Optimal Stiffness Research Articles

Development and Stiffness Optimization for a Flexible-Tail Robotic Fish

The integral flexible tail has the potential advantage of lifelike undulating motion. However, due to the complex manufacturing process and difficult modification of structural parameters, its application in robotic fish encounters many challenges. Combining rigid structure and flexible material, this letter proposes a passive flexible fish tail, which can achieve continuous movement and high swimming frequency with simple but effective structure. First, with the full consideration of bending deformation of the spring steels, a dynamic model is established. Next, a passive fitting method is particularly applied to imitate the traveling wave model of the carangiform fish. More importantly, a calculation model, which can be used to acquire the theoretical ranges of the stiffness of spring steels, is derived based on the bending model of the cantilever beam subjected to the concentrated force and moment. Finally, the extensive simulation and experiments validate the effectiveness of the proposed methods, and the designed robotic fish can achieve 0.77 m/s (i.e., 1.12 BL/s) at a swimming frequency of 2.5 Hz. The obtained results can provide a valuable sight for improving the swimming performance of the robotic fish.

Performance Improvement of a High-Speed Swimming Robot for Fish-Like Leaping

Many aquatic animals are able to leap out of water effortlessly, however, it is still exceedingly challenging for a swimming robot. Inspired by the fast-swimming mechanism of fish, in this letter, we develop an untethered high-speed swimming robot with the integration of high-frequency oscillation and compliant passive components for performance improvements in swimming speed, efficiency and even fish-like leaping motion. The stiffness optimization of compliant components is conducted via experiments. As a result, both swimming speed and efficiency have been significantly improved and the robot surprisingly reaches a swimming speed of 1.88 m/s corresponding to 7.1 body lengths per second (BL/s). That is, the compliant components’ stiffness is of great importance to the performance improvement of a swimming robot at high-frequency oscillation. Additionally, the leaping motion is successfully performed with the optimized swimming robot and the maximum height of center of mass (CM) can reach up to 0.23 m. A simplified model is developed to analyze the leaping height with different speeds and body lengths, which suggests that the robot with small size leaps out of water more easily. The obtained results in this letter will offer some significant insights into the design and optimization for a high-speed swimming robot, as well as the cross-domain motions between water and air.

Isolating Efficiency of Soil Compactor’s Seat Suspension Using Optimal Negative Stiffness Structure under Various Deformable Terrains

Isolating Efficiency of Soil Compactor’s Seat Suspension Using Optimal Negative Stiffness Structure under Various Deformable Terrains

Parameter optimization of a grounded dynamic vibration absorber with lever and inerter

Mechanical vibration is mostly harmful, and it may not only generate noise but also affect the working life of the equipment. Lever, inerter, and grounded stiffness have good performance in the field of vibration control, but the dynamic vibration absorber simultaneously containing lever, inerter, and grounded stiffness is rarely studied. Based on the grounded dampertype dynamic vibration absorber, a dynamic vibration absorber with lever, inerter, and grounded stiffness is presented. And the optimal system parameters are analytically researched in detail. Firstly, the differential equation of motion is established according to Newton’s second law, and the analytical solution of the system is obtained. According to the amplitudefrequency curve of the system, it is obvious that there are two fixed points unrelated to the damping ratio. Meanwhile, the optimal frequency ratio of the dynamic vibration absorber is obtained based on the fixed-point theory. Under the premise of ensuring the system stability, the optimal grounded stiffness ratio is screened out, and the working range of inerter is further calculated. It is found that the inerter ratio has two working ranges when the coupling term values of magnification ratio and mass ratio are different. Furthermore, the approximate optimal damping ratio is derived by minimizing the maximum value of the amplitudefrequency curve. Using MATLAB, the numerical result is analyzed, and the correctness of analytical results is verified. Compared with other dynamic vibration absorbers under harmonic and random excitations, it is known that the model in this paper can evidently reduce the resonance amplitude and broaden the vibration band of the primary system. These results may provide a theoretical basis for the optimal design of similar dynamic vibration absorbers.

