Linear Function Of Displacement Research Articles

Undamped Non-linear Vibration Mechanism of Bolted Flange Joint under Transverse Load

Bolted flange joints are commonly utilized in engineering constructions; nevertheless, their dynamic behavior is complicated when nonlinearity is present. The bolted flange joint structure in this research is considered as a spring-mass system, and the equivalent axial two bi-linear springs-bending beam model, as well as the system’s vibration control equation, is created to examine the joint structure’s vibration characteristics under transverse stress. The results indicate that when the spring stiffness is constant and the stiffness is a linear function of displacement, and the mass matrix of the vibration control equation is coupled, the system produces only transverse vibration upon transverse impact; when the spring axial stiffness is a linear function of displacement and the mass matrix of the vibration control equation is not coupled, the spring stiffness is bilinear and the stiffness is a quadratic function of displacement, the system will vibrate transversely and longitudinally under the transverse impact. Regardless of how the spring stiffness is simplified, the system’s transverse or transverse and axial movement is a stable periodic vibration with a vibration response proportional to the system’s mass and the radius L of the connected cylindrical shell. The theoretical method verifies the nonlinear vibration characteristics of the bolted flange structure under transverse load, which is of great significance for proposing new methods and models.

The elastic distortion problem with large rotation in discontinuous deformation analysis

Conventional discontinuous deformation analysis (DDA) results in a change of block volume, which is known as free expansion, during rotation calculations because of the use of a linear displacement function to simulate the behavior of a block with a rigid body and elastic behaviors. This study demonstrates that the linear displacement function also generates unsolved elastic distortion, especially when the block undergoes large rotation in each calculation step. The distortion disturbs the contact judgment in the open–close iteration and update calculations of vertex coordinates, stresses, velocities, etc. at the end of each calculation step. A new procedure follows the flow chart of the original DDA, but it adopts additional codes for the coordinate-transformation calculations in vertex coordinate, stress, and velocity updates. When the vertex coordinates are updated, vertex displacements caused by strains are calculated before involving the block-rotation term in the displacement function to mitigate the elastic distortion. In addition, new codes compile formulas to transform stresses and velocities with block rotation. The new DDA ensures the correctness of rotating elastic calculations to solve practical falling rock problems with a large rotational angle in each calculation step.

Validation and application of three-dimensional discontinuous deformation analysis with tetrahedron finite element meshed block

In the last decade, three dimensional discontinuous deformation analyses (3D DDA) has attracted more and more attention of researchers and geotechnical engineers worldwide. The original DDA formulation utilizes a linear displacement function to describe the block movement and deformation, which would cause block expansion under rigid body rotation and thus limit its capability to model block deformation. In this paper, 3D DDA is coupled with tetrahedron finite elements to tackle these two problems. Tetrahedron is the simplest in the 3D domain and makes it easy to implement automatic discretization, even for complex topology shape. Furthermore, element faces will remain planar and element edges will remain straight after deformation for tetrahedron finite elements and polyhedral contact detection schemes can be used directly. The matrices of equilibrium equations for this coupled method are given in detail and an effective contact searching algorithm is suggested. Validation is conducted by comparing the results of the proposed coupled method with that of physical model tests using one of the most common failure modes, i.e., wedge failure. Most of the failure modes predicted by the coupled method agree with the physical model results except for 4 cases out of the total 65 cases. Finally, a complex rockslide example demonstrates the robustness and versatility of the coupled method.

Stress recovery procedure for discontinuous deformation analysis

The discontinuous deformation analysis (DDA) has been widely applied for rock engineering problems due to its special features in modeling the discontinuous rock mass. A linear polynomial function is often used in the standard DDA to ease the complex contact determination between the blocks. This linear displacement function generates a constant stress field within a block, which cannot be effectively used to model the stress variation within a block and across the block interface, especially in the region with a large stress gradient. In this paper, a stress recovery procedure is proposed for those DDA blocks which are glued together. Such a procedure can improve the stress accuracy along the block interface, and can be used for more accurate contact determination. Two numerical examples are presented to study the stress accuracy of the proposed method, and the results show that the proposed stress recovery method provides a better accuracy than the direct DDA and the averaging method.

Computing lower and upper bounds on stress intensity factors in bimaterials

This paper presents a finite element approach for finding complementary bounds of stress intensity factors (SIFs) in bimaterials. The SIF is formulated as an explicit computable linear function of displacements by means of the two-point extrapolation method. An appropriate and computable form of the SIF plays a crucial role in the dual problem involved in the computing procedure of both lower and upper bounds. In our discussions, computable forms of stress intensity factors, K 0 and K r , are derived, which are related to the energy release rate, and the ratio of the open mode and shear mode SIFs, respectively. Based on a posteriori finite element error estimation, a bounding procedure is used to compute the bounds on the two stress intensity factors. Finally, bounds on the SIFs in a bimaterial interface crack problem are provided to verify the procedure.

A development of the discontinuous deformation analysis for rock fall analysis

Discontinuous deformation analysis (DDA), a discrete numerical analysis method, is used to simulate the behaviour of falling rock by applying a linear displacement function in the computations. However, when a block rotates, this linear function causes a change in block size called the free expansion phenomenon. In addition, this free expansion results in contact identification problems when the rotating blocks are close to each other. To solve this problem of misjudgment and to obtain a more precise simulation of the falling rock, a new method called Post-Contact Adjustment Method has been developed and applied to the program. The basic procedure of this new method can be divided into three stages: using the linear displacement function to generate the global matrix, introducing the non-linear displacement function to the contact identification, and applying it to update the co-ordinates of block vertices. This new method can be easily applied to the original DDA program, demonstrating better contact identification and size conservation results for falling rock problems than the original program. Copyright © 2005 John Wiley & Sons, Ltd.

