Harmonic balance simulation of the influence of component uniformity and reliability on the performance of a Josephson traveling wave parametric amplifier

A Josephson traveling wave parametric amplifier(JTWPA),which is a quantum-limited amplifier with high gain and large bandwidth,is the core device of large-scale measurement and control systems for quantum computing.A typical JTWPA consists of thousands of Josephson junctions connected in series to form a transmission line and hundreds of shunt LC resonators periodically loaded along the line for phase matching.Because the variation of these capacitors and inductors can be detrimental to their high-frequency characteristics,the fabrication of a JTWPA typically necessitates precise processing equipment.To guide the fabrication process and further improve the design for manufacturability,it is necessary to understand how each electronic component affects the amplifier.In this paper,we use the harmonic balance method to conduct a comprehensive study on the impact of nonuniformity and fabrication yield of the electronic components on the performance of a JTWPA.The results provide insightful and scientific guidance for device design and fabrication processes.

Numerical Analysis of Balanced Methods for the Impulsive Stochastic Differential Equations

Positive results are proved here about the ability of balanced methods to reproduce the mean square stability of the impulsive stochastic differential equations. It is shown that the balanced methods with strong convergence can preserve the mean square stability with the sufficiently small stepsize. Weak variants and their mean square stability are also considered. Several numerical experiments are given for illustration and show that the fully implicit methods are superior to those of the explicit methods in terms of mean-square stabilities for relatively large stepsizes especially.

A novel adaptive harmonic balance method with an asymptotic harmonic selection

The harmonic balance method(HBM)is one of the most widely used methods in solving nonlinear vibration problems,and its accuracy and computational efficiency largely depend on the number of the harmonics selected.The adaptive harmonic balance(AHB)method is an improved HBM method.This paper presents a modified AHB method with the asymptotic harmonic selection(AHS)procedure.This new harmonic selection procedure selects harmonics from the frequency spectra of nonlinear terms instead of estimating the contribution of each harmonic to the whole nonlinear response,by which the additional calculation is avoided.A modified continuation method is proposed to deal with the variable size of nonlinear algebraic equations at different values of path parameters,and then all solution branches of the amplitude-frequency response are obtained.Numerical experiments are carried out to verify the performance of the AHB-AHS method.Five typical nonlinear dynamic equations with different types of nonlinearities and excitations are chosen as the illustrative examples.Compared with the classical HBM and Runge-Kutta methods,the proposed AHB-AHS method is of higher accuracy and better convergence.The AHB-AHS method proposed in this paper has the potential to investigate the nonlinear vibrations of complex high-dimensional nonlinear systems.

A vertical track nonlinear energy sink

Eliminating the effects of gravity and designing nonlinear energy sinks(NESs)that suppress vibration in the vertical direction is a challenging task with numerous damping requirements.In this paper,the dynamic design of a vertical track nonlinear energy sink(VTNES)with zero linear stiffness in the vertical direction is proposed and realized for the first time.The motion differential equations of the VTNES coupled with a linear oscillator(LO)are established.With the strong nonlinearity considered of the VTNES,the steady-state response of the system is analyzed with the harmonic balance method(HBM),and the accuracy of the HBM is verified numerically.On this basis,the VTNES prototype is manufactured,and its nonlinear stiffness is identified.The damping effect and dynamic characteristics of the VTNES are studied theoretically and experimentally.The results show that the VTNES has better damping effects when strong modulation responses(SMRs)occur.Moreover,even for small-amplitude vibration,the VTNES also has a good vibration suppression effect.To sum up,in order to suppress the vertical vibration,an NES is designed and developed,which can suppress the vertical vibration within certain ranges of the resonance frequency and the vibration intensity.