Simplification and further optimization of spatial cable-truss structure without inner ring cables

PurposeIn order to optimize SCSWIRC, the simplification and further optimization method is proposed. SCSWIRC's optimization includes two levels. The first level refers to simplifying structural system from the perspective of components; the second level refers to optimizing components' sectional areas from the perspective of mechanics. The first level aims to remove redundant components, and the second level aims to reduce structural self-weight based on the first level. The purpose of the paper is to simplify SCSWIRC's structural system and optimize structural self-weight and reduce construction forming difficulty.Design/methodology/approachGrid-jumping layout and multi-objective optimization method is used to simplify and further optimize Spatial cable-truss structure without inner ring cables (SCSWIRC). Grid-jumping layout is used to simplify remove redundant components, and multi-objective optimization method is used to reduce structural self-weight. The detailed solving process is given based on grid-jumping layout and multi-objective optimization method.FindingsTake SCSWIRC with a span of 100m as an example to verify the feasibility and correctness of the simplification and further optimization method. The optimization results show that 12 redundant components are removed and the self-weight reduces by 3.128t from original scheme to grid-jumping layout scheme 1. The self-weight reduces from 36.007t to 28.231t and feasible coefficient decreases from 1.0 to 0.627 from grid-jumping layout scheme 1 to multi-objective optimization scheme. The simplification and further optimization can not only remove the redundant components and simplify structural system to reduce construction forming difficulty, but also optimize structural self-weight under considering structural stiffness to reduce project costs.Originality/valueThe proposed method firstly simplifies SCSWIRC and then optimizes the simplified SCSWIRC, which can solve the optimization problem from the perspective of components and mechanics. Meanwhile, the optimal section solving method can be used to obtain circular steel tube size with the optimal stiffness of the same areas. The proposed method successfully solves the problem of construction forming and project cost, which promotes the application of SCSWIRC in practical engineering.

A Study of a Pendulum-Like Vibration Isolator With Quasi-Zero-Stiffness

Abstract This article introduces a pendulum element to a 3-spring vibration isolator to achieve a high-static-low-dynamic (HSLD) stiffness or even quasi-zero stiffness (QZS) around the equilibrium position. The model is first established, the equilibrium point is derived and the optimal stiffness ratio of this novel system at the equilibrium position is also obtained. Numerical simulation is given and the harmonic balance method (HBM) is used to obtain time responses for analysis. Effects of different parameters on the isolation performance are studied and summarized. Approximation force and displacement transmissibility of the system are calculated to evaluate the isolation performance. Comparisons are made with those of an equivalent linear isolator and the typical 1 degree-of-freedom (DOF) QZS isolator. Results show that the novel vibration isolator performs better than existing isolators under selected parameters. The left bent backbone of the novel isolator demonstrates evident softening geometric nonlinearity. Therefore, it achieves a wider frequency range of isolation than the linear 1DOF isolator and typical 3-spring QZS isolator. Moreover, the transmissibility of the novel isolator is smaller at higher frequencies as the jump phenomenon occurs on the left.

Experimental Research on the Coupling Relationship between Fishtail Stiffness and Undulatory Frequency

Fish can swim in a variety of states. For example, they look flexible and perform low-frequency undulatory locomotion when cruising, but they seem very powerful and stiff and perform high-frequency undulatory when hunting. In the process of changing the motion state, the stiffness of the fish body affects the swimming performance of the fish. In this article, we imitated the change of stiffness by superimposing rubber sheets and used experimental methods to test its swimming performance under different swing frequencies. A series of rubber fish tails were made according to the analysis of the swimming movement of real fish, providing different stiffness values and changing the curves of the body. In the prototype experiments, the base of the fish tail was fixed to a platform via a force sensor, which can oscillate at various speeds, so that the fish tail was able to swing and the thrust could be tested at different frequencies. According to the experimental results, we found that with the change of the swing frequency, there were different optimal stiffnesses that could make the thrust reach the maximum value, and with the increase of stiffness, the envelope interval of the swing curve gradually widened, the amplitude increased, and the hysteresis of the tail fin relative to the end decreased.

Parameter Optimization for a Novel Inerter-based Dynamic Vibration Absorber with Negative Stiffness

A novel dynamic vibration absorber(DVA) model with negative stiffness and inerter-mass is presented and analytically studied in this paper. The research shows there are still two fixed points independent of the absorber damping in the amplitude frequency curve of the primary system when the system contains negative stiffness and inerter-mass. The optimum frequency ratio is obtained based on the fixed-point theory. In order to ensure the stability of the system, it is found that inappropriate inerter coefficient will cause the system instable when screening optimal negative stiffness ratio. Accordingly, the best working range of inerter is determined and optimal negative stiffness ratio and approximate optimal damping ratio are also obtained. At last the control performance of the presented DVA is compared with three existing typical DVAs. The comparison results in harmonic and random excitation show that the presented DVA could not only reduce the peak value of the amplitude-frequency curve of the primary system significantly, but also broaden the efficient frequency range of vibration mitigation.

