Calculation Of Stiffness Research Articles

Theoretical Model for the Analysis of Rotational Behavior of Penetrated Mortise-Tenon Joints in Traditional Timber Structures

ABSTRACT Penetrated mortise-tenon joints (PMJs) are typical wood-to-wood connections commonly used in traditional Chinese timber structures. They play a crucial role in the structural behavior of timber constructions. This study derived a method of theoretical estimation for the rotational behavior of PMJs, which is unique and more comprehensive in that it takes both movement of the rotation center and bending deformation of the tenon into consideration. In addition, it experimentally and numerically validated the theoretical model, and quantitatively analyzed the position changes of the rotation center and the effect of the bending deformation. On this base, simplified calculation formulas are proposed for the prediction of equivalent elastic stiffness and peak moment of PMJs, and then applied for the calculation of the lateral stiffness of timber frames. As a result, the predicted lateral stiffness shows good agreement with test results, demonstrating the validity of the estimation method derived and its applicability to the structural analysis. The results of this study show that the rotation center of the tenon is not stationary, and its moving range in the horizontal direction is much larger than that in the vertical direction. Another pattern found is that the bending deformation of the smaller tenon counteracts more than 20% of the compressive deformation caused by the rigid-body motion, and therefore must be considered when analyzing the rotational behavior of PMJs.

Improved analytical model for mesh stiffness calculation of cracked helical gear considering interactions between neighboring teeth

Improved analytical model for mesh stiffness calculation of cracked helical gear considering interactions between neighboring teeth

Influence of Imperfections on the Effective Stiffness of Multilayer Corrugated Board.

There are many possible sources of potential geometrical inaccuracies in each layer of corrugated board during its manufacture. These include, among others, the processes of wetting the corrugated layers during profiling, the process of accelerated drying, the gluing process, and any mechanical impact of the pressure rollers on the cardboard. Work taking into account all the above effects in numerical modeling is not well described in the literature. Therefore, this article presents a simple and practical procedure that allows us to easily account for geometric imperfections in the calculation of the effective stiffness of corrugated board. As a main tool, the numerical homogenization based on the finite element method (FE) was used here. In the proposed procedure, a 3D model of a representative volumetric element (RVE) of a corrugated board is first built. The numerical model can include all kinds of geometrical imperfections and is used to calculate the equivalent tensile and bending stiffnesses. These imperfections were included in the 3D numerical model by appropriate modeling of individual layers, taking into account their distorted shape, which was obtained on the basis of a priori buckling analysis. This paper analyzes different types of buckling in order to find the most representative one. The proposed procedure is easy to implement and fully scalable.

Methods of arterial stiffness calculation and cardiovascular disease events: the multiethnic study of atherosclerosis.

A wide variety of different formulae have been used to calculate local arterial stiffness with little external validation in relationship to cardiovascular events. We compared the associations of several arterial stiffness calculations in a large, multiethnic cohort. The multi-ethnic study of atherosclerosis (MESA) is a longitudinal study of 6814 adults without clinical cardiovascular disease (CVD) at enrollment. MESA participants with CVD surveillance through year 2018 and carotid ultrasound ( n = 5873) or aorta MRI ( n = 3175) at the baseline exam (2000-2002) were included. We analyzed 21 different calculations of local arterial stiffness. Cross-sectional and longitudinal statistical analyses were performed in addition to Cox hazard modeling for associations with CVD events (myocardial infarction, resuscitated cardiac arrest, stroke, adjudicated angina, and cardiovascular death). Carotid artery stiffness calculations had variable correlations with each other ( r = 0.56-0.99); aortic stiffness measures were similar ( r = 0.66-0.99). Nevertheless, for CVD events, the hazard ratio (HR) per standard deviation change were similar for all carotid stiffness calculations with HRs in the range of 1.00-1.10 (equivalence P < 0.001). For the aorta, aortic distensibility coefficient had a stronger association with CVD events (HR 1.18 [1.02-1.37]) compared to aorta Peterson's elastic modulus (HR 0.98 [0.89-1.07]) and aorta pulse wave velocity (HR 1.00 [0.90-1.11]). HRs between all other aortic stiffness calculations were equivalent ( P < 0.01). Different methods of calculating local arterial stiffness largely gave equivalent results, indicating that the variety of different arterial stiffness calculations in use do not cause inconsistent findings.

