Dynamical systems stiffness

J.C. Simo and O. Gonzales. Assessment of energy-monentum and symplectic schemes for stiff dynamical systems. The American Society of Mechanical Engineering, 93-WA/PVP-4, 1993. 

The three primary factors that determine the normal vibration energy levels and the resulting vibration profiles are mass, stiffness, and damping. Every machine-train is designed with a dynamic support system that is based on the following the mass of the dynamic component(s), specific support system stiffness, and a specific amount of damping. 

When the dynamic system is described by a set of stiff ODEs and observations during the fast transients are not available, generation of artificial data by interpolation at times close to the origin may be very risky. If however, we ob-... 

The set of four ordinary differential equations (7.64) to (7.67) for the dynamical system are quite sensitive numerically. Extreme care should be exercised in order to obtain reliable results. We advise our students to experiment with the standard IVP integrators ode... in MATLAB as we have done previously in the book. In particular, the stiff integrator odel5s should be tried if ode45 turns out to converge too slowly and the system is thus found to be stiff by numerical experimentation. 

The stiffness of a dynamical system can be characterised via its timescales. Remember that the ratio l/IRe(A,) I is called the /-th timescale of a dynamical system (see Sect. 6.3). The most widely used stiffness index is the reciprocal of the shortest timescale of the system ... 

Reduced models based on low-dimensional manifolds can usually be simulated faster than full systems of differential equations because the resulting dynamical system contains fewer variables and is usually not stiff (see Sect. 6.7). However, the search and retrieval algorithms required to access the look-up tables can consume significant amounts of computer time. As an example, the simulation of methane combustion based on the ILDM method was eight times faster than that using a detailed mechanism (Riedel et al. 1994). Special algorithms have been developed to speed up the search and retrieval process (Androulakis 2004). In situ tabulation methods have also been developed as discussed in Sect. 7.12 below. 

Goussis, D.A., Valorani, M. An efficient iterative algorithm fin the approximation of the fast and slow dynamics of stiff systems. J. Comput. Phys. 214, 316-346 (2006)... 

An approximate method for the response variability calculation of dynamical systems with uncertain stiffness and damping ratio can be found in Papadimitriou et al. (1995). This approach is based on complex mode analysis where the variability of each mode is analyzed separately and can efficiently treat a variety of probability distributions assumed for the system parameters. A probability density evolution method (PDEM) has also been developed for the dynamic response analysis of linear stochastic structures (Li and Chen 2004). In this method, a probability density evolution equation (PDEE) is derived according to the principle of preservation of probability. With the state equation expression, the PDEE is further reduced to a one-dimensional partial differential equation from which the instantaneous probability density function (PDF) of the response and its evolution are obtained. Finally, variability response functions have been recently proposed as an alternative to direct MCS for the accurate and efficient computation of the dynamic response of linear structural systems with rmcertain Young modulus (Papadopoulos and Kokkinos 2012). 

Usually, in numerically simulated dynamic systems, such as multistory buildings under earthquake excitations, structural impact is considered using force-based methods, also known as penalty methods. These methods allow relatively small interpenetration between the colliding strucmres, which can be justified by the local deformability at the point of impact. The interpenetration depth is used together with an impact stiffness coefficient, which represents an impact spring, to calculate the impact forces that act on the colliding structures and push them apart. 

The transformation of coordinate system is performed for each structural member. This allows one to enforce kinematic compatibility and equilibrium at all nodes connecting structural members. Then, the global dynamic system is constructed following classic direct stiffness procedure ... 

M. E. Tuckerman and B. J. Berne. Molecular dynamics in systems with multiple time scales Systems with stiff and soft degrees of freedom and with short and long range forces. J. Comp. Chem., 95 8362-8364, 1992. 

This method has a simple straightforward logic for even complex systems. Multinested loops are handled like ordinary branched systems, and it can be extended easily to handle dynamic analysis. However, a huge number of equations is involved. The number of unknowns to be solved is roughly equal to six times the number of node points. Therefore, in a simple three-anchor system, the number of equations to be solved in the flexibiUty method is only 12, whereas the number of equations involved in the direct stiffness method can be substantially larger, depending on the actual number of nodes. 

The four primary coolant pumps are connected to the secondary shield wall by three-link snubbers designed to be flexible under static applied loads (thus, allowing thermal expansion) but become stiff under dynamic loads that might occur during an earthquake. Accordingly, the system is coupled to the wall under seismic loading. 

Rotating machines subject to imbalance caused by turbulent or unbalanced media flow include pumps, fans, and compressors. A good machine design for these units incorporates the dynamic forces of the gas or liquid in stabilizing the rotating element. The combination of these forces and the stiffness of the rotor-support system (i.e., bearing and bearing pedestals) determine the vibration level. Rotor-support stiffness is important... 

In a mechanical system, the degrees of freedom indicate how many numbers are required to express its geometrical position at any instant. In machine-trains, the relationship of mass, stiffness, and damping is not the same in all directions. As a result, the rotating or dynamic elements within the machine move more in one direction than in another. A clear understanding of the degrees of freedom is important in that it has a direct impact on the vibration amplitudes generated by a machine or process system. 

Interest in the use of syntactic foam as a shock attenuator led to studies of its static and dynamic mechanical properties. Particularly important is the influence of loading rate on stiffness and crushing strength, since oversensitivity of either of these parameters can complicate the prediction of the effectiveness of a foam system as an energy absorber. 

The studies on adhesion are mostly concerned on predictions and measurements of adhesion forces, but this section is written from a different standpoint. The author intends to present a dynamic analysis of adhesion which has been recently published [7], with the emphasis on the mechanism of energy dissipation. When two solids are brought into contact, or inversely separated apart by applied forces, the process will never go smoothly enough—the surfaces will always jump into and out of contact, no matter how slowly the forces are applied. We will show later that this is originated from the inherent mechanical instability of the system in which two solid bodies of certain stiffness interact through a distance dependent on potential energy. 

The assumption of independent oscillators allows us to study a simplified system containing only one atom, as illustrated in Fig. 14 where x and Xq denote, respectively, the coordinates of the atom and the support block (substrate). The dynamic analysis for the system in tangential sliding is similar to that of adhesion, as described in the previous section. For a given potential V and spring stiffness k, the total energy of the system is again written as... 

Subsequent work by Johansson and Lofroth [183] compared this result with those obtained from Brownian dynamics simulation of hard-sphere diffusion in polymer networks of wormlike chains. They concluded that their theory gave excellent agreement for small particles. For larger particles, the theory predicted a faster diffusion than was observed. They have also compared the diffusion coefficients from Eq. (73) to the experimental values [182] for diffusion of poly(ethylene glycol) in k-carrageenan gels and solutions. It was found that their theory can successfully predict the diffusion of solutes in both flexible and stiff polymer systems. Equation (73) is an example of the so-called stretched exponential function discussed further later. 

The principle of the perfectly-mixed stirred tank has been discussed previously in Sec. 1.2.2, and this provides essential building block for modelling applications. In this section, the concept is applied to tank type reactor systems and stagewise mass transfer applications, such that the resulting model equations often appear in the form of linked sets of first-order difference differential equations. Solution by digital simulation works well for small problems, in which the number of equations are relatively small and where the problem is not compounded by stiffness or by the need for iterative procedures. For these reasons, the dynamic modelling of the continuous distillation columns in this section is intended only as a demonstration of method, rather than as a realistic attempt at solution. For the solution of complex distillation problems, the reader is referred to commercial dynamic simulation packages. 

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