behavior influences the slow torsional dynamics by generating an apparent velocity weakening friction law that has been proposed empirically in earlier work. The analytical expression of the solution of the axial dynamics is used to derive an approximate analytical expression of the velocity weakening friction law related to the physical parameters of the system. This expression can be used to provide recommendations on the operating parameters and the drillstring or the bit design in order to reduce the amplitude of the torsional vibrations. Moreover, it is an appropriate candidate model to replace empirical friction laws encountered in torsional models used for control. © 2009 Society for Industrial and Applied Mathematics.

A study of vibration and vibration control of ship structures

This paper examines the vibration characteristics and vibration control of complex ship structures. It is shown that input mobilities of a ship structure at engine supports, due to out-of-plane force or bending moment excitations, are governed by the flexural stiffness of the engine supports. The frequency averaged input mobilities of the ship structure, due to such excitations, can be represented by those of the corresponding infinite beam. The torsional moment input mobility at the engine support can be estimated from the torsional response of the engine bed section under direct excitation. It is found that the inclusion of ship hull and deck plates in the ship structure model has little effect on the frequency-averaged response of the ship structure. This study also shows that vibration propagation in complex ship structures at low frequencies can be attenuated by imposing irregularities to the ring frame locations in ships. Vibration responses of ship structures due to machinery excitations at higher frequencies can be controlled by structural modifications of the local supporting structures such as engine beds in ships.

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.

Optimal configurations of active fiber composites based on asymptotic torsional analysis - art. no. 641413

Active Fiber Composites (AFC) possess desirable characteristics over a wide range of smart structure applications, such as vibration, shape and flow control as well as structural health monitoring. This type of material, capable of collocated actuation and sensing, call be used in smart structures with self-sensing circuits. This paper proposes four novel applications of AFC structures undergoing torsion: sensors and actuators shaped as strips and tubes; and concludes with a preliminary failure analysis. To enable this, a powerful mathematical technique, the Variational Asymptotic Method (VAM) was used to perform cross-sectional analyses of thin generally anisotropic AFC beams. The resulting closed form expressions have been utilized in the applications presented herein.

Finite element analysis of free vibration of the delaminated composite plate with variable kinematic multilayered plate elements

Composite laminates are prone to delamination. Implementation of delamination in the Carrera Unified Formulation frame work using nine noded quadrilateral MITC9 element is discussed in this article. MITC9 element is devoid of shear locking and membrane locking. Delaminated as well as healthy structure is analyzed for free mode vibration. The results from the present work are compared with the available experimental or/and research article or/and the three dimensional finite element simulations. The effect of different kinds and different percentages of area of delamination on the first three natural frequencies of the structure is discussed. The presence of open-mode delamination mode shape for large delaminations within the first three natural frequencies is discussed. Also, the switching of places between the second bending mode, with that of the first torsional mode frequency is discussed. Results obtained from different ordered theories are compared in the presence of delamination. Advantage of layerwise theories as compared to equivalent single layer theories for very large delaminations is stated. The effect of different kinds of delamination and their effect on the second bending and first torsional mode shape is discussed. (C) 2014 Elsevier Ltd. All rights reserved.

Prediction and measurement of forced response of a composite blade undergoing nonlinear vibration

The main objective of this project is to experimentally demonstrate geometrical nonlinear phenomena due to large displacements during resonant vibration of composite materials and to explain the problem associated with fatigue prediction at resonant conditions. Three different composite blades to be tested were designed and manufactured, being their difference in the composite layup (i.e. unidirectional, cross-ply, and angle-ply layups). Manual envelope bagging technique is explained as applied to the actual manufacturing of the components; problems encountered and their solutions are detailed. Forced response tests of the first flexural, first torsional, and second flexural modes were performed by means of a uniquely contactless excitation system which induced vibration by using a pulsed airflow. Vibration intensity was acquired by means of Polytec LDV system. The first flexural mode is found to be completely linear irrespective of the vibration amplitude. The first torsional mode exhibits a general nonlinear softening behaviour which is interestingly coupled with a hardening behaviour for the unidirectional layup. The second flexural mode has a hardening nonlinear behaviour for either the unidirectional and angle-ply blade, whereas it is slightly softening for the cross-ply layup. By using the same equipment as that used for forced response analyses, free decay tests were performed at different airflow intensities. Discrete Fourier Trasform over the entire decay and Sliding DFT were computed so as to visualise the presence of nonlinear superharmonics in the decay signal and when they were damped out from the vibration over the decay time. Linear modes exhibit an exponential decay, while nonlinearities are associated with a dry-friction damping phenomenon which tends to increase with increasing amplitude. Damping ratio is derived from logarithmic decrement for the exponential branch of the decay.