Euler Theory Research Articles

Stability of tensegrity-inspired structures fabricated through additive manufacturing

This study encompasses experimental and theoretical research on the stability of tensegrity-inspired structures in view of additive manufacturing technology. The aim of the research is to produce a qualitative and quantitative estimation of the post-critical behavior of struts in 3D-printed tensegrity-inspired modules. Presented results of compression tests performed on modules and single struts show that the bars work in the post-critical state before they reach their load bearing capacity. Comparison of theoretical solutions with experimental tests allowed us to conclude that the struts can carry axial loads significantly higher than the limits resulting from the classical Euler theory.

A novel polar mechanical metamaterial with dual deformation characteristics

Polarity refers to the phenomenon of an object exhibiting opposite intrinsic properties in opposing directions, which plays a fundamental role in various fields including mechanical diodes, mechanical logic gates, and shock protection. This paper proposes a new mechanical metamaterial with polar and dual deformation characteristics based on a fishbone-like structure to enable a given surface structure to be hard while its opposite side is soft, while also enabling the task of adapting to different load levels on the soft side. The mechanical metamaterial displays double-stress platforms and exhibits a significant difference in mechanical response when loaded on opposite sides. Through numerical simulations, the polarity characteristic spectrum of the mechanical metamaterial is identified. The stress platform with a soft boundary predicted using Euler's instability theory was verified through both numerical simulation and experimentation. The dimensionless parameters for load differences are defined to evaluate the polar characteristics of polar mechanical metamaterials, and design guidelines are provided for achieving significant polar characteristics of polar mechanical metamaterials by discussing the influence of geometrical parameters. The proposed polar mechanical metamaterial introduces a novel design concept applicable to signal transmission, isolation, and energy absorption for different load levels.

Assembling feather-like reentrant structures with multiple mechanical properties fabricated by laser powder bed fusion

A new assembly design method for structural components is proposed. Inspired by the feathers of the grey crane, which have an energy dissipation effect, novel re-entrant scapus component and novel re-entrant double-crank component are proposed. Novel re-entrant structures are formed by assembling honeycomb structure (HS), poisson structure (PS), and chiral structure (CS). Laser powder bed fusion technology (LPBF) is utilized for the manufacturing of bio-inspired structures. And the impact of structural assembly on mechanical performance is comprehensively analyzed through experimental, numerical, and theoretical approaches. Euler theory is employed to predict the influence of bionic components on structural load trends, while explaining the relationship between compression load and deformation. Specifically, the average load capacity of bio-inspired scapus re-entrant structures and double-crank re-entrant structures exceeds that of their original designs by at least 55% and 13.2%, respectively. Moreover, specific energy absorption increases by a minimum of 8.6% and 3.4%, accompanied by improved heat dissipation performance in these structures. Finally, battery pack impact resistance models as well as thermal conductivity models for electric-powered vehicles are presented as examples to validate the potential application prospects of this novel bio-inspired re-entrant structure in engineering.

Study on mechanical properties of polyurethane-enhanced triply periodic minimal composite structures inspired by rachis microstructure

Inspired by the micro-curved structure and microfiber filling of the feather, four novel polyurethane-enhanced triply periodic minimal composite structures are designed. Based on Euler theory and simulated annealing algorithm, the mechanical relationships of enhanced composite structures at the compression and impact stage are deduced and predicted. The polyurethane-enhanced composite structure is larger than the corresponding triply periodic minimal structure in the average load. The specific energy absorption of the polyurethane-enhanced triply periodic minimal composite structure is at least 10% better than the original. Through the energy expansion of the foam, the energy transfer path is allowed to deviate in the foam. And the load distribution is not only in the vertical direction but also in the transverse direction. All these factors (more energy dissipation and transfer path deviation) work to improve the crashworthiness of polyurethane-enhanced composite structures. The impact videos are captured by high-speed photography FASTCAM Nova R2. The reduction of normalized stress is proved in the design of polyurethane-enhanced structure, and the reduction of the impact acceleration of polyurethane-enhanced composite structure is more than 13.6%. This bio-inspired rigid-flexible coupling design of composite structure is crucial to the novel design of the next generation.

