Modern Practice in Stress and Vibration Analysis VI

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Authors: Hua Jiang Ouyang, John E. Mottershead
Abstract: The vibration of a beam excited by a moving simple oscillator is an extensively studied problem. However, the vibration of a beam excited by an elastic body with conformal contact has attracted much less attention. This is the subject of the present paper. The established model is a big improvement to the moving oscillator model and has many engineering applications. Because the moving body is flexible, the moving loads at the contact interface are not known a priori and must be determined together with the dynamics of the whole system. Considerable mathematical complication arises as a result, compared with the moving-oscillator problem, even if the contact is assumed to be complete. In this paper, the equation of motion of the beam and the moving body are established separately using an analytical-numerical combined approach. The equation for the moving loads is established through the displacement continuity at the contact interface. It is found from the simulated numerical results that the deflection of the beam displays several cycles of oscillation during the passage of the moving body and can exceed the maximum static deflection at moderate speeds, but is close to the static deflection when the speed is either very low or very high.
Authors: T. Butlin, J. Woodhouse
Abstract: The problem of ‘brake squeal’ in the automotive industry remains despite over 70 years of research: the phenomenon is still surprisingly unpredictable and poorly understood. The literature has moved from very simple lumped parameter models to ever more sophisticated finite element models, but testing theory against measurements has been hindered by the difficulty in obtaining repeatable results. It would seem the phenomenon is extremely sensitive to changes in parameters beyond an experimenter’s control. This paper describes recent results from a project to identify and quantify the sources of uncertainty within sliding contact systems and to determine the sensitivity of the friction-coupled system to uncertain parameters. The theoretical approach taken is to use a linear analysis based on the uncoupled transfer functions of two general subsystems to predict stability when they are coupled by a sliding point contact. The model is tested using a pin-on-disc rig whose uncoupled transfer functions can be measured. Using a stability criterion based on the roots of the characteristic equation of the system, the sensitivity of model predictions to parameter variations is investigated numerically. It is shown that using a realistic range of parameters the root locations change considerably and enough to change stability predictions. As the complexity of the model is increased reliable results become harder to achieve as the characteristic equation becomes more ill-conditioned. This is not simply a result of the high order of the system, but is thought to be a result of particular mode combinations. Experimental work shows uncoupled transfer functions vary over time and by enough to significantly affect squeal predictions. These results suggest reasons for the difficulty in obtaining repeatable measurements and for the unreliability of squeal prediction theories developed so far. If the reasons for the sensitivity of squeal can be understood it may be possible to design sliding contact systems that are more robust.
Authors: Seamus D. Garvey, Peter Van Eetvelt, U. Prells
Abstract: Three (m × n) matrices {K, D, M} represent a second-order system in the form (K + Dλ+ Mλ2). If m = n, system eigenvalues are defined as the values of λ for which det(K + Dλ+ Mλ2) = 0. If {K, D, M} are continuous functions of a real scalar parameter, σ, eigenvalues and dimensions of the associated eigenspaces remain constant if and only if the rates of change of {K, D, M} obey certain ODEs called the isospectral flow equations. The integration of these matrix differential equations is of interest here. This paper explains the motivation behind this work in terms of vibrating systems and it reports two related hypotheses concerning how the solutions to these equations may be decoupled. Work underway towards proving and using these hypotheses is presented. No existing known solutions allow this decoupling in general.
Authors: I. Sadaba, Colin H.J. Fox, Stewart McWilliam
Abstract: Residual stress effects in MEMS wafer adhesive bonding were investigated. A waferlevel finite element (FE) modelling approach was used, which accounted for wafer configuration and allowed the deformation of individual devices across a wafer to be considered. A specific test was carried out to determine the adhesive’s viscoelastic response, through an effective timedependent Young`s modulus, so that initial and longer-term post-curing responses could be estimated. These results are compared with predictions based on modulus values obtained from Dynamic Mechanical Analysis (DMA) testing. The predicted wafer deformations are compared with optical surface profile measurements of actual specimens. Initial response predictions using the effective modulus show excellent agreement with measurements, while the DMA modulus gives a significant underestimate. On the other hand, difficulty was found in modelling long-term relaxation, due to non-linear mechanisms such as polymer “ageing” which are harder to model.
