This chapter provides helpful historical context for the methods discussed in this book. Nothing else in the book depends on familiarity with the material in this chapter, so it can be skipped. However, this history makes connections between the various topics that will enrich the big picture of optimization as you become familiar with the material in the rest of the book, so you might want to revisit this chapter.

Results of study in the field and practical use of vibration protection systems with compact mechanisms (removable modules) of negative stiffness are presented, obtained over the years. These are authoring systems considered as an alternative to conventional systems to protect humans against the most dangerous and harmful vibrations. This was proved in the land transport vehicles (electric buses, heavy trucks), construction equipment (wheel cranes and loaders, caterpillar excavators), harvester combines, and in mid-size and heavy helicopters. In some cases, efficiency of vibration protection was increased 100 to 700 times in the infra- and 1500 times in the low-frequency ranges. Using the theory of similarity and dimensions, one can design a lineup of compact systems with payload capacity of 150 to 250,000 N, however, closely approximated in dimensions, for other objects of vibration protection. In active control, operation frequency range of the systems can start from 0.05 to 0.1 Hz. With the advent of fundamentally new structural and functional materials, the possibilities of systems with negative and quasi-zero stiffness seem unlimited. For instance, substituting the parametric elements from spring steels with composites of carbon fibers increases 4−5 times the travel, and we hope to increase longtime durability under nonlinear postbuckling.

Designing and finding a reasonable trade-off between dimensions and performance of structures and mechanisms with parametric elements of negative stiffness in large is a fundamental problem in development and practical use of infra-low-frequency vibration protection systems for humans and engineering. A method is proposed and formulated for modeling the stress-strain under nonlinear postbuckling of the structures and for an optimal dimensioning of the mechanisms. The method is based on the hypotheses and statements of consistent theory of thin shells and includes (a) basic design theory, (b) validation of prediction that parametric elements are to be thin-walled structures to provide viability of the mechanisms and harmony with a vibration protection system, (c) algorithm for modeling geometrically nonlinear deforming the structures and iterative procedure that enables an optimal computable scheme for designing the mechanisms by the FEM, and (d) fundamental relationships between design parameters in terms of compactness and compatibility of the mechanisms with workspace of the systems and for extension the range of stiffness control, where system natural frequencies can be reduced until nearly-zero values. A lineup of geometrically similar mechanisms with negative stiffness in large has been designed for seat suspensions, mountings, and platforms.

As mentioned in , most engineering systems are multidisciplinary, motivating the development of multidisciplinary design optimization (MDO). The analysis of multidisciplinary systems requires coupled models and coupled solvers. We prefer the term component instead of discipline or model because it is more general. However, we use these terms interchangeably depending on the context. When components in a system represent different physics, the term multiphysics is commonly used.

We solve these problems using gradient information to determine a series of steps from a starting guess (or initial design) to the optimum, as shown in Fig. 4.1. We assume the objective function to be nonlinear, $C^2$ continuous, and deterministic. We do not assume unimodality or multimodality, and there is no guarantee that the algorithm finds the global optimum. Referring to the attributes that classify an optimization problem (Fig. 1.22), the optimization algorithms discussed in this chapter range from first to second order, perform a local search, and evaluate the function directly. The algorithms are based on mathematical principles rather than heuristics.

The gradient-based optimization methods introduced in Chapters 4 and 5 require the derivatives of the objective and constraints with respect to the design variables, as illustrated in Fig. 6.1. Derivatives also play a central role in other numerical algorithms. For example, the Newton-based methods introduced in Section 3.8 require the derivatives of the residuals.

