A conjecture on the number of independent vector variables in a multiqubit orthogonal product basis

So far, very little is known about local indistinguishability of multipartite orthogonal product bases except some special cases. We first give a method to construct an orthogonal product basis with n parties each holding a $$\frac{1}{2}(n+1)$$ -dimensional system, where $$n\ge 5$$ and n is odd. The proof of the local indistinguishability of the basis exhibits that it is a sufficient condition for the local indistinguishability of an orthogonal multipartite product basis that all the positive operator-valued measure elements of each party can only be proportional to the identity operator to make further discrimination feasible. Then, we construct a set of n-partite product states, which contains only 2n members and cannot be perfectly distinguished by local operations and classic communication. All the results lead to a better understanding of the phenomenon of quantum nonlocality without entanglement in multipartite and high-dimensional quantum systems.

As we know, unextendible product basis (UPB) is an incomplete basis whose members cannot be perfectly distinguished by local operations and classical communication. However, very little is known about those incomplete and locally indistinguishable product bases that are not UPBs. In this paper, we first construct a series of orthogonal product bases that are completable but not locally distinguishable in a general m ⊗ n (m ≥ 3 and n ≥ 3) quantum system. In particular, we give so far the smallest number of locally indistinguishable states of a completable orthogonal product basis in arbitrary quantum systems. Furthermore, we construct a series of small and locally indistinguishable orthogonal product bases in m ⊗ n (m ≥ 3 and n ≥ 3). All the results lead to a better understanding of the structures of locally indistinguishable product bases in arbitrary bipartite quantum system.

An orthogonal product basis (OPB) of a finite-dimensional Hilbert space is an orthonormal basis of consisting of product vectors . We show that the problem of constructing the OPBs of an n-qubit system can be reduced to a purely combinatorial problem. We solve this combinatorial problem in the case of four qubits and obtain 33 multiparameter families of OPBs. Each OPB of four qubits is equivalent, under local unitary operations and qubit permutations, to an OPB belonging to at least one of these families.

For safe rotor operation it is important to predict the torsional natural frequencies of the full rotor arrangement and not only of its components. The system’s natural frequencies are typically speed-dependent if rotor and blade vibrations are coupled. In this contribution we focus on the torsional rotor-blade interaction, the coupling between torsional vibrations of the shaft and bending vibrations of blade rows attached to the shaft. During the design of a turbine shaft train, rotor blades are modelled using 3D finite elements due to its complex geometry and resulting vibration modes. This kind of model incorporates typically centrifugal loading due to the rotor rotation as well as contact modelling at the rotor-blade interface. Employing the method of substructuring enables to translate any complex blade which is modelled using 3D finite elements with thousands of physical degrees of freedom into a bunch of models with a single modal degree of freedom. Natural frequencies and modal masses are assigned to each modal degree of freedom representing the blade vibrations. These single degree of freedom models are coupled via so-called modal effective moments of inertia to the rotor shaft model. The resulting model resembles the rotor-blade interaction in all its details from the rotor point of view. The efficiency of this process is two-fold. On one hand, the resulting model size of the full rotor dynamic model becomes small and simple enough to allow elaborate parametric studies and design optimisations. On the other hand, translating the complex 3D blade model into a bunch of single degrees of freedom oscillators is extracted straightforwardly from standard output of commercial finite element software packages like Abaqus or Ansys.

In this paper, moment-curvature behavior of reinforced concrete column with constant axial load is determined using finite element method and then it is introduced to a single degree of freedom (SDOF) model based on Euler- Bernoulli theory. Using this SDOF model, dynamic response of the RC column under the blast loading is estimated. The introduced SDOF includes secondary moments (P-δ) effects, nonlinear behavior of the material and effects of strain rate on concrete and steel materials through the time calculation of the model. Results obtained from SDOF model for transverse displacement of RC column under blast loading is compared to analysis by finite element software OPENSEES. Then, introduced SDOF method is used for drawing Pressure-Impulse (P-I) diagram of the column with considering the presence of axial compressive load. According to the results, introduced SDOF model has simple and quick computations and accuracy of predictions is acceptable.

