Jacobi Matrix Research Articles

Type synthesis of a new parallel mechanism with complete separation of constraints and drives

Abstract Currently, there is an increasing demand for parallel mechanisms with fewer inputs and more outputs in aerospace, antennas, and recreational facilities. Therefore, this paper proposes a new parallel mechanism (PM) with a separation of constraints and drives, facilitating centralized drive management. Based on screw theory, the type synthesis of new parallel mechanisms with 3T (T representing translation), 3R (R representing revolute), 2T1R, and 2R1T is proposed. Three-degree-of-freedom (3-DOF) parallel mechanisms with fewer inputs and multiple outputs are proposed. Taking the 3RPS-PPPS parallel mechanism as an example, the closed-loop vector method is employed to solve for the lengths of actuated branches. A complete Jacobi matrix is constructed. The motion/force transmission performance of the 3RRS/PPPS parallel mechanism is better than the 3RRS parallel mechanism. Centralized management of drive branches greatly reduces the number of drives, making maintenance and repair easier. It has many application scenarios in antennas, recreational facilities, and other occasions.

Research on the Control Method of a 2DOF Parallel Platform Based on Electromagnetic Drive

In this paper, a spatial two-degree-of-freedom (2DOF) parallel platform based on electromagnetic redundant drive and its control method are investigated. The platform is redundantly driven by three electromagnetic-spring conforming branched chains, and the design provides better flexibility and responsiveness than conventional parallel structures. The introduction of the electromagnetic drive alleviates the stresses within the conventional rigid redundant drive structure and reduces the detrimental effects associated with rigid redundancy. In this paper, the structure and equivalent SPU model of the platform are first introduced, with S referring to the kinematic sub, P to the spherical sub, and U to the universal joint. The degrees of freedom of the platform are analyzed, and the inverse kinematic model and velocity Jacobi matrix are derived, so as to derive the relationship between the pitch, roll angles, and length of the gimbal chain, and the relational equation between the angle and the current is further established to realize the electromagnetic control of the parallel redundant platform. The control part is realized as follows. Firstly, the angle information of the platform is obtained from the gyroscope to the microcontroller, the filtered angle is derived through the Untraceable Kalman Filter (UKF), and the angle value can be fused with data by both the mathematical model and PID algorithm to introduce the current value required to achieve the balance and realize the balance. In the simulation part, this paper uses Simulink and Simscape in MATLAB for joint simulation, and by giving the simulated trajectory and the desired trajectory of the joints, the driving force diagrams of the three branched chains based on the Least-Second Paradigm method are derived, and the trajectory error and driving force error are given to validate the reliability of the method of this paper.

Engineering static non-reciprocity in mechanical metamaterials

Non-reciprocity has recently been achieved in static fields by mechanical metamaterial that combines large non-linearity and asymmetric geometry. Novel devices are then designed to obtain one-way mechanical functionalities. Non-linearity is the fundamental part that regulates static non-reciprocity, but the influential path remains difficult to quantify. In this work, we employ the geometrically exact beam model to obtain the nonlinear deformation profile of mechanical metamaterials, and build nonlinear equations that govern the deformation behavior. A new two-step method is proposed to solve the nonlinear governing equations, which avoids Jacobi matrix singularity. We establish an accurate model that predicts the non-reciprocal mechanical behavior and verify its accuracy by finite element method. To gain further insight into non-linearity’s influences on mechanical non-reciprocity, we investigate mechanical metamaterials with curved beams, explore the effects of initial curvature on non-reciprocity, and engineer their non-reciprocity to obtain different non-reciprocal force–displacement responses. Our work offers beneficial vehicles to realize static direction-selective functionalities, contributing to shock absorption, vibration damping, and mechanical energy manipulation.

