Multi-degree-of-freedom Systems Research Articles

Transient amplification of maximum vibration amplitudes

AbstractMany low damped structures as turbine blades or drill strings are exposed to high dynamical loads causing high vibration amplitudes. These applications comprise sub‐critical eigenfrequencies. Hereby, the lower eigenfrequencies have to be passed before reaching the operating point. Most investigations of vibration amplitudes caused by a resonance passage deal with the computation of single degree of freedom systems. Thereby, it has been shown that the stationary vibration response provides the highest possible amplitude. Further it can be stated that the maximum vibration response of the resonance passage decreases with an increasing sweep velocity [3]. Isolated modes of linear systems can be represented by single degree of freedom systems. Subsequently a mode shape can be described by the multiplication of the amplification function of the mode and the belonging eigenvector. There are only some recent works that deal with resonance passages of vicinal modes, e. g. [1].In this paper the resonance passage of a three dimensional system with nearby modes is studied. To calculate the transient vibration response an analytical approach is used. It is shown that the maximum amplitude of the stationary vibration response is not the upper limit for the maximum amplitude of the resonance passage. Thus, the maximum amplitude may rise while the sweep velocity increases. Hence, regarding a multi degree of freedom system the maximum amplitude of the resonance passage can exceed the maximum amplitude of the stationary vibration response. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Dynamic properties of a building with viscous dampers in non-proportional arrangement

Any rational approach to define the configuration and size of viscous fluid dampers in a structure should be based on the dynamic properties of the system with the dampers. In this paper we propose an alternative representation of the complex eigenvalues of multi degree of freedom systems with dampers to calculate new equivalent natural frequencies. Analytical expressions for the dynamic properties of a two-story building model with a linear viscous damper in the first floor (i.e. with a non-proportional damping matrix) are derived. The formulas permit to obtain the equivalent damping ratios and equivalent natural frequencies for all the modes as a function of the mass, stiffness and damping coefficient for underdamped and overdamped systems. It is shown that the commonly used formula to define the equivalent natural frequency is not applicable for this type of system and for others where the damping matrix is not proportional to the mass matrix, stiffness matrix or both. Moreover, the new expressions for the equivalent natural frequencies expose a novel phenomenon; the use of viscous fluid dampers can modify the vibration frequencies of the structure. The significance of the new equivalent natural frequencies is expounded by means of a simulated free vibration test. The proposed approach may offer a new perspective to study the effect of viscous dampers on the dynamic properties of a structure.

Dynamic properties of a building with viscous dampers in non-proportional arrangement

Any rational approach to define the configuration and size of viscous fluid dampers in a structure should be based on the dynamic properties of the system with the dampers. In this paper we propose an alternative representation of the complex eigenvalues of multi degree of freedom systems with dampers to calculate new equivalent natural frequencies. Analytical expressions for the dynamic properties of a two-story building model with a linear viscous damper in the first floor (i.e. with a non-proportional damping matrix) are derived. The formulas permit to obtain the equivalent damping ratios and equivalent natural frequencies for all the modes as a function of the mass, stiffness and damping coefficient for underdamped and overdamped systems. It is shown that the commonly used formula to define the equivalent natural frequency is not applicable for this type of system and for others where the damping matrix is not proportional to the mass matrix, stiffness matrix or both. Moreover, the new expressions for the equivalent natural frequencies expose a novel phenomenon; the use of viscous fluid dampers can modify the vibration frequencies of the structure. The significance of the new equivalent natural frequencies is expounded by means of a simulated free vibration test. The proposed approach may offer a new perspective to study the effect of viscous dampers on the dynamic properties of a structure.

Research of complex modal parameters extraction of a multi-degree-of-freedom structure based on similarity search

We present a method for the extraction of modal parameters that could be useful for the parameter identification of complex structural dynamic modes of multi degree of freedom systems that may be subject to environmental load excitations. The approach is linear and in this sense it operates on the basis of the principles suggested by the so-called Rayleigh superposition method. This atoms which have the form of free vibration are constructed. The method combined with MP (Matching Putsuit) and GA (Genetic Algorithm). MP is borrowed to extract every most similar atom to the random decrement of the structure response to identify modal parameters. GA is used to speed up this process. The method is applied on the analysis of model experiments for an FPSO single point mooring system and results are compared against real monitoring data. It is shown that the method is capable of extracting successfully modal characteristics such as complex modal frequency, damping ratio and mode shapes.

