Membranes with thin and heavy inclusions: Asymptotics of spectra

We study the asymptotic behaviour of eigenvalues of 2D vibrating systems with mass density perturbed in a vicinity of closed curves. The threshold case in which the resonance frequencies of the membrane and the frequencies of thin inclusion coincide is investigated. The perturbed eigenvalue problem can be realized as a family of self-adjoint operators acting on varying Hilbert spaces. However the so-called limit operator is non-self-adjoint and possesses the Jordan chains of length 2. Apart from the lack of self-adjointness, the operator has non-compact resolvent. As a consequence, its spectrum has a complicated structure, for instance, the spectrum contains a countable set of eigenvalues with infinite multiplicity. The complete asymptotic analysis of eigenvalues has been carried out.

Modelling of SMA Vibration Systems in an AVA Example

Vibration suppression, as well as its generation, is a common subject of scientific investigations. More and more often, but still rarely, shape memory alloys (SMAs) are used in vibrating systems, despite the fact that SMA springs have many advantages. This is due to the difficulty of the mathematical description and the considerable effortfulness of analysing and synthesising vibrating systems. The article shows the analysis of vibrating systems in which spring elements made of SMAs are used. The modelling and analysis method of vibrating systems is shown in the example of a vibrating system with a dynamic vibration absorber (DVA), which uses springs made of a shape memory alloy. The formulated mathematical model of a 2-DOF system with a controlled spring, mounted in DVA suspension, uses the viscoelastic model of the SMA spring. For the object, a control system was synthesised. Finally, model tests with and without a controller were carried out. The characteristics of the vibrations’ transmissibility functions for both systems were determined. It was shown that the developed DVA can tune to frequency excitation changes of up to ±10%.

Teaching Vibration to university undergraduates

This paper provides a description of an undergraduate course on vibration, given to second and third year students at Bristol university in the UK. The course, and my teaching philosophy, were developed over more than 30 years. The lectures were given in two 20-hour courses which were supported by an equal number of examples classes. Students were provided with a series of question sheets which contained questions from previous examination papers. In addition, laboratory classes were provided to give the students “hands on” experience on how to excite, control, measure, and to interpret various vibrating systems. The first set of lectures began with the analysis of a single degree of freedom system, adding different forcing functions and more degrees of freedom. The second set of lectures introduced continuous systems, consisting of bars, beams, and plates. The limitations of reality, particularly boundary conditions, was emphasized. Wherever possible, some artifact was taken to the lecture amplify the mathematics. I have interleaved into the presentation some of my teaching philosophy and how it is important in a heavily mathematical subject such as vibration to teach rather than to try and impress the students as to how clever is their lecturer. Finally, if the lecturer does not enjoy giving the lectures, the students will not receive that “extra” which distinguishes a good lecture from a bad lecture, and also distinguishes a good lecture from reading a text book.

Support with mechanical vibrations of welding processes ‒ review of own research

The article discusses the most important original achievements in the use of high-power mechanical vibrations with low and ultrasonic frequency in various welding processes such as MIG, MAG, TIG, RW, LW, diffusion welding and brazing in relation to various basic materials such as structural steel and aluminum alloys. Mechanical vibrations were introduced by means of ultrasonic vibrating systems and using the shot-blasting process, as well as acoustic influence. As part of the comparative research, the structure and hardness analysis of HV0.1 was presented. The obtained results indicate that both low-frequency and ultrasonic frequency vibrations significantly change the properties of the resulting structures affected by mechanical vibrations. The scale of these changes varies depending on the frequency and parameters specific to each of the welding processes used. The obtained results allow us to conclude that the skillful application of mechanical vibrations accompanying welding processes can contribute to the reduction of grain sizes, to change the depth or width of penetration and to lowering hardness in the sensitive heat affected zone area.

A hybrid method for broadband vibroacoustic simulations

Many methods for simulating acoustic responses of vibrating systems are only suitable for limited frequency ranges, providing either an accurate low frequency or high frequency response. A hybrid method is presented to combine a low frequency modal response and a high frequency statistical energy response to obtain a unified broadband response. The method is designed to produce an auralizable response. An experimental setup is used to validate the method. Listening tests are conducted to assess the realism of the auralizations compared to measurements. The listening tests confirm that the method is able to produce realistic auralizations, subject to a few limitations.

Robust Assignment of Natural Frequencies and Antiresonances in Vibrating Systems through Dynamic Structural Modification

This paper proposes a novel method for the robust partial assignment of natural frequencies and antiresonances, together with the partial assignment of the related eigenvectors, in lightly damped linear vibrating systems. Dynamic structural modification is exploited to assign the eigenvalues, either of the system or of the adjoint system, together with their sensitivity with respect to some parameters of interest. To handle with constraints on the feasible modifications, the inverse eigenvalue problem is cast as a minimization problem and a solution method is proposed through homotopy optimization. Variables lifting for bilinear and trilinear terms, together with bilinear and double-McCormick’s constraints, are exploited to provide a convexification of the problem and to boost the attainment of the global optimum. The effectiveness of the proposed method is assessed through four numerical examples.