Reasons for the nonlinear analysis

Linearity is just a mathematical construct, an idealization. In reality there is no physical system conforming exactly to the hypotesis of linearization. However, many natural pro- cesses and engineering systems behave in a way very close to linearity, and mathematical models can be constructed under that assumption. Some examples in Mechanics are the constitutive laws of elastic materials such as metals, structures sustaining small deforma- tions or linear viscous dampings. Engineering has traditionally taken a linear approach to design, modelization and analysis that has kept things simple for centuries.

Nevertheless, the Industry is continuously evolving and demanding more efficient designs, lighter materials, better reliability and lower costs in order to remain competitive and fulfill the demands of the materials. In certain fields, such as Aerospace or High Tech, there is a need to attain performances over what is feasible with today’s know-how, turning the concepcion and design of the newest products a big challenge to the old paradigms of Engineering.

Nonlinearity may be present in many ways: hyperelasticity, elastoplasticity, large mechan- ical deformation, nonlinear boundary conditions of service, fluid-structure interactions, self-sustained forces or simply designs of such nature that cannot be subject to any sort of linearization without great loss of information. In those situations, the linear hypote- sis would underestimate the deepest model dynamics and becomes unreliable. Nonlinear analysis is the alternative, even though it is a much more complicated and less devel- oped field, to attain a reliable analysis that could predict the real behavior of a product, structure, mechanism or commodity in the design phase of its life cycle.

Designers can also capitalize on nonlinear dynamics in the concepcion of an engineering system with the purpose of improving performance, reliability, functionality or for other

purposes, and furtheremore, nonlinear analysis tools need to be undertaken for a thorough design according to the product or the system specifications.

Nonlinear modal analysis has been succesfully applied on models of real-life mechanical systems with a large number of dofs, such as on the structures of an aircraft –[22]– and a satellite –[23] and [24]–, with both experimental and computational analysis methods.

2.4.4.1 Practical applications of nonlinearity

In this text I will single out two of the most important practical applications of nonlinear vibration in Mechanics with a huge body of undergone research and published bibliogra- phy. Both have been, at the present moment, been successfully considered on structural and mechanical design:

Vibration control The tradictional approach to vibration control in Engineering has consisted in the use of passive systems of protection, such as dissipative elements or mass dampers. Nonetheless, today’s standard of performance and reliability cannot be fully reached with these techniques. so nowadays ’active’ protection systems are widely used instead. Active control systems rely on the use of actuators that vary stiffness and damping in real time, but they have very high costs and require an energy supply –[31]–. Moreover, those active control systems have unnegligible dimensional constraints leading to a reduction of the mechanism deflections on the linear elastic region –[3]–. A common cause of nonlinearity in those elements is found on constructive clearances between elements or elastic limiting stoppers. On the other hand, linear passive mass dampers have many limitations that hinder their use for certain applications: they fail to damp out several modes of systems with multiple dofs, being limited to the neighborhood of a vibration mode, and they are unable to mitigate vibrations in nonlinear structures. Those drawbacks can be circumvented with the use of ’nonlinear vibration absorbers’ –[31]–. Due to their frequency-energy dependence they are effective in mitigating vibration in a much larger interval of frequencies. The 2dofs oscillator shown on figure 2.4

could be the model of a mechanical structure attached to a nonlinear vibration absorber. To understand this one can compare the frequency response diagrams in figures2.8 and 2.9. If the masses are attached with a nonlinear spring the vibration mitigation spans a larger interval of frequencies around the ’tuning’ frequency than if the attachment was linear. Also, nonlinear phenomena such as energy transfer –or ’mode localization’– and internal resonance can be exploited with the purpose of vibration control –[12] and [2]–.

Some reference books on vibration control with a generalistic approach are [1] and [3].

Nonlinear energy harvesting Energy harvesting is the conversion of environmental energy into electrical energy –[32]–. There is a broad range of applications to energy harvesting, mainly energy powering –in very small scales: miliwatts or microwatts– and monitoring sensors. In what concerns to us, our interest is to pick up the vibrational energy from a mechanical system by means of an oscillator attached to a system of electrical generation –of electromagnatic, piezoelectric or capacitative type, among others– that could convert mechanical energy into electrical energy. A very considerable limitation of vibration harvesting systems is that they are only effective when the mechanical oscillations are in the neighborhood of a resonance of the global structure of attachment of the harvester. In such conditions the sys- tem oscillates close to a local maximum of energy and amplitude and the amount of energy transfered for conversion is optimized, but the neighborhoods of linear resonance are usually very narrow as seen in section 2.3.2.

Consider the two degrees of freedom oscillator represented on2.4. Let the first mass –first dof– be a mechanism or mechanical structure or system, while the second mass –second dof– represents a vibration energy harvester elastically attached to the first mass, constituting a vibrational energy source. This is an integrated system whose dynamics should be analized as a whole for the design and prediction of the so-called ’frequency bandwidths’, which are the frequency intervals around the resonance points maximizing the amount of energy that could be transfered to the harvesting system during a forced regime of vibration. The energy transfered to the

harvester –the second mass– can be converted to electrical power by different means already cited.

Now compare the linear and nonlinear frequency responses of oscillators 2.2 and

2.4 shown in 2.3.2. When a nonlinear stiffness is added to the elastic attachment between the main mechanical structure –mass one– and the harvester –mass two– the frequency bandwith can be greately broadened, that is, more mechanical energy could be effectively harvested in a wider range of frequencies around the resonances. This is a considerable advantage because the most mechanical systems experience multiple excitations at very variable frequencies. Furtheremore, mode localization can also be an advantage if it leads to an increased amount of energy in the harvester. There is a wide amount of research and literature on the topic of nonlinear harvesting and the design of harvesting systems. For a brief introduction one can see references [6], [5] and [33].

(a) Elementary model of the electrodynamic vibration energy harvester –image from [6]–.

(b) Electrodynamic energy harvester Per- petuum PMG17: hppt://www.perpetuum.com/

pmg17.asp.

Figure 2.13: In an electrodynamic vibration energy harvester, the mechanical energy absorbed by a sprung mass is transfered to an electrical circuit through an inductor. Nonlinearities can be considered in the elements that suspend the sprung mass, the

(a) Model of the piezoelectric vibration energy harvester –image from [6]–.

(b) Piezoelectric energy harvester Mid´e Volture:

http://www.mide.com.

Figure 2.14: In a piezoelectric vibration energy harvester, the deformation of a sus- pended beam with a mass on its tip is converted to electrical power by a piezoelectric plate attached to the beam. The models should be deemed as nonlinear for large defor-

mations of the suspended beam or for nonlinear elasticity or piezoelectricity laws.