时域谐波平衡求解器中非物理解出现根源探析

The time domain harmonic balance method is an attractive reduced order method of analyzing unsteady flow for turbomachines. However, the method can admit non-physical solutions. Non-physical solutions were encountered from a three-blade-row compressor configuration in a time domain harmonic balance analysis. This paper aims to investigate the root cause of the non-physical solutions. The investigation involves several strategies, which include increasing the number of harmonics, increasing the number of time instants, including scattered modes,including the rotor-rotor interaction, and the use of a new method-the approximate time domain nonlinear harmonic method. Numerical analyses pertinent to each strategy are presented to reveal the root cause of the non-physical solution. It is found that the nonlinear interaction of unsteady flow components with different fundamental frequencies is the cause of the non-physical solution. The non-physical solution can be eliminated by incorporating extra scattered modes or using the approximate time domain nonlinear harmonic method.

Harmonic balance method with alternating frequency/time domain technique for nonlinear dynamical system with fractional exponential

Comparisons of the common methods for obtaining the periodic responses show that the harmonic balance method with alternating frequency/time （HB-AFT） do- main technique has some advantages in dealing with nonlinear problems of fractional exponential models. By the HB-AFT method, a rigid rotor supported by ball bearings with nonlinearity of Hertz contact and ball passage vibrations is considered. With the aid of the Floquet theory, the movement characteristics of interval stability are deeply studied. Besides, a simple strategy to determine the monodromy matrix is proposed for the stability analysis.

Improving convergence of incremental harmonic balance method using homotopy analysis method

We have deduced incremental harmonic balance an iteration scheme in the （IHB） method using the harmonic balance plus the Newton-Raphson method. Since the convergence of the iteration is dependent upon the initial values in the iteration, the convergent region is greatly restricted for some cases. In this contribution, in order to enlarge the convergent region of the IHB method, we constructed the zeroth-order deformation equation using the homotopy analysis method, in which the IHB method is employed to solve the deformation equation with an embedding parameter as the active increment. Taking the Duffing and the van der Pol equations as examples, we obtained the highly accurate solutions. Importantly, the presented approach renders a convenient way to control and adjust the convergence.

Application of the Material Balance Method in Paleoelevation Recovery: A Case Study of the Longmen Mountains Foreland Basin on the Eastern Margin of the Tibetan Plateau

We applied the material balance principle of the denudation volume and sedimentary flux to study the denudation-accumulation system between the Longmen Mountains （Mts.） and the foreland basin. The amount of sediment in each sedimentation stage of the basin was estimated to obtain the denudation volume, erosion thickness and deposit thickness since the Late Triassic Epoch, to enable us to recover the paleoelevation of the provenance and the sedimentary area. The results show the following： （1） Since the Late Triassic Epoch, the elevation of the surface of the Longmen Mts. has uplifted from 0 m to 2751 m, and the crust of the Longmen Mts. has uplifted by 9.8 km. Approximately 72% of the materials introduced have been denuded from the mountains. （2） It is difficult to recover the paleoelevation of each stage of the Longmen Mts. foreland basin quantitatively by the present-day techniques and data. （3） The formation of the Longmen Mts. foreland basin consisted of three stages of thrust belt tectonic load and three stages of thrust belt erosional unload. During tectonic loading stages （Late Triassic Epoch, Late Jurassic-Early Cretaceous, Late Cretaceous-Miocene）, the average elevation of Longmen Mts. was lower （approximately 700-1700 m）. During erosional unloading stages （Early and Middle Jurassic, Middle Cretaceous and Jiaguan, Late Cenozoic）, the average elevation of Longmen Mts. was high at approximately 2000-2800m.

INCREMENTAL HARMONIC BALANCE METHOD FOR AIRFOIL FLUTTER WITH MULTIPLE STRONG NONLINEARITIES

The incremental harmonic balance method was extended to analyze the flutter of systems with multiple structural strong nonlinearities. The strongly nonlinear cubic plunging and pitching stiffness terms were considered in the flutter equations of two-dimensional airfoil. First, the equations were transferred into matrix form, then the vibration process was divided into the persistent incremental processes of vibration moments. And the expression of their solutions could be obtained by using a certain amplitude as control parameter in the harmonic balance process, and then the bifurcation, limit cycle flutter phenomena and the number of harmonic terms were analyzed. Finally, numerical results calculated by the Runge-Kutta method were given to verify the results obtained by the proposed procedure. It has been shown that the incremental harmonic method is effective and precise in the analysis of strongly nonlinear flutter with multiple structural nonlinearities.

