PSI - Issue 71

Faisal Hussain et al. / Procedia Structural Integrity 71 (2025) 248–255

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1. Introduction The structural integrity of road automobiles and mechanical components is achieved through several methods of joining, including riveting, bolting, welding, or more sophisticated fastening techniques like effective joints (Doranga and Wu, 2021). Engineering systems commonly experience nonlinearity due to various factors such as structural joints with friction or looseness features, limit conditions with changing limit stiffness, materials dependent on amplitude, and the presence of components like dampers, vibration isolation devices, links and bearings. Furthermore, engineering systems commonly exhibit nonlinearity, with joints being the primary source of structure non-linearities. Nomenclature K 1 , K 2 linear translational (N/m) and rotational spring stiffness (N-m/radian) K 1 * , K 2 * non-linear translational spring (N/m) and non-linear rotational spring stiffness (N-m/radian) K 3L, K 4L cubic type stiffness non-linearity considering translational (N/m 3 ) and rotational stiffness (N-m/radian 3 ) B,C subsystem of beam FRF frequency response function (radian/N) [β], [γ] B and C subsystems FRF matrix λ non-dimensional frequency L length of beam (m) EI elastic property (N·m²) ω natural frequency(radian/sec) ρA density mass function (kg/m) x(t) linear translation of the system (m) θ(t) angular displacement of the system (radian) F(t) sinusoidal force function (N) F 0 external force amplitude (N) K 3, K 4 cubic stiffness coefficients ((N/m 3 ) and (N-m/radian 3 ) K c estimated stiffness coefficients (N-m/rad) X X ,θ θ response amplitude of harmonic non-linear systems (m) and (radian) ∆ denominator of the receptance functions (radian/N) β 11 ,β 22 ,β 12 ,β 21 B subsystem direct and cross receptance (rad/N-m) γ 11 , γ 22, γ 12 ,γ 21 C subsystem direct and cross receptance (rad/N-m) F 5 ,F 3 ,F 6 ,F 1 functions of frequency dependent x 3 (t), θ 3 (t) cubic nonlinearities in translation (m 3 ) and rotational (radian 3 ) conditions These non-linearities can greatly alter the structural response on certain occasions (Hussain and Ingole, 2022). Non-linearity can also occur as a result of boundary conditions, such as surfaces that are free of fluids, vibration impacts caused by weak connections or interactions with stiff limitations, spaces, or specific additional non-linear forces exerted by the body (Volkova, 2013). One of the most significant benefits of bolted joints in comparison to other types of connections, such as welding connections, is that they are characterized by acceptable levels of durability and strength (Jamia et al., 2021). Kim et al. presented a frequency domain approach-based method which can be offered for identifying a structure's non-linear parameters that have non-linear joints (Kim et al., n.d.,2004). Hussain et al. in his review work, discussed about the substructure synthesis principle which can be implemented as a mathematical modeling tool, thereby considering a structural joint beam model for studying the non-linear characteristics of various joint parameters and its effectiveness within the nonlinear domain approach, either it may be frequency domain or time domain approach (Hussain and Ingole, 2024). Wang et al. in his research work, successfully reproduced the nonlinear characteristics of stiffness nonlinearity, a model with four parameters was developed to forecast the non-linear stick-slip interface interactions arising at the interfaces of bolted joints (Wang and Zhang, 2020). Binbin et al. proposed a novel friction mechanical model which can accurately characterize the tangential stiffness. The investigated data from the real two-dimensional stress propagation fastened joint can yield an