Multiple Degree Of Freedom Systems Research Articles

Vibration analysis of multiple degrees of freedom mechanical system with dry friction

This article presents a simple procedure for predicting time-domain vibrational behaviors of a multiple degrees of freedom mechanical system with dry friction. The system equations of motion are discretized by means of the implicit Bozzak–Newmark integration scheme. At each time step, the discontinuous frictional force problem involving both the equality and inequality constraints is successfully reduced to a quadratic mathematical problem or the linear complementary problem with the introduction of non-negative and complementary variable pairs (supremum velocities and slack forces). The so-obtained complementary equations in the complementary pairs can be solved efficiently using the Lemke algorithm. Results for several single degree of freedom and multiple degrees of freedom problems with one-dimensional frictional constraints and the classical Coulomb frictional model are obtained using the proposed procedure and compared with those obtained using other approaches. The proposed procedure is found to be accurate, efficient, and robust in solving non-smooth vibration problems of multiple degrees of freedom systems with dry friction. The proposed procedure can also be applied to systems with two-dimensional frictional constraints and more sophisticated frictional models.

Validation Techniques for 6-DOF Vibration Data Acquisition

Multiple Degree of Freedom (MDOF) excitation systems and MDOF vibration control systems continue to improve, and are now standard equipment in many dynamic test laboratories. Determination of an input specification for such MDOF systems is critically dependent on properly acquired field data. Validation of field data will be discussed and demonstrated employing the same transformation tools used in both transformation-based 6-degree-of-freedom (6-DOF) vibration control and generalized MDOF vibration specification development (VSD).

Significance of ground motion time step in one dimensional site response analysis

The discrete nature of the numerical methods utilized in 1D site response analysis and calculation of the response spectra (e.g., frequency domain, Duhamel integral, and Newmark β methods) introduces time-step dependence in the resulting solution. Using an input ground motion with too large of a time-step leads to under-prediction of high-frequency characteristics of the system response due to limitations in the numerical solution of single and multiple degree of freedom systems. In order to reduce potential errors, using a sampling rate at least ten times greater than the maximum considered frequency is recommended. The preferred alternative is selection of input ground motions with a sufficiently small time step to avoid introducing numerical errors. However, where such motions are not available, then the time step of the ground motion can be reduced through interpolation. This paper demonstrates that the use of Fourier transform zero-padded interpolation is the preferred approach to obtain a ground motion with an adequate time step for the calculation of the elastic acceleration response spectra, and to analyze site response using either frequency or time domain methods.

Strength reduction factor for MDOF soil-structure systems

In most of the seismic design provision, the concept of strength reduction factor has been developed to account for inelastic behavior of structures under seismic excitations. Most recent studies considered soil–structure interaction (SSI) in inelastic response analysis are mainly based on idealized structural models of single degree-of-freedom (SDOF) systems. However, an SDOF system might not be able to well capture the SSI and structural response characteristics of real multiple degrees-of-freedom (MDOF) systems. In this paper, through a comprehensive parametric study of 21600 MDOF and its equivalent SDOF (E-SDOF) systems subjected to an ensemble of 30 earthquake ground motions recorded on alluvium and soft soils, effects of SSI on strength reduction factor of MDOF systems have been intensively investigated. It is concluded that generally, SSI reduces the strength reduction factor of both MDOF and more intensively SDOF systems. However, depending on the number of stories, soil flexibility, aspect ratio and inelastic range of vibration, the strength reduction factor of MDOF systems could be significantly different from that of E-SDOF systems. A new simplified equation, which is a function of fixed-base fundamental period, ductility ratio, the number of stories, structure slenderness ratio and dimensionless frequency, is proposed to estimate strength reduction factors for MDOF soil–structure systems. Copyright © 2012 John Wiley & Sons, Ltd.

