Nonlinear Vibration Analysis Using the Method of Direct Normal Forms

In this thesis, an approximative analysis of nonlinear dynamical systems utilising the direct normal forms method (DNF) is addressed. For certain engineering applications, this approach is used to analyse nonlinear equations of motion. Nonetheless, while dealing with its specifics, the DNF approach has several undesirable restrictions, particularly in regard to the enormous algebraic terms and solutions; thus, it is crucial to use computer-based procedures. In this thesis, Maple, a symbolic computing program, plays a crucial role in obtaining step-by-step solutions for difficult nonlinear dynamical engineering problems based on the DNF approach. In this thesis, it is shown that the implementation of such software permits the analysis of more complicated systems, particularly those with higher-order geometric nonlinear stiffness terms, combinations of nonlinear stiffness and viscous damping, and systems with fractional order damping terms.
The analysis in this thesis starts with a comprehensive investigation of single-degree-of-freedom (SDOF) nonlinear systems, beginning with conservative systems and progressing through viscously damped and forced systems with various forms of geometric nonlinearities. Higher order accuracies of the DNF method are then explored in detail; beginning with a reintroduction of the study of $\varepsilon^2$ accuracy using symbolic computations. A unique refinement of the DNF methodology in terms of higher order accuracies is then presented. With the use of symbolic computations tools, the precision of the DNF approach has been increased to any desired level of precision, $\varepsilon^n$, which has been shown with a number of applications. In addition, for viscously damped SDOF systems, a novel approach based on a variation of Burton's method along with a normal form technique to obtain the damped backbone curves is thoroughly discussed with examples.
Furthermore, the analysis is extended to multi-degree-of-freedom (MDOF) nonlinear systems; starting with two verification problems that show the capability of the proposed symbolic DNF technique with Maple software. Then, a more advanced system of 2-DOF cubic-quintic oscillator is briefly discussed, in which analytical expressions of single-mode and double-mode backbone curves are generated.
In conclusion, the overall study findings provide unique enhancements to the technique of DNF for investigating nonlinear SDOF and MDOF systems analytically; this includes creating a tool for the researcher to apply the method of DNF symbolically for systems with high orders of polynomial stiffness nonlinearities, systems with combinations of stiffness and damping nonlinearities, and systems with viscous damping with fractional orders. Moreover, the accuracy of the DNF method is discussed in detail and a general form for any $\varepsilon$ order is obtained. In conclusion, the implementation of symbolic computations of DNF method for such systems is shown to be effective and trustworthy.