A Novel Solution to Find the Dynamic Response of an Euler–Bernoulli Beam Fitted with Intraspan TMDs under Poisson Type Loading

This contribution considers a virtual experiment on the vibrational response of rail and road bridges equipped with smart devices in the form of damping elements to mitigate vibrations. The internal damping of the bridge is considered a discontinuity that contain a dashpot. Exact complex eigenvalues and eigenfunctions are derived from a characteristic equation built as the determinant of a 4 × 4 matrix; this is accomplished through the use of the theory of generalized functions to find the response variables at the positions of the damping elements. To relate this to real world applications, the response of a bridge under Poisson type white noise is evaluated; this is similar to traffic loading that would be seen in a bridge’s service life. The contribution also discusses the importance of smart damping and dampers to sustainability efforts through the reduction of required materials, and it discusses the role played by robust mathematical modelling in the design phase.

On the Reversibility of Discretization

“Discretization” usually denotes the operation of mapping continuous functions to infinite or finite sequences of discrete values. It may also mean to map the operation itself from one that operates on functions to one that operates on infinite or finite sequences. Advantageously, these two meanings coincide within the theory of generalized functions. Discretization moreover reduces to a simple multiplication. It is known, however, that multiplications may fail. In our previous studies, we determined conditions such that multiplications hold in the tempered distributions sense and, hence, corresponding discretizations exist. In this study, we determine, vice versa, conditions such that discretizations can be reversed, i.e., functions can be fully restored from their samples. The classical Whittaker-Kotel’nikov-Shannon (WKS) sampling theorem is just one particular case in one of four interwoven symbolic calculation rules deduced below.

Analysis of Lamellar Structures with Application of Generalized Functions

Theory of differential equations in respect of the functional area is based on the basic concepts on generalized functions and splines. There are some basic concepts related to the theory of generalized functions and their properties are considered in relation to the rod systems and lamellar structures. The application of generalized functions gives the possibility to effectively calculate step-variable stiffness lamellar structures. There are also widely applied structures, in that several in which a number of parallel load bearing layers are interconnected by discrete-elastic links. For analysis of system under study, such as design diagrams, there are applied discrete and discrete-continual models.