Structural Dynamics

Characterization of seismic hazard via (elastic) design spectra and the estimation of linear peak response of a given structure from this characterization continue to form the basis of earthquake-resistant design philosophy in various codes of practice all over the world. Since the direct use of design spectrum ordinates is a preferred option for the practicing engineers, modal combination rules play central role in the peak response estimation. Most of the available modal combination rules are however based on the assumption that nonstationarity affects the structural response alike at the modal and overall response levels. This study considers those situations where this assumption may cause significant errors in the peak response estimation, and preliminary models are proposed for the estimation of the extents to which nonstationarity affects the modal and total system responses, when the ground acceleration process is assumed to be a stationary process. It is shown through numerical examples in the context of complete-quadratic-combination (CQC) method that the nonstationarity-related errors in the estimation of peak base shear may be significant, when strong-motion duration of the excitation is too small compared to the period of the system and/or the response is distributed comparably in several modes. It is also shown that these errors are reduced marginally with the use of the proposed nonstationarity factor models.

Piron, D., Pathak, S., Deraemaeker, A., & Collette, C. (2021). A pole-zero based criterion for optimal placement of collocated sensor-actuator pair. Mechanical Systems and Signal Processing, 155 , 107533.

A new and computationally efficient criterion is developed concerning the optimal placement of a collocated sensor-actuator pair for active control of structural vibrations with a low authority controller. The basic idea behind the proposed criterion is based on the maximization of the pole-zero distance in open-loop which has a direct link with the maximum achievable damping ratio once in closed-loop. Unlike poles, the determination of transmission zeros is computationally tedious because it depends upon the placement of the sensor-actuator pair. Therefore, a new approximation to reliably estimate the transmission zeros is also introduced which remarkably enhances the computational efficiency of the proposed optimization criterion. The effectiveness of the proposed criterion is demonstrated on a cantilever beam and a simply supported plate and is also compared with two other widely used criteria: the Gramian controllability and the spatial H2 norm. It is shown that the proposed criterion is pertinent and ensures higher damping values for the cantilever beam compared to the other two criteria. Nonetheless, the criterion requires adaptations to improve its reliability for structure with large modal density such as a simply supported plate.

Pathak, S., Piron, D., Paknejad, A., Collette, C., & Deraemaeker, A. (2022). On transmission zeros of piezoelectric structures. Journal of Intelligent Material Systems and Structures, 33 (12), 1538–1561.

The evaluation of transmission zeros is of great importance for the control engineering applications. The structures equipped with piezoelectric patches are complex to model and usually require finite element approaches supplemented by model reduction. This study rigorously investigates the influence of mesh size, model reduction, boundary conditions (free and clamped), and sensor/actuator configuration (collocated and non-collocated) on the evaluation of transmission zeros of the piezoelectric structures. The numerical illustrations are presented for a thin rectangular plate equipped with a single pair of piezoelectric voltage sensor/ voltage actuator. Through the examples considered in this study, a link is presented between the static response (or static deflected shape) and the transmission zeros of the piezoelectric structures. This interesting observation forms the basis of: (i) a local mesh refinement strategy for computationally efficient estimation of the transmission zeros and (ii) a physical interpretation of the pole-zero pattern in the case of piezoelectric structures. The physical interpretation developed in this study helps in qualitatively explaining the pole-zero patterns observed for different configurations. It is also shown that this understanding of the relation between the static deformed shape and the transmission zeros can be used by the practitioners to modify the pole-zero pattern through a careful selection of the orientation and the size of the piezoelectric patches. 

Rasa, J., Ahmad, P., Pathak, S., Dimitri, P., & Christophe, C. (2022). Active damping of high modal density of bladed structures with piezoelectric patches. In Proceedings of international conference on noise and vibration engineering and international conference on uncertainty in structural dynamics (12-14 September 2022), Leuven, Belgium.

For more details refer to the journal version Paknejad et al.(2023) Active vibration mitigation of bladed structures with piezoelectric patches by decentralized positive position feedback controller. Journal of Engineering for Gas Turbines and Power, ASME, 145 (2).

Piron, D., Pathak, S., Deraemaeker, A., & Collette, C. (2022). On the link between pole-zero distance and maximum reachable damping in MIMO systems. Mechanical Systems and Signal Processing, 181 , 109519.