Control of structures under uniform hazard earthquake excitation via wavelet analysis and pattern search method

A new methodology for optimal control of structures under uniform hazard earthquake excitation has been presented in this study. The proposed method for generation of uniform hazard earthquake accelerograms is based on the sets of ground motions developed for the SAC Steel Project. The proposed controller employs the decomposing capabilities of wavelet analysis on uniform hazard earthquake accelerograms and the ability of the pattern search method to optimize a cost function to control response of structure. The performance and effectiveness of the presented method are applied to both single degree of freedom and multiple degree of freedom systems for the response control of uniform hazard earthquake-excited systems. The results are compared with the linear quadratic regulator control algorithm; performance of the proposed control system has been found to be better than the linear quadratic regulator controller. Copyright © 2012 John Wiley & Sons, Ltd.

Dynamic Calculation of MDOF Systems Based on GIS

The paper presents an integration of a Geographic Information System (GIS) and Multiple Degree of Freedom (MDOF) model for structural dynamic calculations. The input data of MDOF is stored in the format of spatial data model in GIS with 2D and 3D with the help of Google SketchUp. The output data of MDOF can be viewed in both 2D graph and 3D visualization after simulation calculation in GIS. A case study of dynamic response of campus buildings is presented to illustrate how the GIS-coupled system has been developed. Results show that the proposed GIS-model integrated system can perform a large number of dynamic responses of MDOF affected by earthquakes, promote better earthquake resistant behavior of building structure in urban regions, and thus optimize hazard assessment, vulnerability estimation and seismic risk reduction.

A 6-DOF Vibration Specification Development Methodology

Multiple degree of freedom (MDOF) excitation systems and MDOF vibration control systems continue to improve, and are now standard equipment in many dynamic test laboratories. This paper concentrates on the often overlooked process of determination of an input specification for such MDOF systems. A pair of generalized six-degree-of-freedom (6-DOF) vibration specification development (VSD) techniques are proposed, discussed, and illustrated through an example.

Parametric excitation of a piezoelectrically actuated system near Hopf bifurcation

This paper deals with investigation into the stability analysis for transverse motions of a cantilever micro-beam, which is axially loaded due to a voltage applied to the piezoelectric layers located on the lower and upper surfaces of the micro-beam. The piezoelectric layers are pinned to the open end of the micro-beam and not bonded to it through its length. Application of the DC and AC piezoelectric actuations creates steady and time varying axial forces. The equation of the motion is derived using variational principal, and discretized using modal expansion theorem. The differential equations of the discretized model are a set of Mathieu type ODEs, whose stability analysis is performed using Floquet theory for multiple degree of freedom systems. Considering first two eigen-functions in the modal expansion theorem leads in the prediction of flutter type of instability as a consequence of Hopf bifurcation, which is not seen in the reduced single degree of freedom system. The object of the present study is to passively control the flutter instability in the proposed model by applying AC voltage with suitable amplitude and frequency to the piezoelectric layers. The effect of various parameters on the stability of the structure, including damping coefficient, amplitude of the DC and AC voltages, and the frequency of the applied AC voltage is studied.

Application of Energy Balance Equation on Multiple Degree of Freedom Systems

In recent years, seismic evaluation and design techniques which are able to evaluate cumulative damage have been required. Authors have dealt with this issue from the viewpoint of the energy balance equation. The energy balance equation expresses cumulative information of response, so that the energy balance equation is adequate for the evaluation of the influence of cumulative load such as seismic response. Authors have already clarified qualitative relationships between input energy and fatigue failure of simple single degree of freedom system. However, in order to apply energy balance equation on actual structure for seismic evaluation, expansion of it to multiple degree of freedom systems is required. Therefore this paper proposes three types of energy balance equation for multiple degree of freedom systems, that is a matrix method, an each mass method and a modal method. Energy balance equation by the matrix method indicates energy of a whole system. The equation by the each mass method indicates energy of each mass. The equation by the modal method indicates energy of each mode. In this paper, behaviors and features are investigated by numerical simulation and vibration experiment.