Authors: I. Sadaba, Colin H.J. Fox, Stewart McWilliam
Abstract: Anodic bonding is widely-used in the fabrication of Micro-Electro-Mechanical Systems (MEMS) devices to join silicon and glass components. The process involves the application of temperature, moderate pressure and an electric field. This paper investigates residual stresses arising during anodic bonding, focusing on the resulting induced distortions. Components of a MEMS silicon rate sensor, in which a silicon wafer is anodically bonded to Pyrex™ glass, were used as the vehicle for the investigation. Distortions generated by the anodic bonding process when using two different electrode configurations (point and planar) were measured using a surface optical profiler. These showed a particular pattern across the wafers for both configurations. An efficient FEM study was carried out to model the qualitative effect of the following residual stress sources; thermal stress, glass shrinkage due to structural relaxation and compositional gradients due to ion migration. Importantly, the FE model takes account the actual multi-device wafer-level configuration, as opposed to a single device. The results demonstrate that compositional gradients can make a significant contribution to the observed pattern of distortions.
Authors: R.F. Lennon
Abstract: The buckling resistance of orthogonally stiffened cylinders is investigated for elastic critical buckling and non-linear elasto-plastic buckling tests. The effects of residual stresses arising from cold forming the cylinder and welding frame components are considered in the analysis of two stiffened cylinder models with similar material weights and different geometric spacings. A static axial load is applied to the models to represent loading from the supported structure followed by a non-linear elasto-plastic buckling step representing a wave loading combined with hydrostatic pressure, producing large displacement compartment buckling. Residual stress is shown to cause a reduction in buckling resistance of approximately 25% in the stiffened cylinder segments.
Authors: E. McCulloch, Alan MacBeath, Margaret Lucas
Abstract: The performance of an ultrasonic cutting device critically relies on the interaction of the cutting tool and the material to be cut. A finite element (FE) model of ultrasonic cutting is developed to enable the design of the cutting blade to be influenced by the requirements of the toolmaterial interaction and to allow cutting parameters to be estimated as an integral part of designing the cutting blade. In this paper, an application in food processing is considered and FE models of cutting are demonstrated for toffee; a food product which is typically sticky, highly temperature dependent, and difficult to cut. Two different 2D coupled thermal stress FE models are considered, to simulate ultrasonic cutting. The first model utilises the debond option in ABAQUS standard and the second uses the element erosion model in ABAQUS explicit. Both models represent a single blade ultrasonic cutting device tuned to a longitudinal mode of vibration cutting a specimen of toffee. The model allows blade tip geometry, ultrasonic amplitude, cutting speed, frequency and cutting force to be adjusted, in particular to assess the effects of different cutting blade profiles. The validity of the model is highly dependent on the accuracy of the material data input and on the accuracy of the friction and temperature boundary condition at the blade-material interface. Uniaxial tensile tests are conducted on specimens of toffee for a range of temperatures. This provides temperature dependent stress-strain data, which characterises the material behaviour, to be included in the FE models. Due to the difficulty in gripping the tensile specimens in the test machine, special grips were manufactured to allow the material to be pulled without initiating cracks or causing the specimen to break at the grips. A Coulomb friction condition at the bladematerial interface is estimated from experiments, which study the change in the friction coefficient due to ultrasonic excitation of a surface, made from the same material as the blade, in contact with a specimen of toffee. A model of heat generation at the blade-toffee interface is also included to characterise contact during ultrasonic cutting. The failure criterion for the debond model assumes crack propagation will occur when the stress normal to the crack surface reaches the tensile failure stress of toffee and the element erosion model uses a shear failure criterion to initiate element removal. The validity of the models is discussed, providing some insights into the estimation of contact conditions and it is shown how these models can improve design of ultrasonic cutting devices.
Authors: Heinz Ulbrich, T. Buschmann, S. Lohmeier
Abstract: This paper presents the performance enhanced humanoid robot LOLA which is currently being manufactured. Hardware design, controllers and simulation are based on ex- perience gained during the development of the robot JOHNNIE. The objective of the current research project is to realize a fast, human-like and autonomous walking motion. To enable an optimal design of the robot with respect to lightweight construction, motor and drive sizing, an appropriate simulation model is required. Dynamics simulation is a key tool to develop the hardware and control design properly. For hardware design and detailed dynamic analysis a comprehensive model including motor and gear dynamics is required, while for controller de- sign and stability analysis a simplified model for global system dynamics is sufficient. Both robots are characterized by a lightweight construction. In comparison to JOHNNIE, the new robot LOLA has a modular, multi-sensory joint design with brushless motors. Moreover, the previously purely central electronics architecture is replaced by a network of decentral joint controllers, sensor data acquisition and filtering units and a central PC. The fusion of motor, gear and sensors into a highly integrated mechatronic joint module has several advantages for the whole system, including high power density, good dynamic performance and reliability. Ad- ditional degrees of freedom are introduced in elbow, waist and toes. Linear actuators are used for the knee joints to achieve a better mass distribution in the legs.

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