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This book treats dynamic stability of structures under nonconservative forces. it is not a mathematics-based, but rather a dynamics-phenomena-oriented monograph, written with a full experimental background. Starting with fundamentals on stability of columns under nonconservative forces, it then deals with the divergence of Euler’s column under a dead (conservative) loading from a view point of dynamic stability. Three experiments with cantilevered columns under a rocket-based follower force are described to present the verifiability of nonconservative problems of structural stability. Dynamic stability of columns under pulsating forces is discussed through analog experiments, and by analytical and experimental procedures together with related theories. Throughout the volume the authors retain a good balance between theory and experiments on dynamic stability of columns under nonconservative loading, offering a new window to dynamic stability of structures, promoting student- and scientist-friendly experiments.

### Chapter 1. Fundamentals

The word “column” usually refers to some slender vertical structure. “Column” in this book is used as an engineering term, and means a slender and straight structural member under compression. Here, the definition of a column and some basic concepts related to the dynamic stability of columns are presented.
Yoshihiko Sugiyama, Mikael A. Langthjem, Kazuo Katayama

### Chapter 2. Columns under Conservative Forces

Leonard Euler (1707–1783) was arguably the foremost mathematical scientist of the 18th century. Euler and Daniel Bernoulli established the basic theory of beams and columns. The structural stability of columns under a compressive (conservative) force was first studied by Euler in 1744 [1, 2]. This chapter discusses columns from the viewpoint of dynamic stability.
Yoshihiko Sugiyama, Mikael A. Langthjem, Kazuo Katayama

### Chapter 3. Columns under a Follower Force

An elastic cantilevered beam subjected to a follower force, the so-called Beck’s column , is an ideal/classical structural model in the theory of nonconservative stability problems. The stability of columns associated with follower forces is a relatively new topic in the field of structural stability, at least in comparison with the part dealing with conservative forces. This chapter discusses the basic aspects of a column under a follower force, its positive and negative features.
Yoshihiko Sugiyama, Mikael A. Langthjem, Kazuo Katayama

### Chapter 4. Columns with Damping

Ziegler discovered in 1952 that the introduction of damping in an elastic system under a follower force may have a destabilizing effect [1]. This caused a great deal of interest among structural dynamists, interest that has continued to date [216]. This chapter discusses the effect of internal damping on the flutter limit of a cantilevered column under a follower force. In place of Beck’s column, we consider Pflüger’s column , which has a tip mass [7, 8]. This chapter deals with a Pflüger’s column with internal (Kelvin-Voigt type) and external damping .
Yoshihiko Sugiyama, Mikael A. Langthjem, Kazuo Katayama

### Chapter 5. Energy Consideration on the Role of Damping

In Sect. 4.​3 of the previous chapter, it has been shown that introduction of small internal damping to Beck’s column leads to a considerable reduction in the flutter limit, from $$p_{* } = 20.05$$ (for the undamped case) to $$p_{cr} = 10.94$$ (for the damped case). This effect is referred to as the destabilizing effect of small damping. This chapter presents an energy-based discussion on the role of internal damping in the dynamics of Beck’s column with damping.
Yoshihiko Sugiyama, Mikael A. Langthjem, Kazuo Katayama

### Chapter 6. Cantilevered Pipes Conveying Fluid

As to the reality of a follower force in physical systems, it is said that an end thrust caused by a momentum flux discharged from the free end of a cantilevered pipe conveying fluid is a typical follower force. It is known that the pipe loses its stability by flutter. A cantilevered pipe conveying fluid is a nonconservative system that is applicable in practical uses and realizable in laboratories. Many papers have been published that deal with both theory and experiment.
Yoshihiko Sugiyama, Mikael A. Langthjem, Kazuo Katayama

### Chapter 7. Cantilevered Pipes with a Mechanical Element

Cantilevered pipes conveying fluid are the easiest realizable (in laboratory) mechanical models with a follower thrust. As to cantilevered pipes conveying fluid in practice, they are normally braced by a supporting component. The mechanical functions of such a supporting component are attributable to three mechanical elements: an elastic spring, a lumped mass, and a damper. This chapter discusses, through theory and experiment, the effect of such a mechanical element on the stability of a cantilevered pipe conveying fluid.
Yoshihiko Sugiyama, Mikael A. Langthjem, Kazuo Katayama