Structural design is another strategic point in developing a vibration protection system with mechanisms of negative and quasi-zero stiffness. Missing this stage of the design and errors in designing the structure of mechanisms predisposed to unstable motion can ruin the development idea. A method of structural design of function-generating mechanisms for such systems is proposed. This includes the type and number synthesis of the mechanisms, making this process less empirical and more reasonable and bringing a great number of new candidates. The atlases of the mechanisms for seat suspensions and bogie secondary suspensions for carbody of high-speed trains are elaborated. The method fundamentals are (a) the function-generating mechanism is to be perfectly structured, that is, with a minimal number of redundant constraints; (b) due to unstable motion and transposition of clearances in kinematic pairs, the mechanism with negative stiffness must not directly join the input and output structural elements of function-generating mechanism to avoid structural indeterminacy; (c) mechanisms with negative stiffness shall be joined to the input structural element, and with no more than two kinematic pairs, one of these two is to be higher; (d) an external damping mechanism can be removed from function-generating mechanisms without degradation of the system performance.

Some types of conventional mechanical, pneumatic or other vibration protecting mechanisms with parametric elements of positive stiffness, i.e. having a given load capacity, may reveal the negative or quasi-zero stiffness in small. However, this is considered as a side effect and have no engineering feasibility to be realized in commercial vibration protection systems. This disadvantage can easily be eliminated if join the redundant mechanisms with parametric elements of negative and quasi-zero stiffness in large. Redundant mechanisms can drastically improve the quality of vibration protection in a certain combination and interaction with commercial systems, and without a destroying the system workspace. In this manner, one may arrange a seat suspension, independent wheel suspension, cabin's mounting, table or platform for measuring instrument and in this way protect a man-operator or passenger, power unit, onboard or stationary electronics, and cargo container. It was shown that the mechanisms with negative and quasi-zero stiffness in large, being properly joined to commercial vibration protection systems by using transmissions with short kinematic chain, increased 5 to 57 times the quality of vibration protection in the whole infra-low frequency range including nearly zero values. In some practical cases, this advantage reaches 100 to 300 times and more

Uncertainty is always present in engineering design. Manufacturing processes create deviations from the specifications, operating conditions vary from the ideal, and some parameters are inherently variable. Optimization with deterministic inputs can lead to poorly performing designs. Optimization under uncertainty (OUU) is the optimization of systems in the presence of random parameters or design variables. The objective is to produce robust and reliable designs. A design is robust when the objective function is less sensitive to inherent variability. A design is reliable when it is less prone to violating a constraint when accounting for the variability.*

Stability in large of the systems with mechanisms of negative and quasi-zero stiffness plays an important role for improvement of the infra-low vibration protection. These mechanisms are predisposed to chaotic vibration motion. Analysis of chaotic vibration and comparative selection of the mechanisms are to be reasonable steps before deciding next steps in designing the vibration protection systems. Their dynamic behavior can be diagnosed and predicted by the qualitative and quantitative methods for analysis of chaotic motion. An algorithm has been developed to study chaotic motion of the mechanisms, and the conditions of dynamic stability of the systems with such mechanisms are formulated. The algorithm is based on the Lyapunov largest exponent and Poincare map of phase trajectory methods and includes (a) formulation of chaotic motion models and criterial experiments for the mechanisms and systems, (b) technique of comparative analysis of the models, (c) computation procedure to estimate their dynamic stability in large, (d) formulation of design and functional parameters for providing stable motion of the systems in the infra-frequency range, including near-zero values. Validity of the algorithm is demonstrated through the development of active pneumatic suspensions supplied with passive mechanisms of variable negative stiffness.

Up to this point in the book, all of our optimization problem formulations have had a single objective function. In this chapter, we consider multiobjective optimization problems, that is, problems whose formulations have more than one objective function. Some common examples of multiobjective optimization include risk versus reward, profit versus environmental impact, acquisition cost versus operating cost, and drag versus noise.

Gradient-free algorithms fill an essential role in optimization. The gradient-based algorithms introduced inare efficient in finding local minima for high-dimensional nonlinear problems defined by continuous smooth functions. However, the assumptions made for these algorithms are not always valid, which can render these algorithms ineffective. Also, gradients might not be available when a function is given as a black box.