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In this paper, a modified incremental harmonic balance (IHB) method combined with Tikhonov regularization has been proposed to achieve the semi-analytical solution for the periodic nonlinear system. To the best of our knowledge, the convergence of the traditional IHB method is bound up with the iterative initial values of harmonic coefficients, especially near the bifurcation point. To this end, the Tikhonov regularization is introduced into the linear incremental equation to tackle the ill-posed situation in the iteration. To this end, the convergence performance of the traditional IHB method has been improved significantly. Moreover, convergence proof of the proposed method also has been given in this paper. Finally, a van der Pol–Duffing oscillator with external excitation and a cubic nonlinear airfoil system with the external store are adopted as numerical examples to illustrate the efficiency and the performance of the presented modified IHB method. The numerical examples show that the results achieved by the proposed method are in excellent agreement with the Runge–Kutta method, and the accuracy is not significantly reduced compared with the traditional IHB method. Especially, the modified IHB method also can converge to the exact solution from the initial values that the traditional IHB method cannot converge in both examples.

Nonlinearity is ubiquitously encountered in structural systems, and it may have a great and complicated influence on the dynamic behaviours, including bifurcation, internal resonance, load history dependence, etc. Identifying the nonlinear system parameters is essential for analysis and design of the structure. To this end, a new approach is developed in this paper for nonlinear system parameter identification from frequency response sensitivity analysis. At first, the harmonic balance equation is established to govern the frequency response of the nonlinear system, upon which the frequency response and sensitivity analysis can be conducted. A remarkable feature is that the harmonic balance equation is algebraic so that the sensitivity analysis, pertaining to a linearized equation, is rather simple and straightforward. Then, parameter identification is modelled as a nonlinear least-squares problem, and the sensitivity approach is adopted in conjunction with the trust-region constraint for convergent solution. Numerical examples are conducted to demonstrate the feasibility and performance of the proposed approach.

The present paper develops an enhanced response sensitivity approach for parameter identification of nonlinear time-delay systems from the time-domain data. Accurate and rapid identification of the time-delay system’s parameters is the key to achieve effective control and synchronization, especially for the delay parameters. The work of this paper is mainly threefold. Firstly, taking a general nonlinear time-delay system as the investigated object, the characteristics of the nonlinear time-delay system are introduced, and the response sensitivity analysis with respect to the system parameters is derived. Then, the problem of parameter identification is modeled as a nonlinear least-squares optimization function. To this end, a novel iteration-based approach is developed to solve such problem, and the ill-posed situation is tackled by Tikhonov regularization. Besides, the trust-region constraint is implemented to enhance the convergence of the algorithm. Finally, the feasibility of the proposed approach is verified by two numerical examples and a real electronic circuit experiment. The results proved that the proposed approach can identify parameters of nonlinear time-delay systems accurately, robustly and effectively.

In this paper, a new semi-analytical method, namely the time-domain minimum residual method, is proposed for the nonlinear problems. Unlike the existing approximate analytical method, this method does not depend on the small parameter and can converge to the exact analytical solutions quickly. The method is mainly threefold. Firstly, the approximate analytical solution of the nonlinear system $${\varvec{F\left( \ddot{x},{\dot{x}},x\right) }}={\varvec{0}}$$ is expanded as the appropriate basis function and a set of unknown parameters, i.e., $${\varvec{x(t)}}\approx \sum _{i=0}^{N}{\varvec{a_i\chi _i(t)}}$$ . Then, the problem of solving analytical solutions is transformed into finding a set of parameters so that the residual $${\varvec{R}}={\varvec{F}}\left( \sum _{i=0}^{N}a_i\ddot{\chi }_i,\sum _{i=0}^{N}a_i{\dot{\chi }}_i,\sum _{i=0}^{N}a_i\chi _i\right) $$ is minimum over a period, i.e., $$\underset{{\varvec{a}}\in {\mathscr {A}}}{\min }\int _{0}^T {\varvec{R}}({\varvec{a}},t)^{T} {\varvec{R}}({\varvec{a}},t) \mathrm {d} t$$ . The nonlinear equation $${\varvec{F\left( \ddot{x},{\dot{x}},x\right) }}={\varvec{0}}$$ is regarded as the objective function to optimize, and the process of solving the analytic solution is transformed into a nonlinear optimization process. Finally, the optimization process is iteratively solved by the enhanced response sensitivity approach. Four numerical examples are employed to verify the feasibility and effectiveness of the proposed method.

