The loosening of bolts negatively impacts many industries worldwide. This issue, for example, can be responsible for the collapse of a structure or vehicle, leading to environmental disasters, economic loss, injuries, and death. Current strategies for simulating loosening include high-fidelity finite element models or analytical expressions, based on assumptions of the motion across the join. In both cases, the feasibility of modeling multiple bolts undergoing loosening and their potential interactions with each other are compromised by computation costs and underlying assumptions. To tackle this problem, this research introduces a novel reduced-order modeling approach designed to depict the dynamics of loosening bolts in lap-jointed structures. The study centers on two harmonic oscillators joined by a lap-joint with a single tension-sensing bolt, which experiences loosening due to uniaxial transverse free and forced vibrations. The study is divided into three stages: in the initial stage, the inherent linear characteristics of each oscillator operating independently are identified and modeled; in the second stage, the connection between joint properties (specifically stiffness and damping) and bolt tension is established through lowamplitude modal impacts across various tension levels; and in the third stage, the system is subjected to high-amplitude forcing to induce loosening, and the resulting loss of tension is depicted through a first-order ordinary differential equation. As a result, the findings reveal that the identified model effectively reproduces the structure's measured reaction in response to the bolt's loosening

A data-driven approach has been developed to provide the equations of the Hopf normal form from time-series data. The coefficients of the numerically constructed flow yield the Poincaré-Lyapunov constant, an important quantity determining the criticality of the Hopf bifurcations, as well as the scaling of its amplitude. Error analysis demonstrates the efficacy of the method.

Uncovering the equations of motion and parameter values for vibrating structures is a significant focus in science, engineering, and technology. This study introduces a novel data-driven approach centered on the system's mechanical energy to identify governing dynamics by examining the forces influencing a single degree of freedom Our methodology comprises two stages: model-dissipative and model-stiffness identification. In the initial phase, we develop an approach to directly identify the total energy from kinetic energy, focusing on parameter system identification derived from dissipative forces. In the second phase, once potential energy is calculated, we calculate conservative forces. Subsequently, stiffness parameters are determined by conducting a curve fitting analysis that correlates the conservative force with displacement. The derived governing equations encompass both nonlinear damping and stiffness terms. The proposed Energy-based Dual-Phase Dynamics Identification (EDDI) method is employed to analyze simulated and measured responses of nonlinear single-degree-of-freedom (SDOF) systems.