Primary resonance of fractional-order Duffing–van der Pol oscillator by harmonic balance method

The dynamical properties of fractional-order Duffing–van der Pol oscillator are studied, and the amplitude–frequency response equation of primary resonance is obtained by the harmonic balance method. The stability condition for steady-state solution is obtained based on Lyapunov theory. The comparison of the approximate analytical results with the numerical results is fulfilled, and the approximations obtained are in good agreement with the numerical solutions. The bifurcations of primary resonance for system parameters are analyzed. The results show that the harmonic balance method is effective and convenient for solving this problem, and it provides a reference for the dynamical analysis of similar nonlinear systems.

Auto-Backlund Transformation and Extended Tanh-Function Methods to Solve the Time-Dependent Coefficients Calogero-Degasperis Equation

In this paper, the Auto-B?cklund transformation connected with the homogeneous balance method (HB) and the extended tanh-function method are used to construct new exact solutions for the time-dependent coefficients Calogero-Degasperis (VCCD) equation. New soliton and periodic solutions of many types are obtained. Furthermore, the soliton propagation is discussed under the effect of the variable coefficients.

Data correction of the force balanced accelerometer

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Adjacent mode resonance of a hydraulic pipe system consisting of parallel pipes coupled at middle points

The coupling vibration of a hydraulic pipe system consisting of two pipes is studied.The pipes are installed in parallel and fixed at their ends,and are restrained by clips to one bracket at their middle points.The pipe subjected to the basement excitation at the left end is named as the active pipe,while the pipe without excitation is called the passive pipe.The clips between the two pipes are the bridge for the vibration energy.The adjacent natural frequencies will enhance the vibration coupling.The governing equation of the coupled system is deduced by the generalized Hamilton principle,and is discretized to the modal space.The modal correction is used during the discretization.The investigation on the natural characters indicates that the adjacent natural frequencies can be adjusted by the stiffness of the two clips and bracket.The harmonic balance method(HBM)is used to study the responses in the adjacent natural frequency region.The results show that the vibration energy transmits from the active pipe to the passive pipe swimmingly via the clips together with a flexible bracket,while the locations of them are not node points.The adjacent natural frequencies may arouse wide resonance curves with two peaks for both pipes.The stiffness of the clip and bracket can release the vibration coupling.It is suggested that the stiffness of the clip on the passive pipe should be weak and the bracket should be strong enough.In this way,the vibration energy is reflected by the almost rigid bracket,and is hard to transfer to the passive pipe via a soft clip.The best choice is to set the clips at the pipe node points.The current work gives some suggestions for weakening the coupled vibration during the dynamic design of a coupled hydraulic pipe system.

A novel way for vibration control of FGM fluid-conveying pipes via NiTiNOL-steel wire rope

In this study,a coupling model of fluid-conveying pipes made of functionally graded materials(FGMs)with NiTiNOL-steel(NiTi-ST)for vibration absorption is investigated.The vibration responses of the FGM fluid-conveying pipe with NiTi-ST are studied by the Galerkin truncation method(GTM)and harmonic balance method(HBM).The harmonic balance solutions and the numerical results are consistent.Also,the linearized stability of the structure is determined.The effects of the structure parameters on the absorption performance are also studied.The results show that the NiTi-ST is an effective means of vibration absorption.Furthermore,in studying the effect of the NiTi-ST,a closed detached response(CDR)is first observed.It is noteworthy that the CDR may dramatically change the vibration amplitude and that the parameters of the NiTi-ST may determine the emergence or disappearance of the CDR.This vibration absorption device can be extended to offer more general vibration control in engineering applications.