The model errors are often inevitably encountered when a practical structure is analyzed by a mathematical baseline model, and such model errors would have undesired effects on damage identification. Aiming to alleviate the effect of the model errors, a new Measurement-Changes-Correction strategy that simply uses measurement changes to correct the measured data is developed in this article. To this end, the performance of the Measurement-Changes-Correction strategy under the aleatoric and epistemic model errors is systematically analyzed. The analysis is proceeded along with comparison to the conventional two-step strategy where the intact damage parameters are first updated from the measured data and then, damage identification is conducted upon the updated parameters. Thereafter, it is found that (1) the Measurement-Changes-Correction strategy performs as well as the two-step strategy under small aleatoric and epistemic model errors, (2) the Measurement-Changes-Correction strategy can still work under large scaling model parameter errors, but the two-step strategy may not, and (3) the Measurement-Changes-Correction strategy only requires correction of the measured data by the measurement changes so that the computation cost by the Measurement-Changes-Correction strategy is almost half of that by the two-step strategy. Numerical examples and International Association for Structural Control–American Society of Civil Engineers benchmark problems are studied to verify the performance of the proposed Measurement-Changes-Correction strategy.

The technique to identify the system parameters thereof has attracted extensive research interest, since knowing the parameters would enable effective system control strategy and accurate response prediction. In this paper, a novel approach is developed to identify the parameters of the linear time-delay differential system by analyzing the complex system response in the frequency domain. Firstly, the complex frequency response of the time-delay system is expressed as a function of physical parameters and time-delay parameters, forming a typical optimization problem. Subsequently, the sensitivities with respect to the unknown parameters are derived. A novel sensitivity-based algorithm is adopted in the identification procedure. Trust-region constraint is implemented and hence tackled by Tikhonov regularization, which effectively enhances the efficiency of the algorithm. The feasibility and robustness of the identification procedure are evaluated by identifying the parameters of two numerical time-delay systems and an experimental case.

Breathing cracks are often encountered in engineering structures and detection of such cracks at the earliest stage is important for safety and serviceability of the structures. In this paper, a new breathing crack identification approach based on the time-domain sensitivity analysis is proposed. For the forward problem, the crack is simply treated as a nonlinear oscillator and how to model a structure with breathing cracks is specified. As regards the inverse crack identification, it is formulated as a nonlinear least-squares optimization problem and a new sensitivity-based approach is developed to get the solution. In doing so, the sensitivity analysis of the possibly non-differentiable breathing crack models is necessarily proceeded through a smoothing strategy. Moreover, to enhance the convergence, the trust-region constraint is introduced and the Tikhonov regularization is naturally called to tackle the constraint. Numerical examples are studied to verify the feasibility and efficiency of the proposed approach in breathing crack identification.