Evaluating the Time-Varying Mesh Stiffness of Planetary Gear Set With Gear Crack Considering the Friction Force Under Mixed Elastohydrodynamic Lubrication Condition

Abstract The mesh stiffness calculation of the planetary gear system with gear crack is investigated, and gear crack is modeled into end-face and addendum extended crack. Furthermore, the mesh stiffnesses are calculated under the effect of friction force obtained under the mixed elastohydrodynamic lubrication (EHL) condition. The nonuniformly distributed gear cracks on external and internal gear pairs are comprehensively modeled and analyzed, which is rarely investigated in previous studies. Moreover, the influence of crack propagation conditions on the meshing stiffness of the planetary gear system is analyzed. The obtained results show that the mesh stiffnesses experience abrupt variation in the pitch area during meshing, the extent of change is determined by the amplitude of the friction coefficient; in addition, gear crack position and extent could cause mesh stiffness reduction at varying levels at different stages of gear meshing.

Bending Stiffness of Honeycomb Paperboard

This article analyzes the influence of the initial deflection of the flat layers on the bending stiffness (BS) of honeycomb paperboards and presents two methods for its calculation. Both methods allow for the determination of BS in the main directions in the plane of the paperboard, i.e., the machine direction (MD) and the cross direction (CD). In addition, they have been verified by comparing the calculation results with the results of the BS measurements. The first method allowed for the calculation of the BS of cellular paperboard based on the mechanical properties of the paper used for its production. The second method allowed for the estimation of the BS of cellular paperboard based on the bending stiffness of other honeycomb paperboards with the same raw material composition and the same core cell size but with different thicknesses. In the first analytical method for the calculation of the bending stiffness of cellular paperboard, which does not include the deflections of the flat layers, the calculation results significantly differ from the measurement results, and they are overestimated. The second of the presented BS calculation methods allowed for a much more accurate assessment of paperboard's bending stiffness depending on its thickness.

Design and Analysis of a Novel Variable Stiffness Joint for Robot

The variable stiffness of robot joints plays an important role in improving the robot’s compliance, safety, and energy efficiency. In this paper, a novel type of variable stiffness joint based on a rack and pinion structure (VSJ-RP) is proposed. The structure and the variable stiffness principle of the joint are described in detail. The theoretical stiffness calculation and the dynamic model of the joint are established, and the correctness of the model is validated by simulation. The compliance, safety, energy storage, and release characteristics of the joint are validated by position, bearing capacity, hitting ball, and safety detection experiments, respectively. These experimental results show that the joint stiffness can be adjusted from 14.74 Nm/rad to 726.58 Nm/rad, and the overshoot of the position response is about 5.56–0.5%. The larger the stiffness of the joint, the faster the adjustment response, the smaller the fluctuation, and the more stable the operation are. The maximum output torque of the joint is about 20 Nm, and the torque difference between the minimum and the maximum stiffness of the joint is about 10%. The energy conversion efficiency of the joint is 17.56%~89.86%, and the deformation angle range is 2.66°~4.37°. These phenomena reflect the safety, energy storage, and release capacity of the joint. An effective exploration is performed regarding the miniaturization, safety, and energy utilization of robot variable stiffness joints.

Soil-Pile-Structure Interaction (SPSI) in Bridges: A Review

Response of soil influencing motion of pile supported structures or vice-versa is called Soil-Pile-Structure Interaction (SPSI). A part of SPSI is considered beneficial for the structure wherein it increases damping, and period of vibration. This study provides a comprehensive review addressing SPSI in bridge structures. Past research in this field has typically emphasised on Soil-Structure Interaction (SSI), rather than SPSI. An attempt is made here to compile the work done so far in this field. Detailed study, based on discretising the considered models in various articles, and the method for determining the stiffness of the underlying soil was carried out. The articles in this study were carefully chosen such that even if they have not considered SPSI, their models were worth reporting in this study. Response of soil and structure on the application of earthquake ground motion was also studied and discussed here with emphasis given on calculation of stiffness of soil and the interaction between them. It was observed that the material/spring used for modelling of soil is of utmost importance. The theories for the calculation of stiffness of the soil material for different locations of the pile length are also discussed. Few studies have also considered the vertical component of earthquake ground motion in the past. A detailed justification regarding the selection of method of analysis such as linear or non-linear analysis, and adopted approaches for analysis such as discrete or continuous approaches, and their overall influence on the study was observed, and reported here in a concise manner. This article primarily aims to summarize the previously published researches on SSI and SPSI, including their advantages and issues over the structure.