Suppression effect of bionic guide vanes with different parameters on the hump characteristics of pump-turbines based on entropy production theory

Based on the principles of bionics, this study drew on the protuberances of the leading edge of a humped whale flipper, designed different bionic protuberances applied to stay-guide vanes, and conducted three-dimensional numerical simulation research on the pump-turbine pump mode with an opening of 28.5 mm. The Euler theory and hydraulic loss entropy production analysis method was used to compare and analyze the pump characteristics of different schemes. The results showed that the bionic protuberances could effectively destroy the coherent structure of turbulence in the flow field, thereby improving the internal flow regime of the runner channel and significantly reducing flow separation. The protuberances can reduce the flow angle of the runner outlet by 5%–10%, resulting in an increase in the Euler head by 3%, which significantly increases the hump margin. In addition, bionic protuberances can effectively improve the flow regime in the stay-guide vanes and runner and weaken the flow separation, thereby avoiding the generation of a high local entropy production region and reducing the hydraulic loss by 4%. Therefore, protuberances can effectively improve the hump characteristics.

Buckling in columns. Solution of the indeterminations of Euler's theory and derivation of an equation for inelastic buckling

Buckling in columns under central load has always been a complex problem and until now remains unclear. This work consists of two parts. In the first, a theory is presented to solve the indeterminations of Euler's theory, such as: Moment, slope, deflection, moment and shear reactions. In this sense, the column is analyzed as if it were a prismatic member under pure bending, due to the application of a bending moment and consequently the elastic curve of the column is calculated by the traditional method. Using a second hypothesis, it is established that elastic buckling occurs when the compressive and bending energies become equal; and solving it the Euler's Formula for the critical load is obtained exactly. In the second part, and with the indeterminations already resolved, the maximum-strain-energy failure criterion is raised, in the critic section, where the maximum moment occurs, due to the compressive load and subsequent bending. This allows us to obtain an extraordinary equation for inelastic flexural buckling that predicts highly accurate results consistent with laboratory tests. In the stress diagram versus slenderness, we can see the flat zone of the short columns, and the critical slenderness ratio, that in the case of ASTM-A36 structural steel, points directly to one hundred and twenty (120). And, of course, we can calculate the critical stress according to the type of cross-section and buckling axis and direction. An ideal column is considered, i.e., without initial deflection, with the application point of load on the centroid, and without residual stresses. A lineal stress-strain diagram is used until the yield point.

Novel femur-like multimodal ultrahigh strength structure with superb freedom based on the improvement strategy of curvature fabricated by additive manufacturing

Based on the microstructure of biological femur, the femur-like multimodal surface structures are designed. But they exhibit fracture behavior when the porosity changes, resulting in structural failure. The high curvature surface factor is introduced to improve the fracture behavior of the biomimetic surface structure. Combine the high curvature surface factor to construct novel biomimetic homogenized surface structures. Novel radial gradient surface structures and axial gradient surface structures are designed based on the construction concept of cone functions and linear functions gradients. Theoretical, experimental and numerical methods are used to investigate the mechanical properties of novel biomimetic multimodal surface structures. Euler theory states that the properties of lattice structures are determined by the relative density of structures. The load distribution in the radially gradient structure is uniform, and the deformation mode presents overall failure, which can achieve better resistance and dissipation of the load. The deformation modes of axial gradient structures present layered failure. The underlying mechanism of the difference in mechanical properties of biomimetic multimodal structures is explained in detail. It’s very important significance for targeted application in protective engineering.

Experimental Study and Fragility Analysis of Effective-length Factors in Column Buckling

The design of columns relies heavily on the basis of Leonhard Euler's Theory of Elastic Buckling. However, to increase the accuracy in determining the maximum critical load a column can withstand before buckling, a constant was introduced. This dimensionless coefficient is K, also known as the effective-length factor. This constant is often found in building design codes and varies in value depending on the type of column support that is applied. This paper presents experimental and analytical studies on the determination of the effective-length factor in the buckling stability of columns with partially-fixed support conditions. To this end, the accurate K value of the columns tested by the Instron Testing Machine (ITM) at California State University, Northridge’s (CSUN’s) Mechanics Laboratory is determined. The ITM is used in studying the buckling of columns where the supports are neither pinned nor fixed, and the material cross-section rather rests upon the machine while loading is applied axially. Several column specimens were tested and the experimental data were analyzed in order to estimation of the accurate effective-length factor. The calculations from the tested results as well as the conducted probabilistic analysis shed light on how a fragility curve may aid in predicting the effective-length value of future tests.

