2 Eigenvalue Problems
of Vibrations and Stability
2.1 Introduction
Solving problems of vibrations and stability almost inevitably involves solving
eigenvalue problems (EVPs). In vibration analysis one is faced with an EVP when
free responses are to be determined, or when forced responses are to be expanded in
terms of free mode shapes. Also, EVPs occur when critical buckling loads and
modes are to be determined in problems of elastic stability. EVPs occur repeatedly
in subsequent chapters of this book.
There are two types of EVPs: algebraic (or matrix) EVPs, associated with
discrete finite-DOF systems, and differential EVPs, associated with systems
described by ordinary or partial differential equations. Algebraic and differential
EVPs share many concepts and features, though posed in forms mathematically
quite distinct. However, whereas all algebraic EVPs are alike and rather simple to
solve, differential EVPs come in various forms and the mathematical analysis is
more involved.
This chapter focuses on differential EVPs for ordinary differential equations. To
some extend we shall abstract from the details of specific EVPs. Instead, using the
notion of operators, we present a formal analysis of a rather broad class of EVPs to
which many problems of vibration and stability belongs.
The theory of EVPs is well established, with a formal set of theorems and proofs.
We state those theorems that are important for applications, and a few proofs
illustrating techniques that one should master when coping with EVPs.
Certainly, the complete and true story on differential EVPs cannot be told in a
single chapter of acceptable size. Collatz (1963) devoted an entire textbook to the
subject, the main reference of this chapter. Other references are Flügge (1962),
Inman (2014), Leipholz (1977), and Meirovitch (1967, 2001).
© 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_2
49
of Vibrations and Stability
2.1 Introduction
Solving problems of vibrations and stability almost inevitably involves solving
eigenvalue problems (EVPs). In vibration analysis one is faced with an EVP when
free responses are to be determined, or when forced responses are to be expanded in
terms of free mode shapes. Also, EVPs occur when critical buckling loads and
modes are to be determined in problems of elastic stability. EVPs occur repeatedly
in subsequent chapters of this book.
There are two types of EVPs: algebraic (or matrix) EVPs, associated with
discrete finite-DOF systems, and differential EVPs, associated with systems
described by ordinary or partial differential equations. Algebraic and differential
EVPs share many concepts and features, though posed in forms mathematically
quite distinct. However, whereas all algebraic EVPs are alike and rather simple to
solve, differential EVPs come in various forms and the mathematical analysis is
more involved.
This chapter focuses on differential EVPs for ordinary differential equations. To
some extend we shall abstract from the details of specific EVPs. Instead, using the
notion of operators, we present a formal analysis of a rather broad class of EVPs to
which many problems of vibration and stability belongs.
The theory of EVPs is well established, with a formal set of theorems and proofs.
We state those theorems that are important for applications, and a few proofs
illustrating techniques that one should master when coping with EVPs.
Certainly, the complete and true story on differential EVPs cannot be told in a
single chapter of acceptable size. Collatz (1963) devoted an entire textbook to the
subject, the main reference of this chapter. Other references are Flügge (1962),
Inman (2014), Leipholz (1977), and Meirovitch (1967, 2001).
© 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_2
49