Seismic analysis and connection optimization of the side column on modular prefabricated four‐sided connected composite shear wall

SummaryA modular prefabricated four‐side connected composite (MPFC) shear wall that consists of a composite shear wall modular (CSWM) and steel frame boundary elements is proposed. First, the finite element model (FEM) of the MPFC shear wall, which considers a plastic‐damage constitutive model of both concrete and steel materials, is established based on the finite‐element software ABAQUS. Second, the FEM accuracy of the MPFC shear wall is verified by the experimental results of a modular prefabricated two‐sided connected buckling‐restrained (MTB) steel plate shear wall. Third, the seismic performance of the MPFC shear wall is investigated based on the verified FEM. The connection between the CSWM and side column is optimized. Finally, the initial stiffness calculation formula of the MPFC shear wall that considers the impact of that the thickness ratio between the connection steel plate (CSP) and inner steel plate (ISP) is deduced. The results show that the peak bearing capacity, initial stiffness, total energy, and total strain energy of the MPFC shear wall increased by 42.47%, 44.81%, 113.24%, and 58.97%, respectively, compared with those of the MTB steel plate shear wall. The compressive corner damage of the reinforcement concrete faceplate (RCF) of the MPFC shear wall with a side column and CSWM is connected by a middle steel plate is effectively improved. Compared with those of the MPFC shear wall in which the side column and CSWM are connected by bolts, the shearing force, axial force, and bending moment of the side column of the MPFC shear wall in which the side column and CSWM are connected by the middle steel plate are notably decreased by 25.32%, 26.08%, and 39.51%, respectively. The FEM results are compared with the formula calculation results to establish its accuracy in calculating the initial stiffness of the MPFC shear wall.

Research on the time-varying mesh stiffness method and dynamic analysis of cracked spur gear system considering the crack position

The occurrence of a crack could change the gear mesh stiffness and consequently influence gear vibration properties. Cracks may occur on the tooth root and tooth surface, but the surface crack type has been seldomly concerned. This study proposes a comprehensive time-varying mesh stiffness (TVMS) calculation method in which the crack position is considered and emphasized. Two tooth crack types (Root crack type: the crack occurs on the root fillet; Surface crack type: the crack occurs on the tooth surface) are modeled and analyzed, and corresponding analytical calculation expressions of TVMS are derived. The finite element method is applied to validate the applicability and accuracy of the proposed analytical method for both root and surface crack types. Besides, by establishing a nonlinear dynamic model for a spur gear system, the influences of the crack depth and crack position on the gear dynamic characteristics are investigated. Several statistic parameters are also applied to evaluate the health conditions for the spur gear system. The method and analysis results reported here may be of interest to researchers and engineers attempting to handle the gear fault analysis analytically and effectively.

Improved Mesh Stiffness Method and Vibration Analysis of a Planetary Gear System with a Spatial Tooth Crack

Dynamic modeling and analysis are generally regarded as effective tools to investigate the vibration characteristics and fault mechanisms of planetary gear systems with a tooth crack fault. In actual gearboxes, the tooth crack is always a three-dimensional spatial surface, but it was usually simplified as a two-dimensional domain in most previous studies. In this paper, the tooth crack is modeled as a spatial shape that propagates along the crack depth, length and height directions simultaneously. Based on the potential energy principle, an improved analytical method is proposed to calculate the time-varying mesh stiffness (TVMS) of a planetary gear system with a spatial tooth crack. Furthermore, a coupled translational–torsional dynamic model is established for a planetary gear system including time-varying parameters and nonlinear factors. Numerical simulations are conducted to reveal the influences of the spatial crack propagation on the TVMS and vibration responses. In addition, an experimental study is carried out on a gear transmission test rig to verify the proposed analytical method and dynamic model. The mesh stiffness calculation method of the spatial cracked tooth and corresponding analysis results in this study might provide references to detect tooth crack faults in planetary gear systems.

