Design of tensegrity structures by minimizing static compliance

Tensegrity structures, characterized by enhanced stiffness, slender struts, and superior buckling resistance, have found wide-ranging applications in fields such as engineering, architecture, art, biology, and robotics, attracting extensive attention from researchers. The form-finding process, a critical step in the design of tensegrity structures, aims to discover the self-equilibrated configuration that satisfies specific design requirements. Traditional form-finding methods based on force density often require repeated steps of eigenvalue decomposition and singular value decomposition, making the process complex. In contrast, this paper introduces a new intelligent form-finding algorithm that uses the force density method and combines the Coati optimization algorithm with Graph Neural Networks. This algorithm avoids the complex steps of eigenvalue and singular value decomposition and integrates the physical knowledge of the structure, making the form-finding process faster and more accurate. In this algorithm, various force densities are initially randomized and input into a trained Graph Neural Networks to predict a fitness function’s value. Through optimizing the constrained fitness function, the algorithm determines the appropriate structural force density and coordinates, thereby completing the form-finding process of the structure. The paper presents seven typical tensegrity structure examples and compares various form-finding methods. The results of numerical examples show that the method proposed in this paper can find solutions that align with the super-stable line more quickly and accurately, demonstrating its potential value in practical applications.

Tensegrity structures are three-dimensional networks of truss members loaded in tension or compression. The location of the end points of the truss members, denoted as the nodes, and the associated node-member connectivity matrices are the fundamental descriptors in the modeling and design of tensegrity structures. This paper presents systematic analytical formulas for such node locations and connectivity matrices for tensegrity plates of two different topologies. The formulas apply to plates of any thickness, diameter, and complexity. As application examples, dynamic simulations demonstrating a strategy for morphing the planar plates toward domes are studied. The presented formulas allow for efficient computations and can be employed in the numerical analysis and design of shape-controllable antennas and mirrors, architectural constructions, and other applications based on tensegrity plate and dome-like structures.

<p>The primary objective of this study is to develop new computational methods for robust design optimization (RDO) and reliability-based design optimization (RBDO) of high-dimensional, complex engineering systems. Four major research directions, all anchored in polynomial dimensional decomposition (PDD), have been defined to meet the objective. They involve: (1) development of new sensitivity analysis methods for RDO and RBDO; (2) development of novel optimization methods for solving RDO problems; (3) development of novel optimization methods for solving RBDO problems; and (4) development of a novel scheme and formulation to solve stochastic design optimization problems with both distributional and structural design parameters.</p> <p>The major achievements are as follows. Firstly, three new computational methods were developed for calculating design sensitivities of statistical moments and reliability of high-dimensional complex systems subject to random inputs. The first method represents a novel integration of PDD of a multivariate stochastic response function and score functions, leading to analytical expressions of design sensitivities of the first two moments. The second and third methods, relevant to probability distribution or reliability analysis, exploit two distinct combinations built on PDD: the PDD-SPA method, entailing the saddlepoint approximation (SPA) and score functions; and the PDD-MCS method, utilizing the embedded Monte Carlo simulation (MCS) of the PDD approximation and score functions. For all three methods developed, both the statistical moments or failure probabilities and their design sensitivities are both determined concurrently from a single stochastic analysis or simulation. Secondly, four new methods were developed for RDO of complex engineering systems. The methods involve PDD of a high-dimensional stochastic response for statistical moment analysis, a novel integration of PDD and score functions for calculating the second-moment sensitivities with respect to the design variables, and standard gradient-based optimization algorithms. The methods, depending on how statistical moment and sensitivity analyses are dovetailed with an optimization algorithm, encompass direct, single-step, sequential, and multi-point single-step design processes. Thirdly, two new methods were developed for RBDO of complex engineering systems. The methods involve an adaptive-sparse polynomial dimensional decomposition (AS-PDD) of a high-dimensional stochastic response for reliability analysis, a novel integration of AS-PDD and score functions for calculating the sensitivities of the failure probability with respect to design variables, and standard gradient-based optimization algorithms, resulting in a multi-point, single-step design process. The two methods, depending on how the failure probability and its design sensitivities are evaluated, exploit two distinct combinations built on AS-PDD: the AS-PDD-SPA method, entailing SPA and score functions; and the AS-PDD-MCS method, utilizing the embedded MCS of the AS-PDD approximation and score functions. In addition, a new method, named as the augmented PDD method, was developed for RDO and RBDO subject to mixed design variables, comprising both distributional and structural design variables. The method comprises a new augmented PDD of a high-dimensional stochastic response for statistical moment and reliability analyses; an integration of the augmented PDD, score functions, and finite-difference approximation for calculating the sensitivities of the first two moments and the failure probability with respect to distributional and structural design variables; and standard gradient-based optimization algorithms, leading to a multi-point, single-step design process.</p> <p>The innovative formulations of statistical moment and reliability analysis, design sensitivity analysis, and optimization algorithms have achieved not only highly accurate but also computationally efficient design solutions. Therefore, these new methods are capable of performing industrial-scale design optimization with numerous design variables.</p>

An advanced automatic grouping method for form-finding of tensegrity structures is presented. In the proposed method, properties of self-equilibrium and stability in tensegrity structures can be obtained by using the force density method combined with a genetic algorithm. A constrained minimization problem is formulated using the standard deviation of the force density in the cables. As a result, the minimum number of member groups for tensegrity structures with automatic grouping can be obtained. This elicited regular tensegrity structures with uniform force density values. Moreover, the geometrical and mechanical parameters of tensegrity structures with multiple states of self-stress can be easily obtained by using the proposed method.