Section Size Optimization of Continuum Structure with Bending and Shearing Deformation for Uniform Displacement Criterion

If lateral stiffness distribution of continuum structure is unreasonable, its displacement distribution will be non-uniform under lateral load. The non-uniform displacement distribution will weaken the safety of the structure. The continuum structure is equivalent to the cantilever. The optimal lateral stiffness distribution of the structure is obtained by adjusting the section size distribution. The wind load and earthquake action are simplified into three equivalent static loads: uniform load, inverted triangle load and inertia force related load. The shearing displacement and bending displacement distributions along the structural height are derived. The optimization objective is that the second derivative of the total displacement distribution is zero, that is, the displacement distribution is uniform. The section size of the cantilever is optimized and the optimal scheme is obtained. The optimization results are verified by numerical analysis and finite element analysis. The results show that the uniform displacement criterion can be realized by the cantilever with section size distribution of power function. The optimal parameters of the section under different loads are different. A practical design example is presented to prove the applicability.

Numerical investigation of earthquake response reduction effects by negative stiffness connection for adjacent building structures

Coupled vibration control (CVC) for adjacent buildings can mitigate the seismic response of the system by properly setting the structural parameters, including the connecting elements. However, optimization of the control effect for CVC structures may be limited when using only positive or zero stiffness at the connecting spring. The present study explores the control effectiveness by passive negative stiffness as a connecting element installed between the mainframe and subframe of CVC structures. A two-degrees-of-freedom (2DOF) model representing a coupled building, which consisted of the mainframe and subframe linked by a spring and a viscous damping element, was used. The transfer function (TF) for displacement and acceleration of the mainframe and subframe was evaluated under various structural parameters. The results showed that by setting an optimal negative stiffness and an optimal damping at the connecting portion, the peak amplitude of the TF of the mainframe was significantly reduced compared to that of reference-controlled case without negative stiffness (i.e., with only a damping element). In addition, adopting negative stiffness in CVC structures can extend the range of optimal tuning conditions. Moreover, a time history earthquake response simulation was conducted using 2DOF models with different natural periods and structural parameters subjected to various input motions, including simulated waves and observed records. The results indicate that use of the negative stiffness connection can effectively reduce the seismic response of the mainframe for passive CVC structures.

Structural Weight and Stiffness Optimization of a Midibus Using the Reinforcement and Response Surface Optimization (RSO) Method in Static Condition

Midibuses are medium-sized buses widely used for transportation purposes in Asia and Africa. However, most midibuses are locally built and indirectly regulated through inspecting the end product (finished bus) during licensing for the public transport business in Ethiopia. Due to lack of engineering analysis and testing, low stiffness and overweight of midibus were compromised. This research was aimed at analyzing and optimizing the midibus structure using the reinforcement and response surface optimization (RSO) method for pure bending and torsion loading cases. Results show that the maximum deformation occurred at the roof section of the original structure during both loading cases. Furthermore, the reinforcement design was found by replacing the cross section and layouts of structural members and adding reinforcements for the most suitable location of the original structure. Response surface optimization with the multiobjective genetic algorithm (MOGA) method in ANSYS DesignXplorer was performed on the reinforced structure to maximize the bending and torsional stiffness with reduced weight. The bending stiffness of the reinforced and optimized structure increased by 41.65% (1911.4 N/m) and 10.02% (651.7 N/m), respectively. In addition, the torsional rigidity or stiffness of the bus structure was improved by 12.56% (173.31 Nm/deg) via reinforcement design. Moreover, the torsional stiffness of the optimized (RSO) model was increased by 3.29% (51.07 Nm/deg). Reinforcement design was effectively reduced by 5.23% of the structure’s weight. Moreover, the RSO method has also decreased the weight of the reinforced structure by 2.64%.

