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14-September-2008 12:50:31 - relaxation The dynamic relaxation is a computation modeling, which can be used for the form-finding of cable and fabric structures. The dynamic relaxation method is base on a discretized continuum in which the mass is assumed to be lumped at given nodes. The system oscillates about the equilibrium position under the influence of loads. The iterative process is achieved by simulating a pseudo-dynamic process in time, this iteration are base on an update of the geometry time by time.1 Contents 1 Main equations use 2 Step of the iteration 3 Damping 4 See also 5 Further reading 6 References Main equations use Using the relation of the second Newton's laws by considering the node i, at the time t, in the x direction: R_ixt=M_iA_ixt\frac Where: R is the residual force M is the mass A is the acceleration By realizing a double numerical integration of the acceleration here by central finite difference form2, a relation between the speed V, the geometry X and the residuals is obtained: V_ix\leftt+ \frac \Delta t 2\right = V_ix \leftt- \frac \Delta t 2\right + \frac\Delta tM_iR_ixt X_it+ \Delta t=X_it- \Delta t+\Delta t \times V_ix \leftt+ \frac \Delta t 2\right Where: Δt is the time interval between two updates. Using a sum of the forces at the node, permit to obtain the relation between the residuals and the geometry: R_ixt+ \Delta t=P_ixt+ \Delta t+\sum \frac T_mt+ \Delta tl_mt+ \Delta t \times X_jt+ \Delta t-X_it+ \Delta t where: P is the applied load component T is the tension in link m between nodes i and j l is the length of the link. The sum is realizing on all the link of the node. By repeating the use of the relation between the residuals and the geometry and then the relation between the geometry and the residual, the pseudo-dynamic process is simulated. Step of the iteration 1. Define the initial kinetic energy and all nodal velocity components to zero: E_kt=0=0\frac V_it=0=0\frac 2. Compute the geometry set and the applied load component: X_it=0\frac P_it=0\frac 3. Compute the residual: T_mt\frac R_it\frac 4. Reset the residuals of constrained nodes to zero 5. Update velocity and coordinates: V_it+ \frac \Delta t2\frac X_it+\Delta t\frac 6. Return to step 3 until the structure is in static equilibrium Damping In the dynamic relaxation method, there is a possibility to use damping to enhance the method by reducing the number of iteration3. There are two method of damping: The viscous damping, which assume that cable possess a viscous comportment. The kinetic energy damping, this consist in at each time a kinetic energy peak which means a position of equilibrium is detected, the geometry is update to this position and the speed is reset. The viscous damping possesses the advantage to stay close to the reality of a cable which effectively possesses a viscous comportment. Moreover it is easy to realize because the speed is already computed. The kinetic energy damping is an artificial damping which differs from the reality but offer a drastic reduction of the number of iteration. However it needs to compute the kinetic energy and to detect peak of this energy, after detecting this energy peak the geometry has to be updated to this position. See also Tensile_structure Optimization_mathematics Further reading W J LEWIS, TENSION STRUCTURES: Form and behaviour, London, Telford, 2003 D S WAKEFIELD, Engineering analysis of tension structures: theory and practice, Bath, Tensys Limited, 1999 H.A. BUCHHOLDT, An introduction to cable roof structures, 2nd ed, London, Telford, 1999 References ^ W J LEWIS, TENSION STRUCTURES: Form and behaviour, London, Telford, 2003 ^ D S WAKEFIELD, Engineering analysis of tension structures: theory and practice, Bath, Tensys Limited, 1999 ^ W J LEWIS, TENSION STRUCTURES: Form and behaviour, London, Telford, 2003 Retrieved from http://en..org/wiki/Dynamic_relaxation Categories: Numerical analysis Views Article Discussion this page History Personal tools Log in / create account Navigation Main page Contents Featured content Current events Random article Search Go Search Interaction Community portal Recent changes Contact Donate to Help Toolbox What links here Related changes Upload file Special pages Printable version Permanent link Cite this page Languages Français This page was last modified on 13 August 2008, at 18:40
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