2P2-Q14 多自由度系の低次数化アトラクタ設計と疑似目標値(インフォマティブ・モーションとモーション・メディア-ロボットの身体性と運動-)

Dance teaching, sports teaching and rehabilitation require to transfer motion skills from a "teacher" to a "student". So far, we proposed a "Pseudo-Reference" based on attractor design method. The pseudo-reference is a virtual reference derived from a controller of an autonomous system. It does not always coincide with the measured posture but gives us the timing and amplitude of the input torque which represents knack of motion. However, for multiple degree of freedom systems, the controller is designed with a high order function and it is not easy to obtain an autonomous system. In this paper, we propose a reduced order controller design method of attractor. Based on the correlation of the measured data, the principal component analysis gives us dimension of the motion. The effectiveness of the reduced controller is evaluated by using the tap dancing robot, and pseudo-reference is applied to multiple degree of freedom system.

Seismic-Resistant Investigation of Multi-Storey Building by Response Spectrum Method

The present article attempts to evaluate the earthquake resistance of a four-storey building frame. It also compares the performance of buildings with soft ground floors and different infill parameters, such as brick wall and shear wall. The Indian standard code (IS-code) of practice (IS: 1893, Part-1:2002) guidelines and methodology are used to analyze the problem. Response spectrum method, involving various modal participations, is employed to evaluate the nodal forces in the typical beam-column junctions. The frame members, infill walls and floors are tackled as rigid end joints, panel element, and diaphragms rigid in-plane, respectively. Linear elastic analysis is performed with individual floor mass and stiffness, taking into account the building as a multiple degrees of freedom system. The effect of damping and the soil criteria are also studied as per the provision of IS-code. The investigation concludes that the proper mass and stiffness distribution in multi-storied buildings will minimize the hazardous effects of seismic excitation. The mathematical model of the building finally recognized the hazardous features of Indian reinforced concrete buildings and recommends some measures to improve the seismic performance.

P–Δ Effect in Inelastic Seismic Demands

Structures are designed to dissipate input seismic energy by performing hysteresis cycles through repetitive inelastic excursions. P–Δ effect becomes prominent during inelastic seismic excursions and may considerably increase the ductility demand. Furthermore, ductility demand, prima facie, is regulated by lateral strength and hysteresis behaviour of structural material. In this context, this paper presents an attempt to capture inelastic seismic response incorporating P–Δ effect for single degree of freedom (SDOF) and multiple degrees of freedom (MDOF) systems with representative short period, medium period and long period structures. Four hysteresis behaviours, namely: (a) elasto-plastic, (b) only stiffness degrading, (c) only strength deteriorating and (d) both strength and stiffness degrading hysteresis behaviours are considered to cover all possible material characteristics. Rigorous nonlinear dynamic analysis is performed and the ratios of displacement ductility demands and overturning moments obtained incorporating P–Δ effect to the same obtained without P–Δ effect are computed for various response reduction factors (Rμ). The results indicate that SDOF systems are relatively more vulnerable when compared to MDOF systems with the same fundamental periods. Further, long period systems are more vulnerable than short and medium period systems, irrespective of their degrees of freedom (DOF). The study further indicates that P–Δ effect may cause an instantaneous magnification of responses at some limiting values of Rμ depending on material hysteresis behaviour. Such observation helps to fix the maximum limits of Rμ to yield similar ductility demands intended by code-specified Rμ values normally computed by ignoring P–Δ effect. The study as a whole aims to provide all round guidelines for modifying the codal provisions for performance-based design to incorporate the extra demand due to P–Δ effect. In the absence of consideration of the degrading behaviour so far, the applicability of the results in this area was restricted only to steel structures. The present study is an effort to overcome this limitation for enabling the design provisions to be extended for concrete structures as well.

