# Wavelet-Based Estimation of Damping from Multi-Sensor, Multi-Impact Data
###### Abstract
Furthermore, recent studies on wavelet-based damping identification methods, such as [27; 28; 29; 30], have not sufficiently addressed effective strategies for combining multiple impulse responses in a way that mitigates noise while preserving meaningful damping information. This research paper investigates the extraction of damping from multiple non-synchronized impulse response measurements in mechanical systems, focusing on the process's reliability. Despite the availability of various methods, accurately calculating damping characteristics, especially in noisy environments, remains a challenge [17; 22]. Additionally, traditional Gaussian noise reduction techniques, such as simple averaging in the time domain, are ineffective when recordings are misaligned and inconsistent. Moreover, damping extraction is inherently nonlinear, meaning that the extracted feature may deviate from the Gaussian distribution, leading to a loss of additive noise cancellation. Furthermore, averaging FFTs directly without considering phase information can lead to misleading results due to phase variations among the signals. This limitation further complicates efforts to extract reliable damping estimates in real-world conditions. Therefore, this paper aims to explore the effects of non-alignment and identify an averaging point within the well-established damping extraction algorithms, while maintaining the nature of additive noise cancellation. The need for an algorithm that can process non-synchronized or mathematically non-aligned data arises from application limitations and Canadian legislation, which prohibits operators from being physically connected to moving machinery. This situation makes wireless connections essential for conducting machine inspections. Typically, wireless sensors are synchronized using the Network Time Protocol (NTP). Although the NTP can provide clock synchronization with nanosecond precision, the structure of the analog-to-digital (A/D) converter introduces a jitter of one to two samples. At a 1 kHz sampling rate, this jitter causes a phase error in a 15 Hz signal, rendering it unsuitable for many applications. To address this issue, we propose aligning the recordings with a correlatable event, such as a hammer impact. We take into account the speed of a mechanical wave in metal, which is approximately 5900 m per second, resulting in a travel time of 0.0005 s over a distance of 3 m. Previous attempts using known damping algorithms did not yield satisfactory results, which has motivated the research presented in this paper. In this paper, we identify key limitations of existing damping extraction methods, specifically their dependence on perfect impact alignment and their sensitivity to noise. To address these challenges, we propose two novel wavelet-based algorithms that utilize wavelet energy for improved alignment and averaging. This advancement allows for robust damping estimation from multiple non-synchronized recordings. We evaluate our approach against four traditional algorithms using both numerical simulations and experimental data, taking into account variations in noise levels and time shifts. For a rigorous assessment, we employ statistical validation through an analysis of variance (ANOVA) test, which highlights significant differences in performance across different recording conditions. The results demonstrate that our proposed methods exhibit excellent resilience to misalignment while maintaining accuracy in noisy environments. These findings have broad implications for applications such as structural health monitoring and machinery diagnostics, where accurate damping estimation is crucial for predictive maintenance and early fault detection. Furthermore, this research underscores the need for ongoing efforts to optimize the performance of these methods. The remainder of this paper is structured as follows. Section 2 presents the problem formulation, including the mathematical model of the mechanical system, the observation model, and the challenges associated with non-aligned impulse responses.