Fujing Xie,1,* Lianhai Zhang,2,* Qing Yao,3 Liyu Shan,3 Jike Liu,2 Nanhai Dong,4 Jun Liang1 1Department of Pediatrics, Liaocheng People’s Hospital, Liaocheng 252000, Shandong Province, People’s Republic of China; 2Department of Pediatric Surgery, Liaocheng People’s Hospital, Liaocheng 252000, Shandong Province, People’s Republic of China; 3Medical College of Xi’an Jiaotong University, Xi’an 710061, Shaanxi Province, People’s Republic of China; 4Department of Clinical Laboratory, Liaocheng People’s Hospital, Liaocheng 252000, Shandong Province, People’s Republic of China*These authors contributed equally to this workCorrespondence: Jun Liang; Nanhai Dong West Dongchang Road 67, Liaocheng 252000, People’s Republic of ChinaTel +86 13346255358; +86 13563016897Email 251217955@qq.com; dnhjyk@163.comIntroduction: HB presents with the highest frequency of CTNNB1 mutations, resulting in activation of Wnt signaling pathway. A number of studies have demonstrated CTNNB1 mutation contributed to the development of HB. However, limited research explored the function of lncRNAs in HB with CTNNB1 mutation.Methods: We screened lncRNA expression profiles in CTNNB1-mutated HB samples and identified lncRNAs associated with malignant phenotype in HB. The association between lncRNA and immune microenvironment was investigated. The biological function of lncRNA was further explored using in vitro experiments.Results: TUG1 was identified as onco-lncRNA in CTNNB1-mutated HB. TUG1 was shown to be associated with the infiltration of pro-tumor immunocytes via regulating the expression of CXCR4, a chemokine receptor playing a critical role in regulation of immune microenvironment. Inhibiting TUG1 could increase endogenous levels of miR-335-5p and consequently downregulating CXCR4, a direct target of miR-335-5p.Conclusion: Our findings provide evidence for TUG1 mediating infiltration of pro-tumor immunocytes in HB patients carrying CTNNB1 mutation. TUG1-miR-335-5p-CXCR4 axis might be a promising immunological target for the treatment of HB patients.Keywords: hepatoblastoma, TUG1, CXCR4, immunocyte

In nonlinear problems, the Hasofer–Lind–Rackwitz–Fiessler algorithm of the first order reliability method sometimes is puzzled by its non-convergence. A new Hasofer–Lind–Rackwitz–Fiessler algorithm incorporating Barzilai–Borwein step is investigated in this paper to speed up the rate of convergence and performs in a stable manner. The algorithm is essentially established on the basis of the global Barzilai–Borwein gradient method, which is dealt with two stages. The first stage, implemented by the traditional steepest descent method with specific decayed step sizes, prepares a good initial point for the global Barzilai–Borwein gradient algorithm in the second stage, which takes the merit function as the objective to locate the most probable failure point. The efficiency and convergence of the proposed method and some other reliability analysis methods are presented and discussed in details by several numerical examples. It is found that the proposed method is stable and very efficient in the nonlinear problems except those super nonlinear ones, even more accurate than the descent direction method with step sizes following the fixed exponential decay strategy.

Identifying the damages from test data is central to assuring the structural safety. The static model is the simplest model to describe the mechanical behavior of the structure where only the stiffness is involved and it is independent of the mass and the complex damping. As a result, damage identification based on the static data will not be deteriorated by the inexact damping and the possible error in the mass. Notwithstanding, the major difficulty regarding damage identification with static test data is that the amount of the static data is quite limited and insufficient with respect to the amount of damage parameters, rendering the identification very sensitive to the measurement noise. Attempting to circumvent this difficulty, a novel damage identification approach is developed in this article where the sparse regularization is introduced to implicitly enforce the sparsity constraint of the damage locations. Moreover, in order to work well with the sparse regularization, a new goal function is established by resorting to the eigenparameter decomposition for which the decoupling feature would make the sparse regularization be tackled immediately with closed-form solutions. Then, the alternating minimization approach is used to get the solution of the new goal function and the threshold setting method is simply called to determine a proper regularization parameter. Numerical and experimental examples are studied to testify the feasibility, accuracy, and robustness of the proposed damage identification approach.

This paper presents an investigation on parameters identification of fractional order system based on the enhanced response sensitivity approach. Compared with the integer order differential operator, fractional order operator can describe the characteristics of some phenomena more accurately such as hereditary and memory properties. In order to achieve the solution of the fractional order system, a new computational scheme is presented in this paper based on Adams-type scheme and the Newmark -β method. Such an identification problem is classed as an inverse problem and can be modeled as an nonlinear optimization problem. In this paper, the enhanced response sensitivity approach is derived and used to identify the parameters of the fractional order system. To this end, the time domain response of the fractional order system is obtained by the proposed computational scheme, then the response sensitivity with respect to the parameter is achieved as well. Numerical examples illustrated the computing and identification procedure and shown the accuracy and the convergence of identification scheme.

