Per Lidström

The Influence of shearing and rotary inertia on the resonant properties of gold nanowires

In a previous publication [ P. A. T. Olsson, J. Appl. Phys. 108, 034318 (2010) ], molecular dynamics (MD) simulations have been performed to study the resonant properties of gold nanowires. It has been documented in the aforementioned publication that the eigenfrequencies of the fundamental mode follows the continuum mechanically predicted behavior when Bernoulli–Euler beam theory is used, whereas the higher order modes tend to be low in comparison to Bernoulli–Euler beam theory predictions. In this work, we have studied the resonant properties of unstressed and prestressed nanowires to explain why the eigenfrequencies of the fundamental mode follows the behavior predicted by Bernoulli–Euler beam theory while those of higher order modes are low in comparison. This is done by employing Timoshenko beam theory and studying the nanowire deformations for different modes. We find good agreement between the MD results and Timoshenko predictions due to the increasing importance of shearing and rotary inertia for higher order resonant modes. Furthermore, we argue that this type of behavior is merely a geometric effect stemming from low aspect ratio for the considered structures as a converging type of behavior is found when the aspect ratios fall between 15 and 20. Finally, we have found that classical Timoshenko beam theory that neglects nanoscale surface effects is able to, simply through utilization of the size dependent Young’s modulus, capture the dynamic properties of the gold nanowires as calculated through MD. (Less)

Moving regions in Euclidean space and Reynolds’ transport theorem

This paper gives new demonstrations of Reynolds’ transport theorems for moving regions in Euclidean space. For moving volume regions the proof is based on differential forms and Stokes’ formula. Moving curves and surface regions are defined and the intrinsic normal time derivative is introduced. The corresponding surface transport theorem is derived using the partition of unity and the surface divergence theorem. A proof of the surface divergence theorem is also given.

On the equations of motion in multibody dynamics

The equations of motion for a multibody system are derived from a continuum mechanical point of view. This will allow for the presence of rigid, as well as deformable, parts in the multibody system. The approach, using the principle of virtual power, leads to the classical Lagrange equations of motion. The generalized forces appearing in the equations are given as expressions involving contact, internal and body forces in the sense of continuum mechanics. A detailed analysis of the interaction between parts and their implication for the equations of motion is presented. Transformation properties, covariance and invariance under changes of configuration coordinates, are elucidated and a power theorem for the multibody system is proved. The equivalence between the standard balance equations for momentum and moment of momentum and the principle of virtual power is demonstrated.

A Joint-Space Parametric Formulation for the Vibrations of Symmetric Gough-Stewart Platforms

Natural frequencies of a Symmetric Gough-Stewart Platform (SGSP) mechanically limit its bandwidth and precision e.g. in CNCs or optical collimation systems. Hence, the required vibrational behavior at the neutral configuration of an SGSP can be regarded as an essential property to be optimized. However, due to the complexity of its geometry, the analysis of the vibrational behavior, using analytical methods, is quite challenging and in the literature a complete joint-space formulation of SGSP vibrations has not yet been addressed. In this paper, we present an analytical and parametric formulation of this problem in the joint space. We parametrically formulate the Jacobian matrix, the linearized equations of motion and calculate the eigenvectors and eigenfrequencies in terms of the design variables of the system. The parametric model presented in this study can be directly employed for design, optimization and control of SGSPs. It is concluded that for SGSPs, the joint-space formulation gives additional insights to the modal properties complementing the Cartesian-space analysis.

On the equations of motion in constrained multibody dynamics

The equations of motion for a constrained multibody system are derived from a continuum mechanical point of view. This will allow for the presence of rigid, as well as deformable, parts in the multibody system together with kinematical constraints. The approach leads to the classical Lagrange–d’Alembert equations of motion under constraint conditions. The generalized forces appearing in the equations of motion are given as expressions involving contact, internal and body forces in the sense of continuum mechanics. A detailed analysis of physical constraint conditions and their implication for the equations of motion is presented. A precise distinction is made between constraints on the motion on the one hand and resistance to the motion on the other. Transformation properties – covariance and invariance under changes of configuration coordinates – are elucidated. The elimination and calculation of the so-called Lagrangian multipliers is discussed and some useful reformulations of the equations of motion are presented. Finally a Power theorem for the constrained multibody system is proved.