Optimal self-stress determination of tensegrity structures

Design of Tensegrity Structures with Static and Dynamic Modal Requirements

본 논문은 위상최적화와 치수최적화 해석기법을 이용한 샌드위치 복합재와 알루미늄 압출재로 구성되는 하이브리드 철도차량 차체 구조물의 경량 모듈화 설계에 관한 연구이다. 위상최적설계는 초기설계단계에 차체 구조물의 최적의 재료적용부위를 선정하기 위해 사용되었으며, 치수최적화설계는 하이브리드 차체 구조물의 최적 설계 변수를 찾기 위해 사용되었다. 이때, 최적화해석은 일계법과 부분문제근사법을 사용하였다. 위상최적설계해석을 통해 차체 구조물의 강성 및 제작성을 향상시키기 위한 언더프레임, 로우 사이드 패널 모듈에 1차 부재인 알루미늄 압출재의 적용이 적절하고, 샌드위치 복합재는 무게를 최소화하기 위한 2차 부재로서 루프 및 미들 사이드 패널 모듈에 적용이 적합하다는 결론을 얻었다. 또한, 알루미늄 압출재와 샌드위치 복합재로 구성되는 하이브리드 차체 구조물의 무게는 샌드위치 복합재로만 만들어진 차체 구조물과 비교하여 최대 17.7%까지의 경량화 효과를 얻을 수 있음을 확인하였다.

On the stability of equilibria of nonholonomic systems with nonlinear constraints

In order to solve the problem that a traditional articulated arm coordinate measuring machine (AACMM) cannot measure automatically, a self-driven AACMM is proposed. Based on the self-driven AACMM design indicators, the length of the connecting rods of the self-driven AACMM was allocated. The self-driven AACMM virtual prototype was designed and assembled based on joint module selection and joint component design, and its measurement space range was also verified. The key components of the structural optimized design and dynamic performance analysis of motion parameters are adopted, respectively, to correct the positioning error of the self-driven AACMM’s probe caused by static and dynamic deformations. The static deformation of the structure caused by the self-driven AACMM’s weight was analyzed. The influence of the dynamic flexible deformation on the positioning error of the probe of the self-driven AACMM during the variable-speed approach and the constant-speed touch measurement stage was studied. The results show that the measurement space range of the self-driven AACMM designed in this paper can meet the design index of the measuring radius. The probe position error caused by static deformation of the measuring machine after structural optimization is reduced by an order of magnitude, and the equivalent stress of the mechanical structure of the self-driven AACMM is within the allowable stress of the material. The positioning error of the probe caused by the dynamic deformation of the self-driven AACMM structure meets the positioning accuracy index. During the stage of self-driven AACMM constant-speed touch measurement, the instantaneous position error of the probe changes linearly with the touch time and the optimal touch speed (6.6 mm s−1, 6.4 mm s−1) exists to minimize the probe positioning error. During the variable-speed approach stage, the influence of the angular acceleration and the angular velocity of each joint on the positioning error of the probe is negligible when the self-driven AACMM is in the typical posture. In the extreme posture, with the optimal joint angular acceleration ( 0.00055∘s−2 ) and angular velocity ( 0.0185∘s−1 ), the inertial force of the measuring machine structure and the instantaneous position error of the probe are the smallest, and the movement stability is the best. The structural optimization design and motion parameter performance analyses of the self-driven AACMM can provide a theoretical research foundation for subsequent motion parameter optimization and dynamic error compensation modeling.

Multi-product, multi-venders inventory models with different cases of the rational function under linear and non-linear constraints via geometric programming approach

Some achievements have been made in the study on mechanical properties of antler, but they have not been applied in engineering practice, especially in the study of thin-walled tubes similar to antler crashworthiness. In order to improve the crashworthiness and energy absorption of the thin-walled tube structures, a bionic thin-walled tube with the same inner diameter and equal gradient of outer diameter was designed based on the structural characteristics of antler bone and the principle of structural bionics. The finite element method was used to simulate the energy absorption characteristics of 75 kinds of bionic thin-walled tube structures under the oblique impacts with the impact angles of 10°, 20° and 30°. The polynomial regression element model and multi-objective particle swarm optimization algorithm were used to optimize, and the Pareto front optimization principle was used to obtain the optimal allocation scheme of each target. The minimum distance selection method was used in optimization analysis to obtain the optimal structural design parameters of each scheme. The optimization method used in this study can provide reference for the follow-up research on the crashworthiness of thin-walled tubes, and the optimal structure of bionic thin-walled tubes can provide reference for practical engineering application. The results show that when only considering a single impact angle, the optimal number of biomimetic layers n is 6, and the parameter combination of maximum wall thickness and thickness gradient tmax-a is 2.84 mm-0.38 mm, 2.89 mm-0.29 mm, 2.91 mm-0.34 mm, respectively under 10°, 20° and 30° impact angles. Considering various impact angle weight factors and different configuration schemes, the optimal number of biomimetic layers n is 6, and the parameter combination of maximum wall thickness and thickness gradient tmax-a is 2.95 mm-0.28 mm, 2.92 mm-0.30 mm and 2.85 mm-0.33 mm, respectively.