This paper presents a new semi-analytical approach, namely the enhanced time-domain minimum residual method to solve the semi-analytical quasi-periodic solution for nonlinear system. This approach does not require multiple numerical integration and can be applied to strongly nonlinear systems. The approach is mainly three-fold. Firstly, the semi-analytical solution of the nonlinear quasi-periodic system is expanded into a set of trigonometric series with unknown coefficients, i.e., x ( t ) ≈ ∑ k = 1 N [ b k cos ( ω k t ) + c k sin ( ω k t ) ] . Then, the problem of solving quasi-periodic solution can be expressed as: determining the coefficients of the trigonometric series so that the residual objective function R = M x ¨ + C x ˙ + Kx + N ( x ¨ , x ˙ , x , t ) − F ( t ) is minimum over a period, i.e., min a ∈ A ∫ 0 T R ( a , t ) T R ( a , t ) d t . Finally, the nonlinear minimum optimization problem is solved iteratively through the enhanced response sensitivity approach. Moreover, the “adaptive zero-setting curve” is introduced to accelerate the convergence. Two numerical examples, a van der Pol-Duffing system and a nonlinear energy sink system are adopted to verify the feasibility of the proposed approach.

Correct modelling of the foundation of rotating machinery is an invaluable asset in model-based rotor dynamic study. One attractive approach for such purpose is to identify the relevant modal parameters of an equivalent foundation using the motion measurements of rotor and foundation at the bearing supports. Previous research showed that, a complex quasi-modal parameter based system identification technique could be feasible for this purpose; however, the technique was only validated by identifying simple structures under harmonic excitation. In this paper, such identification technique is further extended and evaluated by identifying the foundation of a numerical rotor-bearing-foundation system and an experimental rotor rig respectively. In the identification of rotor foundation with multiple bearing supports, all application points of excitation forces transmitted through bearings need to be included; however the assumed vibration modes far outside the rotor operating speed cannot or not necessary to be identified. The extended identification technique allows one to identify correctly an equivalent foundation with fewer modes than the assumed number of degrees of freedom, essentially by generalising the technique to be able to handle rectangular complex modal matrices. The extended technique is robust in numerical and experimental validation and is therefore likely to be applicable in the field.

This paper presents a systematic and rigorous analysis on the convergence rates of the Harmonic balance Method (HB) for general smooth and non-smooth systems. In doing so, the convergence rates of Fourier truncation are established at first for functions with different smoothness, and then, the errors of HB are estimated with the help of a coercive condition and the established results on Fourier truncation errors. As is found in this work, when the restoring forces are discontinuous or of some special low smoothness, the convergence rates of HB become different from those of Fourier truncation. Numerical examples are studied and the results well verify the present theoretic convergence rates.

Hysteresis is a ubiquitous phenomenon describing the special nonlinear memory-based relation between the input and the output in many physical systems. Identifying the hysteretic parameters is the first step towards practical application of hysteretic models. In this paper, a general framework for parameter identification of nonlinear hysteretic models is developed based on the enhanced response sensitivity approach. To do so, three typical hysteretic models—Bouc–Wen model, bilinear model with kinematic hardening and bilinear model with equal yielding force are analyzed at first and the general way to model a structure with such hysteretic components is established thereafter. Then, the enhanced response sensitivity approach is presented for inverse parameter identification where the key lies in the sensitivity analysis and the trust-region constraint. Particularly, smoothing procedure is introduced to overcome the non-differentiability of bilinear hysteretic functions for sensitivity analysis of bilinear models. Numerical examples are studied to testify the feasibility and performance of the proposed approach.

