Modelling of static structural stability problems is considered. Focus is set on problems where passive physical constraints affect the response to applied forces, and where more than one free parameter describes the setting. The existence of vibration frequencies at equilibrium states is investigated, as an indication of stability. The relevant Jacobian matrix is developed, with an emphasis on the necessity to formulate the constraint equations from an energy form in a conservative problem. The corresponding mass matrix is introduced, with zero mass contribution from constraint equations. Three different forms of the relevant Jacobians are considered, and alternative methods for the eigenvalue extraction given. Stability is discussed in a context of generalized equilibrium problems, where auxiliary parameters and equations can be included in a continuation setting. Examples show the formulation, implementation and interpretation of stability.

Modelling of structural instability problems is considered for thin square membranes subjected to hydrostatic pressure, with a focus on the effects from symmetry conditions considered or neglected in the model. An analysis is performed through group-theoretical concepts of the symmetry aspects present in a flat membrane with one-sided pressure loading. The response of the membrane is described by its inherent differential eigensolutions, which are shown to be of five different types with respect to symmetry. A discussion is given on how boundary conditions must be introduced in order to catch all types of eigensolutions when modelling only a subdomain of the whole. Lacking symmetry in a FEM model of the whole domain is seen as a perturbation to the problem, and is shown to affect the calculated instability response, hiding or modifying instability modes. Numerical simulations verify and illustrate the analytical results, and further show the convergence with mesh fineness of different aspects of instability results.

KTH, School of Engineering Sciences (SCI), Mechanics.

Nordmark, Arne

KTH, School of Engineering Sciences (SCI), Mechanics.

A simple model for the falling cat problem2018In: European journal of physics, ISSN 0143-0807, E-ISSN 1361-6404, Vol. 39, no 3, article id 035004Article in journal (Refereed)

Abstract [en]

We introduce a specific four-particle, four degree-of-freedom model and calculate the rotation that can be achieved by purely internal torques and forces, keeping the total angular momentum zero. We argue that the model qualitatively explains much of the ability of a cat to land on its feet even though released from rest upside down.

KTH, School of Engineering Sciences (SCI), Mechanics.

Essén, Hanno

KTH, School of Engineering Sciences (SCI), Mechanics.

An impacting linear three body system2018In: European journal of physics, ISSN 0143-0807, E-ISSN 1361-6404, Vol. 39, no 1, article id 015001Article in journal (Refereed)

Abstract [en]

We study a system of three identical bodies that can move freely on a horizontal track. Initially one body moves and two are at rest. The moving body impacts with one of the resting bodies which then impacts with the third and so on. The impacts are assumed to be characterised by a coefficient of restitution. We investigate the total number of impacts, the final velocities of the bodies, and the final energy of the system in terms of the initial velocity and the coefficient of restitution. The problem, which originates from mechanics textbooks, can be analysed as a discrete dynamical system with three degrees of freedom. The full solution is more subtle that one might expect.

This paper analyzes in detail the dynamics in a neighborhood of a Genot-Brogliato point, colloquially termed the G-spot, which physically represents so-called dynamic jam in rigid body mechanics with unilateral contact and Coulomb friction. Such singular points arise in planar rigid body problems with slipping point contacts at the intersection between the conditions for onset of lift-off and for the Painleve paradox. The G-spot can be approached in finite time by an open set of initial conditions in a general class of problems. The key question addressed is what happens next. In principle, trajectories could, at least instantaneously, lift off, continue in slip, or undergo a so-called impact without collision. Such impacts are nonlocal in momentum space and depend on properties evaluated away from the G-spot. The answer is obtained via an analysis that involves a consistent contact regularization with a stiffness proportional to 1/epsilon(2) for some epsilon. Taking a singular limit as epsilon -> 0, one finds an inner and an outer asymptotic zone in the neighborhood of the G-spot. Matched asymptotic analysis then enables continuation from the G-spot in the limit epsilon -> 0 and also reveals the sensitivity of trajectories to epsilon. The solution involves large-time asymptotics of certain generalized hypergeometric functions, which leads to conditions for the existence of a distinguished smoothest trajectory that remains uniformly bounded in t and epsilon. Such a solution corresponds to a canard that connects stable slipping motion to unstable slipping motion through the G-spot. Perturbations to the distinguished trajectory are then studied asymptotically. Two distinct cases are distinguished according to whether the contact force becomes infinite or remains finite as the G-spot is approached. In the former case it is argued that there can be no such canards and so an impact without collision must occur. In the latter case, the canard trajectory acts as a dividing surface between trajectories that momentarily lift off and those that do not before taking the impact. The orientation of the initial condition set leading to each eventuality is shown to change each time a certain positive parameter beta passes through an integer. Finally, the results are illustrated in a particular physical example, namely the frictional impact oscillator first studied by Leine, Brogliato, and Nijmeijer.

A fluid-filled truncated spherical membrane fixed along its truncated edge to a horizontal, rigid and frictionless plane and spinning around a center axis was investigated. A two-parameter Mooney-Rivlin model was used to describe the material of the membrane. The truncated sphere was modeled in 3D using finite element meshes with different symmetry properties. A quadratic function was used for interpolating hydro-static pressure, giving a symmetric tangent stiffness matrix, thereby reproducing the conservative problem. Various problem settings were considered, related to the spinning, and different instability behaviors were observed. Multi-parametric problems were defined, generalized paths including primary and secondary paths were followed. Stability of the multi parametric problem was evaluated using generalized eigenvalue analysis based on the total differential matrix for the constrained problem. Numerical results showed that mesh symmetry affected the simulated stability behavior. Fold line evaluations showed the parametric effects on critical solutions.