An Efficient Wiener Path Integral Technique Formulation for Stochastic Response Determination of Nonlinear MDOF Systems

The recently developed approximate Wiener path integral (WPI) technique for determining the stochastic response of nonlinear/hysteretic multi-degree-of-freedom (MDOF) systems has proven to be reliable and significantly more efficient than a Monte Carlo simulation (MCS) treatment of the problem for low-dimensional systems. Nevertheless, the standard implementation of the WPI technique can be computationally cumbersome for relatively high-dimensional MDOF systems. In this paper, a novel WPI technique formulation/implementation is developed by combining the “localization” capabilities of the WPI solution framework with an appropriately chosen expansion for approximating the system response PDF. It is shown that, for the case of relatively high-dimensional systems, the herein proposed implementation can drastically decrease the associated computational cost by several orders of magnitude, as compared to both the standard WPI technique and an MCS approach. Several numerical examples are included, whereas comparisons with pertinent MCS data demonstrate the efficiency and reliability of the technique.

Vertical seismic response analysis of straight girder bridges considering effects of support structures

Vertical earthquake ground motion may magnify vertical dynamic responses of structures, and thus cause serious damage to bridges. As main support structures, piers and bearings play an important role in vertical seismic response analysis of girder bridges. In this study, the pier and bearing are simplified as a vertical series spring system without mass. Then, based on the assumption of small displacement, the equation of motion governing the simply-supported straight girder bridge under vertical ground motion is established including effects of vertical deformation of support structures. Considering boundary conditions, the differential quadrature method (DQM) is applied to discretize the above equation of motion into a MDOF (multi-degree-of-freedom) system. Then seismic responses of this MDOF system are calculated by a step-by-step integration method. Effects of support structures on vertical dynamic responses of girder bridges are studied under different vertical strong earthquake motions. Results indicate that support structures may remarkably increase or decrease vertical seismic responses of girder bridges. So it is of great importance to consider effects of support structures in structural seismic design of girder bridges in near-fault region. Finally, optimization of support structures to resist vertical strong earthquake motions is discussed.

Entangled W state of multi degree of freedom system based on orbital angular momentum

We propose a method of generating the three-photon W state. The method uses parametric down-conversion process and hybrid entanglement swapping from multiphoton spin-entangled states to multiphoton orbital angular momentum (OAM) entangled states, with the aid of a pair of polarization photons. They generate W state entangled in different degrees of freedom of polarization and OAM with a high-dimensional Hilbert space. By simply changing the methods of generating a polarization-OAM-linear momentum entangled W state. Our method produces two mutually symmetric W states with strong entanglement and high dimension, which is expected to realize the secure communication of extending quantum bits.

Ductility reduction factors for steel buildings considering different structural representations

The global ductility parameter $$(\mu _{G})$$ , commonly used to represent the capacity of a structure to dissipate energy, and its effects, considered through the ductility reduction factor $$(R_{\upmu })$$ , are studied for buildings with moment resisting steel frames (MRSF) which are modeled as complex multi degree of freedom systems. Results indicate that the $$\mu _{G}$$ value of 4, commonly assumed, cannot be justified, a value between 2.5 and 3 is suggested. The ductility reduction factors associated to global response parameters may be quite different than those of local response parameters, showing the limitation of the commonly used equivalent lateral force procedure (ELFP). The ratio $$(Q)$$ of $$R_{\upmu }$$ to $$\mu _{G}$$ is larger for the models with spatial MRSF than for the models with perimeter MRSF since their ductility demands are smaller and/or their ductility reduction factors larger. According to the simplified Newmark and Hall procedure, the $$Q$$ ratio should be equal to unity for the structural models under consideration. Based on the results of this study, this ratio cannot be justified. The reason for this is that single degree of freedom systems were used to derive the mentioned simplified procedure, where higher mode and energy dissipation effects cannot be explicitly considered. A value of 0.5 is suggested for $$Q$$ for steel buildings with perimeter MRSF in the intermediate and long period regions. The findings of this paper are for the particular structural systems and models used in the study. Much more research is needed to reach more general conclusions.