Bifurcation boundaries analysis of the thin-walled internal resonance energy harvester

The dynamic instability of the thin-walled 1: 2 internal resonant piezoelectric vibration energy harvester is investigated. Previously work (Nie et al., 2019) derived the nonlinear electromechanical-coupled governing equations and validated them experimentally. The modulation equations are performed by the method of multiple scales. The system stability is obtained from the eigenvalues of the Jacobi matrix. Then, three types of bifurcation boundaries of the system are determined analytically and simulated numerically via Routh–Hurwitz criterion. Results show that internal resonance complicates the dynamic properties of the bifurcation region. For a large external excitation, new saddle node bifurcation regions appear in the stability diagram. Moreover, the new saddle node and Hopf bifurcation regions may intertwine. Furthermore, as the acceleration excitation increases, the system can be observed to transition to stable–unstable–stable states. The system’s high and low branch responses are fed back by the system’s large and small initial conditions. The system’s periodic and quasi-periodic motions are characterized by time response, FFT, phase trajectory, and Poincaré map. The study of the stability boundary of the piezoelectric energy harvester contributes to the harvesting of energy from ambient vibrations over a broader frequency range.

Polynomial and Rational Measure Modifications of Orthogonal Polynomials via Infinite-Dimensional Banded Matrix Factorizations

AbstractWe describe fast algorithms for approximating the connection coefficients between a family of orthogonal polynomials and another family with a polynomially or rationally modified measure. The connection coefficients are computed via infinite-dimensional banded matrix factorizations and may be used to compute the modified Jacobi matrices all in linear complexity with respect to the truncation degree. A family of orthogonal polynomials with modified classical weights is constructed that support banded differentiation matrices, enabling sparse spectral methods with modified classical orthogonal polynomials. We present several applications and numerical experiments using an open source implementation which make direct use of these results.

An Inverse Matrix Eigenvalue Problem for Constructing a Vibrating Rod

Abstract The free longitudinal vibrations of a rod are described by a differential equation of the form ( P ( x ) y ′ ) ′ + λ P ( x ) y ( x ) = 0 {(P(x)y\prime)^{\prime}+\lambda P(x)y(x)=0} , where P ⁢ ( x ) {P(x)} is the cross section area at point x and λ is an eigenvalue parameter. In this paper, first we discretize this differential equation by using the finite difference method to obtain a matrix eigenvalue problem of the form 𝐀 ⁢ Y = Λ ⁢ 𝐁 ⁢ Y {\mathbf{A}Y=\Lambda\mathbf{B}Y} , where 𝐀 {\mathbf{A}} and 𝐁 {\mathbf{B}} are Jacobi and diagonal matrices dependent to cross section P ⁢ ( x ) {P(x)} , respectively. Then we estimate the eigenvalues of the rod equation by correcting the eigenvalues of the resulting matrix eigenvalue problem. We give a method based on a correction idea to construct the cross section P ⁢ ( x ) {P(x)} by solving an inverse matrix eigenvalue problem. We give some numerical examples to illustrate the efficiency of the proposed method. The results show that the convergence order of the method is O ⁢ ( h 2 ) {O(h^{2})} .

Action of derivations on polynomials and on Jacobian derivations

Let $\mathbb K$ be a field of characteristic zero, $A := \mathbb K[x_{1}, x_{2}]$ the polynomial ring and $W_2(\mathbb K)$ the Lie algebra of all $\mathbb K$-derivations on $A$. Every polynomial $f \in A$ defines a Jacobian derivation $D_f\in W_2(\mathbb K)$ by the rule $D_f(h)=\det J(f, h)$ for any $h\in A$, where $J(f, h)$ is the Jacobi matrix for $f, h$. The Lie algebra $W_2(\mathbb K)$ acts naturally on $A$ and on itself (by multiplication). We study relations between such actions from the viewpoint of Darboux polynomials of derivations from $W_2(\mathbb K)$. It is proved that for a Jordan chain $T(f_1)=\lambda f_1+f_2$, ..., $T(f_{k-1})=\lambda f_{k-1}+f_k$, $T(f_k)=\lambda f_k$ for a derivation $T\in W_2(\mathbb K)$ on $A$ there exists an analogous chain $[T,D_{f_1}]=(\lambda -\mathop{\mathrm{div}} T)D_{f_1} + D_{f_2}$, ..., $[T,D_{f_{k}}]=(\lambda -\mathop{\mathrm{div}} T)D_{f_{k}}$ in $W_2(\mathbb K)$. In case $A:=\mathbb K[x_1, \ldots , x_n]$, the action of normalizers of elements $f$ from $A$ in $W_n(\mathbb K)$ on the principal ideals $(f)$ is considered.

