A study of axial joint behaviour in a high power ultrasonic device through transfer function synthesis

Further development of an open loop control strategy for High Power Ultrasonic (HPU) devices requires investigation of the dynamic behaviour of its interfaces. These interfaces exist between components that are assembled through a screwed threaded axial joint. An unknown prestress permeates from this bearing surface due to the tension in the stud. It is well known that the vibration of bolted interfaces is highly nonlinear. It is assumed that the axial joint will not loosen to the extent that the entire contacting surfaces will slip. Therefore the characteristic behaviour of this interface descends from the microscopic motions, of asperities in and out of contact, as slipping occurs locally. A review of the literature suggests that this physical mechanism will contribute to second harmonics in the response of the jointed system. This complex constitutive behaviour is not amenable to dynamical modelling. Instead the Bouc-Wen (BW) model for hysteresis is adopted to capture the phenomenology. The force-displacement behaviour of the axial joint is required to identify the parameters to this model. Dynamic force behaviour of a component that is subjected to high power ultrasonic loading cannot be obtained directly through conventional measurement systems. Force estimation of a set of ultrasonic test assemblies is attempted through a hybrid analytical-experimental scheme. The test assemblies consist of a Commercial Piezoelectric Actuator (CPA) or an In-House Piezoelectric Actuator (IHPA) with a stepped ultrasonic horn attached. Stepped ultrasonic horns are produced with and without an additional axial joint that is set to three different tightnesses. These are named the Jointed Horns (JH) or the Monolithic Horn (MH) respectively. The model of an actuator with the MH attached is named the linear calibration model. This is formulated through a Distributed Transfer Function Method that describes waveguides, with discontinuous parameters, that behave according to Love Rod Theory. This is too limited to represent the behaviour of a piezoelectric rod, so the electromechanical nature of the piezoelectric actuator is not considered in detail. Instead the parameters of the model are updated based on results from Experimental Modal Analysis through the Genetic Algorithm. The force at the foremost point of the piezoelectric stack is deconvolved from the distributed parameter rod model. Stable solutions to the ill-posed inverse problem are achieved by means of the Truncated Singular Value Decomposition or Tikhonov regularisation schemes. Electrical impedance analysis of the piezoelectric actuators, with a JH attached, demonstrate that the introduction of the joint to an equivalent assembly, made with the MH attached, will significantly modify the impedance behaviour. Input forces are deconvolved from the linear calibration assembly when the output is the measured response of the IHPA with the MH attached. The response of the IHPA with the JH attached is then assumed as the output and the joint force is estimated for the input force that corresponds to the voltage that was applied for the JH test. This force will be out of phase with the true linear input force to the JH assembly. The responses of the test assemblies are measured at 2 or 3 locations through Laser Doppler Vibrometry (LDV). Simultaneous images of a portion of the ultrasonic horn are recorded through an ultra-high speed camera. Digital Image Correlation (DIC) is applied to estimate a displacement field that can be compared to a LDV measurement. Good agreement in phase is found, but there are significant errors in the amplitude. It is suggested that this is the result of projection error. The strain fields are obtained from the displacement field through bi-cubic interpolation. It is not possible to achieve comparable results through the force estimation scheme. The BW model of hysteresis is defined by four shape parameters that are not straightforwardly related to physical quantities. The influence of these parameters on the hysteresis loop, and the frequency response of the model, is demonstrated through a sensitivity analysis. This suggests that the model will fit to the softening overhang behaviour that was associated with the looseness of the axial joint in the literature. Two interpretations of identifying the model are discussed. Firstly the limit cycle is fitted to experimental data through a set of MATLAB functions that make use of analytical solutions to each branch of the loop. Secondly a minimisation is carried out between the output of the model and the measured response of the IHPA with a JH attached. This is achieved through Differential Evolution. It is not possible to identify a reliable model without an improved estimation of the input to the model. To make progress with this problem a number of contributions have been made. Simulations of the BW model have demonstrated that it is capable of describing the softening overhang behaviour that has been observed in the response of HPU devices and the subharmonic generation that descends from Contact Acoustic Nonlinearity. A methodical attempt to identify the parameters to this model from measurements of HPU test assemblies has been presented. A new force estimation scheme has been developed, which is based on a recent formulation of the Distributed Transfer Function Method. This provides a closed form route to estimating the force at an axial joint in a rod-like system. Regularisation methods have been applied to stabilise these estimations. An experimental configuration has been presented to test the force estimation scheme with observations from HPU test assemblies. These test assemblies have been analysed through electrical impedance analysis and Experimental Modal Analysis. This demonstrates changes in the behaviour of the assembly due to the introduction of the joint and its set tightness. Digital Image Correlation is presented as an alternative to finding the hysteresis behaviour at the axial joint which is subject to ultrasonic loading. Recommendations are made towards improving this set up.