In the introductory chapter, we discussed function characteristics from the point of view of the function’s output—the black-box view shown in Fig. 1.16. Here, we discuss how the function is modeled and computed. The better your understanding of the model and the more access you have to its details, the more effectively you can solve the optimization problem. We explain the errors involved in the modeling process so that we can interpret optimization results correctly.

The infra-low-frequency vibrations are most dangerous and harmful for humans. They affect a person as an operator or passenger of the vehicles and other vibrating machines, and his environment the rest of a day and night. These are longtime and thus devastating effects since human natural frequencies dramatically coincide with forced vibrations of the machines in the same spectra, resulting in permanent broadband resonances. The general and industrial standards regulate exposure time of vibration impacts to humans, since conventional vibration protection systems, to put it mildly, do not quite cope with the functions assigned to them. Moreover, they operate as vibration amplifiers rather than vibration systems just in these frequency spectra. Besides, new potentially hazard vibrations, for example, in the near-zero frequencies appear with advent of the machines of next generation under intensive development, such as high-speed railroad trains for long distances and multiple-purpose helicopters. Fundamentally other design methods and proper technology are required to provide the infra-low-frequency vibration protection of humans inside and outside operating transport vehicles, construction equipment, and other machines, especially since a gap increases between efficiency of the conventional systems and vibration limits required for health, activity, and comfort of humans

The systems with negative and quasi-zero stiffness can provide perfect vibration protection in a wide frequency range including near-zero values. However, it is possible under a certain structure of system damping. Intelligent design approaches are presented to control the structural (hysteretic) components in thin-walled elastic elements and slip ones in movable joints of the mechanisms structuring the vibration protection systems with small stiffness. These involve (a) models for analysis and optimization of the components, (b) novel antifriction materials and composites providing reducing the hysteretic and kinetic frictions up to 0.02 to 0.005, which are adozen times less than using most prevalent materials. For instance, synthesized incompressible lubricants and solid lightweight composites provided a little to no hysteretic friction in the elements designed as the packages of thin plates, and no kinetic friction in the insert liners and guiderails. The use of approaches reduced at least 3.5 to 4 times unwanted level of system damping. These approaches can provide even more drastic, 0.01 to 0.02, decrease in the system damping. Assuming active motion control, this will reduce the system natural frequencies up to 0.15 to 0.2 Hz and improve vibration protection 50 to 300 times and more.

Optimization is a human instinct. People constantly seek to improve their lives and the systems that surround them. Optimization is intrinsic in biology, as exemplified by the evolution of species. Birds optimize their wings’ shape in real time, and dogs have been shown to find optimal trajectories. Even more broadly, many laws of physics relate to optimization, such as the principle of minimum energy. As Leonhard Euler once wrote, “nothing at all takes place in the universe in which some rule of maximum or minimum does not appear.”

Most routine strategies of motion control in vibration protection systems are based on attenuation of resonant responses using external semi-active or active dampers. In the systems with negative and quasi-zero stiffness, it is simpler because there is no need to know and continually process random signal data from an external vibration source. The control focuses on maintaining a certain balance between the positive and negative stiffness of parametric elements varied in predetermined ranges to keep separation of the system natural frequency spectra and the frequencies of forced vibrations including near-zero values. The control criteria, formulated and quantitatively estimated, provide extremely small stiffness, immobility in a steady state motion, and stabilization in a transient motion of the system without an external damper. The control strategy is validated through designing the systems supplied with active pneumatic suspensions and passive mechanisms of variable negative stiffness. The control algorithms were realized with the help of a two-channel control system and actuators made of commercial hardware and operating in parallel. Efficiency of the algorithms has been estimated through comparison of results of computer simulation and development test of seat suspensions for vibration protection of drivers of heavy trucks and buses and for helicopter pilots.

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.