Cyber physical systems (CPS), in the event of a cyber attack, can have a serious impact on the operating physical equipment. In order to improve the attack detection capability of CPS, an support vector machine (SVM) attacks detection model based on particle swarm optimization (PSO) is proposed. First, the box plot anomaly detection method is used to detect the characteristic variables, and the characteristic variables with abnormal distribution are discretized. Secondly, the number of attack samples was increased by the SMOTE method to solve the problem of data imbalance, and the linear combination of characteristic variables was performed on the high-dimensional CPS network traffic data using principal component analysis (PCA). Then, the penalty coefficient and the hyperparameter of the kernel function in the SVM model are optimized by the PSO algorithm. Finally, Experiments on attack detection of CPS network traffic data show that the proposed model can detect different types of attack data and has higher detection accuracy compared with general detection models.

This paper applies a recently proposed numerical algorithm to discuss the deflection of viscoelastic micro-beams in the time domain with direct access. A nonlinear-fractional order model for viscoelastic micro-beams is constructed. Before solving the governing equations, the operator matrices of shifted Chebyshev polynomials are derived first. Shifted Chebyshev polynomials are used to approximate the deflection function, and the nonlinear fractional order governing equation is expressed in the form of operator matrices. Next, the collocation method is used to discretize the equations into the form of algebraic equations for solution. It simplifies the calculation. MATLAB software was used to program this algorithm to simulate the numerical solution of the deflection. The effectiveness and accuracy of the algorithm are verified by the numerical example. Finally, numerical simulations are carried out on the viscoelastic micro-beams. It is found that the viscous damping coefficient is inversely proportional to the deflection, and the length scale parameter of the micro-beam is also inversely proportional to the deflection. In addition, the stress and strain of micro-beam, the change of deflection under different simple harmonic loads, and potential energy of micro-beam are discussed. The results of the study fully demonstrated that the shifted Chebyshev polynomial algorithm is effective for the numerical simulations of viscoelastic micro-beams.

In this study, an efficacious method for solving viscoelastic dynamic plates in the time domain is proposed for the first time. The differential operator matrices of different orders of Bernstein polynomials algorithm are adopted to approximate the ternary displacement function. The approximate results are simulated by code. In addition, it is proved that the proposed method is feasible and effective through error analysis and mathematical examples. Finally, the effects of external load, side length of plate, thickness of plate and boundary condition on the dynamic response of square plate are studied. The numerical results illustrate that displacement and stress of the plate change with the change of various parameters. It is further verified that the Bernstein polynomials algorithm can be used as a powerful tool for numerical solution and dynamic analysis of viscoelastic plates.

This paper primarily investigates the dynamics response of viscoelastic orthotropic plates under a fractional-order derivative model, which is efficiently simulated numerically using the FKV (Fractional Kelvin–Voigt) model and the shifted Legendre polynomial algorithm. By establishing the fractional-order governing equation and directly solving it in the time domain using a shifted Legendre polynomial, the approach achieves low error and high accuracy. The analysis shows that the load, plate thickness, and creep time all affect the plate displacement, and the fractional-order model outperforms the integer-order model to better capture the dynamics response of the material.

A numerical scheme utilizing shifted Bernstein polynomials is developed to address the variable fractional-order governing equation in viscoelastic fluid-conveying pipes. The pipe’s mechanical response is characterized through a variable fractional-order Kelvin–Voigt (FKV) model, which effectively captures the time-dependent and memory properties of viscoelastic materials. By coupling the FKV constitutive model with the motion equation, the governing equation for a viscoelastic pipe is obtained. The deformation field is approximated using shifted Bernstein trial functions, leading to the construction of a derivative matrix with a variable fractional order. The obtained governing relation is expressed in matrix form, and after discretization, an algebraic system is formulated that is solvable in the time domain to evaluate the pipe displacement. Moreover, a convergence investigation is carried out to examine the reliability and effectiveness of the proposed framework. Computational results demonstrate that the introduced method delivers outstanding precision and performance, while the viscoelastic pipe’s response under different scenarios—including applied loads and fluid flow rates—is comprehensively investigated.