An easy start

The classical way to represent nonlinear resonances graphically is by the construction of so-called resonance curves. A resonance curve is the locus of pairs of wavevectors interacting resonantly with a given wavevector. Its construction, very popular in the 1960s, gives some useful, though partial, information about possible resonances in a wave system. For instance, in [152] and [195] resonance curves are regarded for three- and four-wave resonance systems, respectively. The form of the resonant curve changes drastically according to the magnitudes of the wavevectors. A locus might be an ellipse, be shaped like an hour-glass, or even degenerate into a couple of lines. Examples of resonance curves for ocean planetary waves, with a dispersion function of the form (2.21), are shown in Fig. 3.1.

Without knowledge of the wave numbers corresponding to the complete resonance set, all this information is only qualitative, of course. Eventually, the characteristic form of loci is useful to know before planning any laboratory experiments. For instance, in [195] a special type of resonant quartet of surface gravity waves is studied in which two wavevectors coincide. This special configuration was chosen because it “is convenient for experimental study”.

An easy start

As with every new theory, nonlinear resonance analysis has not come ex caelo. On the contrary, as Newton said: “If I have seen further it is only by standing on the shoulders of giants.” The giants to be grateful to now are Galileo Galilei (1564–1642), Jean Baptiste Joseph Fourier (1768–1830), and Jules Henri Poincaré (1854–1912). The father of modern physics, an Italian, Galileo Galilei, fascinated by the movement of a simple pendulum, identified one of the most important natural phenomena – resonance. French politician and mathematician Fourier, trying to understand what happens when a hot piece of metal rod is put into water, developed the mathematical apparatus – Fourier analysis – for describing solutions to linear partial differential equations (PDEs), without which no area of contemporary science is conceivable. Another French mathematician, Henri Poincaré, used Fourier analysis in order to give a strict mathematical definition of resonance and developed a method – Poincaré transformation – allowing, under some assumptions, to reduce the search for solutions to a nonlinear PDE to the search for resonances. This way the foundations of nonlinear resonance analysis were laid.

This book is intended for a vibration course in an undergraduate Mechanical Engineering curriculum. It is based on my lecture notes of a course (ME370) that I have been teaching for many years at The Pennsylvania State University (PSU), University Park. This vibration course is a required core course in the PSU mechanical engineering curriculum and is taken by junior-level or third-year students. Textbooks that have been used at PSU are as follows: Hutton (1981) and Rao (1995, First Edition 1986). In addition, I have used the book by Thomson and Dahleh (1993, First Edition 1972) as an important reference book while teaching this course. It will be a valid question if one asks why I am writing another book when there are already a large number of excellent textbooks on vibration since Den Hartog wrote the classic book in 1956. One reason is that most of the books are intended for senior-level undergraduate and graduate students. As a result, our faculties have not found any book that can be called ideal for our junior-level course. Another motivation for writing this book is that I have developed certain unique ways of presenting vibration concepts in response to my understanding of the background of a typical undergraduate student in our department and the available time during a semester. Some of the examples are as follows: review of selected topics in mechanics; the description of the chapter on single-degree-of-freedom (SDOF) systems in terms of equivalent mass, equivalent stiffness, and equivalent damping; unified treatment of various forced response problems such as base excitation and rotating balance; introduction of system thinking, highlighting the fact that SDOF analysis is a building block for multi-degree-of-freedom (MDOF) and continuous system analyses via modal analysis; and a simple introduction of finite element analysis to connect continuous system and MDOF analyses.

As mentioned before, there are a large number of excellent books on vibration. But, because of a desire to include everything, many of these books often become difficult for undergraduate students. In this book, all the basic concepts in mechanical vibration are clearly identified and presented in a simple manner with illustrative and practical examples. I have also attempted to make this book self-contained as much as possible; for example, materials needed from previous courses, such as differential equation and engineering mechanics, are presented. At the end of each chapter, exercise problems are included. The use of MATLAB software is also included.

In Chapter 2, the response has been calculated when the excitation is either constant or sinusoidal. Here, a general form of periodic excitation, which repeats itself after a finite period of time, is considered. The periodic function is expanded in a Fourier series, and it is shown how the response can be calculated from the responses to many sinusoidal excitations. Next, a unit impulse function is described and the response of the single-degree-of-freedom (SDOF) system to a unit impulse forcing function is derived. Then, the concept of the convolution integral, which is based on the superposition of responses to many impulses, is developed to compute the response of an SDOF system to any arbitrary type of excitation. Last, the Laplace transform technique is presented. The concepts of transfer function, poles, zeros, and frequency response function are also introduced. The connection between the steady-state response to sinusoidal excitation and the frequency response function is shown.

Response of an SDOF System to a Periodic Force

The procedure of a Fourier series expansion of a periodic function is described first. The concepts of odd and even functions are introduced next to facilitate the computation of the Fourier coefficients. It is also shown how can a Fourier series expansion be interpreted and used for a function with a finite duration. Last, the particular integral of an SDOF system subjected to a periodic excitation is obtained by computing the response due to each term in the Fourier series expansion and then using the principle of superposition.