Genetic-Chicken Swarm Algorithm for Minimizing Energy in Wireless Sensor Network

Wireless Sensor Network(WSN)technology is the real-time applica-tion that is growing rapidly as the result of smart environments.Battery power is one of the most significant resources in WSN.For enhancing a power factor,the clustering techniques are used.During the forward of data in WSN,more power is consumed.In the existing system,it works with Load Balanced Cluster-ing Method(LBCM)and provides the lifespan of the network with scalability and reliability.In the existing system,it does not deal with end-to-end delay and deliv-ery of packets.For overcoming these issues in WSN,the proposed Genetic Algo-rithm based on Chicken Swarm Optimization(GA-CSO)with Load Balanced Clustering Method(LBCM)is used.Genetic Algorithm generates chromosomes in an arbitrary method then the chromosomes values are calculated using Fitness Function.Chicken Swarm Optimization(CSO)helps to solve the complex opti-mization problems.Also,it consists of chickens,hens,and rooster.It divides the chicken into clusters.Load Balanced Clustering Method(LBCM)maintains the energy during communication among the sensor nodes and also it balances the load in the gateways.The proposed GA-CSO with LBCM improves the life-span of the network.Moreover,it minimizes the energy consumption and also bal-ances the load over the network.The proposed method outperforms by using the following metrics such as energy efficiency,ratio of packet delivery,throughput of the network,lifetime of the sensor nodes.Therefore,the evaluation result shows the energy efficiency that has achieved 83.56%and the delivery ratio of the packet has reached 99.12%.Also,it has attained linear standard deviation and reduced the end-to-end delay as 97.32 ms.

A semi-analytical multi-harmonic balance method on full-3D contact model for dynamic analysis of dry friction systems

Dry friction damping structures are widely-used in aero-engines to mitigate vibration.The nonlinear nature of friction and the two-dimensional in-plane motion on the contact interface bring challenges to accurately and efficiently predict the forced response of frictionally damped structures.The state-of-the-art Multi-Harmonic Balance Method(MHBM)on quasi-3D contact model in engineering cannot precisely capture the kinematics on the friction interface although the efficiency is high.The full-3D contact model can describe the constitutive relationship of the interface in a more accurate manner;however,the efficiency and convergence are not guaranteed for large-scale models.In this paper,a semi-analytical MHBM on full-3D contact model is proposed.The original Trajectory Tracking Method(TTM)for evaluating the contact force is reformulated to make the calculation more concise and the derivation of the Analytical Jacobian Matrix(AJM)feasible.Based on the chain rule of derivation,the AJM which is the core to upgrade the performance is deduced.Through a shrouded blade finite element model,the accuracy and efficiency of the proposed method are compared with both the MHBM on full-3D contact model with numerical Jacobian matrix and the MHBM on quasi-3D contact model with AJM.The results show that the AJM improves significantly the efficiency of the MHBM on full-3D contact model.The time cost of the proposed method is in the same order of magnitude as that of the MHBM on quasi-3D contact model.We also confirm that the full-3D contact model is necessary for the dynamic analyses of shrouded blades.If one uses the quasi-3D model,the estimation relative error of damping can even reach 31.8%in some cases.In addition,the AJM also brings benefits for stability analysis.It is highly recommended that engineers use the MHBM on full-3D contact model for the dynamic analysis and design of shrouded blades.

Nonlinear dynamic analysis of cantilevered piezoelectric energy harvesters under simultaneous parametric and external excitations

The nonlinear dynamics of cantilevered piezoelectric beams is investigated under simultaneous parametric and external excitations. The beam is composed of a substrate and two piezoelectric layers and assumed as an Euler-Bernoulli model with inextensible deformation. A nonlinear distributed parameter model of cantilevered piezoelectric energy harvesters is proposed using the generalized Hamilton＇s principle. The proposed model includes geometric and inertia nonlinearity, but neglects the material nonlinearity. Using the Galerkin decomposition method and harmonic balance method, analytical expressions of the frequency-response curves are presented when the first bending mode of the beam plays a dominant role. Using these expressions, we investigate the effects of the damping, load resistance, electromechanical coupling, and excitation amplitude on the frequency-response curves. We also study the difference between the nonlinear lumped-parameter and distributed- parameter model for predicting the performance of the energy harvesting system. Only in the case of parametric excitation, we demonstrate that the energy harvesting system has an initiation excitation threshold below which no energy can be harvested. We also illustrate that the damping and load resistance affect the initiation excitation threshold.