Estimating floor spectra in multiple degree of freedom systems

As the desire for high performance buildings increases, it is increasingly evident that engineers require reliable methods for the estimation of seismic demands on both structural and non-structural components. To this extent, improved tools for the prediction of floor spectra would assist in the assessment of acceleration sensitive non-structural and secondary components. Recently, a new procedure was successfully developed and tested for the simplified construction of floor spectra, at various levels of elastic damping, atop single-degree-of-freedom structures. This paper extends the methodology to multi-degree-of-freedom (MDOF) supporting systems responding in the elastic range, proposing a simplified modal combination approach for floor spectra over upper storeys and accounting for the limited filtering of the ground motion input that occurs over lower storeys. The procedure is tested numerically by comparing predictions with floor spectra obtained from time-history analyses of RC wall structures of 2- to 20-storeys in height. Results demonstrate that the method performs well for MDOF systems responding in the elastic range. Future research should further develop the approach to permit the prediction of floor spectra in MDOF systems that respond in the inelastic range.

Reliable estimation of the mean annual frequency of collapse by considering ground motion spectral shape effects

Ground motion record (GMR) selection is an important issue in the nonlinear dynamic analysis procedure. Most of the current design codes recommend to use GMRs in which their mean spectrum be matched to a design spectrum e.g. uniform hazard spectrum. However recent research results have shown that the code methodology is neither robust nor realistic. On the other hand, GMR selection, based on spectral shape, is recently proposed in order to deal with this problem. $$\varepsilon $$ and $$\eta $$ are two powerful spectral shape indicators which are used for GMR selection purposes. A comparison between $$\varepsilon $$ and $$\eta $$ was made in order to access their capability to predict the linear spectral shape and the structural nonlinear response. The $$\eta $$ -based conditional mean spectrum (E-CMS), which has been recently emerged as a new design spectrum, was also investigated in this study. The E-CMS formulation format is fully compatible with the existing CMS definition which makes E-CMS quite easy to be implemented. The resulted E-CMS was used as a target spectrum for the record selection. Analysis of a set of multi degree of freedom systems shows that the mean annual frequency of collapse is achievable, with more reliability, based on the new emerged $$\eta $$ indicator. Therefore, the bias is decreased by employing the $$\eta $$ concept into the record selection procedure. The bias reduction is more significant in higher hazard levels and in the case of structures with low natural periods or with significant higher mode effects.

Influence of High Mode Effects on Ductility Reduction Factors for MDOF Shear-Type Structures

The influences of high mode effect on ductility reduction factors for multi-degree-of-freedom (MDOF) systems are studied by modifying ductility reduction factors for equivalent single-degree-of-freedom (SDOF) systems. Based on MDOF lumped-mass shear-type models, nonlinear dynamic time history analysis are performed to investigate the influence of ductility demand increase owing to high mode effect on ductility reduction factors. An empirical estimating model of MDOF modification factor is proposed. The results demonstrate that ductility reduction factors for MDOF systems are clearly smaller than those for SDOF systems. The modification factor is mainly affected by the fundamental period and ductility.

Nonstationary Stochastic Response Determination of Nonlinear Systems: A Wiener Path Integral Formalism

AbstractA novel approximate analytical technique is developed for determining the nonstationary response probability density function (PDF) of randomly excited nonlinear multidegree-of-freedom (MDOF) systems. Specifically, the concept of the Wiener path integral (WPI) is used in conjunction with a variational formulation to derive an approximate closed-form solution for the system response PDF. Notably, determining the nonstationary response PDF is accomplished without the need to advance the solution in short time steps as it is required by existing alternative numerical path integral solution schemes, which rely on a discrete version of the Chapman-Kolmogorov (C-K) equation. In this manner, the analytical WPI-based technique developed by the authors is extended and generalized herein to account for hysteretic nonlinearities and MDOF systems. This enhancement of the technique affords circumventing approximations associated with the stochastic averaging treatment of the previously developed technique. Hop...

