Calculation Of Stiffness Research Articles

On gear time-varying meshing stiffness calculation considering indexing feeds and processing characteristics of gear shaping processing

Gear shaping causes processing characteristics on gear tooth surfaces (PCGTS), e.g., gear tooth surface deviation (GTSD) and processing textures (PT), which can affect gear meshing performances. Gear shaping simulation is operated to obtain PCGTS with different indexing feeds. An improved contact stiffness method of gear tooth surfaces is developed based on a wedge-shaped contact form to obtain tooth contact stiffness with PT caused by gear shaping. An improved time-varying meshing stiffness (TVMS) method for gears manufactured by gear shaping is developed to calculate TVMS of gears with GTSD and PT under different indexing feed conditions. Effectiveness of the improved TVMS method is validated using the finite element method. Analysis results of the proposed contact stiffness method of gear tooth surfaces indicate that PT caused by gear shaping can aggravate fluctuations of gear tooth contact stress. Analysis results of the proposed TVMS method for gears also indicate that GTSD of gear tooth surfaces caused by gear shaping can reduce their TVMS. The proposed TVMS method of gears can effectively analyze TVMS of gear tooth surfaces via gear shaping and evaluate processing settings of gear shaping to improve gear meshing performances.

Evaluation of the Flexural Behavior of One-Way Slabs by the Amount of Carbon Grid Manufactured by Adhesive Bonding

Fiber-reinforced polymers (FRPs), which are resistant to corrosion, are used as reinforcement material for concrete. However, the flexural behavior of concrete members reinforced with FRPs can vary depending on the properties of FRPs. In this study, the flexural behavior of one-way concrete slab specimens reinforced with a new grid-type carbon-fiber-reinforced polymer (CFRP) (carbon grid) manufactured by bonding pultruded CFRP strands to an adhesive was investigated. The experimental results indicated differences in the load–deflection relationships of the specimens depending on the carbon grid reinforcement amount. Specimens in which the carbon grids were over-reinforced or reinforced close to the balanced reinforcement ratio reached the maximum load due to concrete crushing and exhibited ductile failure. The specimen under-reinforced with the carbon grid exhibited brittle failure. Specimens with carbon grid reinforcement close to a balanced reinforcement ratio exhibited maximum loads ranging from 0.43 to 0.61 times the calculated flexural strength, which resulted in becoming 0.86–1.00 lower in the specimens with a wider width of the CFRP strands. This study proposes coefficients to estimate the stiffness of carbon-grid-reinforced concrete flexural members after cracking. Applying these coefficients resulted in stiffness calculations that reasonably simulated the behavior of the specimens reinforced with carbon grids after crack formation.

Innovative load identification with Res-UNet: Integrating phase space reconstruction and physics-informed deep learning

This study developed a specialized load identification model for a specific ocean platform that integrates advanced deep learning techniques with the unique physical characteristics of marine engineering. Key achievements include high-dimensional mapping of data features using phase space reconstruction (PSR) techniques to enhance the predictive capacity of the model through chaos system theory; optimization of ResNet and UNet deep learning networks with attention mechanisms to boost performance; calculation of stiffness, mass, and damping matrices using finite element software; establishment of corresponding state-space equations; and integration of structural dynamics equations with the deep learning network loss function to significantly improve load identification accuracy. In addition, proprietary structural equations for ocean platform structures were developed, underscoring the uniqueness and applicability of the algorithm. The proposed physics-informed Res-UNet model exhibited excellent performance in identifying both periodic and random loads. These improvements were validated through training with finite element simulated data and subsequent experimental applications, demonstrating the effectiveness of the proposed innovative methods.

