Dynamic stiffness and response analysis

Dynamic stiffness analysis of toroidal shells

Dynamic stiffness analysis of follower force

Dynamic Stiffness Analysis of Follower Moments

Dynamic stiffness analysis for free vibrations of axially loaded laminated composite beams

Harmonic response analysis of coupled plate structures using the dynamic stiffness method

Current and displacement stiffness are important parameters of axial magnetic bearing (AMB) and are usually considered as constants for the control system. However, in actual dynamic work situations, time-varying force leads to time-varying currents and air gap with a specific frequency, which makes the stiffness of appear decrease and even worsens control performance for the whole system. In this paper, an AMB dynamic stiffness model considering the flux variation across the air gap due to frequency is established to obtain the accurate dynamic stiffness. The dynamic stiffness characteristics are analyzed by means of the dynamic equivalent magnetic circuit method. The analytical results show that the amplitude of current and displacement stiffness decreases with frequency increasing. Moreover, compared with the stiffness model without considering the variation of flux density across the air gap, the improved dynamic stiffness results are closer to the actual results. Through the dynamic stiffness measurement method of AMB, experiments of AMB in magnetically suspended molecular pump (MSMP) are carried out and the experimental results are consistent with theoretical analysis results. This paper proposes the dynamic stiffness model of axial magnetic bearing considering the variation of flux density across the air gap, which improves the accuracy of the AMB stiffness analysis.

Two new methods for calculating vibration problems—the dynamic stiffness method and the mobility method—have been introduced in recent years. Each of these methods offers certain advantages over the older vibration theory. This article is an at tempt to unify these two methods so that the best features of each are included in one theory. By means of this theory it is possible to determine the resonance and antiresonance frequencies and the relative amplitudes of vibration of a mechanical system. The effect of the position of torque application on the vibration characteristics of a system are discussed. The application of the theory is demonstrated by a few examples including the problem of the pendulum damper.

The paper addresses the in-plane free vibration analysis of rotating beams using an exact dynamic stiffness method. The analysis includes the Coriolis effects in the free vibratory motion as well as the effects of an arbitrary hub radius and an outboard force. The investigation focuses on the formulation of the frequency dependent dynamic stiffness matrix to perform exact modal analysis of rotating beams or beam assemblies. The governing differential equations of motion, derived from Hamilton's principle, are solved using the Frobenius method. Natural boundary conditions resulting from the Hamiltonian formulation enable expressions for nodal forces to be obtained in terms of arbitrary constants. The dynamic stiffness matrix is developed by relating the amplitudes of the nodal forces to those of the corresponding responses, thereby eliminating the arbitrary constants. Then the natural frequencies and mode shapes follow from the application of the Wittrick–Williams algorithm. Numerical results for an individual rotating beam for cantilever boundary condition are given and some results are validated. The influences of Coriolis effects, rotational speed and hub radius on the natural frequencies and mode shapes are illustrated.

The dynamic stiffness method is used to investigate the free vibration behaviour of a functionally graded beam (FGB). The material properties of the FGB are assumed to vary in the thickness direction based on a power-law. The kinetic and potential energies of the beam are formulated using the Timoshenko beam theory. The governing differential equations of motion in free vibration for the FGB are derived using Hamilton’s principle. The analytical expressions for axial force, shear force and bending moments at any cross-section of the FGB are obtained as a by-product of the Hamiltonian formulation. The differential equations are solved in closed analytical form for harmonic oscillation. The dynamic stiffness matrix of the FGB is then formulated by relating the amplitudes of forces and displacements at the ends of the beam. The Wittrick-Williams algorithm is used as solution technique to yield natural frequencies and mode shapes of the FGB. A parametric study is carried out by varying significant beam parameters and boundary conditions. The investigation required a substantial amount of validation exercise to confirm the predictable accuracy of the dynamic stiffness method. The results are discussed and some concluding remarks are made.