Evaluation of training efficiency of table tennis players based on computer video processing technology

In this paper to improve the training efficiency of table tennis players, this paper applies computer video processing technology to the evaluation of training efficiency of table tennis players, and uses intelligent video processing to identify the training process of table tennis players. Moreover, this paper introduces the curve coordinates to mathematicalize the projection problem, establishes the spherical coordinate system and the plane coordinate system, obtains the spherical quadrilateral and the plane quadrilateral, obtains Euler's idea of establishing a projection suitable for table tennis, and obtains the road map of Euler's projection theory. In addition, this paper builds a training efficiency of table tennis players evaluation system based on projection algorithm and athlete feature recognition. Through the experimental research, it can be seen that the training efficiency evaluation system of table tennis based on computer video processing technology proposed in this paper can effectively improve the training efficiency of table tennis players.

Evaluation of Buckling of 2024-T3 under High Temperatures

The mechanical and buckling behavior of AA 2024 – T3 at high temperatures has been presented. The material was examined by thermal tensile test rig with 400℃ capacity. While buckling tests were carried out using a thermal rotating buckling test machine. Several observations were drawn from the experimental results, such as the mechanical and buckling properties are reduced by application of high temperatures. The experimental results of (UTS), (YS), (BHN) and (E) were decreased by 13.77%, 19.76%, 28.8% and 24.65% respectively due to application of 250c comparing to that at room temperature. The critical buckling load (Pcr) is increased when the column length and (SR) are reduced. The critical buckling load (Pcr) results were reduced from 910N to 610N when the applied temperature increased from (RT) to 250℃. Therefore, using a high temperature true of 250℃ gives a reduction percentage of 33% in critical buckling load result. The estimation of Euler theory was overestimated the buckling properties, but when using a safety factor the estimation seems to be resemble.

On the role of nonlinear piezoelectric constitutive parameters in the dynamic response of clamped–clamped axially loaded beams

In this paper the influence of nonlinear constitutive piezoelectric parameters in the electromechanical coupled response for a clamped–clamped axially loaded energy harvester is studied. Due to the high slenderness ratio of the device, a three section beam according to Bernoulli–Euler theory, is proposed. The second section corresponds to the MFC 2814 bonded piezoelectric sheet. The complete partial differential equations that govern the axial–flexional–electric coupled problem are derived by using the Hamilton’s principle. A reduced version of these expressions are obtained by assuming the axial displacement is quite smaller than transverse. The discretization of the equations is made according to the Galerkin method. An approximated solution for the single mode representation is obtained by the Method of Multiple Scales (MMS) and compared with the direct numerical integration of the equations showing excellent results. A prototype device is built, and experimentally tested to be compared with the proposed model. The numerical solutions and experimental results confirmed the need to consider nonlinear piezoelectric relationships for the proposed configuration to correctly predict the generated voltage.

Flow induced bifurcation and phase-plane stability analysis of branched nanotubes resting on two parameter foundation in a magnetic environment

This paper explores the nonlinear partial differential equations (PDEs) for transverse and longitudinal vibrations of slightly curved fluid-conveying embedded branched nanotubes operating in a magnetic environment. Using Bernoulli–Euler, Nonlocal and Hamilton theories, equations of motion governing the vibrations of the nanotubes are developed. Thereafter, the developed inimitable equations are solved using the numerical PDE solver and PDE-tool in MATLAB. With the numerical solutions, visualizations and parametric studies were carried out. The results indicate that increasing the branch angle at the downstream decreases quantitative stability. Furthermore, the stability results acquired via simulation point out that the magnetic field has a damping or attenuating influence of about 20%. The solutions as presented in this study match with existing results from previous studies, hence the verifications and validations of this work. It is envisioned that the results obtained will provide better physical insights into the stability criteria for carbon nanotubes and will enhance the designs of nanomechanical carbon nanotubes-based devices that convey fluid and operate in elastic and magnetic environments.