Magnetic stiffness calculation for the corresponding force between two current-carrying circular filaments arbitrarily oriented in the space

In this article, sets of analytical formulas for calculation of nine components of magnetic stiffness of corresponding force arising between two current-carrying circular filaments arbitrarily oriented in the space are derived by using Babic’s method and the method of mutual inductance (Kalantarov–Zeitlin’s method). Formulas are presented through integral expressions, whose kernel function is expressed in terms of the elliptic integrals of the first and second kinds. Also, we obtained an additional set of expressions for calculation of components of magnetic stiffness by means of differentiation of Grover’s formula of the mutual inductance between two circular filaments with respect to appropriate coordinates. The derived sets of formulas were mutually validated and results of calculation of components of magnetic stiffness agree well to each other.

Lamb wave mode conversion and multiple-reflection mechanisms for simply and reliably evaluating delamination in composite laminates

Lamb wave propagation must be understood comprehensively for simply evaluating delamination during ultrasonic testing. However, the difference between wave propagation, visualized using laser Doppler vibrometer and pulsed-laser scanners, has not been sufficiently investigated, and knowledge of optimal conditions for evaluating delamination is limited. Thus, in this study, the mode conversion and multiple reflections of Lamb waves propagating in a delaminated cross-ply laminate were visualized using different laser scanners, delamination depths, and wave incident angles. Delamination was characterized using maximum-amplitude map postprocessing under specific conditions. Further numerical analysis revealed that owing to multiple reflections of the antisymmetric mode in incident and mode-converted waves, standing waves were generated in the delaminated sublaminate. Dispersion curve and flexural stiffness calculations confirmed the conditions required for high-amplitude standing waves, thereby providing guidelines for simply and reliably evaluating delamination during inspections.

Performance of timber board models for prediction of local bending stiffness and strength - with application on Douglas fir sawn timber

Efficient utilization of structural timber requires accurate methods for machine strength grading. One of the most accurate methods presented this far is based on data of local fiber orientation on board surfaces, obtained from laser scanning. In this paper, two potential improvements of this method are examined. The first one consists of replacing a model based on simple integration over cross sections of boards for calculation of local bending stiffness by a 3D solid finite element (FE) model from which local bending stiffness is derived. The second improvement concerns replacement of a simple model for the fiber orientation in the interior of board by a more advanced one taking location of pith and growth direction of knots into account. Application of the alternative models on a sample of more than 200 Douglas fir boards, size 40 mm X 100 mm X 3000 mm, cut from large logs, show that each of the evaluated model improvements contributes to improved grading accuracy. When local bending stiffness is calculated utilizing the herein suggested FE model in combination with the improved model of fiber orientation in the interior of boards, a coefficient of determination to bending strength as high as 0.76 is obtained. For comparison, a coefficient of determination of 0.71 is obtain using the simpler original models.

Dynamic modelling and analysis of cracked gear system with tip relief based on proposed variable-angle deformation energy integration method

Due to ignoring the effects of the change of the tooth attachment position caused by the cracks, traditional time vary mesh stiffness (TVMS) calculation models and dynamic simulations for cracked gears will lose their precision in the body crack case. To address this shortcoming, a new analytical TVMS calculation model of cracked gear considering tip relief (TR) is developed based on a proposed variable-angle deformation energy integration method. On this basis, a dynamic model of the gear system for the analysis of fault vibration characteristics is established. The effectiveness and accuracy of the proposed TVMS calculation model are verified by the finite element method. A comprehensive investigation is finally carried out to reveal the effects of the parameters of TR, load and crack on the TVMS and dynamic characteristics of the cracked gears. The study results indicate that the proposed models can meet the accurate TVMS calculation and dynamic simulation for both the tooth- and body-cracked gears, and the influences of the tooth attachment position change caused by the crack cannot be ignored. This study could provide a systematic methodology and meaningful reference for the dynamic modelling, simulation and fault diagnosis of gear systems with crack failures.