Dynamic stiffness method and CUF-based component-wise theories applied to free vibration analysis of solid beams, thin-walled structures and reinforced panels

Free vibration of the complex cable system − An exact method using symbolic computation

The dynamic stiffness method is introduced to investigate the free vibration of laminated composite beams based on a third-order shear deformation theory which accounts for parabolic distribution of the transverse shear strain through the thickness of the beam. The exact dynamic stiffness matrix is found directly from the analytical solutions of the basic governing differential equations of motion. The Poisson effect, shear deformation, rotary inertia, in-plane deformation are considered in the analysis. Application of the derived dynamic stiffness matrix to several particular laminated beams is discussed. The influences of Poisson effect, material anisotropy, slenderness and end condition on the natural frequencies of the beams are investigated. The numerical results are compared with the existing solutions in literature whenever possible to demonstrate and validate the present method.

With the increase of the span and height of modern engineering structures, the design length and complexity of the cable structure are constantly increasing, whose dynamic problem has become the key to structural design, performance monitoring and maintenance, and vibration control. Therefore, it is necessary to study and develop a unified dynamic analysis theory for complex cable system to meet the requirements of calculation accuracy and efficiency. In view of this, a unified dynamic analysis method for dynamic analysis of complex cable system is proposed in this paper on the basis of the classical dynamic stiffness method. In this paper, the dynamic stiffness method has been enhanced in multiple aspects, which not only retains the advantages of high computational efficiency and wide application range but also overcomes its technical bottlenecks in analysis of shallow-sag cable system considering the damping effect, composite cable systems, and multisegment cable systems. The proposed method can not only be used to the structural design of the complex cable system but also be extended to the vibration control of the system during the service period.

Free vibration of a rotating tapered Rayleigh beam: A dynamic stiffness method of solution

This study focuses on investigating the seismic performance of a reference 15 MW offshore wind turbine (OWT) supported by a large-diameter monopile, under various scour conditions. Fully coupled nonlinear finite element models considering 3D soil continuum for seismic response analysis of monopile-supported OWTs are very complicated and computationally expensive. Therefore, it is important to have good insight on the dynamic behaviour of monopile-supported OWTs subjected to earthquake excitations before performing high-fidelity fully nonlinear 3D simulations, which will not generally be practical in cases considering various seismic loading and scouring scenarios. The purpose of this study is to provide a fast and efficient tool to understand the interaction between the seismic behaviour of emerging large-capacity OWTs when scour erosion affects the dynamic properties of the structure. The dynamic stiffness method (DSM) is applied for the free vibration analysis of the structure. Modal superposition is used to calculate the seismic responses of the OWT by using the exact mode shapes obtained from the dynamic stiffness formulations. The rotor-nacelle-assembly (RNA) is idealized as a lumped mass at the tower top. The tower and monopile are modelled as Timoshenko beam-columns under axial compressive loading. The soil-structure interaction (SSI) is modelled using the Winkler framework, whereby the soil contribution is assumed as elastic to take advantage of the modal superposition method. Two different approaches are considered for the soil stiffness; a constant profile, and a linearly-varying subgrade modulus along the embedded depth of the pile, and both are appraised in this paper. Firstly, natural frequencies obtained from dynamic stiffness formulations are validated using experimental data from literature for the case of controlled dynamic SSI experiments, and for operating OWTs. Additionally, the natural frequency provided by a technical document for the reference 15 MW OWT is compared to the result from the DSM, where a very good agreement is observed. The seismic response time-histories of the OWT model are obtained under different input earthquake conditions, and subsequently verified by comparison with the results of finite element simulations. The effects of different scour scenarios and foundation modelling approaches on the free vibration and seismic responses are subsequently investigated. A parametric analysis is also performed using different soil properties to observe the interaction between the soil stiffness, scour condition, and seismic response. Finally, the influence of the frequency content of seismic records on the dynamic response of the OWT, specifically the tower top acceleration, the mudline rotation, and the mudline stress, is investigated considering various scour scenarios. The specific numerical case study undertaken for a reference 15 MW OWT, which is supported by a large monopile in very dense sand, shows that the seismic responses are very limited and that the elastic soil behaviour assumption is reasonable. Additionally, considering the computational efficiency, the proposed combination of DSM and modal superposition can be used for rapid seismic performance assessment of large diameter monopile-supported OWTs at the preliminary design stage to gain insight to the global dynamic response of the whole vibrating system.