Study on the interior equilibrium point of a special class of 2 × 2 × 2 asymmetric evolutionary games.

Many behavioural interactions in real life involve three individuals. When each individual has two alternative strategies, they can be abstracted into mathematical models by means of asymmetric games. In this paper, we explore a special class of asymmetric games satisfying fixed conditions. Firstly, we analyse two solitary interior equilibrium points and provide the judgement condition for their instability based on the Jacobi matrix local stability analysis method. Secondly, we analyse the interior equilibrium points that are continuously distributed within a line and probe into their stability conditions based on generalized Hamiltonian systems theory. Under the circumstances, the stable interior equilibrium point is surrounded by closed orbits in phase space, which presents an observable stable state where two strategies coexist and fluctuate in each of the three game populations. This work enriches the study of asymmetric games' evolutionary dynamics.

Brauer and Geršgorin Locations of Zeros of Perturbed Chebyshev Polynomials of the Second Kind by Dilation

We consider some perturbations of Chebyshev polynomials of the second kind obtained by modifying by dilation one of its recurrence coefficients at an arbitrary order. By applying Brauer and Geršgorin theorems to Jacobi matrices associated with such perturbed sequences we obtain some locations of their zeros.

A useful formula for periodic Jacobi matrices on trees

We introduce a function of the density of states for periodic Jacobi matrices on trees and prove a useful formula for it in terms of entries of the resolvent of the matrix and its "half-tree" restrictions. This formula is closely related to the one-dimensional Thouless formula and associates a natural phase with points in the bands. This allows streamlined proofs of the gap labeling and Aomoto index theorems. We give a complete proof of gap labeling and sketch the proof of the Aomoto index theorem. We also prove a version of this formula for the Anderson model on trees.

Darboux transformation of symmetric Jacobi matrices and Toda lattices

Let J be a symmetric Jacobi matrix associated with some Toda lattice. We find conditions for Jacobi matrix J to admit factorization J = LU (or J = 𝔘𝔏) with L (or 𝔏) and U (or 𝔘) being lower and upper triangular two-diagonal matrices, respectively. In this case, the Darboux transformation of J is the symmetric Jacobi matrix J(p) = UL (or J(d) = 𝔏𝔘), which is associated with another Toda lattice. In addition, we found explicit transformation formulas for orthogonal polynomials, m-functions and Toda lattices associated with the Jacobi matrices and their Darboux transformations.

Development of a fully implicit ODE-solver for containment analysis code

The thermal–hydraulic dynamics in containment are governed by a system of stiff ordinary differential equations (ODEs). A fully implicit discretization scheme is adopted to discretize these ODEs in order to mitigate the effects of stiffness. In comparison with explicit or semi-implicit discretization schemes that are subject to Courant limits on time steps, the fully implicit discretization scheme is more suitable for a containment analysis code that focuses on predicting both short-term and long-term thermal–hydraulic parameters after an accident. This study introduces a general-purpose ODE solver for the containment analysis code. The outline of the solver is as follows: The fully implicit discrete equations lead to a large set of nonlinear equations that need to be solved using Newton’s iterative method. The partial derivative components in the Jacobi matrix are calculated by the perturbation method using finite difference approximation, which avoids the complicated derivation of partial derivatives. The scaling modification technique is incorporated into this ODE solver to deal with significant differences in unknown variable magnitudes, and the line search method is introduced to address the difficulty of obtaining an accurate root estimate with Newton’s method when the initial guess is far from the actual root. This proposed ODE solver was applied to two typical stiff ODE problems to test its stiffness-suppressed ability and to demonstrate that this proposed solver can perform calculations with a very large time step. Then, the CASSIA code, a containment analysis code developed by China Nuclear Power Technology Research Institute Co., Ltd (CNPRI), equipped with this ODE solver, was applied to the CSNI (Committee on the Safety of Nuclear Installations) benchmark problem and the Carolinas Virginia Tube Reactor (CVTR) test 3 problem to preliminarily demonstrate that the proposed ODE solver can perform containment thermal–hydraulic analysis correctly. This study could provide references for the development of a home-made containment analysis code.

