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Using the state space method, a novel analytical method is proposed to investigate the dynamic behavior of multi-cracked reinforced concrete (RC) beams strengthened with fiber-reinforced polymer (FRP) plates, while taking damping effects into account. The flexural cracks in RC beams are modeled as rotational springs, with their stiffness dependent on the crack depths. A numerically stable technique and a frequency-scanning strategy are employed to solve the homogeneous state equations associated with mode function vectors and the frequency equation, respectively. This allows for the determination of natural frequencies and corresponding modal shapes of FRP-strengthened multi-cracked RC beams under generalized boundary conditions. The orthogonality relation of vibration modes is established based on the state space formulae and the concept of symplectic inner product. Analytical solutions for the dynamic responses of FRP-strengthened multi-cracked RC beams subjected to arbitrary dynamic loads are derived using the orthogonality relation and the mode superposition method. Numerical examples are provided to predict the dynamic responses of strengthened cracked beams under a dynamic uniformly distributed step load and a concentrated triangular impulse load. The effectiveness of the proposed analytical method is validated through comparisons with finite element simulations and experimental results. Furthermore, parametric studies are performed to analyze the effects of damping ratio, crack depth, crack position and the number of cracks on the dynamic responses of these strengthened cracked beams. The results demonstrate that the strengthening strategy using externally bonded FRP plates is effective, while the effects of cracks and the first-order damping ratio on the dynamic responses are significant and cannot be ignored. Additionally, the proposed method can efficiently and accurately analyze the dynamic behavior of FRP-strengthened RC beams with arbitrary distributions of cracks and dynamic loads.

We presented a pure-shear film bulk acoustic resonator (FBAR) and investigated its sensing characteristics in viscous liquids. In the resonator, the electrodes were located on the surface of c-axis-oriented AlN film to generate the lateral electric field and excite the shear acoustic resonance. Compared with the typical quasi-shear film bulk acoustic resonator based on inclined c-axis-oriented AlN or ZnO piezoelectric film, the proposed device exhibits significantly higher Q-factors and a notably improved detection limit, particularly in water and viscous liquids. The frequency shifts show a linear dependency on the square root of the product of the liquid viscosity and density of the glycerol solution in the viscosity range of 1–5 mPa$\cdot$s. Furthermore, we measured the mass sensitivity through real-time monitoring of the frequency change during the volatilization process of the loaded saline solutions. The proposed device shows the mass sensitivity of 465 Hz$\cdot$cm²/ng and the mass resolutions of 0.17 ng/cm² in air, 0.25 ng/cm² in water and 2.08 ng/cm² in 50% glycerol solution, respectively. The obtained results clearly indicate that the proposed device is capable of using in liquid phase detection with high sensitivity requirements.

Fully submerged sphere and cylinder point absorber (PA), wave energy converters (WECs) are analyzed numerically based on linearized potential flow theory. A boundary element method (BEM) (a radiation–diffraction panel program for wave-body interactions) is used for the basic wave-structure interaction analysis. In the present numerical model, the viscous damping is modeled by an equivalent linearized damping which extracts the same amount of wave energy over one cycle as the conventional quadratic damping term. The wave power capture width in each case is predicted. Comparisons are also made between the sphere and cylinder PAs which have identical geometrical scales and submerged depths. The results show that: (i) viscous damping has a greater influence on wave power performance of the cylinder PA than that of the sphere PA; (ii) the increasing wave height reduces wave power performance of PAs; (iii) the cylinder PA has a better wave power performance compared to the sphere PA in larger wave height scenarios, which indicates that fully submerged cylinder PA is a preferable prototype of WEC.

