A surrogate model, also known as a response surface model or metamodel, is an approximate model of a functional output that represents a “curve fit” to some underlying data. The goal of a surrogate model is to build a model that is much faster to compute than the original function, but that still retains sufficient accuracy away from known data points.

Within a short run, a novel class of mechanisms and systems has been created with parametric (elastic-dissipative) elements of sign-changing stiffness controlled in a range from positive to negative or quasi-zero values. A great deal of natural and hand-made designs on different physical bases appeared that could reveal such a phenomenon. These mechanisms and systems can cut the stiffness and provide a perfect vibration protection in a frequency range required. However, only some of them either are ready for to substitute or could be used in advanced hybrids in parallel with conventional vibration protection mechanisms and systems in certain types of machines and equipment. The main reason is very small travel where the negative or quasi-zero stiffness can be realized. A small error in passive control or a soft fault in an active one is enough to move such mechanisms and systems to performance degradation. A generic model of the parametric elements with negative and quasi-zero stiffness in small and a transition model to provide these effects in large are formulated. The model analysis led to important predictions on how to obtain an optimal trade-off between the dimensions and performance of the mechanisms and systems of novel class.

General nonlinear optimization problems are difficult to solve. Depending on particular optimization algorithm, they may require tuning parameters, providing derivatives, adjusting scaling, and trying multiple starting points. Convex optimization problems do not have any of those issues and are thus easier to solve. The challenge is that these problems must meet strict requirements. Even for candidate problems with the potential to be convex, significant experience is usually needed to recognize and utilize techniques that reformulate the problems into an appropriate form.

Engineering design optimization problems are rarely unconstrained. In this chapter, we explain how to solve constrained problems. The methods in this chapter build on the gradient-based unconstrained methods fromand also assume smooth functions. We first introduce the optimality conditions for a constrained optimization problem and then focus on three main methods for handling constraints: penalty methods, sequential quadratic programming (SQP), and interior-point methods.

Simulation and instrumental measurement are the most reliable methods to examine, prove, and enhance adequacy of theoretical models and prototypes and predict their practical use. A trade-off complex of testing and measuring instruments is presented for designing vibration protection systems. This includes both the standard test equipment and add-on devices to recognize specifics in behavior of the systems with negative and quasi-zero stiffness. This covers the computer-aided tensile machines and special adapters for static and low-cycle testing, optical aids for holographic interferometry and structural testing, electro- and/or hydrodynamic exciters, sets of special infra-frequency accelerometers and external filters, AD/DA-boards, FFT- and/or wavelet analyzers, recording equipment, and standard and special software. The complex developed provides a full cycle of the system experiment, including (a) path-generation and optimization of elastic responses, (b) strain state analysis of parametric elements and mechanism units, (c) simulation and online analysis of dynamic behavior of scaled models of the systems with extremely small stiffness and damping under vibrations in a frequency range starting from near-zero values. The method of laboratory experiment is an integral part of the methodology to investigate the systems in the field.

Most algorithms in this book assume that the design variables are continuous. However, sometimes design variables must be discrete. Common examples of discrete optimization include scheduling, network problems, and resource allocation. This chapter introduces some techniques for solving discrete optimization problems.

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.