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

### **Alberto Di Matteo <sup>1</sup> , Iain Peter Dunn 1,\*, Giuseppe Failla <sup>2</sup> and Antonina Pirrotta <sup>1</sup>**


Received: 15 February 2020; Accepted: 30 April 2020; Published: 7 May 2020

**Abstract:** 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.

**Keywords:** Euler Bernoulli beam; poissonian loading; tuned mass damper