Nonlinear stiffness optimization with prescribed deformed geometry and loads

Nonlinear stiffness optimization with prescribed deformed geometry and loads

The family of elastically isotropic stretching-dominated cubic truss lattices

We present a family of elastically isotropic stretching-dominated cubic truss lattices. Their Young’s, shear, bulk moduli approach nearly 1/3 of the theoretical Hashin-Shtrikman upper bounds at low density limit, which are the maximum isotropic stiffness that truss lattices can achieve. The proposed lattices are designed by combination of two elementary anisotropic lattices based on theoretical criteria for elastically isotropic stretching-dominated truss lattices. The elementary truss lattices are identified by exhaustive search of the Euclidean Patterns in Non-Euclidean Tilings database. Matrix analysis formulas for periodic pin-jointed assemblies are developed to prove that all combined lattices exhibit stretching-dominated behaviors under arbitrary macroscopic loads, i.e., their corresponding periodic pin-jointed assemblies have no macroscopic strain-producing mechanisms. The stiffness of the combined lattices is further evaluated through C3D10 solid element based finite element models and numerical homogenization. Numerical results show good agreement with theoretical calculations at low densities, thus validating the stretching-dominated behavior and elastic isotropy. The lattices exhibit different elastic moduli and anisotropy at moderate relative densities due to their different joint connections, among which the sqc1 + sqc5432 lattice possesses both nearly isotropic stiffness and relatively high elastic moduli at 30% relative density. This work presents a systematic method to design elastically isotropic cubic truss lattices with optimal isotropic stiffness, and reveals that the optimal designs are infinite.

A partition and microstructure based method applicable to large-scale topology optimization

Topology optimization (TO) of large-scale structures is a computationally demanding process that challenges its widespread adoption in industrial design. We present a new partition based TO framework applicable to the design of large-scale problems and apply it to mechanical stiffness optimization problems. The method employs, first, a data-driven physical partitioning of strain energy contours to partition the design domain, and then a partition based TO. In contrast to conventional topology optimization in which the number of design variables is typically equal to the number of elements in the discretized domain, the proposed method assigns density design variables to each spatial partition leading to significant computational cost reduction and convergence acceleration. The constitutive matrix required for finite element analysis is iteratively determined according to the Hashin-Shtirkman upper bounds based on partition densities. Once the optimized partition densities are achieved, the manufacturable binary structure is realized by mapping a set of generated high-performance isotropic microstructure cells onto partition elements. To validate the capability, effectiveness, and efficiency of the proposed method, several numerical examples are provided. The optimized structures using macrostructural analysis exhibit comparable performance to the conventional SIMP method for small-size problems. Besides, the TO results for the large-scale problems suggest significant computational cost efficiency.

Optimal negative stiffness inertial-amplifier-base-isolators: Exact closed-form expressions

This paper introduces the concept of negative stiffness inertial-amplifier-base-isolators to achieve enhanced broadband vibration control. Three physically different novel isolators, namely inertial amplifier base isolator (IABI), negative stiffness inertial amplifier base isolator (NSIABI), and negative stiffness base isolator (NSBI), are achieved only by tuning a single system parameter, namely the mass tuning ratio for novel isolators γ. The exact closed-form expressions for the optimal system parameters, such as the frequency and damping ratio of the novel isolators, have been derived analytically using H2 and H∞ optimization methods. The response reduction capacity of each optimized novel isolator has been compared with the optimized traditional base isolator (TBI). Results showed that the vibration reduction capacity of H2 optimized NSBI, NSIABI, and IABI is significantly 73.02%, 75.55%, and 76.48 %, superior to the TBI. While, the vibration reduction capacity of H∞ optimized NSBI, NSIABI, and IABI is significantly 69.69%, 73.67%, and 77.37% superior to the TBI. Overall all three novel base isolators have at least 69% more vibration reduction capacity than traditional base isolators, respectively. These novel isolators are cost-effective and can provide superior vibration reduction capacity than other existing traditional base isolators, respectively.

Upper limb involuntary tremor reduction using cantilever beam TMDs

Tuned mass dampers (TMDs) are proposed as a solution to reduce the involuntary tremor at the upper limb of a patient with postural tremor. The upper limb is modeled as a three-degrees-of-freedom rotating system in the vertical plane, with a flexion-extension motion at the joints. The measured extensor carpi radialis signal of a patient is used to excite the dynamic model. We propose a numerical methodology to optimize the parameters of the TMDs in the frequency domain combined with the response in the time domain. The objective function for the optimization of the dynamic problem is the maximum angular displacement of the wrist joint. The optimal stiffness and damping of the TMDs are obtained by satisfying the minimization of the selected objective function. The considered passive absorber is a cantilever beam–like TMD, whose length, beam cross-sectional diameter, and mass position reflect its stiffness for a chosen additional mass. A parametric study of the TMD is conducted to evaluate the effect of the TMD position along the hand segment, the number of TMDs, and the total mass of TMDs. The sensitivity of the TMD to a decrease of its modal damping ratio is studied to meet the range of stainless steel. TMDs are manufactured using stainless steel beams of the same length (9.1 cm) and cross-sectional diameter (0.79 mm), for which the mass (14.13 g) position is adjusted to match the optimal frequency. Three TMDs holding a mass of 14.13 g each cause 89% reduction in the wrist joint angular displacement.