The bellows of a vacuum interrupter (VI) become the most vulnerable components when the VI is introduced to comprise a dead-tank fast vacuum circuit breaker (FVCB). However, there is no work revealing the optimal design of VI bellows subjected to high gas pressure and high operating velocity in achieving a high mechanical endurance. The objective of this paper is to confirm the optimum design parameters of bellow structure, which would be subjected to a high gas pressure and high operating velocity. We adopted a transient fluid-structure coupled simulation method, as well as a fatigue failure simulation method, to obtain the dynamic strain-stress distribution and the fatigue life of the bellows. The U-shaped VI bellows used in the simulation were kept the same as those in the FVCB, which was with stainless-steel. The compressed length of the bellows was configured to be 30.00 mm corresponding to the rated contact gap of 30 mm for the VI. The gas pressure was set as 0.50 MPa and the operating velocity was set as 5.00 m/s. The equivalent stress the U-shaped VI bellows' first wave crest being suffered was changed as follows: it was reduced by 27.46% when the thickness was increased from 0.16 mm to 0.18 mm; when the wave height was decreased from 8.34 mm to 7.74 mm, it was reduced by 25.37%; as for the radius of curvature of wave crest and the radius of curvature of wave trough, the equivalent stress was reduced by 16.15%, 19.36% respectively when the first parameter was changed from 2.00 mm to 1.80 mm and the second one was changed from 2.00 mm to 2.20 mm. Excessive reduction of the wave height contributes to higher superimposed frequency of the stress waves, the preferred optimum parameters were as follows: the thickness was 0.18 mm; the wave height was 8.04 mm; the radius of curvature of wave crest was 2.00 mm; the radius of curvature of wave trough was 2.00 mm. Two whole bellow structure models were built to verify the optimum design parameters. The equivalent stress the U-shaped VI bellows' first wave crest being suffered was reduced by 17.13% when the optimum design parameters of the bellow structure were adopted. The fatigue life of the improved model is 942 cycles while the initial model is 720 cycles with the operating speed of 5 m/s under the pressure of 0.50 MPa. The mechanical endurance of the bellows was increased by 30.83% when the optimum design parameters of bellow structure was adopted.

The present paper aims to analyze the surface finish of polished silicon wafer by chemically etching with potassium hydroxide (KOH) and mechanical polishing by Double Disk Magnetic Abrasive Finishing (DDMAF) process. The study is emphasized to study the effect of process parameters i.e. polishing speed, working gap, abrasive mesh number and percentage weight of KOH on the surface roughness (Ra). Response surface methodology (RSM) has been used to plan the experiments and Analysis of variance (ANOVA) has been used to analyze the impact of each process parameter on surface roughness. Regression equation for surface roughness in terms of significant process parameters has been developed to determine the surface roughness of polished silicon wafer. The equation has been further optimized using optimizer available with Minitab 17 and Genetic Algorithm (GA) tool box available with MatLab 16, to obtain the optimum process parameters and to prefigure the minimum surface roughness. The confirmatory experiment was carried out at optimum parameters and the prefigured results were found to be closely matched with the experimental findings. The communication further vocalizes the study of surface integrity for the unpolished sample to polished sample at optimum parameter using SEM and AFM images.

Geometric Programming is a new technique developed to solve nonlinear engineering design problems including linear or nonlinear constraints. This paper illustrates the use of Geometric Programming in obtaining optimal design parameters for a class of welded beam structures. The procedure is illustrated through the solution of a particular welded beam design formulation. In G/P format the problem solved consists of 9 nonlinear constraints, 24 terms, 7 variables, with 16 degrees of difficulty and a nonlinear objective function. Geometric Programming is compared to several other solution techniques, and found to be very efficient. Computational experience suggests that other problems of this class may be solved with similar efficiency. The welded beam problem given is a real world design situation typical of many encountered in actual practice. The solution is given for the first time in this paper.

With the widespread application of 3D printing technology in the field of orthopedic implants, 3D-printed interbody fusion cages have attracted increasing attention. This article systematically reviews the latest research progress from two main aspects: materials and manufacturing processes, as well as structural design. Titanium alloys and polyetheretherketone are the most widely used materials, while novel composite and bioresorbable materials have further expanded the diversity of applications. In the future, optimization of material properties and improvements in manufacturing processes will be needed to achieve better clinical outcomes. 3D printing technology has promoted the diversification of fusion cage structural design, showing a trend toward "de-solidification". However, the optimal design parameters for porous structures still require further research to balance mechanical properties and osseointegration requirements.