PSI - Issue 78

Anna Brunetti et al. / Procedia Structural Integrity 78 (2026) 1729–1736

1734

m/s², 2.0 m/s², 1.8 m/s², and 0.5 m/s², respectively. These values exceed the 0.5 m/s² threshold associated with maximum comfort, although they never go beyond the levels that guarantees minimum comfort. Conversely, accelerations in the transverse and longitudinal directions are negligible. To ensure optimal comfort, the design of TMDs is recommended to mitigate the dynamic effects associated with these frequencies.

Fig. 5. Results of the steady-state analyses.

4. Design of the Tuned Mass Damper system A TMD is a device designed to reduce the dynamic response of the structure to which it is attached. It consists of a secondary mass connected to the main structure through a spring and typically incorporates a damping element. The TMD's properties are tuned so that its natural frequency matches that of the structure at a specific frequency. When the structure is excited at this frequency, the TMD oscillates in antiphase, contributing to reduce the dynamic effects ( Dall’Asta et al . 2016, Li et al. 2023). The standard design approach, proposed by Den Hartog (1985), allows for the optimal calibration of the key parameters of a TMD (mass, natural frequency and damping ratio) in the case of a single-degree-of-freedom (SDOF) system. The proposed approach allows these parameters to be derived from the mass ratio between the TMD and the structure. It can be demonstrated that the design of a TMD for a MDOF system can be reduced to the same problem applied to an equivalent SDOF system. The dynamic problem of a MDOF equipped with a TMD, in the direction of the TMD's action, can thus be defined as follows (Connor 2003, Tophøj et al. 2018): ̈( ) + ( )̇ + ( ) = ( ) + ⌈ ( ) + ̇ ( ) ⌉ ( ) = [ ( ) ( ) ] ( ) = [ ( ) ( ) ] (2a,b,c) where , and represent the mass, damping and stiffness matrices of the MDOF system, respectively, and and are the stiffness and the damping of the TMD device. The displacement vector ( ) can be partitioned into ( ) and ( ) , where ( ) is the displacement vector of the nodes of the system where TMD is not applied and ( ) is the displacement of the node where the device is applied. The same applies to the external force vector, ( ) , which is composed of ( ) , the vector of forces acting on the nodes without TMD, and ( ) , the force acting on the node where the device is attached. Finally, ( ) represents the displacement of the device itself. The displacement ( ) can be expressed in the principal coordinates through: ( ) = ( ) (3) where ( ) is the vector of principal coordinates and is the modal matrix. By substituting (3) into (1), standard algebra allows decoupling the dynamics of the MDOF into N single-degree-of-freedom (SDOF) problems, where