Modeling of normal contact stiffness for surface with machining textures and analysis of its influencing factors

A new contact stiffness model for the rough surface with different machining textures is established. The studied surface consists of regular surface machining texture in the global scale and stochastic micro-asperities in the local scale. The proposed model improves upon previous work that considers only the effect of stochastic asperities and completely ignores that of the machining textures generated by different machining methods. The analytical expression to calculate the contact stiffness of the studied textured surface is derived and numerical results are obtained via the finite element method. A high-precision test bench to measure the interface contact stiffness of the workpiece with different surface textures is designed. The measured results and simulated results are directly compared to demonstrate the validity of the proposed method. The effects of different texture types and surface roughness values on the contact stiffness of the proposed textured surface are analyzed. The proposed method provides a research basis for the accurate calculation of nonlinear contact stiffness and dynamic characteristics of the surface with complex texture topography.

Performance and numerical modelling of full-scale demountable bolted PC wall panels subjected to cyclic loading

This paper proposes a lightly reinforced precast concrete (PC) wall panel building system with bolted connections, in which the bolted joints are designed to resist both tensile and shear forces. The proposed system offers a solution for sustainable low-rise prefabricated buildings which can shorten the construction period and realizing easy assembly and disassembly for sustainability concerns. Four wall specimens with different axial load ratios, concrete compressive strength, and aspect ratio, were experimentally and numerically investigated. The test results demonstrated that the panels provided a reliable bearing capacity. A finite element analysis (FEA) model was also developed by considering the interaction of the main components to study the effects of main variables on the seismic performance of the bolted PC panels, and to quantify the sensitivity of structural response and failure modes. Based on these results, a simplified design method was proposed for the calculation of the initial stiffness and bearing capacity of the bolted PC wall panels, which was proved with good assessment accuracy. The study suggested that the structural stiffness, ultimate capacity, and damage patterns of the PC wall panels can be reasonably controlled through a well design of the bolted joints.

A stiffness-based method for damage detection in 3-D offshore jacket platforms using inverse vibration problem (IVP) technique

More attention has been paid to Damage detection in offshore jacket platforms (OJP) because of repair and maintenance issues. Damage detection using modal data from a few sensors is one of the main important subjects that researchers have paid attention to it. The inverse Vibration Problem (IVP) technique does not need modal data in all nodes of OJP but first modal data in different levels of platform and principal degrees of freedom could be sufficient. Since the complex structural model becomes a reduced shear building by implementing this technique, there must be a reduced equivalent stiffness matrix for the healthy structure to make the comparison for damage detection. This paper deals with the formulation and implementation of IVP technique in corporation of the proper method for calculation of equivalent level stiffness. The proper method was selected based on a comparison of modal data obtained from full and reduced models using comparison tools like the MAC coefficient. Also, damage detection in a typical OJP model under a specific damage scenario was investigated numerically and experimentally and the results show the reliable performance of the methodology. A damage detection flowchart is suggested to detect damage location properly. It is developed mainly based on geometrical couplings in OJP structure in damaged conditions and the method leads to damage location exactly and maximum error in damage prediction is obtained at about 12%.

Dynamic response mechanism analysis of hinged FBG acceleration sensor

Considering the influence of the adhesive layer and hinge structure on the dynamic response effect of the hinged fiber Bragg grating (FBG) acceleration sensor, a multi-degree-of-freedom series vibration model and a transfer function were established based on the system dynamic equation. The stiffness calculation formulas of the hinge and adhesive layer were derived based on Castigliano’s second theorem. The results demonstrated that the response time of the sensor could significantly lag when the elastic modulus of the adhesive layer is lower than a certain critical value. The most crucial parameter affecting the dynamic response performance of the sensor is hinge thickness. The jagged oscillations in the sensor response signals can be reduced with the appropriate increase in the thickness, width, and semi-minor axis of the hinge as well as by reducing the semi-major axis of the hinge. However, it could also lead to an increase in the resonance frequency and a decline in sensitivity.

A Novel Semi-Analytical Approach to the Assessment of the Equivalent Stiffness of a Beam

Abstract This paper introduces several new approaches to stiffness calculation of the composite beam with a circular cross-section. The paper poses the problem of calculating the stiffness of the fibre-reinforced winded beam and introduces a new way of stiffness calculation. The novel semi-analytical calculation represents an approach that is less time consuming than conventional calculation methods or FEA methods. The expected application of the new method of calculation is in preliminary design and design calculations.