Stability of cold-formed steel stud walls subjected to vertical compression

Cold-formed steel (CFS) stud walls are the primary vertical load-bearing units in CFS residential systems. These walls need to meet both strength requirements and good stability. In this study, a CFS stud wall was simplified into a mechanical model of an orthotropic plate rib system using the orthotropic plate (OP) method. A formula for calculating the equivalent stiffness of the stud wall was obtained, and the correctness of the simplified calculation model and equivalent stiffness calculation formula was verified by finite element analysis. Various factors that affected the equivalent stiffness of the wall were investigated. Additionally, the stability of the stud walls under a vertical load was derived by considering the stability theory of the plate. The results indicated that the stability of the stud walls was mainly related to the height-thickness ratio of the walls. Finally, the correction factor of the allowable height-thickness ratio of the walls was obtained by analyzing the parameters of the wall thickness, stud spacing, sheathing materials, and wall opening, and a case study was used to illustrate the procedure of this simplified method.

A new approach to the calculation of variable tangent bending stiffness for helical strands

As the cross-section of the helical strand increases, the effect of its nonlinear bending characteristics becomes significant. To accurately simulate the response of helical strands under such conditions using the finite element method, determining the tangent bending stiffness of helical strands is indispensable. However, the approach of calculating the tangent bending stiffness of helical strands based on the stick-slip state of wires is not universally applicable across all strand models. In this study, we propose an approach for calculating the tangent bending stiffness of strands based on two mechanical models of helical strands. This proposed approach uses the ratio of the “actual increment” to the “theoretical maximum increment” of the nonlinear component of the wire axial force to calculate the contribution of one wire to the tangent bending stiffness of the strand. Significantly, this approach can be applied to different strand models, each of which may adopt different slip criteria. By comparing the results with the numerical solutions, we validate the effectiveness of the proposed approach for calculating the tangent bending stiffness of helical strands on different strand models. Furthermore, it is pointed out that the analytical results of wire sliding may vary significantly depending on the adopted slip criteria.

The effect of testing rate on biomechanical measurements related to stalk lodging

BackgroundStalk lodging (the premature breaking of plant stalks or stems prior to harvest) is a persistent agricultural problem that causes billions of dollars in lost yield every year. Three-point bending tests, and rind puncture tests are common biomechanical measurements utilized to investigate crops susceptibility to lodging. However, the effect of testing rate on these biomechanical measurements is not well understood. In general, biological specimens (including plant stems) are well known to exhibit viscoelastic mechanical properties, thus their mechanical response is dependent upon the rate at which they are deflected. However, there is very little information in the literature regarding the effect of testing rate (aka displacement rate) on flexural stiffness, bending strength and rind puncture measurements of plant stems.ResultsFully mature and senesced maize stems and wheat stems were tested in three-point bending at various rates. Maize stems were also subjected to rind penetration tests at various rates. Testing rate had a small effect on flexural stiffness and bending strength calculations obtained from three-point bending tests. Rind puncture measurements exhibited strong rate dependent effects. As puncture rate increased, puncture force decreased. This was unexpected as viscoelastic materials typically show an increase in resistive force when rate is increased.ConclusionsTesting rate influenced three-point bending test results and rind puncture measurements of fully mature and dry plant stems. In green stems these effects are expected to be even larger. When conducting biomechanical tests of plant stems it is important to utilize consistent span lengths and displacement rates within a study. Ideally samples should be tested at a rate similar to what they would experience in-vivo.

Nonlinear Dynamics and Meshing State Transition Analysis of Spalling Defect Gears with Multi-Lubrication Conditions

The purpose of this paper is to analyze the dynamic characteristics and the meshing state transition characteristics of spalling defect gears under various lubrication conditions. The time-varying geometric parameters, motion parameters and load distribution of the gears in the state of positive meshing and reverse impact are analyzed. The time-varying meshing stiffness considering the time-varying friction coefficient under elastohydrodynamic lubrication (EHL), mixed lubrication and boundary lubrication are calculated, and the spalling defect is introduced in the stiffness calculation process. The equivalent lubrication model of the gear is established, the switching condition of the lubrication state is derived, and the distribution of the lubrication state in the torque-speed plane is discussed. The multimeshing state dynamic model of the spalling defect gear is established, and the evolution characteristics of the coexistence response and the transition of the meshing state under different lubrication conditions are simulated by using the attraction domain, phase trajectory, meshing force and bifurcation diagram. The results show that the phase trajectory of the gear with the spalling defect is deformed compared with that of the healthy gear. The change of lubrication state has little effect on the phase trajectory and attraction domain of the gear with spalling defect, but has a great influence on the meshing force. At the spalling boundary, the meshing force is changed abruptly, and the worse the lubrication state, the more obvious the mutation. Due to the influence of spalling on tooth stiffness, it will also affect the meshing force in the nonspalling area.

