1 Vibration Basics
1.1 Introduction
This chapter surveys some fundamental concepts, methods, and phenomena associated with vibrations. Included only as a reference for subsequent chapters, it
assumes the reader to be reasonably familiar with most of the topics described. The
presentation will be brief, with few examples and no proofs.
For further reference regarding basic vibration theory, you may consult, e.g.,
Den Hartog (1985), Ginsberg (2001), Harris (1996), Inman (2001, 2014), Kelly
(1993, 2007), Meirovitch (2001), Shabana (1996, 1997), Thomson and Dahleh
(1998), or Timoshenko et al. (1974). Numerical tools are described in, e.g., Bathe
and Wilson (1976), Cook et al. (1989), Morton and Mayers (1994), Press et al.
(2002), and Zienkiewicz (1982) – and experimental methods and data analysis in,
e.g., Brandt (2011), Broch (1984), Ewins (2000), Holman (1994), and Kobayashi
(1993). You might even enjoy reading some of the great ancestors in this field, e.g.,
Lord Rayleigh’s The Theory of Sound (Rayleigh 1877).
As should be well known, vibrations of physical objects are described in terms
of degrees of freedom (DOFs). A model of a mechanical system has as many DOFs
as are required for uniquely specifying its state of deformation with respect to some
fixed reference configuration. The specific choice of DOFs depends on the problem
to be solved. Thus, an aircraft wing may be adequately modeled as a continuous
structure (having infinitely many DOFs), as a multiple-DOF system (having a finite
number of DOFs), or even as a single-DOF system. Excitations may exist in the
form of non-zero initial conditions only, giving rise to free vibrations of the object.
Or there may be time-varying loads, in which case the object performs forced
vibrations.
The survey is organized accordingly, that is: we consider systems having single,
multiple, and infinitely many DOFs, and for each DOF-class we consider different
types of excitation. Then we summarize two important energy methods for setting
up equations of motions: Lagrange’s equations and Hamilton’s principle, both of
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
J. J. Thomsen, Vibrations and Stability,
https://doi.org/10.1007/978-3-030-68045-9_1
1
1.1 Introduction
This chapter surveys some fundamental concepts, methods, and phenomena associated with vibrations. Included only as a reference for subsequent chapters, it
assumes the reader to be reasonably familiar with most of the topics described. The
presentation will be brief, with few examples and no proofs.
For further reference regarding basic vibration theory, you may consult, e.g.,
Den Hartog (1985), Ginsberg (2001), Harris (1996), Inman (2001, 2014), Kelly
(1993, 2007), Meirovitch (2001), Shabana (1996, 1997), Thomson and Dahleh
(1998), or Timoshenko et al. (1974). Numerical tools are described in, e.g., Bathe
and Wilson (1976), Cook et al. (1989), Morton and Mayers (1994), Press et al.
(2002), and Zienkiewicz (1982) – and experimental methods and data analysis in,
e.g., Brandt (2011), Broch (1984), Ewins (2000), Holman (1994), and Kobayashi
(1993). You might even enjoy reading some of the great ancestors in this field, e.g.,
Lord Rayleigh’s The Theory of Sound (Rayleigh 1877).
As should be well known, vibrations of physical objects are described in terms
of degrees of freedom (DOFs). A model of a mechanical system has as many DOFs
as are required for uniquely specifying its state of deformation with respect to some
fixed reference configuration. The specific choice of DOFs depends on the problem
to be solved. Thus, an aircraft wing may be adequately modeled as a continuous
structure (having infinitely many DOFs), as a multiple-DOF system (having a finite
number of DOFs), or even as a single-DOF system. Excitations may exist in the
form of non-zero initial conditions only, giving rise to free vibrations of the object.
Or there may be time-varying loads, in which case the object performs forced
vibrations.
The survey is organized accordingly, that is: we consider systems having single,
multiple, and infinitely many DOFs, and for each DOF-class we consider different
types of excitation. Then we summarize two important energy methods for setting
up equations of motions: Lagrange’s equations and Hamilton’s principle, both of
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
J. J. Thomsen, Vibrations and Stability,
https://doi.org/10.1007/978-3-030-68045-9_1
1