On the number of closed gaps of discrete periodic one-dimensional operators

From the general inverse theory of periodic Jacobi matrices, it is known that a periodic Jacobi matrix of minimal period p≥2 may have at most p−2 closed spectral gaps. We discuss the maximal number of closed gaps for one-dimensional periodic discrete Schrödinger operators of period p. We prove nontrivial upper and lower bounds on this quantity for large p and compute it exactly for p≤6. Among our results, we show that a discrete Schrödinger operator of period four or five may have at most a single closed gap, and we characterize exactly which potentials may exhibit a closed gap. For period six, we show that at most two gaps may close. In all cases in which the maximal number of closed gaps is computed, it is seen to be strictly smaller than p−2, the bound guaranteed by the inverse theory. We also discuss similar results for purely off-diagonal Jacobi matrices.

Approximate analytical solutions of the 1:3 internal resonance response of piezoelectric energy harvester considering two types of coupled frequency components

The 1:3 internal resonance effect in the L-shaped piezoelectric vibration energy harvester leads to an expanded frequency band for energy harvesting and better matching of external excitation frequency in the environment. The electromechanical-coupled governing equations for this vibration energy harvester have been validated in our previous work (Nie et al., 2019). The effect of load resistance on the energy harvester’s natural frequency and damping ratio is addressed. The quadratic and cubic nonlinearities in this system leads to two types of coupled frequency components appear, they ultimately affect the approximate solution of the voltage response. The method of multiple scales is used to derive the two types of coupled frequency components of the system along with their contribution to the output responses. The revised approximate analytical solutions of the output responses are then presented. The system’s stability is assessed through the Jacobi matrix of modulation equations. The nonlinear softening phenomenon that favors the broadening of the bandwidth of the energy harvester is analyzed. The vibration patterns in regions with nonlinear softening and non-softening behavior are investigated through frequency spectra, phase trajectories, and Poincaré cross-sections. The findings reveal that the load resistance significantly affects the damping ratio of the harvester. Inclusion of quadratic and cubic nonlinearities results in three frequency components in the output responses spectrum, with an approximate ratio of 1:2:3. Two types of coupled frequency components appeared in the spectrum of the system’s output responses. The contribution of coupled frequency components to the output response proves crucial, and the revised approximate solutions are in good agreement with that of numerical solutions. The output responses are significantly affected by the initial conditions in the nonlinear softening region, the large and small initial conditions yield solutions in the upper and lower branches, respectively.

Enhancing Green Port Construction through Advanced Computing: An Evolutionary Game Model under Social Accountability and Government Supervision

Ports, as pivotal hubs for global resource consumption and pollution emissions, have garnered considerable attention in the pursuit of sustainability. Despite the Chinese government’s efforts in promoting the green construction of ports and a growing public awareness of environmental protection, progress remains sluggish, with pollution and energy consumption issues still rampant. This paper introduces an evolutionary game model that leverages advanced computing techniques, including big data analytics and simulation, to optimize green port construction strategies, supervision, and enforcement under the lens of social accountability. By applying the Jacobi matrix for analyzing equilibrium paths at diverse initial settings and conducting simulations under various scenarios of punishment intensity and social accountability probabilities, this study unveils that high social accountability likelihoods naturally incentivize ports towards green practices. Conversely, in low accountability scenarios, differentiated outcomes emerge from imposing penalties on ports and local governments. Notably, increasing penalties on ports substantially elevates green construction standards, whereas local government penalties exhibit only transient efficacy. Furthermore, leveraging artificial intelligence for predictive analysis and simulation demonstrates the critical role of technological advancements in enhancing green port construction. The study proposes augmenting penalties on ports, amplifying social oversight, and diminishing green construction costs via port consortia as strategic measures to expedite China’s transition to green ports.

Estimates of Green matrix entries of selfadjoint unbounded block Jacobi matrices

In a wide class of block Jacobi matrices, the norms of Green matrix (resolvent) entries are estimated. This estimate depends on the rate of growth of the norms of the off-diagonal entries and on the distance from the spectral parameter to the essential spectrum if the latter is nonempty. The sharpness of this estimate is shown by an example.