It is well known that the response modification factor (R) takes into account the ductility, over‐strength, redundancy and damping of structural systems. The ductility factor has played an important role in seismic design, as it is a key component of R. In this study, the ductility factors (Rμ,MDOF) of special steel moment‐resisting frames are calculated by multiplying the ductility factor of single degree of freedom (SDOF) systems (Rμ,SDOF) with the multi‐degree of freedom (MDOF) modification factors (RM). The ductility factors (Rμ,SDOF) of SDOF systems are computed from non‐linear dynamic analysis undergoing different levels of displacement ductility demands and periods when subjected to a large number of recorded earthquake ground motions. To compute the Rμ,SDOF, a group of 1,860 ground motions recorded from 47 earthquakes were considered. RM factors are proposed to account for the MDOF systems, based on previous studies. A total of 108 prototype steel frames were designed to investigate the ductility factors, considering design parameters such as the number of stories (4, 8 and 16), framing systems (perimeter frames and distributed frames), failure mechanisms (strong column‐weak beam and weak column‐strong beam), soil profiles (SA, SC and SE in Uniform Building Code 1997) and seismic zone factors (Z = 0·075, 0·2, and 0·4 in UBC 1997). The effects of these design parameters on the Rμ,MDOF of special steel‐moment‐resisting frames were investigated. Copyright © 2009 John Wiley & Sons, Ltd.

Ductility capacity is comprehensively studied for steel moment-resisting frames. Local, story and global ductility are being considered. An appropriate measure of global ductility is suggested. A time domain nonlinear seismic response algorithm is used to evaluate several definitions of ductility. It is observed that for one-story structures, resembling a single degree of freedom (SDOF) system, all definitions of global ductility seem to give reasonable values. However, for complex structures it may give unreasonable values. It indicates that using SDOF systems to estimate the ductility capacity may be a very crude approximation. For multi degree of freedom (MDOF) systems some definitions may not be appropriate, even though they are used in the profession. Results also indicate that the structural global ductility of 4, commonly used for moment-resisting steel frames, cannot be justified based on this study. The ductility of MDOF structural systems and the corresponding equivalent SDOF systems is studied. The global ductility values are very different for the two representations. The ductility reduction factor F-mu is also estimated. For a given frame, the values of the F-mu parameter significantly vary from one earthquake to another, even though the maximum deformation in terms of the interstory displacement is roughly the same for all earthquakes. This is because the F-mu values depend on the amount of dissipated energy, which in turn depends on the plastic mechanism, formed in the frames as well as on the loading, unloading and reloading process at plastic hinges. Based on the results of this study, the Newmark and Hall procedure to relate the ductility reduction factor and the ductility parameter cannot be justified. The reason for this is that SDOF systems were used to model real frames in these studies. Higher mode effects were neglected and energy dissipation was not explicitly considered. In addition, it is not possible to observe the formation of a collapse mechanism in the equivalent SDOF systems. Therefore, the ductility parameter and the force reduction factor should be estimated by using the MDOF representation.

Traditionally limited to high-security military and diplomatic facilities, the market for blast resistant windows and curtain walls has greatly expanded in the last decade to include courthouses, government office buildings, military housing, commercial buildings and research institutions. These buildings often have architectural, operational and budgetary requirements which are a challenge to satisfy when combined with the need to achieve high levels of protection. Single degree of freedom (SDOF) analytical methods typically used in the design of blast resistant glazed facade systems have been repeatedly shown to produce conservative designs when compared to results from explosive testing. The development of advanced analysis software has allowed engineers to more closely calculate the actual performance of glazed facade systems, achieving more cost efficient and aesthetically pleasing designs without compromising protection. In particular, non-linear, explicit, dynamic, finite element software that has the capability to model coupled behavior between the glass and framing members has been developed and validated through explosive testing. This paper examines recent advances in analytical tools used to predict the behavior of windows and curtain walls when subjected to blast loadings. Comparisons of designs developed using SDOF methodologies and advanced analytical approaches are presented. SINGLE-DEGREE-OF-FREEDOM METHODS Single-degree-of-freedom (SDOF) methods can be used to develop conservative, dynamic design solutions for many facade configurations on new building projects. For the most basic glazed facades, which consist of punched windows or storefront broken up into symmetrical patterns of glass with aluminum or steel mullions, designs developed using SDOF methods provide reasonably cost-effective window systems that meet the client’s performance requirements. The typical approach in SDOF design of windows includes determination of the blast design load, design of the glazing lay-up, and design of the supporting window mullions, framing and connections. The blast pressure (in pounds per square inch, psi) and the corresponding blast impulse (in pounds per square inch x milliseconds, psi-msec) are calculated based on the specified explosive charge weight and the distance to the potential detonation. The blast pressure and impulse are idealized into a linearly decaying, triangular blast-loading function and input into glazing analysis software, along with the window dimensions, to determine the glazing make-up and bite required to meet the client-specified glass hazard. The glazing edge reactions