Displacement estimation of nonlinear structures using the force analogy method

SUMMARYSignificant effort has gone toward developing accurate and efficient displacement estimation procedures for the nonlinear multi‐degree‐of‐freedom (MDOF) system. Although the dynamic nonlinear analysis is capable of providing the high computational precision through the step‐by‐step time integration method, the simplified method is still expected and imperative for seismic design practices. The work presented in this paper focuses on the implementation of using the modal superposition method to estimate displacement responses of the nonlinear MDOF system based on the force analogy method (FAM). The current research demonstrated that the equation of motion for the nonlinear MDOF system can be decoupled, but other two governing equations in the FAM about the internal force, such as the moment and force of structural members, are not decomposable. Thus, the FAM is incorporated with the modal pushover analysis (MPA) method to determine the basic parameters of each mode such that the modal superposition method can be suitable for the solution of the nonlinear MDOF system. The procedure presented here is an approximately estimation method due to the application of MPA method. However, the value and potential for the maximum displacement estimation of the nonlinear MDOF system were demonstrated through the application in a framed structure. Copyright © 2014 John Wiley & Sons, Ltd.

Structural Dynamic Analysis of Freight Railway Wagon Using Finite Element Method

This paper describes the development of a virtual freight wagon vehicle using virtual prototyping computer tools. The freight wagon considered as open type wagon “BOXN25” of the Indian Railways. The freight wagon vehicle comprises of car body structure and two bogies. Solidworks is used for modeling the freight wagon and the geometry is exported to finite element tool, ANSYS. A multi degree of freedom system has been reviewed and compared with system having infinite degree of freedom (continuous structure). The current problem falls in the category of large models (Block Lanczos Algorithm is used) and has high degrees of freedom (PCG Solver is used). The structural dynamic response for the virtual freight wagon is determined. It is seen that the car body deformation is influenced by its elastic underframe and sidewalls. The influence of vibration modes, which describe local deflection, on the comfort level and stability of laden goods is discussed. Mode shapes up to a frequency of 30Hz are considered.

Numerical and Experimental Investigation on Parameter Identification of Time-Varying Dynamical System Using Hilbert Transform and Empirical Mode Decomposition

This paper proposes an approach to identifying time-varying structural modal parameters using the Hilbert transform and empirical mode decomposition. Definition of instantaneous frequency and instantaneous damping ratio based on Hilbert transform for single-degree-of-freedom (SDOF) system is first introduced. The following is the definition of Hilbert damping spectrum from which the time-varying damping ratio of multi-degree-of-freedom (MDOF) system can be calculated. Identification procedures for both instantaneous frequency and damping ratios based on their definition are then introduced. Applicability of the proposed identification algorithm has been validated through several numerical examples. The instantaneous frequency and damping ratios of SDOF system under free vibration and under sinusoidal and white noise excitation have been identified. The proposed method is also applied to MDOF system with slow and sudden changing structural parameters. The results demonstrate that when the system modal parameters are slowly changing, the instantaneous frequency could be easily and well identified with satisfied accuracy for all cases. However, the instantaneous damping ratio could be extracted only when the system is lightly damped. The damping results are better for free vibration situation than for the forced vibration cases. It is also shown that the suggested method can easily track the abrupt change of system modal parameter under free vibration. The proposed method is then applied to a 12-story short-lag shear wall structure model tested on a shaking table. The instantaneous dynamic properties of the structure were identified and were then introduced as known parameters into a finite element model. Comparisons with the numerical results using constant structural parameters demonstrate that the calculated structural responses using the identified time-varying parameters are much closer to the experimental results.

Lateral vibration analysis of continuous bridges utilizing equal displacement rule

The application of equal displacement rule simplifies the evaluation of lateral displacement demand forSDOF system. For complex multi-degree-of-freedom (MDOF) structures such as continuous bridge systems, however, it requires more investigations. In this paper, a comprehensive parametric study of the ratio of maximum inelastic displacement to maximum elastic displacement for typical continuous bridges is performedto advance the application of equal displacement rule to MDOF systems. Particurlarly for the bridges with long periods, this adapted methodlogy is further simplified. It is concluded that equal displacement rule of MDOF is applicable to continuous bridges when the periods of the main modes are no less than the limiting period, which usually serves as an indication to the level of inelastic deformation for a bridge subjected to an earthquake.