Static strain energy and dynamic amplification factor on multiple degree of freedom systems

Many references and design codes propose the use of a dynamic amplification factor applied to the static response of a structure for estimating its dynamic response. This is the case of the present recommendations on cable stayed bridges or other references for evaluating the dynamic effects produced by the accidental loss of a segment during the cantilever construction of a precast concrete bridge. This paper deals with the dynamic amplification factor of multiple degree of freedom systems under the action of a rectangular pulse load of infinite duration. A theoretical study of the possible maximum values that the dynamic amplification factor can reach has been carried out, establishing the relation between this factor and the strain energy of the system under the load statically applied. This relation allows us to determine if the dynamic amplification factor can exceed the value of 2.0 and to derive a simple method for comparing the dynamic effects of different pulse loads. Finally, a numerical example illustrates this method applied to determine which rectangular pulse load, produced by the accidental loss of a segment of a bridge, can produce a larger and a smaller dynamic amplification factor on a cantilever precast segmental bridge under construction. In this example, the possible maximum dynamic amplification factor obtained has been 3.24.

A Methodology for the Modeling of Forced Dynamical Systems From Time Series Measurements Using Time-Delay Neural Networks

The main goal of this research was to develop and present a general, efficient, mathematical, and theoretical based methodology to model nonlinear forced-vibrating mechanical systems from time series measurements. A system identification modeling methodology for forced dynamical systems is presented based on a dynamic system theory and a nonlinear time series analysis that employ phase space reconstruction (delay vector embedding) in modeling dynamical systems from time series data using time-delay neural networks. The first part of this work details the modeling methodology, including background on dynamic systems, phase space reconstruction, and neural networks. In the second part of this work, the methodology is evaluated based on its ability to model selected analytical lumped-parameter forced-vibrating dynamic systems, including an example of a linear system predicting lumped mass displacement subjected to a displacement forcing function. The work discusses the application to nonlinear systems, multiple degree of freedom systems, and multiple input systems. The methodology is further evaluated on its ability to model an analytical passenger rail car predicting vertical wheel∕rail force using a measured vertical rail profile as the input function. Studying the neural modeling methodology using analytical systems shows the clearest observations from results, providing prospective users of this tool an understanding of the expectations and limitations of the modeling methodology.

Nonlinear Numerical Prediction of Gas Foil Bearing Stability and Unbalanced Response

One of the main interests of gas foil bearings lies in their superior rotordynamic characteristics compared with conventional bearings. A numerical investigation on the stability limit and on the unbalanced response of foil bearings is presented in this paper. The main difficulty in modeling the dynamic behavior of such bearings comes from the dry friction that occurs within the foil structure. Indeed, dry friction is highly nonlinear and is strongly influenced by the dynamic amplitude of the pressure field. To deal with these nonlinearities, a structural dynamic model has been developed in a previous work. This model considers the entire corrugated foil and the interactions between the bumps by describing the foil bearing structure as a multiple degrees of freedom system. It allows the determination of the dynamic friction forces at the top and at the bottom of the bumps by simple integration of ordinary differential equations. The dynamic displacements of the entire corrugated sheet are then easily obtained at each time step. The coupling between this structural model and a gas bearing prediction code is presented in this paper and allows performing full nonlinear analyses of a complete foil bearing. The bearing stability is the first investigated problem. The results show that the structural deflection enhances the stability of compliant surface bearings compared with rigid ones. Moreover, when friction is introduced, a new level of stability is reached, revealing the importance of this dissipation mechanism. The second investigated problem is the unbalanced response of foil bearings. The shaft trajectories depict a nonlinear jump in the response of both rigid and foil bearings when the value of the unbalance increases. Again, it is evidenced that the foil bearing can support higher mass unbalance before this undesirable step occurs.

Parametric Resonance in Mechanics: Classical Problems and New Results

We formulate and solve parametric resonance problems for one- and multiple degrees of freedom systems in three-dimensional space of physical parameters: excitation frequency, amplitude, and viscous damping coefficient assuming that the last two parameters are small. The main result obtained here is that we find the instability domains (simple and combination parametric resonances) as half-cones in three-parameter space with the use of eigenfrequencies and eigenmodes of the corresponding conservative system.