Vibration energy harvesting can be used as a sustainable power source for various applications. Usually, the generators are designed as devices with a single degree of freedom (SDoF) along the direction of the driving motion. In this research, harvesting from multi-directional (translational) motion sources will be investigated. Three strategies are assessed: a reference SDoF generator, a SDoF generator using an orientation strategy, and a Multi Degree of Freedom (MDoF) system. This led to the development of a design metric by which any 2D design problem can be described by two dimensionless parameters: the relative strength of vibrations, $p_{v}$, and the relative dimension of the design space, $p_{l}$. It was shown that the relative power density (RPD) of a 2DoF system compared to a reference SDoF system only depends on the product $p^{\ast}=p_{v}p_{l}$, and has a maximum of 1.185 for $p^{\ast}=1$. The application of powering a hearing aid is investigated as a case study. It was found that the vibrations in the area of the human head while walking can be represented by a two-directional vibration source with $p_{v}=0.55$. Three different design spaces are assessed for a miniaturized generator and three different optimal embodiments are found. For one of the considered situations where $p^{\ast}=1.1$, a 2DoF system was found to have a 16% higher power output compared to a SDoF reference. The aim of future work will be the validation of the developed metric.

A jack up rig, such as those used for oil drilling in the Gulf of Mexico, is a dynamic sensitive system subjected to random and periodic environmental loads (wave, wind, current, etc.), in which the inertia forces cannot be ignored. However, a static load analysis approach can be justified only if an extra inertial load set, due to the dynamic effect response, is included in the analysis. The Society of Naval Architects and Marine Engineers, SNAME T&R 5-5A, “Guidelines for Site Specific Assessment of Mobile Jack-Up Units”[1], addresses the calculation of the inertial load set by using the classical Single Degree of Freedom (SDOF) analogy to calculate such dynamic effect. This study evaluates how far apart crucial structural members’ stresses/loads obtained by using the SDOF analogy method are from those values obtained from a more realistic modal dynamic analysis. The analysis is performed for wave loads at different heights and frequencies such as those existing on extreme or severe design conditions as a storm. Although a jack up rig structure presents some non-linearities, especially in the legs-hull contact areas and the spud-cans interaction with the soil; a modal superposition analysis can be used if proper linearization is considered. The results of this study for a jack show that the SDOF analogy method tends to underestimate the base shear loads for high wave periods and conversely overestimate some stresses on crucial members, such as the legs’ chords close to the hull for all the wave periods. This study shows that the stresses on one of the legs’ chord of the most loaded leg, due to the dynamic effect produced by the harmonic loads calculated with the SDOF analogy developed in this study tend to be overestimated as the wave period decreases. Conversely, the calculation of the base shear of the structure employing a quasi-static analysis with inertial load set (ILS) as calculated in this study shows that the base shear forces difference between the modal dynamic analysis and the quasi static analysis tends to decrease as the periods of the wave decreases.

This study looks at the influence of superelastic Shape Memory Alloys (SMAs) on the response of a single degree of freedom (SDOF) oscillator. SMAs are a metallic class of materials which possess unique properties that include stress plateaus, hysteretic damping, and large ductility with small residual strains. Superelastic SMAs have the ability to return to their original shape from strains of up to 8%. This study investigates the effects of the unique stress-strain behavior of SMAs on displacement demand, absolute acceleration, and residual deformation of a SDOF system under code level earthquakes. First, the paper investigates the differences in the response of an elastoplastic (EP) and a superelastic system. A double trigger-line model is used to capture the SMA’s hysteretic properties. Previous research has noted that SMAs have varying mechanical properties depending on thermomechanical processing. Therefore, the effects of several different SMA properties on the structural response are investigated. This paper investigates the importance of damping on the response of the SDOF for a suite of ground motions. The results show that shape memory alloys are much less effective in reducing the peak response of a SDOF system in the short period range, as compared to an EP system. As the strength reduction factor increases, peak displacements increase for both the SMA system and the EP system. However, the EP system, in some cases, has large residuals, compared with perfect recentering for the SMA system. Finally, the paper takes an initial look into the effects of using a parallel system consisting of a SMA and an EP element. The parallel system has over three times the energy dissipation in the hysteresis as the SMA system while maintaining significant recentering capabilities compared to the EP system.