Robust optimization of a hybrid control system for wind-exposed tall buildings with uncertain mass distribution

In this paper is studied the influence of the uncertain mass distribution over the floors on the choice of the optimal parameters of a hybrid control system for tall buildings subjected to wind load. In particular, an optimization procedure is developed for the robust design of a hybrid control system that is based on an enhanced Monte Carlo simulation technique and the genetic algorithm. The large computational effort inherent in the use of a MC-based procedure is reduced by the employment of the Latin Hypercube Sampling. With reference to a tall building modeled as a multi degrees of freedom system, several numerical analyses are carried out varying the parameters influencing the floors' masses, like the coefficient of variation of the distribution and the correlation between the floors' masses. The procedure allows to obtain optimal designs of the control system that are robust with respect to the uncertainties on the distribution of the dead and live loads.

Dynamic reduction method for frame structures

This paper proposes a new method for reducing MDOF (Multi Degree of Freedom) system of frame structures into SDOF (Single Degree of Freedom) system and recovering back the results of dynamic analysis from the SDOF to the MDOF system.In most of dynamic condensation methods, the reduction of MDOF into an SDOF system is usually done by means of modal analysis. As a result, simplification against the MDOF is made forward-wise and not reversible. In other words, after the responses of SDOF system evaluated by the modal and dynamic analyses, then it is not possible to recover back the responses of the other condensed degree of freedoms accurately. In the new Strain Reduction Method, displacement and rotational DOFs of all nodal points in the frame structures except an arbitrarily selected representative node are transformed into their strains field before the reduction process. By adopting previous authors’ work in improving the dynamic reduction method, small order in magnitude of strains field can be separated as secondary DOFs, hence leaving the representative node as an SDOFsystem. After conducting dynamic analysis to the SDOF system, the time historical responses of the SDOF system can be used to recover back the time historical responses in the MDOF system of frame structures with negligible error.

Response transformation factors for deterministic-based and reliability-based seismic design

One of the main requirements of the seismic design codes must be its easy application by structural engineers. The use of practically-applicable models or simplified models as single-degree-of-freedom (SDOF) systems is a good alternative to achieve this condition. In this study, deterministic and probabilistic response transformation factors are obtained to evaluate the response in terms of maximum ductility and maximum interstory drifts of multi-degree-of-freedom (MDOF) systems based on the response of equivalent SDOF systems. For this aim, five steel frames designed with the Mexican City Building Code (MCBC) as well as their corresponding equivalent SDOF systems (which represent the characteristics of the frames) are analyzed. Both structural systems are subjected to ground motions records. For the MDOF and the simplified systems, incremental dynamic analyses IDAs are developed in first place, then, structural demand hazard curves are obtained. The ratio between the IDAs curves corresponding to the MDOF systems and the curves corresponding to the simplified models are used to obtain deterministic response transformation factors. On the other hand, demand hazard curves are used to calculate probabilistic response transformation factors. It was found that both approaches give place to similar results.

Output frequency response function based design of additional nonlinear viscous dampers for vibration control of multi-degree-of-freedom systems

This paper is concerned with the development of a new technique for the optimal placement and design of additional nonlinear viscous dampers for vibration control of Multi-Degree-Of-Freedom (MDOF) systems. The MDOF systems described by a multi-storey shear building model and subjected to harmonic loadings are first considered in the present study; then, the basic ideas of the design are extended to the cases of earthquake loadings. The studies are based on the concept of Output Frequency Response Function (OFRF) and associated methods proposed recently by the authors in control engineering and extend, for the first time, the well-established linear viscous damper optimal placement and design techniques for MDOF system vibration control to the nonlinear case. The design examples verify the effectiveness of the new technique, and demonstrate the beneficial effects of the designed additional nonlinear dampers on the MDOF system vibration control and the advantages of nonlinear viscous dampers over linear dampers in such applications.