Elliptic harmonic balance method for two degree-of-freedom self-excited oscillators

Elliptic harmonic balance (EHB) method as an analytical method is widely used for strongly non-linear oscillators with a single degree-of-freedom (DOF). The oscillation equations are expressed by elliptic functions, and then the expressions are expanded as harmonics of elliptic type while only the first harmonic is retained. To the best of our knowledge, however, it seems that the EHB method has not been found applications in two or multiple DOFs systems. One possible reason is that the EHB method may cause a difficult problem that the number of equations obtained by harmonic balancing is not equal to that of unknowns. To this end, in the present paper, the EHB method is therefore extended to study a class of strongly self-excited oscillators with two DOFs. Prior to harmonic balancing, an additional equation is established to tackle the aforementioned problem. Illustrative examples show that solutions of limit cycles obtained by the proposed method are in good agreement with the numerical solutions.

Influence of site effects on inelastic displacement ratios for SDOF and MDOF systems

Lateral displacements’ control of structures subjected to earthquake ground motion has now been recognized as a key factor in the assessment of system performance, leading to design approaches that use displacements rather than forces as the starting point for the seismic evaluation of structures. In fact performance-based approaches offer significant advantages in comparison with traditional force-based approaches, since the former are capable of focusing on nonlinear behaviour and consequent damage to the structure, in contrast to the latter. Lateral displacement demand, particularly in structures that exhibit nonlinear behaviour, can be significantly affected by the features of strong ground motion, i.e., amplitude, frequency content and duration. Such characteristics are in turn profoundly influenced by the irregularity and changeability in earthquake ground motions, which should therefore be taken into account appropriately. The great number of strong motion records gathered throughout the last decades in the most widely varying soil-site conditions has made accounting for soil-site effects in the characterization of elastic and inelastic displacement demands feasible. The aim of this paper is to present the results of numerical investigations on the response of both single-degree-of-freedom (SDOF) and multiple-degree-of-freedom (MDOF) systems, through nonlinear time-history analyses performed on the basis of a wide data set of strong motion records. Constant ductility spectra of the ratios of the maximum inelastic displacement to the corresponding maximum elastic demand were derived for this purpose. In particular, the influences of earthquake magnitude, source-to-site distance, local soil-site conditions, ductility and hysteretic behaviour were quantified. Finally, simplified expressions for the ratio of the maximum inelastic to the maximum elastic displacement were established, in order to allow the evaluation of inelastic displacements for new or rehabilitated structures for which the global displacement ductility can be estimated, directly from the knowledge of the corresponding elastic demands.

Enhanced, Unconditionally Stable, Explicit Pseudodynamic Algorithm

In performing a pseudodynamic test, an explicit method is generally preferred over an implicit method since it involves no iteration procedure or extra hardware that is often needed for an implicit method. However, its integration time step is usually limited by stability. Hence, it is very promising for the pseudodynamic testing if an explicit method can have unconditional stability, which might eliminate the limitation on time step for the testing of a multiple degree of freedom system or a substructure system. Although an explicit pseudodynamic algorithm with unconditional stability has been successfully implemented and its superior characteristics have been identified, an enhanced unconditionally stable explicit pseudodynamic algorithm is further proposed. In this study, it is verified that both explicit pseudodynamic algorithms possess the same numerical characteristics in the step-by-step integration. However, the newly developed explicit pseudodynamic algorithm shows better error propagation properties when compared to that developed previously.

On precise time integration method for non-classically damped MDOF systems

In the complex mode superposition method, the equations of motion for non-classically damped multiple-degree-of-freedom (MDOF) discrete systems can be transferred into a combination of some generalized SDOF complex oscillators. Based on the state space theory, a precise recurrence relationship for these complex oscillators is set up; then a delicate general solution of non-classically damped MDOF systems, completely in real value form, is presented in this paper. In the proposed method, no calculation of the matrix exponential function is needed and the algorithm is unconditionally stable. A numerical example is given to demonstrate the validity and efficiency of the proposed method.