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Viscoelastic Behaviour
Published in B. R. Gupta, Rheology Applied in Polymer Processing, 2023
The superposition principle applies to any linear systems, however the physical systems are only approximately linear (including the so called linear polymeric systems) and the application of superposition principle is only an approximation of true physical behaviour.
Deterministic Models
Published in William Delaney, Erminia Vaccari, Dynamic Models and Discrete Event Simulation, 2020
William Delaney, Erminia Vaccari
state XiCto) = 0 and x2(t0) = 0 since in this case x(t) = 0 for any k! and k2. The superposition principle is very useful since determining the solution y to a linear model for one initial state x(t) and input signal u, solutions for an infinite number of other situations [initial states kx(t) and inputs ku] can be obtained as ky without having to solve the model. More generally, having determined two solutions corresponding to two different initial states and input signals, an infinite number of other solutions corresponding to all linear combinations of the two initial states and the two input signals can be determined without having to solve the model explicitly for these new situations .
Basics of Electrical Communication Systems
Published in P. S. Neelakanta, ATM Telecommunications, 2018
The system depicted in Fig. 2.3 responding to a signal can be characterized as follows: ✓ Linear or nonlinear: If a system is linear, then the superposition principle applies. That is, if f(t) = [a1f1(t) + a2f2(t) + … ], then g[a1f1(t) + a2f2(t) +…] = a1g[f1(t)] + a2 g(f2(t) + …] where a1, a2, … are constants✓ Time-invariant or time-varying: A system is time-invariant, if a time-shift in the input results in a corresponding time-shift in the output. That is, if fin(t) = f(t – τ), then fout(t) = g[f(t – τ)] where τ is a constant time-shift✓ Realizable or nonrealizable: A physically realizable system cannot have an output response before an arbitrary input function is applied. That is, the output of a physical system at t = to, namely, g[f(to)] must depend only on values of the input f(t) for t < to. A system having this property is known as a realizable or a causal system.
Optimal containment preview control for continuous-time multi-agent systems using internal model principle
Published in International Journal of Systems Science, 2023
Yanrong Lu, Xiaomeng Zhang, Zhiwen Wang, Liang Qiao
In this section, the sufficient conditions and the distributed control design procedure will be given to solve the local output regulation problems. It needs to be pointed out that there are multiple leaders existed in the network. Moreover, according to Assumption 2.5, the preview lengths of these leaders' outputs may be different. Therefore, the main conclusion in Katayama and Hirono (1987) and Liao et al. (2011) cannot be generalised directly. However, one of the basic properties of linear systems is that they satisfy the superposition principle (Zheng, 2002). On this basis, the first formula in systems (19) can be decomposed into the superposition of M independent subsystems as shown in Figure 1, where we set , . Moreover, denote . For each independent subsystem shown in Figure 1, the following result provides the gain matrices of under the local performance index functions (20).
Shear Flat Jack Test for Evaluating Shear Characteristics on Unreinforced Masonry Structures
Published in International Journal of Architectural Heritage, 2021
For what concerns the shear modulus consider, as first approximation, a linear, elastic and isotropic wall with a generic vertical load , with the cuts already realized but with zero pressure in the FJs. To simplify the problem it is possible to consider the wall as a 2D problem where the cuts are crossing the whole section of the wall. With these hypotheses, the wall can be treated as a plane stress problem. The stress-strain state of the wall in these hypotheses is considered as the initial one. Due to linearity, it is possible to use the superposition principle. The final state of the sample (with ) can be obtained adding the initial state to the one created by the FJs only.