A fuzzy approach for the analysis of unbalanced nonlinear rotor systems

The components of most structural and mechanical systems exhibit considerable variations or uncertainties in their properties. Thus, the performance characteristics of such systems are also subject to uncertainties. In the case of a rotor-bearing system, the nonlinear bearing restoring force is usually represented as a third or fourth power of displacement or as a piecewise linear function of displacement. The coefficients of these models are acquired from experiments and approximations, and will vary considerably during the operation of the bearing. Hence, it is more reasonable to treat them as uncertain values. Other bearing parameters such as the inertial properties of concentrated disks, distributed mass and damping of the rotating assemblies are also uncertain due to manufacturing and assembly errors and imprecise operating conditions. It is known that the vibration response of a rotor is highly sensitive to small fluctuations or variations in the bearing parameters. Therefore, any realistic analysis and design of rotor-bearing systems must take the uncertainties into account. In this paper, a methodology is presented for the fuzzy analysis of nonlinear rotor-bearing systems along with numerical results to demonstrate the computational feasibility of the methodology.

剛体回転に関する不連続変形法 (DDA) の研究

The separation of rigid body rotation term, r0, from the other displacement variables is a breakthrough idea in original discontinuous deformation analysis (DDA) formula. Although the original linear displacement function is a fast and efficient method in computation, it loses accuracy when blocks undergo large rigid body rotations. It can be a significant problem especially when solving the problem with high speed rigid body rotation, such as rock fall problem, or the larger time interval is introduced to the problem with rigid body rotation to obtain the results within less computation steps. Up to date, it is only known that the large rigid body rotation can cause free expansion and change the weight of block in the analysis. However, the authors consider that the problem can also affect the contact judgments in open-close iterations and cause wrong contact forces in the computation. In this paper, this problem will be discussed, and a new method named “post contact adjustment” is developed and applied to the contact computation terms of DDA. After the improvements, the simulation results show better contact computations and block area preservations even when the large rigid body rotation is carried out.

Rigid body rotation and block internal discretization in DDA analysis

In the original formulation of DDA by Shi, a linear displacement function term is used. This has the limitations of uncontrolled block and stress distortion due to rigid body rotation. In the present paper, the authors propose a new iterative method which can avoid the distortion due to the rotation even when the rotation or number of time-step is large. Furthermore, the authors propose a simple internal discretization scheme which is applicable for both concave and convex polygon which is particularly important for a large block. The stress and strain distribution with a large block can be obtained with ease under this scheme. The numerical examples as shown have demonstrated the advantages of the present proposal in DDA analysis. Copyright © 2000 John Wiley & Sons, Ltd.

A model for the effect of roughness on the load capacity of journal bearings

The flow generated by a rotating circular cylinder inside a corrugated outer cylinder is considered. The corrugations are small and are used to model roughness. The force on the inner cylinder, as a linear function of displacement, is found. This requires a Stokes expansion to determine a first-order correction in the Reynolds number R R . The result implies as expected that transverse roughness acts to increase the load capacity of the bearing at low R R , which is confirmed by previous numerical experiments. In addition, roughness can induce reversed flow at much smaller displacements than for smooth bearings.

Timoshenko beam finite element with four dof and convergence of O( h4)

In situations where transverse shear deformations and rotary inertia in beams are important, elements based on the Timoshenko beam theory are useful. Among the two-noded, four DOF elements derived from the minimum total potential energy principle, the HTK. element proposed by Hughes et al. using linear displacement functions for both w and θ and the T1CC4 element proposed by Tessler et al. using quadratic displacement function for w and linear displacement function for θ are well known in the literature. The convergence of the HTK element in the thin beam situation has been too poor due to shear locking but by using selective integration this element can be shown to be equivalent to the T1CC4 element which has a rate of convergence of O( h 2). In this paper a five DOF element with w and θ at the end nodes and θ at the middle node and based on the cubic displacement function for w and the quadratic displacement function for θ is first developed. Statically condensing the middle rotational DOF, the well-known (4 × 4) stiffness matrix using the φ-factor defined as φ = 12 EI/ kGAL 2 and hitherto obtained only through a flexibility approach or closed-form solution of the governing equations of the Timoshenko beam theory is derived. This element based on cubic displacement function for w has rate of convergence of O( h 4), is completely free of shear locking and performs equally well in thin as well as thick beam situations.

Fundamental analysis of deep sea mooring line in static equilibrium

The integrated equations of deep sea mooring lines in static equilibrium under consideration of gravity force and line elongation are obtained. The most useful forms of the equations for designing deep sea mooring system are those in which horizontal displacement and horizontal and vertical components of tension are obtained as functions of line length, water depth, critical tension and unit weight of line. When the dynamic simulation of a buoy system is carried out, line tensions acting on the buoy should be represented as a linear function of displacement. Therefore the linearisation method is introduced by use of linearisation coefficient matrix. Translation formulae for the matrix from two to three dimensions is also presented. The influence of line elongation on the horizontal component of tension in a taut mooring condition is important, therefore the elongation of stretched deep sea mooring line must be considered carefully. Also, the scaling law under consideration of line elongation in static equilibrium, which is necessary in estimation from a small sized experiment, is discussed.