Aki Mikkola

A non-incremental finite element procedure for the analysis of large deformation of plates and shells in mechanical system applications

In this investigation, a non-incremental solution procedure for the finite rotationand large deformation analysis of plates is presented. The method, whichis based on the absolute nodal coordinate formulation, leads to plateelements capable of representing exact rigid body motion. In thismethod, continuity conditions on all the displacement gradients areimposed. Therefore, non-smoothness of the plate mid-surface at the nodalpoints is avoided. Unlike other existing finite element formulationsthat lead to a highly nonlinear inertial forces for three-dimensionalelements, the proposed formulation leads to a constant mass matrix, andas a result, the centrifugal and Coriolis inertia forces are identicallyequal to zero. Furthermore, the method relaxes some of the assumptionsused in the classical and Mindlin plate models and automatically satisfiesthe objectivity requirements. By using a generalcontinuum mechanics approach, a relatively simple expression for theelastic forces is obtained. Generalization of the formulation to thecase of shell elements is discussed. As examples of the implementationof the proposed method, two different plate elements are presented; oneplate element does not guarantee the continuity of the displacementgradients between the nodal points, while the other plate elementguarantees this continuity. Numerical results are presented in order todemonstrate the use of the proposed method in the large rotation anddeformation analysis of plates and shells. The numerical results, whichare compared with the results obtained using existing incrementalprocedures, show that the solution obtained using the proposed methodsatisfies the principle of work and energy. These results are obtainedusing explicit numerical integration method. Potential applications ofthe proposed method include high-speed metal forming, vehiclecrashworthiness, rotor blades, and tires.

Flexible multibody simulation approach in the analysis of tibial strain during walking

Strains within the bone tissue play a major role in bone (re)modeling. These small strains can be assessed using experimental strain gage measurements, which are challenging and invasive. Further, the strain measurements are, in practise, limited to certain regions of superficial bones only, such as the anterior surface of the tibia. In this study, tibial strains occurring during walking were estimated using a numerical approach based on flexible multibody dynamics. In the introduced approach, a lower body musculoskeletal model was developed by employing motion capture data obtained from walking at a constant velocity. The motion capture data were used in inverse dynamics simulation to teach the muscles in the model to replicate the motion in forward dynamics simulation. The maximum and minimum tibial principal strains predicted by the model were 490 and -588 microstrain, respectively, which are in line with literature values from in vivo measurements. In conclusion, the non-invasive flexible multibody simulation approach may be used as a surrogate for experimental bone strain measurements and thus be of use in detailed strain estimations of bones in different applications.

Use of the finite element absolute nodal coordinate formulation in modeling slope discontinuity

A large rigid body rotation of a finite element can be described by rotating the axes of the element coordinate system or by keeping the axes unchanged and change the slopes or the position vector gradients. In the first method, the definition of the local element parameters (spatial coordinates) changes with respect to a body or a global coordinate system. The use of this method will always lead to a nonlinear mass matrix and non-zero centrifugal and Coriolis forces. The second method, in which the axes of the element coordinate system do not rotate with respect to the body or the global coordinate system, leads to a constant mass matrix and zero centrifugal and Coriolis forces when the absolute nodal coordinate formulation is used. This important property remains in effect even in the case of flexible bodies with slope discontinuities. The concept employed to accomplish this goal resembles the concept of the intermediate element coordinate system previously adopted in the finite element floating frame of reference formulation. It is shown in this paper that the absolute nodal coordinate formulation that leads to exact representation of the rigid body dynamics can be effecitively used in the analysis of complex structures with slope discontinuities. The analysis presented in this paper also demonstrates that objectivity is not an issue when the absolute nodal coordinate formulation is used due to the fact that this formulation automatically accounts for the proper coordinate transformations.

Two Simple Triangular Plate Elements Based on the Absolute Nodal Coordinate Formulation

In this paper, two triangular plate elements based on the absolute nodal coordinate formulation (ANCF) are introduced. Triangular elements employ the Kirchhoff plate theory and can, accordingly, be used in thin plate problems. As usual in ANCF, the introduced elements can exactly describe arbitrary rigid body motion when their mass matrices are constant. Previous plate developments in the absolute nodal coordinate formulation have focused on rectangular elements that are difficult to use when arbitrary meshes need to be described. The elements introduced in this study have overcome this problem and represent an important addition to the absolute nodal coordinate formulation. The two elements introduced are based on Spechts and Morleys shape functions, previously used in conventional finite element formulations. The numerical solutions of these elements are compared with previously proposed rectangular finite element and analytical results.

Development of Elastic Forces for a Large Deformation Plate Element Based on the Absolute Nodal Coordinate Formulation

Dynamic analysis of large rotation and deformation can be carried out using the absolute nodal coordinate formulation. This formulation, which utilizes global displacements and slope coordinates as nodal variables, make it possible to avoid the difficulties that arise when a rotation is interpolated in three-dimensional applications. In the absolute nodal coordinate formulation, a continuum mechanics approach has become the dominating procedure when elastic forces are defined. It has recently been perceived, however, that the continuum mechanics based absolute nodal coordinate elements suffer from serious shortcomings, including Poissons locking and poor convergence rate. These problems can be circumvented by modifying the displacement field of a finite element in the definition of elastic forces. This allows the use of the mixed type interpolation technique, leading to accurate and efficient finite element formulations. This approach has been previously applied to two- and three-dimensional absolute nodal coordinate based finite elements. In this study, the improved approach for elastic forces is extended to the absolute nodal coordinate plate element. The introduced plate element is compared in static examples to the continuum mechanics based absolute nodal coordinate plate element, as well as to commercial finite element software. A simple dynamic analysis is performed using the introduced element in order to demonstrate the capability of the element to conserve energy.