This paper studies the viscously damped free and forced vibrations of longitudinal and torsional bars. The method is exact and yields closed form solution for the vibration displacement in contrast with the well-known eigenfunction superposition (ES) method, which requires expression of the distributed forcing functions and displacement response functions as infinite series sums of free vibration eigenfunctions. The viscously damped natural frequency equation and the critical viscous damping equation are exactly derived for the bars. Then the viscously damped free vibration frequencies and corresponding damped mode shapes are calculated and plotted, aside from the undamped free vibration and corresponding mode shapes typically computed and used in vibration problems. The longitudinal or torsional amplitude versus forcing frequency curves showing the forced response to distributed loadings are plotted for various viscous damping parameters. It is found that the viscous damping affects the natural frequencies and the corresponding mode shapes of longitudinal and torsional bars, especially for the fundamental frequency.

The key aim of this research is to develop an analytical model for the forced vibration of graphene-reinforced composite (GRC) cylindrical shell with viscous damping and thermal environment effects under several boundary conditions. The prescribed mechanical and thermal characteristics of the GRC layer are evaluated utilizing a micromechanical extended Halpin–Tsai technique. Further, the governing equations are determined to be grounded on the shear deformation theory (SDT) with the von Kármán-form in terms of the geometric nonlinearity. The basic nonlinear partial differential governing equation is transformed into a nonlinear ordinary differential equation with the Galerkin-based method. The resulting ordinary governing differential equations are systematically solved based on the multiple scales scheme in order to obtain the nonlinear forced vibration frequency response of the laminated GRC cylindrical shell in the presence of damping impact and subjected to multiple boundary conditions. The validation of the obtained expressions is achieved by comparing them with the available literature data where the comparison revealed a clear consistency between them. Moreover, a parametric investigation is provided to illustrate the impacts of the distribution of graphene layers, elastic foundation coefficients, damping ratios, temperature effects and dimensionless radial excitation amplitude on the forced nonlinear amplitude-frequency ratio responses for a functionally-graded graphene-reinforced composite (FG-GRC)-layered cylindrical shells. The analytical outcomes indicate that the different distribution types of graphene, elastic foundation coefficients, damping ratios, temperature, and radial excitation parameters have a noteworthy impact on the frequency–amplitude behaviors of an FG-GRC-layered cylindrical shell. In addition, the new insights gained from this research might contribute to a deeper understanding of the nonlinear forced vibration responses in subsequent analysis and design techniques for giving appropriate benchmark findings.

Nonlinear time domain site response analysis is used to capture the soil hysteretic response and nonlinearity due to medium and large ground motions. Soil damping is captured primarily through the hysteretic energy dissipating response. Viscous damping, using the Rayleigh damping formulation, is often added to represent damping at very small strains where many soil models are primarily linear. The Rayleigh damping formulation results in frequency dependent damping, in contrast to experiments that show that the damping of soil is mostly frequency independent. Artificially high damping is introduced outside a limited frequency range that filters high frequency ground motion. The extended Rayleigh damping formulation is introduced to reduce the overdamping at high frequencies. The formulation reduces the filtering of high frequency motion content when examining the motion Fourier spectrum. With appropriate choice of frequency range, both formulations provide a similar response when represented by the 5% damped elastic response spectrum.

The proposed formulations used in non-linear site response analysis show that the equivalent linear frequency domain solution commonly used to approximate non-linear site response underestimates surface ground motion within a period range relevant to engineering applications. A new guideline is provided for the use of the proposed formulations in non-linear site response analysis.

The characterisation of viscous damping in time history analysis is discussed in this paper. Although it has been more common in the past to use a constant damping coefficient for single-degree-of-freedom time history analyses, it is contended that tangent-stiffness proportional damping is a more realistic assumption for inelastic systems. Analyses are reported showing the difference in peak displacement response of single-degree-of-freedom systems with various hysteretic characteristics analysed with 5% initial-stiffness or tangent-stiffness proportional damping. The difference is found to be significant, and dependent on hysteresis rule, ductility level and period. The relationship between the level of elastic viscous damping assumed in time-history analysis, and the value adopted in Direct Displacement-Based Design is investigated. It is shown that the difference in characteristic stiffness between time-history analysis (i.e. the initial stiffness) and displacement based design (the secant stiffness to maximum response) requires a modification to the elastic viscous damping added to the hysteretic damping in Direct Displacement-Based Design.