Influence of Longitudinal Inertia of Sliding Mass on Nonlinear Transverse Vibrations of Hydraulic Cylinder Piston Rod

In this work, transverse vibrations of the piston rod of a hydraulic cylinder, which is connected with a system characterized by its own high weight, e.g. a tank gate, were considered. The case is considered when the actuator is fully extended and the piston rod is not affected by external static axial forces (the actuator acts in a horizontal position, as a result of which the weight load at its end does not compress the piston rod). To analyze this issue, a beam model was developed taking into account the longitudinal inertia of the mass element associated with one of the ends of the system. The boundary problem of vibrations was formulated using the Bernoulli–Euler theory. Taking into account mass inertia (directed longitudinally) results in the appearance of nonlinear terms in equations describing the behavior of the system during vibrations. The small parameter method was used to finally formulate the problem of nonlinear vibrations. In the description thus adopted, the vibrations of the considered system strictly depend on the dead weight of the gate valve as well as the amplitude of the piston rod oscillations. The results of numerical simulations are presented taking into account the impact of piston rod stiffness and stiffness of mounting to the valve on its vibrations. In the considered range of masses, the effect of amplitude on the value of the natural frequency of the system is presented. Theoretical considerations have been confirmed to some extent by experimental studies.

Calibration of the length scale parameter for the stress-driven nonlocal elasticity model from quasi-static and dynamic experiments

The available experimental results in the literature on the quasi-static bending and free flexural vibration of microcantilevers and nanocantilevers are used to calibrate the length scale parameter of the stress-driven nonlocal elasticity model. The Bernoulli–Euler theory is used to define the kinematic field. The closed form solution derived for the bending problem is used to calibrate the length scale parameter by fitting the load–displacement curves to the experimental results. For the vibration problem, the calibration is done using the least-squares curve fitting method for the natural frequencies. The stress-driven nonlocal theory can adequately capture the size-dependent experimental results.

Early-age buildability-rheological properties relationship in additively manufactured cement paste hollow cylinders

Ability to extrude and to achieve shape stability of layer-wise additively manufactured cement-based elements at large scale depends upon the early-age rheological properties (shear moduli, yield stress, viscosity) of the deposited materials. Upon successful extrusion, buildability challenges can manifest themselves through two common failure mechanisms: yielding of lower layers or buckling of the element. However, it is yet unclear which of the various rheological properties control the early-age materials' deformation, specifically buckling, during the printing processes and thus influence the resulting buildability of the elements. This paper focuses on how buildability is dependent upon rheological properties as well as on predicting the buildability using a buckling theoretical framework. Specifically, the relationship between early-age rheological properties of various cement pastes and the buildability of hollow cylinders with failure dominated by buckling mechanism was investigated. It was found that certain types of shear moduli of the fresh pastes (G, G*, and G′) obtained from oscillatory shear stress sweep tests, performed within the first 30 min of hydration, correlate better with the buildability of hollow elements than some other rheological properties (loss modulus G″, yield stress σyield, yield strain γyield, and complex viscosity η*). Measured values of shear modulus (G) were used to calculate elastic modulus (E) of the pastes using the assumed value of 0.5 for Poisson's ratio (ν) of fresh cement paste. Euler's buckling theory was used to predict buildability (height of the element) of hollow cylinders. It was found that Euler's theory overestimates the buildability by 93%–194%, mainly due to assumption of ideal geometry (i.e., absence initial or printing imperfection), rate-independent behavior and linear elasticity.

Effects of Nonlinear Damping on Vibrations of Microbeam

The present paper develops a new Bernoulli–Euler theory of microbeams for the consideration of small-scale effects and nonlinear terms, which are induced by the axial elongation of the beam and Kelvin–Voigt damping. The non-resonance and primary resonance of microbeams are researched through the application of Galerkin and multiple scale methods to the new model. The results suggest the following: (1) Nonlinear damping slightly affects the vibration amplitudes under the non-resonance condition; (2) nonlinear damping can significantly change the bifurcation points that induce a jump in the vibration amplitudes under the primary resonance condition. The current researches indicate that nonlinear damping is necessary for an accurate description of microbeam vibrations.