This paper presents a study on amplitude death in nonlinear coupled oscillators containing fractional derivatives. Analytical criterion for amplitude death region is obtained by eigenvalue analysis and verified by numerical results. It is found that amplitude death regions can be enlarged to a large extent by fractional derivatives. For this reason, amplitude death can be detected in fractional Stuart–Landau systems with weak coupling strength and low frequency, whereas it never appears in integer-order systems. Interestingly, the widening of amplitude death region induced by fractional derivative is shared by a variety of oscillators with different types of coupling mechanisms. An interpretation for the underlying mechanism of this phenomenon is briefly addressed, based on which we further suggest a coupling organization leading to amplitude death only in fractional oscillators.

This paper develops an incremental response sensitivity approach for parameter identification of chaotic and hyperchaotic systems. In doing so, there hinge on two points. The first point is the establishment of the incremental objective function; this origins from the idea that long-term behavior of (hyper-)chaotic systems is often sensitive to the change of system parameters, but the short-term behavior is not. The advantage over the general objective function resides in that the incremental objective function is often well-defined, that is, having only one local minimal in the parametric space. The second is the utilization of the response sensitivity approach to get the minimal or identified parameters of the incremental objective function. In order to guarantee the performance, the trust-region constraint is additionally enforced to enhance the response sensitivity approach and it is practically dealt with by the Tikhonov regularization. Numerical tests are performed to verify the proposed approach.

This paper introduced a modal parameter based identification procedure to identify the equivalent system of structures under harmonic excitations. The developed identification technique assumed non-proportional hysteretic damping in the equivalent system, which would be applicable in identifying more general structures. By introducing quasi-modal parameter, modal analysis equation was decoupled under physical coordinate; hence, the modal parameters of each vibration mode are identified independently. Double iteration algorithm was developed to solve the derived non-linear identification equation with complex unknowns. The developed identification procedure was applied to identify the equivalent system of a numerical model in order to evaluate the feasibility of the technique in practice. The identification procedure was also applied to identify an experimental mass and bar rig for validation purpose. Identification results showed that the identification procedure could identify accurately and robustly the equivalent system with non-proportional hysteretic damping assumption; hence, it is likely to be applicable in the field.

A bandlimited force identification approach is proposed in this paper. It consists of two main ingredients. On the one hand, the sinc-dictionaries are established for bandlimited force representation. This motivates from the sampling formula that a bandlimited signal can be well interpolated by sinc functions. It is also noteworthy that using the sinc-dictionaries, the amount of variables is dramatically reduced. On the other hand, the Tikhonov regularization is introduced to circumvent the possible ill-posedness of force identification. In proceeding so, the generalized cross validation criterion is adopted to determine a proper regularization parameter. Numerical examples are conducted to testify the accuracy and efficiency of the proposed approach for bandlimited force identification.

This paper proposed a numerical algorithm based on precise integration method to investigate the aeroelastic system of an airfoil with a freeplay. The system was split into three linear sub-systems separated by switching points related with the freeplay. A predictor–corrector algorithm was constructed to tackle the key computational obstacle in accurately searching system responses passing the switching points. With the aid of the algorithm, the precise integration method can solve the sub-systems one by one and provide solutions to any desired accuracy compared with exact solutions. Moreover, it can keep high precision with the step length increasing. The precise integration method is more accurate and efficient than the Runge–Kutta method with the same time step. In addition, the Runge–Kutta sometimes provides limit cycle oscillations, bifurcation charts or chaotic responses falsely even though the step length is much smaller than that adopted in precise integration method. Due to the high precision and efficiency, the presented approach has potential to become a benchmark for solution techniques for piecewise nonlinear dynamical systems.

A two-stage approach based on residual force vector and response sensitivity analysis is proposed for structural damage identification in this study. Local damage is represented by a perturbation in the elemental stiffness parameter of the structural finite element model. Primarily, the difference between the virtual residual force vector of an intact and damaged structure is used to localize the damage. Then, a response sensitivity-based method is adopted to identify the perturbation of the stiffness parameter (i.e., the extent of the local damage) from the measured dynamic responses. Numerical simulations are conducted to identify both single and multiple structural damages under sinusoidal or impulsive excitation. The proposed approach gives satisfactory results with artificial measurement noise.

