On coupled transversal and axial motions of two beams with a joint (original) (raw)
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We report some results of an ongoing research related to a model consisting of an assembly of two beams, coupled to a simple joint through two legs. The motivation for this problem comes from the need to simulate the dynamic behavior of the next generation of inflatable/rigidizable space structures and to accurately account for damping and joint effects. We assume Kelvin-Voigt damping in the two beams and coupling through a joint which includes an internal moment. The resulting equations of motions consist of four, second order in time, partial differential equations, four second order ordinary differential equations, and certain compatibility boundary conditions. An explicit form for the energy of the system is found and its dissipativeness is proved. The system can be written as a second order ODE in an appropriate Hilbert space, in which well posedness, exponential stability as well as other regularity properties of the solutions can be obtained. A standard finiteelement approximation leads to a second-order differential-algebraic system including joint-forces and an algebraic constraint. A projection approach is used to eliminate joint-forces and enforce compatibility. In a second approach, the compatibility constraint is enforced in the construction of the finite element basis. Several numerical results are shown.
Mathematical and Computer Modelling, 2009
An important class of proposed large space structures features a triangular truss backbone. In this paper we study thermomechanical behavior of a truss component; namely, a triangular frame consisting of two thin-walled circular beams connected through a joint. Transverse and axial mechanical motions of the beams are coupled though a mechanical joint. The nature of the external solar load suggests a decomposition of the temperature fields in the beams leading to two heat equations for each beam. One of these fields models the circumferential average temperature and is coupled to axial motions of the beam, while the second field accounts for a temperature gradient across the beam and is coupled to beam bending. The resulting system of partial and ordinary differential equations formally describes the coupled thermomechanical behavior of the joint-beam system. The main work is in developing an appropriate state-space form and then using semigroup theory to establish well-posedness and exponential stability.
Analysis and Approximation of Viscoelastic and Thermoelastic Joint-Beam Systems
Rigidizable/Inflatable space structures have been the focus of renewed interest in recent years due to efficient packaging for transport. In this work, we examine new mathematical systems used to model small-scale joint dynamics for inflatable space truss structures. We investigate the regularity and asymptotic behavior of systems resulting from various damping models, including Kelvin-Voigt, Boltzmann, and thermoelastic damping. Approximation schemes will also be introduced. Finally, we look at optimal control for the Kelvin-Voigt model using a linear feedback regulator.
Buckling and nonlinear dynamics of elastically coupled double-beam systems
International Journal of Non-Linear Mechanics, 2016
This paper deals with damped transverse vibrations of elastically coupled double-beam system under even compressive axial loading. Each beam is assumed to be elastic, extensible and supported at the ends. The related stationary problem is proved to admit both unimodal (only one eigenfunction is involved) and bimodal (two eigenfunctions are involved) buckled solutions, and their number depends on structural parameters and applied axial loads. The occurrence of a so complex structure of the steady states motivates a global analysis of the longtime dynamics. In this regard, we are able to prove the existence of a global regular attractor of solutions. When a finite set of stationary solutions occurs, it consists of the unstable manifolds connecting them. & 2016 Elsevier Ltd. All rights reserved. the flexural rigidity of the beams and the viscosity of the external environment, respectively. Finally, γ is a positive constant, whereas the parameter ℓ summarizes the effect of the axial force acting at one end of each beam and is positive when both beams are stretched, negative when compressed. Contents lists available at ScienceDirect
Steady states of elastically-coupled extensible double-beam systems
arXiv (Cornell University), 2016
1 0 |u ′ (s)| 2 ds u ′′ + k(u − v) = 0 v ′′′′ − β + ̺ 1 0 |v ′ (s)| 2 ds v ′′ − k(u − v) = 0 describing the equilibria of an elastically-coupled extensible double-beam system subject to evenly compressive axial loads. Necessary and sufficient conditions in order to have nontrivial solutions are established, and their explicit closed-form expressions are found. In particular, the solutions are shown to exhibit at most three nonvanishing Fourier modes. In spite of the symmetry of the system, nonsymmetric solutions appear, as well as solutions for which the elastic energy fails to be evenly distributed. Such a feature turns out to be of some relevance in the analysis of the longterm dynamics, for it may lead up to nonsymmetric energy exchanges between the two beams, mimicking the transition from vertical to torsional oscillations.
Exponential stability of damped Euler-Bernoulli beam controlled by boundary springs and dampers
arXiv (Cornell University), 2023
In this paper, the vibration model of an elastic beam, governed by the damped Euler-Bernoulli equation ρ(x)u tt + µ(x)u t + (r(x)u xx) xx = 0, subject to the clamped boundary conditions u(0, t) = u x (0, t) = 0 at x = 0, and the boundary conditions (−r(x)u xx) x=ℓ = k r u x (ℓ, t) + k a u xt (ℓ, t), (− (r(x)u xx) x) x=ℓ = −k d u(ℓ, t) − k v u t (ℓ, t) at x = ℓ, is analyzed. The boundary conditions at x = ℓ correspond to linear combinations of damping moments caused by rotation and angular velocity and also, of forces caused by displacement and velocity, respectively. The system stability analysis based on well-known Lyapunov approach is developed. Under the natural assumptions guaranteeing the existence of a regular weak solution, uniform exponential decay estimate for the energy of the system is derived. The decay rate constant in this estimate depends only on the physical and geometric parameters of the beam, including the viscous external damping coefficient µ(x) ≥ 0, and the boundary springs k r , k d ≥ 0 and dampers k a , k v ≥ 0. Some numerical examples are given to illustrate the role of the damping coefficient and the boundary dampers.
Journal of Sound and Vibration, 2011
A novel state-space form for studying transverse vibrations of double-beam systems, made of two outer elastic beams continuously joined by an inner viscoelastic layer, is presented and numerically validated. As opposite to other methods available in the literature, the proposed technique enables one to consider (i) inhomogeneous systems,(ii) any boundary conditions and (iii) rate-dependent constitutive law for the inner layer. The formulation is developed by means of Galerkin-type approximations for the fields of transverse ...
Polynomial stability of a joint-leg-beam system with local damping
Mathematical and Computer Modelling, 2007
Recent advances in the design and construction of large inflatable/rigidizable space structures and potential new applications of such structures have produced a demand for better analysis and computational tools to deal with the new class of structures. Understanding stability and damping properties of truss systems composed of these materials is central to the successful operation of future systems. In this paper, we consider a mathematical model for an assembly of two elastic beams connected to a joint through legs. The dynamic joint model is composed of two rigid-bodies (the joint-legs) with an internal moment. In an ideal design all struts and joints will have identical material and geometric properties. In this case we previously established exponential stability of the beam-joint system. However, in order to apply theoretical stability estimates to realistic systems one must deal with the case where the individual truss components 1 are not identical and still be able to analyze damping. We consider a problem of this type where one beam is assumed to have a small Kelvin-Voigt damping parameter and the second beam has no damping. In this case, we prove that the component system is only polynomially damped even if additional rotational damping is assumed in the joint.
Dynamics of Non-viscously Damped Distributed Parameter Systems
46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, 2005
Linear dynamics of Euler-Bernoulli beams with non-viscous non-local damping is considered. It is assumed that the damping force at a given point in the beam depends on the past history of velocities at different points via convolution integrals over exponentially decaying kernel functions. Conventional viscous and viscoelastic damping models can be obtained as special cases of this general damping model. The equation of motion of the beam with such general damping model results in a linear partial integro-differential equation. Exact closed-form expressions of the natural frequencies and mode-shapes of the beam are derived. The analytical method is capable of handling complex boundary conditions. Numerical examples are provided to illustrate the new results.