Determining the Rayleigh damping parameters of flexible pavements for finite element modeling

The dynamic analysis has been successfully used to predict the pavement response based on the finite element modeling, during which the stiffness and mass matrices have been established well, whereas the method to determine the damping matrix based on Rayleigh damping is still under development. This article presents a novel method to determine the two parameters of the Rayleigh damping for dynamic modeling in pavement engineering. Based on the idealized shear beam model, a more reasonable method to calculate natural frequencies of different layers is proposed, by which the global damping matrix of the road pavement can be assembled. The least squares method is simplified and used to calculate the frequency-independent damping. The best-fit Rayleigh damping is obtained by only determining the natural frequencies of the two modal. Finite element model and in-situ field test subjected by the same falling weight deflectometer pulse loads are performed to validate the accuracy of this method. Good agreements are noted between simulation and field in-situ results demonstrating that this method can provide a more accurate approach for future finite element modeling and back-calculation.

Particle swarm optimization–based characterization technique of nonproportional viscous damping parameter of a cantilever beam

A complex eigenvector is a result of nonproportional damping present in a structural system. However, it is difficult to identify the accurate damping matrix considering the modal sparsity and coordinate sparsity. A nonproportional viscous damping parameter identification is formulated as an unconstrained optimization problem in the present study. The damping coefficient of each element is considered as the design variable for the optimization problem. The objective function is defined using the incomplete complex eigenvectors, which are generated because of the presence of external damping devices in the structure. This objective function is then minimized using standard particle swarm optimization to identify the damping coefficient of the damping matrix. The accuracy and efficiency of the particle swarm optimization are investigated by solving a few numerical problems with simulated measured data. The proposed method works well with the incomplete measured modal data. The current methodology performs satisfactorily with and without noisy data. A comparison study is performed with the existing gradient-based method, and the results show that the proposed method performs better than the existing gradient-based method for the present problem with and without noisy measurement data.

Active Fractional-Order Sliding Mode Control of Flexible Spacecraft Under Actuators Saturation

In this article, a novel multi-purpose modified fractional-order nonsingular terminal sliding mode (MFONTSM) controller is designed for the flexible spacecraft attitude control and appendages passive vibration suppression, assuming the control torque saturation in the system dynamics. Furthermore, an active FONTSM controller is proposed separately to perform active vibration suppression of the flexible appendages using piezoelectric actuators. The fixed-time stability of the closed-loop system for both the passive and active controllers is analyzed and proved using the Lyapunov theorem. Finally, the performance of the proposed controllers has been tested in the presence of uncertainties, external disturbances, and the absence of the damping matrix in order to study the effectiveness of the proposed method.

Determination of the exact mode frequencies of multi-storey structures by state-space method and a comparison with mode superposition method

A comparative research has been carried out for obtaining the time-consuming exact solution (state-space) and approximate solution (mode superposition) of transient and steady-state vibrations of linearly damped linear frame buildings. In the mode superposition method, the proportional damping matrix has been constructed by different approaches such as modal combination of mass and stiffness matrixes (Rayleigh) and disregarding the off-diagonal elements of the non-classical damping matrix, while in the state-space method the non-proportional damping matrix is constructed in exact situation. These observations are individually investigated, which the most suitable parameter to render the approximate results as close as possible to the exact results. Harmonic forces are applied on the different storeys of three and five storey frame buildings, and the responses are displayed in comparative tables and figures. The maximum responses are calculated by square root of sum of the squares (SRSS) method. A MATLAB code is generated and the equations of exact and approximate methods are solved.

Optimal Method for Determination of Rayleigh Damping Co-efficients for Different Materials using Modal Analysis

Rayleigh damping co-efficients are the essential parameters to determine the damping matrix of a system in dynamic analysis. For the systems with multiple degrees of freedom, it is difficult to arrive for suitable Rayleigh damping co-efficients. This paper represents a simple and effective method for the determination of Rayleigh co-efficients α and β for the system with multiple degrees of freedom. An unrealistic constant damping ratio for all modes is assumed to get rational value of α and β, which leads the determination of progressively varying damping ratio for all modes. By comparing the damping ratio arrived from assumed α and β with assumed unrealistic damping ratio, the suitable and most precise values are determined. This method is implemented for different materials with different boundary conditions by considering different significant modes and the effect of above parameters on α and β values are also discussed.

Modal analysis and insights into damping phenomena of a special vibration chain

Vibration chains are of interest in many fields of practical applications. In this contribution, a modal analysis of the rather special Mikota’s vibration chain is performed. Herein, focus is set on the mode shapes of this multibody oscillator, which was firstly introduced by Mikota as a solid body compensator in hydraulic systems for filtering out fluid flow pulsations. The mode shapes show interesting properties, e.g. an increase in the polynomial representing the coordinates of each mode shape with an increasing eigenfrequency associated with the respective mode shape. This and other properties are discussed exemplary. Some of these properties still have to be proven, which is the task of future work. Additionally, modal damping of Mikota’s vibration chain is discussed. Moreover, an approach for determining the damping matrix for given Lehr’s damping measures without knowing the mode shapes in advance is introduced. This approach involves the determination of a matrix root.

Model Updating for Systems with General Proportional Damping Using Complex FRFs

In this paper, we propose a model updating method for systems with nonviscous proportional damping. In comparison to the traditional viscous damping model, the introduction of nonviscous damping will not only reduce the vibration of the system but also change the resonance frequencies. Therefore, most of the existing updating methods cannot be directly applied to systems with nonviscous damping. In many works, for simplicity, the Rayleigh damping model has been applied in the model updating procedure. However, the assumption of Rayleigh damping may result in large errors of damping for higher modes. To capture the variation of modal damping ratio with frequency in a more general way, the diagonal elements of the modal damping matrix and relaxation parameter are updated to characterize the damping energy dissipation of the structure by the proposed method. Spatial and modal incompleteness are both discussed for the updating procedure. Numerical simulations and experimental examples are adopted to validate the effectiveness of the proposed method. The results show that the systems with general proportional damping can be predicted more accurately by the proposed updating method.