The equivalent linearization method (ELM) is modified to investigate the nonlinear flutter system of an airfoil with a cubic damping. After obtaining the linearization quantity of the cubic nonlinearity by the ELM, an equivalent system can be deduced and then investigated by linear flutter analysis methods. Different from the routine procedures of the ELM, the frequency rather than the amplitude of limit cycle oscillation (LCO) is chosen as an active increment to produce bifurcation charts. Numerical examples show that this modification makes the ELM much more efficient. Meanwhile, the LCOs obtained by the ELM are in good agreement with numerical solutions. The nonlinear damping can delay the occurrence of secondary bifurcation. On the other hand, it has marginal influence on bifurcation characteristics or LCOs.

Which factor determines animal aggressivity? Wynn-Edwards proposed the hypothesis that aggressive level increases with population density; Adams and Mesterton-Gibbons proposed the hypothesis that body weight is an indicator of animal aggressivity; however, Smith and Price hypothesized that aggression level varied with external conditions; that is, the population lived in the most unfavorable environment demonstrated the highest average aggression level, and the population that lived under the most favorable external conditions demonstrated the lowest average aggression level. In this paper, we tested these three hypotheses by manipulating enclosed root vole (Microtus oeconomus) populations under different food and predation treatments and observed their aggressive behavior. Aggressive behavior was measured by matching mice in a neutral arena. The experimental results supported Smith and Price’s hypothesis and Adams and Mesterton-Gibbons’s hypothesis; however, they did not support Wynn-Edwards’ hypothesis. Moreover, we found that reproductively active individuals were more aggressive. We concluded that the growth of population density did not cause or otherwise bring about increased aggressive behavior of root voles, but the external factors (predation and food supply) and physical factors (body weight and reproductive condition) were significantly correlated with aggression levels in a root vole population.

This study carries out numerical simulations to identify the magnitude of prestress force in a highway bridge by making use of the dynamic responses from moving vehicular loads. The prestressed bridges are modeled using four-node isoparametric flat shell element taking into account the transverse shearing deformation in the finite element model. The vehicle is modeled as a multiple degrees-of-freedom system. An approach based on dynamic response sensitivity-based finite element model updating is proposed to identify the elemental prestress force. The identified results are obtained iteratively with the penalty function method with regularization from the measured structural dynamic responses. A single-span prestressed Tee beam and two-span prestressed box-girder bridge are studied as two numerical examples. The effects of road surface roughness, measurement noise, and speed of moving vehicle on the identification results are investigated. Studies indicate that the proposed method is efficient and robust for prestress force identification. Good identified results can be obtained from several measured acceleration responses.

This paper is focused on the uncertainty of simulation results in accident reconstruction. The Upper and Lower Bound Method (ULM) and the Finite Difference Method (FDM), which can be easily applied in this field, are introduced firstly; the Response Surface Methodology (RSM) is then introduced into this field as an alternative methodology. In RSM, a sample set is firstly generated via uniform design; secondly, experiments are conducted according to the sample set with the help of simulation methods; thirdly, a response surface model is determined through regression analysis; finally, the uncertainty of simulation results can be analyzed using a combination of the response surface model and existing uncertainty analysis methods. It is later discussed in detail how to generate a sample set, how to calculate the range of simulation results and how to analyze the parameter sensitivity in RSM. Finally, the feasibility of RSM is validated by five cases. Moreover, the applicability of RSM, ULM and FDM in analyzing the uncertainty of simulation results is studied; the phenomena that ULM and FDM can hardly work while RSM can is found in the latter two cases. After an analysis of these five cases and the number of simulation runs required for each method, both advantages and disadvantages of these uncertainty analysis methods are indicated.