Bridge damage identification based on synchronous statistical moment theory of vehicle–bridge interaction

AbstractConsidering the weak noise resistance and low identification efficiency of traditional bridge damage identification methods, a data‐driven approach based on synchronous statistical moment theory and vehicle–bridge interaction vibration theory is proposed. This method involves two main steps. First, a two‐axle test vehicle is used to collect acceleration response signals synchronously from adjacent designated measurement points while stationary. This operation is repeated to calculate the second‐order statistical moment curvature (SOSMC) difference of entire bridge points corresponding signals in different states. By comparing with the reference value, the preliminary damage location of the bridge can be obtained. Second, the first‐order modal shape curve is constructed using the second‐order statistical moment (SOSM). The refined identification of bridge damage is then based on an improved direct stiffness back calculation of the bridge's stiffness. This article proposes the synchronization theory for the first time and combines it with the statistical moment clustering method, forming an innovative approach to obtaining structural vibration modes. The effectiveness of this method has been well validated through numerical simulations with different parameters and on‐site bridge tests. The research results indicate that SOSMC indicators have better noise resistance and higher recognition efficiency in identifying damage locations, compared to modal curvature and flexibility curvature indicators. Additionally, compared to transfer rate and random subspace methods, the SOSM method results in smaller error and higher identification efficiency.

Identification of force chains in wet coal dust layer and the effect of porosity on three-body contact stiffness

Aiming at the three-body contact problem of mechanical rough surface containing wet coal dust interface, the three-body contact model of rough surface containing wet coal dust interface is constructed by comprehensively considering the contact deformation of rough surface and contact characteristics of wet coal dust, and based on the crushing theory. By analysing the contact force, load-bearing particle size and adjacent contact angle thresholds of the wet coal dust layer, the force chain identification criterion is formulated. Finally, quantitative calculations of the force chain characteristics are performed to reveal the effect of different initial porosities on the three-body contact stiffness, which is verified experimentally. The results of the study show that the average contact force of the wet coal dust layer can be used as the force chain contact force threshold, the average particle size can be used as the force chain particle size threshold, and the force chain angle threshold is determined by the particle coordination number. As the initial porosity decreases, the number, length and stiffness of force chains in the wet coal dust layer increase significantly, and the stiffness reaches a maximum value of 2.007 × 108 pa/m at the moment of downward pressure to stabilisation, while the trend of force chain bending varies in the opposite direction, and its minimum bending degree decreases to 20°. The maximum relative error between the simulation and experimental results of three-body contact stiffness is 9.64%, which proves the accuracy of the force chain identification criterion and the quantitative calculation of three-body contact stiffness by force chain.

A shape-independent analytical method for gear mesh stiffness with asymmetric spalling defects

Spalling, a common failure mechanism of gear systems, greatly affects the dynamics of gears operation, which is reflected in the time-varying mesh stiffness (TVMS). Current TVMS models often overestimate the asymmetric spalling phenomenon and may lead to inaccuracy in identifying and predicting the spalling failure. To address this problem, in this paper, a new stiffness, namely torsional stiffness, is introduced to quantify the effect of asymmetric spalling defects, and an equivalent stiffness calculation method for different asymmetric shapes is proposed. Based on this, a shape-independent TVMS model is constructed, which can realize the fast calculation of TVMS for spalling defects with different shapes at arbitrary asymmetric locations. Furthermore, a FEM-based validation method is developed by considering diverse loading states and improving the current result extraction method. Case studies are presented to illustrate the proposed model and to analyze the effects of different types of asymmetric spalling defects on gear dynamics. The FEM validation has shown that the proposed model has a good effectiveness.