Coordinate representations for rigid parts in multibody dynamics

The paper is concerned with coordinate representations for rigid parts in multibody dynamics. The discussion is based on the general theory of the dynamics of multibody systems under constraints. General configuration coordinates are introduced and the requirement of their regularity is discussed. The use of Euler angles, quaternions and linear coordinates, where quaternions and linear coordinates require constraint conditions, is analysed in detail. These coordinate systems are all shown to be regular. Equations of motion are formulated, using Lagrange’s as well as Euler’s equations, and they are supplemented by the appropriate constraint conditions in the cases of quaternions and linear coordinates. Mass matrices are derived, and in terms of the Euler angles the mass matrix components are products of trigonometric functions whereas in terms of quaternions the matrix components are quadratic polynomials. Using linear coordinates gives rise to a constant mass matrix. Thus, there is a decreasing degree of complexity, regarding mass matrix components, when going from Euler angles to linear coordinates. This is obtained at the expense of an increasing gross number of degrees of freedom and the necessary introduction of constraint conditions. The different equations of motion obtained are compared with respect to their structural complexity. In all representations the components of the angular velocity are explicitly calculated. This is not always the case in previous investigations of this subject. The paper also gives a new proof of the well-known relation between angular velocity and unit quaternions and their time derivative.

Simulation of Ductile Fracture of Slabs Subjected to Dynamic Loading Using Cohesive Elements

Dynamic loading of elastic-plastic slabs is studied numerically using a finite element approach, where introduction of cohesive elements enables fracture. For the bulk material, J2-flow theory is chosen as constitutive model, whereas the behavior of the cohesive elements is described by an irreversible cohesive law. The fracture behavior is governed solely by the cohesive elements, which are incorporated in the original mesh after a certain deformation state is reached. It is found that cohesive elements that are introduced early in the simulation are able to sustain the shape of the slab, so that necking and specimen fracture may occur at a location that does not coincide with the position of the first developed cohesive zone. This is an important feature, especially in the case of multiple necking. The influence of introducing cohesive elements on the deformation behavior is studied by comparison of contour plots describing the time development of normalized strain rates. Numerical simulations where the insertion of cohesive elements is enabled are compared with corresponding results where the material model only consists of J2-flow theory. Differences in the simulated material response, such as delay in the onset of necking, indicate that depending on the aim of the study, introduction of cohesive elements will influence the observed results. In addition, the proportion of kinetic energy of the post-fracture fragment compared to the amount of input work as function of imposed loading velocity is studied along with the effect of changing specimen aspect ratio.

Analysis of volume-average relations in continuum mechanics

In this paper, volume-average relations related to the multilevel modelling process in continuum mechanics are analysed and the concept of average consistency is investigated both analytically and numerically. These volume averages are used in the computational homogenization technique, where a transition of the mechanical properties from the local, microscopic, to the global, macroscopic, length scale is obtained. The representative volume element (RVE) is used as a reference placement and the solution, in terms of volume-averaged stress, will depend on which boundary conditions are chosen for the RVE. Three types of boundary conditions – periodic, affine and anti-periodic – are analysed with respect to the average consistence for the kinematical and stress relations used in continuum mechanics. The inconsistence is quantified by introducing the inconsistence ratio. It is shown analytically that some average stress relations are fulfilled, assuming the periodic boundary condition and anti-periodic traction vector, whereas the average relations connected to the deformation are in general not average consistent. The inconsistence is investigated in a plane model using the finite element technique. The numerical investigation has shown that the inconsistence ratios related to the deformation are also average consistent in the examples considered.

On the relative rotation of rigid parts and the visco-elastic torsion bushing element

In this paper the interaction between two rigid parts of a multibody, connected by an ideal spherical joint equipped with a visco-elastic torsion bushing element, is derived. The model allows for arbitrary relative rotations of the parts and involves a non-linear torsion stiffness of the bushing. An expression for the interaction between the parts is derived and a specialization to the isotropic bushing element is presented.

On the principle of virtual power in continuum mechanics

The principle of virtual power and its equivalence to the Euler laws expressing the global balance of momentum and moment of momentum is demonstrated. The demonstration includes bodies subjected to jump discontinuities (shocks) and external and internal constraints.