### Chapter 8. Columns under a Follower Force with a Constant Line of Action

This chapter discusses the relation between Beck’s column and Reut’s column from the viewpoint of boundary value problems. It is shown that Reut’s column can be derived through variational calculus as the adjoint boundary value problem of Beck’s column.
Yoshihiko Sugiyama, Mikael A. Langthjem, Kazuo Katayama

### Chapter 9. Generalized Reut’s Column

This chapter discusses a generalized Reut’s column, one of the models in the nonconservative problems of structural stability, showing that a mechanical model of the column is realizable in laboratories. A method for realizing the Reut force with a constant line of action, and a generalized Reut force as well, is proposed. The force is produced by an impinging air-jet method. Some experiments with columns under forces with different magnitudes of nonconservativeness are presented. It is experimentally shown that the column under this type of force loses stability by divergence and flutter, depending on the magnitude of nonconservativeness of the applied force.
Yoshihiko Sugiyama, Mikael A. Langthjem, Kazuo Katayama

### Chapter 10. Columns under a Rocket-Based Follower Force

Beck’s column has been subject to criticism regarding its reality. Is Beck’s column realizable in the laboratory? How valid are the flutter limits, the one obtained by neglecting damping and the one obtained by taking damping into account?
Yoshihiko Sugiyama, Mikael A. Langthjem, Kazuo Katayama

### Chapter 11. Columns under a Rocket-Based Follower Force and with a Lumped Mass

The experiment described in Chap. 10 demonstrated that the cantilevered column under a rocket thrust lose its stability by flutter. This chapter will take a similar but more quantitative experimental approach to the column subjected to a rocket thrust and equipped with a lumped mass. The stability analysis is carried out by applying the finite element method to discuss the effect of a rigid body on the flutter bound.
Yoshihiko Sugiyama, Mikael A. Langthjem, Kazuo Katayama

### Chapter 12. Columns under a Rocket-Based Subtangential Follower Force

The stability of columns under conservative forces has been the basis of structural stability theory, while the stability of columns under nonconservative forces has been only of recent interest in regard to structural stability. The stability of columns under the combined action of conservative and nonconservative forces has been an interesting topic in the field of structural stability problems, as it bridges the gap between the stability with conservative forces and the stability with nonconservative forces [18].
Yoshihiko Sugiyama, Mikael A. Langthjem, Kazuo Katayama

### Chapter 13. Pinned-Pinned Columns under a Pulsating Axial Force

There are many examples of physical systems, including structural systems, that are subject to time-varying excitations. One example is a string subjected to a pulsating axial tension. The string loses its stability by so-called parametric resonance , which occurs primarily when the excitation frequency $$\theta$$ is twice the string’s eigenfrequency $$\omega_{\text{o}}$$, that is, $$\theta = 2\omega_{\text{o}}$$.
Yoshihiko Sugiyama, Mikael A. Langthjem, Kazuo Katayama

### Chapter 14. Parametric Resonances of Columns

This chapter aims to give a general overview of the parametric resonances of columns under a harmonically pulsating force . In addition to simple resonance, combination resonances of sum and difference type are introduced, with columns having various kinds of boundary condition, other than pinned-pinned ends . For simplicity, it is assumed that damping is absent. Hsu’s conditions for resonances are introduced to give the first estimate of the principal regions of instability. Experiments with columns having clamped-clamped and clamped-pinned ends are presented, to demonstrate combination resonances of sum type. Combination resonances of difference type are introduced through an analog computer-based experiment of a cantilevered column subjected to a pulsating follower force .
Yoshihiko Sugiyama, Mikael A. Langthjem, Kazuo Katayama

### Chapter 15. Parametric Resonances of Columns with Damping

This chapter discusses the mathematical aspects of parametric resonances of columns with damping, dealing generally with dynamical systems with damping. In addition to the first-order approximation for the resonance boundaries of Mathieu-Hill equations, an approach to the second-order approximation is suggested.
Yoshihiko Sugiyama, Mikael A. Langthjem, Kazuo Katayama

### Chapter 16. Columns under a Pulsating Reut Force

Combination resonances of difference type may possibly be one of the most interesting topics in the field of structural dynamics. This chapter describes an experimental approach to this topic.
Yoshihiko Sugiyama, Mikael A. Langthjem, Kazuo Katayama

### Chapter 17. Remarks about Approaches to the Dynamic Stability of Structures

Different approaches to the dynamic stability of structures, and of columns in particular, are considered in the present book. The typical approaches are analytical (mathematical analysis-based), computational (numerical), and experimental approaches.
Yoshihiko Sugiyama, Mikael A. Langthjem, Kazuo Katayama