A comparison of rolling element bearing stiffness calculation methods

PurposeThis study examines two distinct bearing stiffness calculation methods, both of which are based on the displacement-load function. Previous research typically incorporated one type of bearing stiffness into their system mechanics or vibration analysis. However, these two methods of calculating stiffness lead to different vibration models. This implies that the choice for vibration investigation is not merely about selecting one of the two types of stiffness, but also about how to appropriately implement that chosen stiffness within a model. The primary objective of this work is to compare these two methods of bearing calculation and to discuss the suitable applications of each method in both static and dynamic analyses.Design/methodology/approachThis study compares two distinct methods for calculating bearing stiffness. It explores the relationships between varying bearing stiffnesses, their internal structures, and contact features. Furthermore, it examines the impact of external loads on the static properties and dynamic characteristics of different bearing stiffnesses. Finally, based on the outcomes observed under various operating conditions, the study discusses the suitability of each method for static and dynamic analysis.FindingsMean stiffness is more suitable for calculating load transmissibility in a static state or capturing the delivery performance at instantaneous equilibrium positions in a dynamic state. Since the variation of the equilibrium positions is ignored, the alternating stiffness model is better suited for capturing the fluctuating properties of the vibration behaviors, especially under variable external load conditions.Originality/valueWe compare the two bearing calculation methods and discuss the appropriate applications of each method for static and dynamic analysis.

Research on Simplified Calculation of Initial Bending Stiffness of T-Stubs

Research on Simplified Calculation of Initial Bending Stiffness of T-Stubs

Effect of oil film stiffness on vibration of full ceramic ball bearing under grease lubrication

PurposeThis paper aims to understand the vibration characteristics of full ceramic ball bearings under grease lubrication, reduce the vibration of the bearings and improve their service life.Design/methodology/approachThe Hertz contact stiffness formula for full ceramic ball bearings is constructed; the equivalent comprehensive stiffness calculation model and vibration model of full ceramic ball bearings are established. The dynamic characteristic test of full ceramic ball bearing under grease lubrication was carried out by using the bearing life testing machine, and its vibration was measured, and its vibration acceleration root-mean-square was obtained by software calculation and compared with the simulation results.FindingsAt the rotational speed of 12,000 r/min, the root-mean-square value of vibration acceleration is maximum 10.82 m/s2, and the error is also maximum 7.49%. As the rotational speed increases, the oil film stiffness decreases. In the radial load of 600 N, the vibration acceleration root-mean-square is minimum 6.40 m/s2, but its error is maximum 6.56%. As the radial load increases, the vibration of the bearing decreases and then increases, so under certain conditions increasing the radial load can reduce the bearing vibration. With different types of grease, the best preload is also different; low-speed heavy load should be used when the viscosity of the grease is large, and high-speed light load should be used when the choice of smaller viscosity grease is made.Originality/valueIt provides a theoretical basis for the application of full ceramic ball bearings under grease lubrication, which is of great significance for reducing the vibration of bearings as well as enhancing the service life of bearings.Peer reviewThe peer review history for this article is available at: https://publons.com/publon/10.1108/ILT-03-2024-0094/

Meshing stiffness characteristics of modified variable hyperbolic circular-arc-tooth-trace cylindrical gears