Optimization of stiffness performance of six-axis industrial robots based on posture stiffness evaluation index

Aiming at the problem that the stiffness performance of industrial robots depends on the posture, a method to improve the stiffness performance by optimizing the robot posture is proposed based on the robot posture stiffness evaluation index. The robot joint stiffness identification model is established by using the principle of virtual work and Jacobi matrix, and the stiffness values of six joints are obtained by least-squares. On the basis of revealing the mapping relationship between robot joints and end, the robot end flexibility ellipsoid is obtained, and the volume of the end flexibility ellipsoid is utilized as the global stiffness coefficient of the robot, so as to obtain the robot posture stiffness evaluation index. According to the posture stiffness evaluation index, the robot posture optimization model is established, and the optimized posture is obtained by using genetic algorithm. Taking the QJR6-1 robot as the experimental object, under the maximum load, the deformation of the robot end center is reduced from 0.6238 to 0.3984 mm, which is 36.13%, and the deformation of the end is greatly reduced compared with that before the optimization, which indicates that the robot’s stiffness is improved after the posture optimization, and it verifies the feasibility of the optimization model, and provides a reference to improve the stiffness performance of industrial robots further.

Evolution of stakeholder behavior strategies in the water quantity eco-compensation mechanism of water use coordination inside and outside the river

Eco-compensation is an important means to implement the rights and responsibilities of ecological protection and alleviate the water conflicts inside and outside the river. Previous studies focused on water conflicts among users, overlooking the government's regulatory role in the eco-compensation mechanism and the coordinating function of water transfer compensation fees among users. This study constructed a tripartite evolutionary game model involving government regulatory agencies (GRA), in-channel ecological environment water demanders (EEWD), and off-channel socio-economic water users (SEWU). The evolutionary stability strategies (ESS) of the GRA, EEWD, and SEWU and the evolutionary trend of the tripartite system were analyzed using the finite rationality evolutionary game model, and the sensitivity analysis of the influencing factors was conducted using the eigenvalues of the Jacobi matrix and numerical simulation. Taking the Bai River Basin in Nanyang, China, as the case study, the results showed that the GRA's rewards and penalties were controlled within a reasonable range can promote the enthusiasm of the participating subjects in the water quantity compensation mechanism; water quantity compensation standard for the SEWU was an important indicator for the smooth implementation of the compensation mechanism, the loss coefficient and the overall ecological benefit allocation coefficient of the EEWD will influence the formulation of water quantity compensation standards. This suggested that the GRA should enhance the management of the eco-compensation-related policies and laws and regulations, limit the SEWU's eco-compensation standard, and improve the net benefit of EEWD, to achieve the optimal eco-compensation strategy of ''supervision, protection, compensation''. The research results provide a reference value for the formulation of the water quantity eco-compensation mechanism for water use coordination inside and outside the river.

Reactionless control of free‐floating space manipulators with parameter uncertainty and input disturbance

AbstractThis paper studies the control problem of free‐floating space manipulators, and a control scheme is proposed to solve reactionless control of the end‐effector pose tracking with parameter uncertainty and input disturbance. First, based on dual modeling to treat the end‐effector as a virtual base spacecraft, the dynamics with uncertainties are established which map the joints' torque to the end‐effector pose and base spacecraft attitude, while the inverse kinematics and the derivative of the generalized Jacobi matrix can be avoided in controller design. Then, the reference acceleration stabilization schemes satisfying prescribed performance constraints are carefully designed for tracking errors, and based on these schemes the steady‐state and transient performance of the tracking control can be guaranteed. Further, the radial basis function neural network is adopted to estimate modeling errors caused by parameter uncertainty and input disturbance. In addition, a concurrent learning method is introduced in the network weights matrix update law, which allows the estimation errors to converge a neighborhood of zeros without the need for satisfying the persistent excitation condition. The simulation results verify the effectiveness of the proposed control scheme.

A generalized Hermite–Biehler theorem and non-Hermitian perturbations of Jacobi matrices

The classical Hermite–Biehler theorem describes the zero configuration of a complex linear combination of two real polynomials whose zeros are real, simple, and strictly interlace. We provide the full characterization of the zero configuration for the case when this interlacing is broken at exactly one location. We apply this result to solve the direct and inverse spectral problem for non-Hermitian rank-one multiplicative perturbations and rank-two additive perturbations of finite Hermitian and Jacobi matrices.