In order to investigate the applicability of accident velocity reconstruction model in different cases, a method used to evaluate accuracy degree of accident velocity reconstruction model is given. Based on pre-crash velocity in theory and calculation, an accuracy degree evaluation formula is obtained. With a numerical simulation case, Accuracy degrees and applicability of two accident velocity reconstruction models are analyzed; results show that this method is feasible in practice.

Through theoretical analysis and introduction of some empirical parameters, the relationship between post-braking-distance and throw distance was studied concentratedly. Here, the post-braking-distance is the distance a vehicle will travel from the impact position to when it comes to a complete stop. Two useful formulas which are meaningful in vehicle-pedestrian accident reconstruction were finally obtained. The first one can be used to calculate impact speed according to throw distance, while the other one can describe the relationship between post-braking-distance and throw distance. Their feasibility has been validated by comparing with other scholars' empirical formulas and simulation results of software Pc-Crash, respectively. The relationship between post-braking-distance and throw distance has very bright application perspective in vehicle-pedestrian accident reconstruction such as helping policemen obtain more useful evidences, validating credibility of the throw distance, judging whether the vehicle is fully braked or not, determining the impact position etc. Finally its application was demonstrated by three cases, in which the impact speed was also calculated. All results until now have shown that conclusions obtained in this article are feasible and helpful in vehicle-pedestrian accident reconstruction.

For the parameter sensitivity estimation with implicit limit state functions in the time-invariant reliability analysis, the common Monte Carlo simulation based approach involves multiple trials for each parameter being varied, which will increase associated computational cost and the cost may become inevitably high especially when many random variables are involved. Another effective approach for this problem is featured as constructing the equivalent limit state function (usually called response surface) and performing the estimation in FORM/SORM. However, as the equivalent limit state function is polynomial in the traditional response surface method, it is not a good approximation especially for some highly non-linear limit state functions. To solve the above two problems, a new method, support vector regression based response surface method, is therefore presented in this paper. The support vector regression algorithm is employed to construct the equivalent limit state function and FORM/SORM is used in the parameter sensitivity estimation, and then two illustrative examples are given. It is shown that the computational cost of the sensitivity estimation can be greatly reduced and the accuracy can be retained, and results of the sensitivity estimation obtained by the proposed method are in satisfactory agreement with those computed by the conventional Monte Carlo methods.

This paper presents a study of the relationship between the homotopy analysis method (HAM) and harmonic balance (HB) method. The HAM is employed to obtain periodic solutions of conservative oscillators and limit cycles of self-excited systems, respectively. Different from the usual procedures in the existing literature, the HAM is modified by retaining a given number of harmonics in higher-order approximations. It is proved that as long as the solution given by the modified HAM is convergent, it converges to one HB solution. The Duffing equation, the van der Pol equation and the flutter equation of a two-dimensional airfoil are taken as illustrations to validate the attained results.

This study proposes an iterative method, in which all iterations are linear, for solving the nonlinear dynamical system of an electrostatically actuated micro-cantilever in MEMS. The presented method is further improved, so that the CPU time needed is significantly reduced. This method provides solutions in excellent agreement with numerical ones.

The homotopy analysis method for limit cycle of the van der Pol equation is computationally extensive. Aiming at reducing the computational efforts, we construct the deformation equations using different initial conditions from those in the existing literature. The computational efforts are significantly reduced. And, the calculated amplitude and frequency of the limit cycle are in excellent agreement with the exact solutions.

We have deduced an iteration scheme in the incremental harmonic balance (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.

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.

The stress concentration problem in structures with a circular or elliptic hole can be investigated by analytical methods. For the problem with a rectangular hole, only approximate results are derived. This paper deduces the analytical solutions to the stress concentration problem in plates with a rectangular hole under biaxial tensions. By using the U-transformation technique and the finite element method, the analytical displacement solutions of the finite element equations are derived in the series form. Therefore, the stress concentration can then be discussed easily and conveniently. For plate problem the bilinear rectangular element with four nodes is taken as an example to demonstrate the applicability of the proposed method. The stress concentration factors for various ratios of height to width of the hole are obtained.