Abstract. Stiffness excitation is one of the important excitations for the variable hyperbolic circular-arc-tooth-trace (VH-CATT) cylindrical gear system. Accurate calculation of the gear meshing stiffness is of great significance to investigating dynamic characteristics of the VH-CATT cylindrical gear system. Firstly, based on the forming theory of the modified tooth surface, the modified tooth surface equation of the VH-CATT cylindrical gear was deduced, and the 3D reconstruction was realised. Next, the load tooth contact analysis (LTCA) model of the VH-CATT cylindrical gear was developed to calculate the meshing stiffness of the VH-CATT cylindrical gear, and it was verified by the finite-element calculation. Finally, the influence of the load and modification parameters on the VH-CATT cylindrical gear stiffness was investigated. Research shows that the stiffness calculation method of the VH-CATT cylindrical gear based on LTCA is accurate. The meshing stiffness of the VH-CATT cylindrical gear in the double-tooth meshing area is large, and the meshing stiffness of the VH-CATT cylindrical gear in the single-tooth meshing area is small. The stiffness of the VH-CATT cylindrical gear increases with an increase in the load and cutter inclination angle, the stiffness of the VH-CATT cylindrical gear only in the double-tooth meshing area decreases with an increase in the parabolic coefficient, and the stiffness of the VH-CATT cylindrical gear increases with a decrease in the blade parabolic vertex position value. The research results provide a basis for improving the bearing capacity of the VH-CATT cylindrical gear and optimising design.

Stiffness analysis and structural optimization design of an air spring for ships

An air spring (AS) for ships must have the structural strength of its bellows enhanced considerably to ensure its reliability under high internal pressure and strong impact. In this case, the stiffness of the bellows gradually dominates the overall stiffness of the AS. Nevertheless, the parameterization calculation of stiffness for an AS mainly focuses on its pneumatic stiffness. The bellows stiffness is normally analyzed by virtue of equivalent simplification or numeric simulation. There is not an effective parameterization calculation model for the stiffness of the bellows, making it difficult to achieve the structural optimization design of the bellows. In this paper, the shell theory was borrowed to build a mechanical model for the bellows. Subsequently, the state vector of the bellows was solved by precision integration and boundary condition. Iteration was conducted to identify the complex coupling relationship between the vector of the bellows and other parameters. On this basis, the parameterization calculation method was introduced for the stiffness of the bellows to obtain the vertical and horizontal stiffness of the AS. After that, a dual-membrane low-stiffness structure was designed to analyze the dominating factors affecting the strength and stiffness of the AS, which highlighted the way to the low-stiffness optimization design of high-strength ASs. In the end, three prototypes and one optimized prototype were tested to verify the correctness of the parameterization design model for stiffness as well as the effectiveness of the structural optimization design.

The anomalous crack growth behaviour of an elastic-brittle octet-truss architected solid

Strongly rising R-curves are observed for octet-truss architected material specimens comprising ∼1 million unit cells and made from an elastic-brittle parent material. The measurements are supported by finite element (FE) calculations where the octet-truss is modelled discretely and show that the pseudo-ductility is not related to the usual toughening mechanisms such as crack bridging or crack-tip plasticity. Rather, the toughening is attributed to the three-dimensional (3D) structure of the octet-truss specimen: remarkably, no R-curve effect is observed in a quasi two-dimensional (2D) octet-truss specimen. This anomalous behaviour is clarified via FE calculations of the indentation stiffness of the octet-truss which reveal a strong indentation size effect in 3D with the stiffness decreasing with decreasing indenter size relative to the cell size of the octet-truss. In the 3D octet-truss, the size independent continuum stiffness is only attained for indenter sizes greater than about 20 unit cells while the size effect is almost absent in the quasi-2D octet-truss. The strong elastic softening in the 3D octet-truss in the presence of strain gradients gives rise to a large elastic crack-tip process zone. This in turn implies that a specimen geometry independent fracture toughness (i.e., a material property) can only be measured in specimens with large numbers of unit cells. We report a fracture mechanism map that reveals that measurements of specimen geometry independent fracture toughness can only be made in specimens with more than ∼10 million unit cells. This map is also used to rationalise the measured rising R-curves in the 3D specimens with ∼ 1 million unit cells. We end with a discussion on the implication of these findings for the laboratory measurement of the fracture toughness of both architected materials as well as biological cellular materials such as bone.