This paper studies the possibility of extending the already proved link between the pole-zero distance and the maximum reachable damping ratio in single input single output (SISO) systems to multiple inputs multiple outputs (MIMO) ones. This extension is shown to be possible when the considered system presents specific properties: (i) it is equipped with collocated transducers with small authority, (ii) the system has a small modal density in the frequency band of interest and (iii) a low authority control law is used. It is indeed demonstrated that when these three conditions are satisfied, the analytical development of the closed-loop poles convergence is equivalent to the one observed with SISO cases, except that the anti-resonances are replaced by the transmission zeros (TZs). Consequently, it is concluded that the maximum reachable damping ratio is directly proportional to the pole-transmission zero distance for such MIMO systems. This conclusion is demonstrated with two numerical examples (a cantilever beam and a simply supported plate) and experimentally validated on a cantilever beam where all the studied systems are equipped with two collocated pairs of piezoelectric patches.

Paknejad, A., Jamshidi, R., Pathak, S., & Collette, C. (2023). Active vibration mitigation of bladed structures with piezoelectric patches by decentralized positive position feedback controller. Journal of Engineering for Gas Turbines and Power, ASME, 145 (2).

This paper proposes an active damping system to mitigate the vibration of bladed assemblies. The damping system consists of multiple pairs of piezoelectric patches accompanied by a decentralized control configuration. To maximize the control authority, the size and the location of the patches are optimized based on maximizing the strain energy. In each pair, one patch is used as a sensor and the other one as an actuator. As the control plants of such configuration have no high-frequency roll-off, a second-order low-pass filter known as a positive position feedback (PPF) controller is considered as the control law. The parameters of the controller are tuned based on maximizing the closed-loop damping of the first family of modes. This active damping system is implemented on a monobloc bladed rail which is representative of a portion of bladed drum, i.e., BluM. Numerical simulations are performed to assess the performance of the designed control system and experimental tests are carried out to validate the numerical design.

Shrivastava, H., & Pathak, S. (2024). On seismic demand of near-field ground motions. Sadhana, 49 (2), 111.

Near-fault ground motions (NFGM) may have a significant component of the long period velocity pulse (LPVP) due to the directivity effect. The structures subjected to such ground motions are prone to experience large displacements. In this paper, the damage potential of such motions is compared with NFGM (without LPVP) and far-field motions in terms of seismic demand (inter-storey drift and inelastic displacement). Due to the lack of site-specific recorded NFGM, the synthetic ground motions are generated and used in this study. For this purpose, first the bedrock motions are obtained through stochastic simulation approach in which the Specific Barrier Model (SBM) is used as source spectrum and then these motions are transferred to the ground surface using one-dimensional ground response analysis. In order to obtain the NFGM with pulse, the bedrock motions are superimposed with the LPVPs and then transferred to the ground surface. To illustrate, the case of a ten-storey reinforced concrete (RC) building is considered in Delhi region (India) which falls under the seismic zone-IV. The main finding of this study is the observation that the NFGM with LPVP impose almost double seismic demand compared to the NFGM without LPVP, specially for the structures lying in the velocity-region of the response spectrum. In this study, the effect of depth of bedrock on seismic demand is highlighted and recommended to be included in the design seismic demand of NFGM. 

Pathak, S. (2024). Fractional differential equations: A primer for structural dynamics applications. In M. Ruzhansky & K. Van Bockstal (Eds.), Extended Abstracts 2021/2022: Ghent Analysis and PDE Seminar, Trends in Mathematics (2) (pp. 271–281). Birkhauser, Cham.

This paper presents a very brief overview of the framework for the application of fractional calculus (FC) in structural dynamics with the target audience from two scholarly communities, namely, structural dynamists and applied mathematicians. For the former, this paper provides a quick and simplified introduction to fractional differential equations (FDE) to explore the dynamics of structures consisting of rate-dependent materials. A systematic link has been shown between conventional structural dynamics (CSD) and its fractional generalisation. Whereas, for the latter, it highlights the powerful applications of fractional equations in the broad field of structural dynamics with some potential problems that may need their attention in the near future.