Stiffness modelling

As a bit of an aside, one can think of the algebraic constraint as an infinitely stiff problem. Referring to the stiff model problem (Section 15.2), stiff problems are characterized by a fast transient and a slowly varying solution. Regardless of the initial condition, a stiff problem will always decay to the slowly varying solution, and the stiffer the problem, the faster will be the decay (e.g., Fig. 15.1). The situation in a problem like that in Fig. 7.5 is that there is no transient in the y2 component because it is a constraint, and not a differential equation. If, however, the y2 equation is modeled as y 2 = — X(y2 — 1), then y2 = (y2(0) — l)e Xl. As A. becomes larger, the differential equation becomes stiffer, and as X —> oo, the differential equation becomes an algebraic constraint. 

So-called stiff differential equation models are particularly challenging to solve. Stiff models have dynamic behavior that encompasses a wide range of time scales. An example would be fast kinetics combined with long fluid-residence times in a chemical reactor. Gear s method is perhaps the most commonly used technique for solving these types of problems. 

We begin with a description of the fast dynamics of the stiff model in Equation (4.5). This is readily obtained in the form of Equation (4.20) by considering the dynamic model (4.5) in the limit as e —> 0 ... 

Equation (4.36) is a non-stiff model that approximates the dynamics of the reactor-condenser system in Figure 4.1 in the original (fast) time scale t. 

We then proposed a method for deriving reduced-order, non-stiff models for the behavior of such systems in each time scale. In the general case, the slow dynamics of the systems was shown to be ID and directly associated with the total impurity holdup. 

From a mathematical point of view, we can see that Equation (5.10) is in a (nonstandard) singularly perturbed form. This suggests that the integrated processes under consideration will feature a dynamic behavior with at least two distinct time scales. Drawing on the developments in Chapters 2, 3, and 4, the following section demonstrates that these systems evolve in effect over three distinct time scales and proposes a method for deriving reduced-order, non-stiff models for the dynamics in each time scale. 

If none of the aforementioned information is available, then construction and inspection of models can give a rough idea of the angle strain involved. For example, cyclodecyne can be built with stiff models without problems, but cyclo-nonyne and cyclooctyne can only be constructed with a flexible C—C=C—C moiety. 

C0 yield stress constant, Robertson-Stiff model (s 1)... 

Bulge appears stiff model assumes local motions at probe same for both duplexes temperature-dependent, longer DNA, longer times wobbling in hairpin faster by 40%... 

In the next section, the proposed updating approach is presented which provides estimates of the system modal frequencies and system mode shapes, as well as estimates of the stiffness model parameters, based on incomplete modal data. Examples with a twelve-story building and a three-dimensional braced frame wiU be used to demonstrate the method with applications to structural health monitoring. 

Beirute, R.M. and Flumerfelt, R.W., An Evaluation of the Robertson - Stiff Model Describing Rheological Properties of Drilling Fluids and Cement Slurries,... 

A disadvantage of real-time observer-based residual generation is that for large behavioural models, or for stiff model equations, the numerical computation of the estimated response to system inputs may take a significant part of the sampling interval [34]. 

Back in 1993, Asher proposed to assist an ideal switch by a resistor he called causality resistor that adapts its causality to causality changes at the switch port so that the rest of the bond graph remains causally unaffected [23]. As long as the simulated dynamic behaviour is not significantly affected, the parameter value of a causality resistor can be chosen within reasonable limits but may lead to stiff model equations and thus may give rise to an increase of computational costs. 

Equations can be formulated so that parameters of auxiliary elements can be set to zero. Also, the ON-resistance of switches can be set to zero turning them into ideal switches so that small time constants and thus a set of stiff model equations can be avoided. In the case residual sinks are used, which is similar to the use of Lagrange multipliers, the resulting mathematical model is a DAE system of index 2. 

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