Experimental and analytical evaluation of semi-rigid timber connection with screwed-in threaded rods and steel coupling part

A major challenge for mid and high-rise timber buildings is fulfilling serviceability requirements. Moment-resisting timber frames with semi-rigid moment-resisting connections as the primary lateral load-resisting system can be an alternative to buildings with wind bracing or shear walls, allowing for more free space, enhanced building performance, and architectural flexibility. Moreover, introducing semi-rigid timber connections can improve the performance of floors against human-induced vibrations and potentially decrease the material used. The performance of moment-resisting frames depends mainly on the properties of their beam-to-column connections. The paper presents an experimental investigation of semi-rigid moment-resisting connections with long screwed-in threaded rods and a steel coupling part. Long threaded rods (i.e., screwed-in threaded rods with wood screw threads) feature high axial stiffness and resistance and allow for immediate load transfer without initial slip. Due to these properties, threaded rods are preferred over conventional dowel-type fasteners such as bolts and dowels. Five full-scale moment-resisting timber connections were tested under cyclic and monotonic loading to investigate moment resistance, rotational stiffness, and energy dissipation. The stiffness obtained by cyclic and monotonic tests was compared with numerical and analytical methods. The beam-to-column moment-resisting connections demonstrated a high moment resistance of 92.2 kNm and rotational stiffness of 8049 kNm/rad without initial slip. The experimental results and analytical calculations of rotational stiffness differed only by 5 %, whereas the finite element model exhibited a discrepancy of approximately 35 %. The equivalent viscous damping was determined based on cyclic load tests at service load levels and was found to be in the range of 5.5 %–10.0 %. Finally, a parametric study is presented to discuss the effect of semi-rigid connections on the performance of the timber floors and material savings. The increase in rotational stiffness of the beam-to-column connections could lead to up to 13 % material savings.

Calculation of magnetic stiffness over torque between two current-carrying circular filaments arbitrarily positioning in the space

The sets of expressions for calculation of sixteen components of magnetic stiffness over torque between two current-carrying circular filaments arbitrarily positioning in the space are derived by using mutual inductance and Grover’s method. The expressions are presented in the analytical form via integral equations, whose kernels include the elliptic functions of first and second kinds. The derived sets of formulas were mutually validated and results of calculation of components of magnetic stiffness agree well to each other.

Time-varying mesh stiffness calculation and dynamic modeling of spiral bevel gear with spalling defects

Time-varying mesh stiffness calculation and dynamic modeling of spiral bevel gear with spalling defects

Loaded tooth contact analysis and meshing stiffness calculation for cracked spiral bevel gears

Tooth cracks may occur in spiral bevel gear transmission system of the aerospace equipment. In this study, an accurate and efficient loaded tooth contact analysis (LTCA) model is developed to predict the contact behavior and time-varying meshing stiffness (TVMS) of spiral bevel gear pair with cracked tooth. The tooth is sliced, and the contact points on slices are computed using roll angle surfaces. Considering the geometric complexity of crack surface, a set of procedures is formulated to generate spatial crack and determine crack parameters for contact points. According to the positional relationship between contact point and crack path, each sliced tooth is modeled as a non-uniform cantilever beam with varying reduced effective load-bearing tooth thickness. Then the compliance model of the cracked tooth is established to perform contact analysis, along with TVMS calculations utilizing three different models. By employing spiral bevel gear pairs with distinct types of cracks as examples, the accuracy and efficiency of the developed approach are validated via comparative analyses with finite element analysis (FEA) outcomes. Furthermore, the investigation on effects of cracks shows that tooth cracks can induce alterations in meshing performance of both entire gear pair and individual tooth pairs, including not only cracked tooth pair but also adjacent non-cracked tooth pairs. Hence, the proposed model can serve as a useful tool for analyzing the variations in contact behavior and meshing stiffness of spiral bevel gears due to different cracks.