Abstract
Models consisting of chains of particles that are coupled to their neighbours appear in many applications in physics or engineering, such as in the study of dynamics of mono-atomic and multi-atomic lattices, the resonances of crystals with impurities and the response of damaged bladed discs. Analytical properties of the dynamic responses of such disturbed chains of identical springs and masses are presented, including when damping is present. Several remarkable properties in the location of the resonances (poles) and anti-resonances (zeros) of the displacements in the frequency domain are presented and proved. In particular, it is shown that there exists an elliptical region in the frequency–disturbance magnitude plane from which zeros are excluded and the discrete values of the frequency and disturbance at which double poles occur are identified. A particular focus is on a local disturbance, such as when a spring or damper is modified at or between the first and last masses. It is demonstrated how, notably through normalization, the techniques and results of the paper apply to a broad category of more complex systems in physics, chemistry and engineering.
Highlights
The dynamics of collinear chains of particles has been studied for a long time
These systems exhibit remarkable properties. Their interest is not limited to the academic realm as they model the fundamental vibrations of crystals in solidstate physics [1,2], the atomic and molecular dynamics of chains of molecules in physics, chemistry and biology [3,4,5,6,7], as well as the behaviour of real-life objects such as structures with repetitive components or rods and beams that are widely used in engineering
The expressions of lemma 3.1 are completely general for any regular collinear system that has identical point elements and connectors between two neighbour elements, independently of their physical nature or the real or complex value of their parameters
Summary
Their interest is not limited to the academic realm as they model the fundamental vibrations of crystals in solidstate physics [1,2], the atomic and molecular dynamics of chains of molecules in physics, chemistry and biology [3,4,5,6,7], as well as the behaviour of real-life objects such as structures with repetitive components or rods and beams that are widely used in engineering. The focus here is on the location of poles and zeros of the transfer functions of a disturbed system in the frequency–disturbance magnitude plane These loci are important system properties as they notably indicate if a system is stable, observable or controllable [22,23,24].
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