In order to improve the performance of the tuned mass damper (TMD) with a smaller physical mass for machining vibration suppression and energy harvesting, a dual-functional inerter-based damper, called electromagnetic tuned inerter damper (ETID), is proposed. To evaluate the performance of the ETID, the model of coupled ETID and a single degree of freedom (SDOF) system has been established. The H2 optimal design of the ETID-SDOF system has been conducted, whose goal is to minimize the value of the root mean square (RMS) of the displacement and absolute acceleration of the SDOF system. The analytical solutions of the design parameters of the ETID-SDOF system, namely, frequency ratio and damping ratio, have been derived. The control performance and robustness for the undamped SDOF system with ETID have been evaluated via parametric study compared with the undamped SDOF system with the TMD system. The potential other layouts of the ETID are also discussed. The influence of the structural damping on design parameters and performance has also been investigated.

A new effective model for calculation of the equivalent uniform blast load for non-uniform blast load such as close-in explosion of a one-way square and rectangle reinforced concrete slab is proposed in this paper. The model is then validated using single degree of freedom (SDOF) system with the experiments and blast tests for square slabs and rectangle slabs. Test results showed that the model is accurate in predicting the damage level on the tested RC slabs under the given explosive charge weight and stand-off distance especially for close-in blast load. The results are also compared with those obtained by conventional SDOF analysis and finite element (FE) analysis using solid elements. It is shown that the new model is more accurate than the conventional SDOF analysis and is running faster than the FE analysis.

Pressure-impulse (P-I) diagrams, which relates damage with both impulse and pressure, are widely used in the design and damage assessment of structural elements under blast loading. Among many methods of deriving P-I diagrams, single degree of freedom (SDOF) models are widely used to develop P-I diagrams for damage assessment of structural members exposed to blast loading. The popularity of the SDOF method in structural response calculation in its simplicity and cost-effective approach that requires limited input data and less computational effort. The SDOF model gives reasonably good results if the response mode shape is representative of the real behaviour. Pressure-impulse diagrams based on SDOF models are derived based on idealised structural resistance functions and the effect of few of the parameters related to structural response and blast loading are ignored. Effects of idealisation of resistance function, inclusion of damping and load rise time on P-I diagrams constructed from SDOF models have been investigated in this study. In idealisation of load, the negative phase of the blast pressure pulse is ignored in SDOF analysis. The effect of this simplification has also been explored. Matrix Laboratory (MATLAB) codes were developed for response calculation of the SDOF system and for repeated analyses of the SDOF models to construct the P-I diagrams. Resistance functions were found to have significant effect on the P-I diagrams were observed. Inclusion of negative phase was found to have notable impact of the shape of P-I diagrams in the dynamic zone.

To approximate nonlinear response of structures subjected to an earthquake excitation, the displacement method in U.S. or the energy method in Japan has been practically used. However, unless these methods include the nature of dynamics of nonlinear structure well, the nonlinear response is not adequately calculated. Applying the equivalent linearization technique to a Single Degree Of Freedom (SDOF) system with bilinear hysteresis subjected to white noise base acceleration, this paper mathematically quantifies deterioration in a spring constant and increase in a damping coefficient with the progress of nonlinearity in the restoring force system as a function of the ensemble ductility ratio. As the nonlinearity progresses, the spring constant rapidly deteriorates and the damping substantially increases. Increments of damping of the lightly damped linear SDOF system are more than that of the moderately damped one. A comparison of the response of the equivalently linearized system to that of the corresponding linear system reveals the capability for growth in the nonlinear response. The nonlinear response predicable by either method is identified by values of the ensemble ductility ratio and damping ratio of the linear SDOF system. In addition, in a range of the ensemble ductility ratio where the bulk of the engineered structures are included, neither method can properly evaluate the nonlinear response. Although the results presented herein give the mean nature of the nonlinear response and phase and amplitude characteristics of accelerograms make the nonlinear response vary around the mean, uniform application of either displacement method or energy method to approximating the nonlinear response may be reconsidered.