Which muscles compromise human locomotor performance with age

Ageing leads to a progressive decline in human locomotor performance. However, it is not known whether this decline results from reduced joint moment and power generation of all lower limb muscle groups or just some of them. To further our understanding of age-related locomotor decline, we compare the amounts of joint moments and powers generated by lower limb muscles during walking (self-selected), running (4 m s−1) and sprinting (maximal speed) among young, middle-aged and old adults. We find that age-related deficit in ankle plantarflexor moment and power generation becomes more severe as locomotion change from walking to running to sprinting. As a result, old adults generate more power at the knee and hip extensors than their younger counterparts when walking and running at the same speed. During maximal sprinting, young adults with faster top speeds demonstrate greater moments and powers from the ankle and hip joints, but interestingly, not from the knee joint when compared with the middle-aged and old adults. These findings indicate that propulsive deficit of ankle contributes most to the age-related locomotor decline. In addition, reduced muscular output from the hip rather than from knee limits the sprinting performance in older age.

A Linear Beam Finite Element Based on the Absolute Nodal Coordinate Formulation

In this paper, a new two-dimensional shear deformable beam element based on the absolute nodal coordinate formulation is proposed. The nonlinear elastic forces of the beam element are obtained using a continuum mechanics approach, without employing a local element coordinate system. In this study, linear polynomials are used to interpolate both the transverse and longitudinal components of the displacement. This is different from other absolute nodal-coordinate-based beam elements where cubic polynomials are used in the longitudinal direction. The use of linear interpolation polynomials leads to the phenomenon known as shear locking. This defect is avoided through the adoption of selective integration within the numerical integration method. The proposed element is verified using several numerical examples. The results of the proposed element are compared to analytical solutions and the results for an existing shear deformable beam element. It is shown that by using the proposed element, accurate linear and nonlinear static deformations, as well as realistic dynamic behavior including the capturing of the centrifugal stiffening effect, can be achieved with a smaller computational effort than by using existing shear deformable two-dimensional beam elements.

A Non-Incremental Nonlinear Finite Element Solution for Cable Problems

In this investigation, a nonlinear finite element method for the large deformation and rotation of cable problems is presented. This method is based on finite element absolute nodal coordinate formulation that guarantees the continuity of all displacement gradients and leads to a constant mass matrix. The classical cable theory is first reviewed and the assumptions used in this linear theory are defined in order to demonstrate the basic differences between the linear theory and the nonlinear finite element formulation proposed in this paper for cable applications. The elastic cable forces in the absolute nodal coordinate formulation are obtained using a general continuum mechanics approach that accounts for the effect of all geometric nonlinearities. It is shown in this investigation that the use of the general continuum mechanics approach leads to a simpler and more efficient formulation as compared to the use of the assumptions of the linear theory that employs a local finite element coordinate system. The results obtained using the absolute nodal coordinate formulation show a good agreement with the results obtained using the classical cable theory when linear cable problems are considered. In particular it is shown that the use of perturbation methods to linearize the finite element equations of motion leads to modal characteristics results that are in a good agreement with the linear theory. The results of this investigation obtained using explicit numerical integration also show the potential of the proposed finite element formulation in the nonlinear analysis of cables that experience large rotations and deformations. The generalization of the procedure presented in this paper to three-dimensional cable problems is demonstrated and the computer implementation in multibody algorithms is discussed.

Simple and Versatile Dynamic Model of Spherical Roller Bearing

Rolling element bearings are essential components of rotating machinery. The spherical roller bearing (SRB) is one variant witnessing increasing use because it is self-aligning and can support high loads. It is becoming increasingly important to understand how the SRB responds dynamically under a variety of conditions. This study introduces a computationally efficient, three-degree-of-freedom, SRB model that was developed to predict the transient dynamic behaviors of a rotor-SRB system. In the model, bearing forces and deflections were calculated as a function of contact deformation and bearing geometry parameters according to the nonlinear Hertzian contact theory. The results reveal how some of the more important parameters, such as diametral clearance, the number of rollers, and osculation number, influence ultimate bearing performance. One pair of calculations looked at bearing displacement with respect to time for two separate arrangements of the caged side-by-side roller arrays, when they are aligned and when they are staggered. As theory suggests, significantly lower displacement variations were predicted for the staggered arrangement. Following model verification, a numerical simulation was carried out successfully for a full rotor-bearing system to demonstrate the application of this newly developed SRB model in a typical real world analysis.

Dynamic Analysis of Rotor System With Misaligned Retainer Bearings

Active magnetic bearings present a technology that has many advantages compared to traditional bearing concepts. Active magnetic bearings, however, require retainer bearings in order to prevent damages in the event of a component, power, or a control system failure. In the drop-down, when the rotor drops from the magnetic field on the retainer bearings, the design of the retainer bearings has a significant influence on the dynamic behavior of the rotor. In this study, the dynamics of an active magnetic bearing supported rotor during the drop on retainer bearings is studied employing a simulation model. The retainer bearings are modeled using a detailed ball bearing model while the flexibility of the rotor is described using the finite element method with component mode synthesis. The model is verified by comparing measurements carried out using an existing test rig and simulation results. In this study, the verified simulation model is employed studying the effect of misalignment of retainer bearings during the rotor drop-down on the retainer bearings. It is concluded in this study that the misalignment of the retainer bearings is harmful and can lead to whirling motion of the rotor.