Nonlinear vibration of functionally graded graphene platelet-reinforced composite truncated conical shell using first-order shear deformation theory

In this study,the first-order shear deformation theory(FSDT)is used to establish a nonlinear dynamic model for a conical shell truncated by a functionally graded graphene platelet-reinforced composite(FG-GPLRC).The vibration analyses of the FG-GPLRC truncated conical shell are presented.Considering the graphene platelets(GPLs)of the FG-GPLRC truncated conical shell with three different distribution patterns,the modified Halpin-Tsai model is used to calculate the effective Young’s modulus.Hamilton’s principle,the FSDT,and the von-Karman type nonlinear geometric relationships are used to derive a system of partial differential governing equations of the FG-GPLRC truncated conical shell.The Galerkin method is used to obtain the ordinary differential equations of the truncated conical shell.Then,the analytical nonlinear frequencies of the FG-GPLRC truncated conical shell are solved by the harmonic balance method.The effects of the weight fraction and distribution pattern of the GPLs,the ratio of the length to the radius as well as the ratio of the radius to the thickness of the FG-GPLRC truncated conical shell on the nonlinear natural frequency characteristics are discussed.This study culminates in the discovery of the periodic motion and chaotic motion of the FG-GPLRC truncated conical shell.

NES cell

A broadband adaptive vibration control strategy with high reliability and flexible versatility is proposed.The high vibration damping performance of nonlinear energy sink(NES)has attracted attention.However,targeted energy transfer may cause severe vibration of NES.Besides,it is difficult to realize pure nonlinear stiffness without the linear part.As a result,the reliability of NES is not high.The low reliability of NES has hindered its application in engineering.In addition,the performance of NES depends on its mass ratio of the primary system,and NES lacks versatility for different vibration systems.Therefore,this paper proposes the concept of NES cell.The advantages of the adaptive vibration control of NES are applied to cellular NES.By applying a large number of NES cells in parallel,the reliability of NES and its versatility to complex vibration structures are improved.An elastic beam is used as the primary vibration structure,and a limited NES is used as the cell.The relationship between the vibration suppression effect of NES cells and the number of NES cell is studied.In addition,the effect of the distribution of NES cells on the multi-mode resonance suppression of the beam is also studied.In summary,the mode of the primary structure can be efficiently controlled by a large number of lightweight NES cell.Therefore,the lightweight NES cell is flexible for vibration control of complex structures.In addition,it improves the reliability of NES applications.Therefore,the distributed application of NES cells proposed in this paper is a valuable vibration suppression strategy.

Effects of Ga,Al,Ag,and Ce multi-additions on the wetting characteristics of Sn-9Zn lead-free solder

An orthogonal method was used to evaluate the effects of Ga, Al, Ag, and Ce multi-additions on the wetting characteristics of Sn-9Zn lead-free solders by wetting balance method. The results show that the optimal loading of Ga, Al, Ag, and Ce was 0.2 wt.%, 0.002 wt.%, 0.25 wt.%, and 0.15 wt.%, respectively. Intermetallic compounds （IMCs） formed at the interface between Sn-9Zn-0.2Ga-0.002Al-0.25Ag- 0.15Ce solder and Cu substrate were investigated by scanning electron microscope （SEM） and energy dispersive spectroscopy （EDS） analysis. The SEM images illustrate that the IMCs can be divided into two portions from the substrate side to the solder side： a planar CusZns layer and an additional continuous scallop-like AgZn3 layer. The EDS analysis also shows that Ga segregates in the solder abutting upon the interface. X-ray photoelectron spectroscopy （XPS） and Auger electron spectroscopy （AES） of the surface components of Sn-9Zn-0.2Ga-0.002Al- 0.25Ag-0.15Ce solder indicate that Al aggregates at the surface in the form of Al203 protective fdm, which prevents the further oxidation of the solder surface. On the other hand, Ce aggregates at the subsurface, which may reduce the surface tension of the solder and improve the wettability in consequence.