The true potential of nonlinear stiffness for point absorbing wave energy converters

A spherical point absorbing wave energy converter was simulated in floating and submerged conditions for regular and irregular waves. A nonlinear stiffness was included to augment a linear stiffness and damper control system. The linear control parameters were optimised for each regular and irregular wave condition. The optimal linear control stiffness for the floating system was negative to counteract the hydrostatic stiffness. The nonlinear stiffness was shown to increase the converted power if no linear control stiffness was used, and reduce the converted power if optimal linear control parameters were used for regular waves. Similarly, for irregular conditions, nonlinear stiffness degrades the performance of the system when compared to optimised parameters, but enhances the robustness of the system to changing sea states and stochastic phases. Therefore, a nonlinear stiffness mechanism may improve the power output for poorly tuned control systems arising from plant uncertainty for point absorbing wave energy converters.

Topological optimization of the stiffness of an irregular structure based on an element size independent filter

Because of the overly averaged element sensitivity in the topological optimization of an irregular structure with a grid independent filter, a topological optimization model was built for the structural domain. The maximization of stiffness was first taken as the goal for the topological optimization of irregular structure stiffness. Subsequently, an element size filter was proposed to address the overly averaged local element sensitivity with the grid independent filter when the designed domain element size varied dramatically. Finally, the element sensitivity of the objective function was derived under the given constraints. A case study was then conducted on a naval gun mount with the maximization of structural flexibility as the objective function and the volume of structural material as a constraint. A stiffness optimization model based on the bi-directional evolutionary structural optimization algorithm was adopted for the topological optimization of the gun mount. Structural optimization was conducted for the gun mount with different shooting angles to realize its optimal stiffness and strength under the constraint of consistent material volume. The optimization results proved that the element independent filter proposed in this paper could be effectively applied in the topological optimization of an irregular structure and used to explore the topological optimization of the supporting structure under impact.

Stiffness Optimization Through a Modified Greedy Algorithm

Due to the 3D printing techniques, it is possible to create and design components with soft and rigid combinations of materials with different elastic properties which may satisfy multiple material requirements. A soft (blue) and a rigid (purple) material with moduli of elasticity in a ratio 1:10 can be innkjet printed together following an initial pattern. The objective function was formulated as to maximize the effective stiffness on both in-plane directions. Finite element analyses were performed by using the PyAnsys software under an MIT License in a domain with 16 Q8 elements which represents by symmetry a representative volume element (RVE). The objective function to be maximized can also reduce the anisotropy of the printed topology. We proved that the formulation of three objective functions affects the results obtained with a Greedy algorithm.

Study on Vibration Characteristics of Stay Cable-Nonlinear Viscous Damper System

Tension cables are easily prone to generating varied vibrations under the action of external loads, which adversely affects the safety of bridges. Therefore, it is necessary to take effective measures to suppress the vibrations of tension cables. Cable end dampers are widely used in vibration reduction for cable-stayed bridges due to their convenient installation and low costs. However, the previous studies on the tension cable-viscous damper systems mostly adopt the linear method, and the weakening effect of the flexibility of mounting brackets on the damper vibration reduction is not sufficiently taken into account. Therefore, this paper adopts the improved Kelvin model to conduct the derivation, solution, and parametric analysis of vibration equations for the stay cable-nonlinear viscous damper systems. The results of parametric analysis show that the maximum modal damping ratio that can be obtained by cables and the corresponding optimal damping coefficient of dampers are correlated with the damping nonlinear coefficient α, stiffness nonlinear coefficient β, vibration order n, installation position a/L, and stiffness coefficient μ, etc.; among them, n damping nonlinear coefficient α and stiffness nonlinear coefficient β are the key parameters that affect the parameter design of dampers, where damping nonlinear coefficient α mainly controls the optimal damping coefficient and stiffness nonlinear coefficient β mainly controls the maximum damping ratio. Based on the parametric analysis, the design principles of dampers and value requirements of key parameters under different vibration suppression objectives are presented.