Tangent stiffness matrix

About: Tangent stiffness matrix is a research topic. Over the lifetime, 1031 publications have been published within this topic receiving 21140 citations.

Non Linear Models for Cable Stayed Bridges Analysis

Abstract: In this paper we analyze the non linear behaviour of elastic cables acted upon by dead load and tensile forces. Such cables are often employed in long-span cable-stayed bridges. We also study the stay‘s stiffening obtained by a counterstays system and used to reduce the overall defor-mability of such bridges. The analysis is made by using two non linear finite element models here proposed and suitable to grasp the geometric nonlinearity of these structures.

Study of Size Effects on Stiffness Matrix

Abstract: The surface effect can be significant for nanoscale structures, and the surface energy is expected to be prominent in governing the geometric size-dependent deformation and strength mechanisms of single crystals at the nanoscale. In a new numerical method which combines surface energy and three-dimensional finite element analysis, size effects on the stiffness matrix with surface effects was studied numerically. Results show the surface stiffness matrix is more and more important relative to the bulk stiffness matrix with the size of elements decreasing.

Robot’s Cartesian Stiffness Adjustment Through the Stiffness Ellipsoid Shaping

Abstract: One of the key properties that define the behaviour of the compliant robot with its environment is end-effector (EE) Cartesian stiffness. Typically, EE stiffness is represented as a stiffness matrix whose design can be impossible due to the lack of degrees of freedom to adjust all the elements within the stiffness matrix. Therefore, a tradeoff between matrix elements must be made. This paper proposed an approach for offline shaping of the stiffness matrix using stiffness ellipsoids as an alternative, where stiffness is shaped by adjusting the ellipsoid orientation and axis magnitudes. Shaping of the ellipsoid requires fewer parameters that need to be adjusted relative to the stiffness matrix. A criteria function for shaping the ellipsoid stiffness has been proposed. Optimal values of joint positions and stiffness were computed using the algorithm based on SLSQP (Sequential Least SQuare Programming). Further analysis is conducted on parts of the trajectory that cannot satisfy EE ellipsoid shaping criteria. This is overcome by starching the critical parts of the trajectory by dividing it into smaller increments. Results show improvement in ellipsoid shaping and reduced orientation error. The algorithm is tested in a simulation environment on the KUKALWR model.

External stiffness approach for thin-walled frames with elastic-plasticity

Abstract: This work presents a one-dimensional finite element formulation for non-linear analysis of framed structures with thin-walled cross-section. Using the updated Lagrangian (UL) incremental formulation, the assumption of isotropic and linear-elastic material behaviour and the non-linear displacement field of thin-walled cross-section based on inclusion of second-order terms of large rotations, a tangent stiffness matrix of a two-node space beam element is firstly developed. Due to the non-linear displacement field, all internal moments occurring in the geometric stiffness are obtained as those of semitangential behaviour. In this way the joint equilibrium of non-collinear elements is provided. External stiffness approach (ESA) is applied in the force recovery phase. Material non-linearity is introduced for an elastic-perfectly plastic material through the plastic hinge formation at finite element ends and for this a plastic reduction matrix of the element is determined. The interaction of element forces at a hinge and the possibility of elastic unloading are taken into account. The effectiveness of the numerical algorithm discussed is validated through the test problem.

Exploiting frame-invariant operators for the efficient numerical simulation of flexible multibody systems

Abstract: Recently, the authors have presented a library of generic elements formulated in the special Euclidean group SE(3) formalism for the numerical simulation of flexible multibody systems. This library includes an implicit time integration method [1], a rigid body and kinematic joints [2], a geometrically exact flexible beam [3, 4], a geometrically exact flexible shell [5] and a geometrically exact super-element [6]. The geometric description of the elements is based on the representation of frame transformations as 4×4 homogeneous transformation matrices H = [ R x 01×3 1 ] (1) where x is a 3 × 1 vector and R is a 3 × 3 rotation matrix whose meaning is related to the element kinematics. For instance, for a beam, x accounts for the position of the neutral axis and R for the orientation of the cross-sections whereas, for a kinematic joint, x accounts for the relative displacements and R for the relative rotations in the joint. The proposed framework exhibits reduced non-linearities compared to classical formulations for two reasons • The equations of motion are expressed, both at position and rotation level, in local frames. Accordingly, the numerical expressions are invariant with respect to superimposed rigid body transformations. As a consequence, the tangent matrices and the constraint gradient are insensitive to large amplitude motions and depend on local transformations only, i.e. deformations and relative motions. • The equations of motion, which take the form of second order differential-algebraic equations on a Lie group, are solved without introducing a global parametrization of the motion, and in particular of the rotation. Several important properties also appear as a consequence of the formalism. For instance, the representation of large rotations is naturally singularity-free, the flexible elements are naturally locking-free and the initially curvature of flexible elements is trivially accounted for at any additional costs. In this work, computational strategies taking advantages of these convenient numerical properties are presented and several applications in flexible multibody systems are considered. As an example of the importance of the reduction of non-linearites, the 45◦-bend (Fig. 1a) presented in [7] and modelled with flexible beam elements can be solved in one load step using the material part of tangent stiffness matrix only. In the case of small deformations, most parts of the iteration matrix used in the implicit integration method do not have to be updated thanks to the invariance with respect to superimposed rigid body transformation, which leads to a significant reduction in computational time. Indeed, both the evaluation and the inversion of these parts can be done once for an entire simulation. This feature is applied to the analysis of the dynamic behaviour of a tape-spring (see representation in Fig. 1b) modelled with flexible shell elements and to the shape-optimization of a flexible robot.