Determinations of Compression Buckling for Long and Intermediate Aluminum Alloy 2014-T4 Columns

Experimental tests were carried out on long and intermediate columns of 2014-T4 aluminum alloy stressed to compression buckling. 24 samples were used, divided into two groups (12 specimens before shot peening and ultrasonic impact peening (SP +UIP) and the rest after (SP +UIP)). The ultimate tensile strength (UTS) and yield strength (YS) were improved due (SP +UIP) by 2.60% and 3.030%, respectively. ANSYS 18.2 (APDL) with the theories of Euler and Johnson was theories was used to estimate the critical buckling load (Pcr) and compare the results with the experimental work. It was found that there was strong agreement between the theories of Euler and Johnson and theories and the experimental work, with safety factors of (1.6, 2.3) and (1.7, 2.5) for long and medium columns before and after the experiment, respectively (SP +UIP). In contrast, ANSYS 18.2 (APDL) provided good predictions with a safety factor of (2.1 and 2.6) and (1.8 and 2.6) for long and medium columns before and after (SP +UIP), respectively.

Effect of shot peening on the critical buckling load of stainless steel 304 columns immersed in sea water

Abstract A machine part subjected to an axial compressive load is called strut. But, a vertical strut is known as a column. The machine members that must be investigated for the column action are connecting rods, piston rods, screw jack, etc. When a column is subjected to a compressive load and this load is gradually increased, a stage will be reached when the column will be subjected to ultimate load. Beyond this, the column will fail by crushing, and the load will be known as crushing load. When the column is short or intermediate, sometimes this column fails by bending, i.e. buckling. When the column is long, the value of buckling load is low for long columns and is relatively high for intermediate columns. The present investigation focuses on the testing and evaluation of the mechanical end buckling columns (samples) using 304 stainless steel under dry, corrosion, and combined dry with shot peening (SP) process. The buckling behavior of the axial compressive load has been studied experimentally and theoretically using Euler and Rankin theories. The results obtained from the above study manifested that the column, whose slenderness ratio (SR) is more than 120, is denoted as the long column, and the Euler theory can be successfully used. But, when the SR is less than 120, the column is known as an intermediate one. The mechanical and buckling properties exhibited a reduction due to the corrosion media and a reasonable improvement when using SP.

A Simulation of an Elastic Filament Using Kirchhoff Model

This paper presents numerical simulations and comparisons between different approaches concerning elastic thin rods. Elastic rods are ideal for modeling the stretching, bending, and twisting deformations of such long and thin elastic materials. The static solution of Kirchhoff's equations [2] is produced using ODE45 solver where Kirchhoff and reference system equations are combined instantaneously. Solutions using formulations are based on Euler's elastica theory [1] which determines the deformed centerline of the rod by solving a boundary-value problem, on the Discreet Elastic Rod method using Bishop frame (DER) [5,6] which is based on discrete differential geometry, it starts with a discrete energy formulation and obtains the forces and equations of motion by taking the derivative of energies. Instead of discretizing smooth equations, DER solves discrete equations and obeys geometrical exactness. Using DER we measure torsion as the difference of angles between the material and the Bishop frame of the rod so that no additional degree of freedom is needed to represent the torsional behavior. We found excellent agreement between our Kirchhoff-based solution and numerical results obtained by the other methods. In our numerical results, we include simulation of the rod under the action of the terminal moment and illustrations of the gravity effects.

Modeling and Analyzing the Free Vibration of Simply Supported Functionally Graded Beam

Euler, Timoshenko and high shear deformation theories to analyze the free vibration of the functionally graded (FG) beam were developed. The mechanical properties of this beam were assumed to differ in thickness direction according to the model of a power-law distribution. The principle of Hamilton was used to find equations of motion. For free vibration, the analytical solution of these equations was presented using the Navier method. The effect of power index, aspect ratio, modulus ratio, and deformation theories on dimensionless frequency were studied numerically by Ansys software and analytically according to different beam theories using the Fortran program. The obtained results from these programs were compared with each other and with some previous research. Results showed an excellent agreement with the previous research. The numerical and analytical results showed that the use of this new FG beam model especially based on first and high shear deformation theories leads to the reduction of dimensionless frequency. It may be concluded that, the including of shear’s effect leads to a decrease in the dimensionless frequency. From the modeling and analysis of this model, it is possible to know what is the appropriate design for this FG beam model to reduce the vibration.