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An analytical method for calculating the dynamic stiffness of rubber isolator with multi-rigid-flexible bodies and various coupling deformation modes is proposed. The three-dimensional transfer matrices are reduced to two-dimensional representation for calculations, based on the excitation direction of the isolator and the deformation modes of the rubber components, which reduces computational costs. The metal and rubber components in the isolator are treated as rigid and flexible bodies, respectively, and the transfer matrices of each component are derived in conjunction with continuous system theory. The deformation of the rubber components involves shear, bending, longitudinal, and torsional motions. The frequency-dependent complex elastic modulus of the rubber material is obtained from dynamic shear tests and fitted to a five-parameter fractional derivative model. The extended transfer matrix method is employed to solve for the frequency-dependent nonlinear dynamic stiffness of the isolator in three translational directions. Finally, the proposed method is validated through a combination of finite element analysis and dynamic stiffness testing, and the influences of material and geometric parameters are analyzed.

The coupled free vibration analysis of the thin-walled laminated composite I-beams with bisymmetric and monosymmetric cross sections considering shear effects is developed. The laminated composite beam takes into account the transverse shear and the restrained warping induced shear deformation based on the first-order shear deformation beam theory. The analytical technique is used to derive the constitutive equations and the equations of motion of the beam in a systematic manner considering all deformations and their mutual couplings. The explicit expressions for displacement parameters are presented by applying the power series expansions of displacement components to simultaneous ordinary differential equations. Finally, the dynamic stiffness matrix is determined using the force–displacement relationships. In addition, for comparison, a finite beam element with two-nodes and fourteen-degrees-of-freedom is presented to solve the equations of motion. The performance of the dynamic stiffness matrix developed by study is tested through the solutions of numerical examples and the obtained results are compared with results available in literature and the detailed three-dimensional analysis results using the shell elements of ABAQUS. The vibrational behavior and the effect of shear deformation are investigated with respect to the modulus ratios and the fiber angle change.

The exact vibration frequencies of continuous beams with internal releases are found using the dynamic stiffness method. Two types of releases are considered: hinge and sliding discontinuities. First, the exact dynamic stiffness matrix for a beam element with a release is derived and then used in the assembly of the structure dynamic stiffness matrix. The natural frequencies are found as the values of frequency that make this matrix singular. Then the mode shapes are found exactly. Examples are given for continuous beams with different releases.

The interaction between underground pipelines and soils is crucial to the design and maintenance of underground pipeline network systems. In this paper, the dynamic stiffness matrix in the frequency-domain of the buried pipeline is obtained by the improved scaled boundary finite element method (SBFEM) coupled with the finite element method (FEM) at the interface between the far and near fields. A new coordinate transformation together with a scaled line is introduced in the improved SBFEM. Combined with the mixed variable algorithm, the time-domain solution of the buried pipeline under dynamic loads is then obtained. The accuracy of the proposed algorithm was verified by numerical examples. A parametric study is performed to assess the influence of the anisotropic characteristics of the layered soils on the dynamic response of the pipeline, the result of which provides a reliable basis for engineering practice. The results show that these parameters have a significant impact on the pipeline. The understanding of this impact can contribute to the design, construction, and maintenance of the corresponding engineering projects.

Recently, applying damped outriggers in high-rise buildings to reduce vibration due to earthquake and wind has attracted a lot of attention. By placing energy dissipated devices vertically between the end of outriggers and perimeter columns, the damped outrigger systems emphasize the supplementary damping rather than stiffness. This paper investigates the applicability of viscously damped outrigger systems using Timoshenko beam theory. First, the building is modeled as a cantilevered Timoshenko beam, and the damped outriggers are simplified as equivalent complex rotational spring. With the assembly of dynamic stiffness of the damped outrigger system, the Wittrick–Williams algorithm and Newtonian iteration are then combined to calculate the dynamic characteristic of the system. The proposed approach has been verified by finite element method through a case study. The shear deformation and rotatory inertia of the core, later blended as slenderness ratio of the height-radius following Timoshenko beam theory, stiffness ratios of core-column and core-outrigger, and the number of damped outriggers are subsequently evaluated associated with supplementary damping as applicability analysis. This study shows the following: (1) adding damping amplifies twice to triple as the slenderness ratio increases from 5 to 25 for the first mode; (2) configurating more damped outriggers could provide stable damping contribution across modes with diminishing benefit; (3) Timoshenko beam analysis is better suited while Euler–Bernoulli beam analysis is an acceptable approximation only when the slenderness ratio of one-damped outrigger system is bigger than 25 and the first mode is predominant; and finally (4) the stiffness ratios of core-column, core-outrigger and stiffness of dampers have substantial influence on structural damping, particularly within the range smaller than 1, 10 and 4, respectively, but the presence of properly designed negative stiffness of dampers could bring substantial improvement. This study could serve as guidelines to apply damped outrigger systems in high-rise buildings.

This study aims to elucidate the nonlinear characteristics (dynamic stiffness and damping) of a thin-walled hallowed spherical rubber isolator subjected to harmonic excitations using harmonic balance method (HBM). Firstly, harmonic excitation experiments with different loads and different acceleration levels were carried out to test the softening effect of the spherical rubber isolator. The dependence of the dynamic stiffness and damping of the spherical rubber isolator on the displacement amplitude was established. Secondly, the $N$th harmonic of the experimental time signal was extracted, and the nonlinear contribution of the second and third harmonics to the experimental results was verified. Finally, the simulations were performed and the results were in line with the experimental results. Our study identifies that the nonlinear model is feasible to predict the vibration characteristic of the thin-walled hallowed spherical rubber isolation systems.

In a railroad wagon, the vertical excitation on the wheels originated from the track irregularities; the air spring model is reliably used to restrict these abnormal vibrations at a required level and improve traveler comfort. When air springs deflate during running, too much vibration occurs and the ride is less comfortable, which are common complaints among high-speed rail operators. In the situation that the air spring deflates, the laminated rubber spring acts as a temporary emergency spring to reduce the dynamic load. An approximation method for measuring laminated rubber’s static stiffness is presented and examined the effect on the air suspension. In this study the equations of motion for the pitch, roll, and the bounce motion of the bogies and the car’s body were developed with deflated state of air spring to systematically study the effect of deflated air spring on the ride comfort of passengers. The study show that passengers’ ride comfort was predicted accurately by the proposed model. It also observed that in the presence of a laminated rubber spring, the air spring’s dynamic stiffness is moved to higher frequencies. The air spring’s dynamic stiffness is reduced at lower frequency when the diameter of the surge pipe is increased. According to the results of the research, the vehicle’s speed should be reduced while the air spring is deflated in order to keep the passengers’ comfort.

Double-slab ballastless track use the WJ-8C fastening system. Because the fastener assume an important role in the regulation track deformation. Study performance especially dynamic and static stiffness of fasteners for stress analysis of track structure has important theoretical and practical value. Through the dynamic load and static load experiment on track structure. Rational utilization of the experiment of WJ-8C fastener performance appraisal. Determine the WJ-8C fastener reasonable value of dynamic and static stiffness. Designed for the double-slab ballastless track fasteners of dynamic and static stiffness values to provide the reference data and performance testing