Dynamics of structures. 3rd ed.

*(English)*Zbl 1246.74001
Boca Raton, FL: CRC Press (ISBN 978-0-415-62086-4/hbk). xxvii, 1028 p. (2012).

This book is written for engineers and scientists to understand the dynamic response of structures and analytical tools required for determining such a response. First, the book explains the basic calculus of engineering mechanics. The essential step in the dynamic analysis of a system is the mathematical modelling, the formulation of equations of motion, and solutions of these equations. Modelling techniques can be divided into two categories. In the first one the system is modeled as an assembly of rigid bodies and massless deformable elements. In the other technique both mass and deformation are coupled, and the system is called continuous or distributed parameter system. Generally, a continuous model better represents the behaviour of dynamical systems. However, the equations of motion of a continuous system are very difficult or impossible to solve. Therefore, the dynamic analysis of engineering structures is often based on a representation of the structure by a discrete parameter model.

Chapter 1 describes how to study the dynamic response of engineering structures, and explains the importance of such a study in its design. The dynamic forces acting on a structure may be the result of different causes such as rotating machinery, wind, blast, earthquake, and etc. The forces can be classified according to the nature of their vibration with time as periodic, non-periodic, or random. To classify the dynamic forces as deterministic or nondeterministic, can be useful in statistical problems. The above classification is also the topic of Chapter 1. Chapter 2 deals with the single-degree-of-freedom systems. In this chapter, the equations of motion are formulated by using the Newton vectorial mechanics and the principle virtual displacements. However, the procedures of Newtonian mechanics are adequate only for simple structures. For more complex single-degree as well as multi-degree-of-freedom systems the derivatives of equations of motion employ also the energy principle or the principles of analytical mechanics. Chapter 2 presents additionally the methods of representing a continuous system or a discrete multi-degree-of-freedom system by an equivalent single-degree-of-freedom system. Chapter 3 describes the formulation of the equations of motion for a multi-degree-of-freedom system through the principles of vectorial mechanics. Ritz’ method is applied here to the modelling of continuous and discrete systems. Chapter 4 introduces the principles of analytical mechanics and their application to the formulation of equations of motion. The author describes here the generalized coordinates, the generalized forces, the important principle of virtual work, and derives the Lagrange equations of motion. Chapter 5 describes vibrations of a single-degree-of-freedom system. Various damped systems, in the presence of viscous damping, structural damping, and Coulomb damping, are considered. The application of the phase plane to the analysis of vibrational response of damped and free systems is described. Chapter 6 deals with the response of single-degree-of-freedom systems under a harmonic excitation. Resonant response of undamped systems is considered. The author also considers the Nyquist plots for a single-degree-of-freedom system subjected to a harmonic force. Unfortunately, the text does not give information how to find the coefficients of energy dissipation, see [Yu. N. Sankin, Tr. Sredn. Mat. Obshch. 8, No. 2, 22–33 (2006; Zbl 1150.74385)].

Chapter 7 presents the response of single-degree-of-freedom systems to general dynamic loading, such as an impulsive force and shock loading. In Chapter 8, the author gives a review of approximate and numerical methods for the analysis of single-degree and multi-degree-of-freedom systems. Frequency determination of such systems is often complex. On the other hand, if a reasonable estimate can be made on the vibration shape, the system can be represented by an equivalent single-degree-of-freedom model whose frequency of vibration can be easily determined by the Rayleigh method. Chapter 9 deals with transform methods for single-degree-of-freedom systems, in particular with Fourier transform and Laplace transform. Unfortunately, the author does not explain how to use initial conditions [loc. cit.].

Chapter 10 considers free vibrations of multi-degree-of-freedom systems and the application of eigenvalue solutions to the determination of free vibration response. Various methods for the solution of eigenvalue problems of structural dynamics are given in Chapter 11. The methods include transformation methods, iteration methods, and the determinant search methods. Chapter 12 considers the force dynamic response based on normal coordinate transformation, called here the mode superposition method. In Chapter 13 the author presents approximate and numerical methods for multi-degree-of-freedom systems. He considers the Rayleigh-Ritz method and the direct numerical integration in frequency domain. Chapter 14 describes the equations of motion for simple continuous systems such as transverse vibration of a beam, axial and torsional vibrations of a rod, and transverse vibrations of a string and shear beams. Chapter 15 deals with free-vibration response of a simple continuous system. Here, the author considers the associated eigenvalue problem and mode superposition method. Chapter 16 examines the forced response of a simple continuous system under the action of time-varying forces. In Chapter 17, one-dimensional wave propagation is considered. The mode superposition method is not applicable to a system of semi-infinite or infinite extent. In contrast to a system of finite extent, a system of infinite extent has a continuous band of frequencies and the term “mode shape” loses its meaning. Thus the author presents new methods to study the propagation of travelling and standing waves, harmonic waves, wave velocity, wave length and frequency, reflection and refraction of waves and wave dispersion. Chapter 18 gives an introductory treatment of finite element method considered as a variational Bubnov-Galerkin method. Chapter 19 presents the component mode synthesis methods. These methods are effective in structural dynamic analysis of large complex structures, such as an aircraft, an aerospace structure, or an automobile. The final Chapter 20 analyzes the nonlinear response of single-and multi-degree-of-freedom systems.

On balance, this is a fundamental textbook on the vibration theory. It provides the classical principles and modern methods of analysis, modelling and simulation of mechanical systems. The book can be very useful for specialists in the theory of vibrations, for students, and for postgraduate students.

Chapter 1 describes how to study the dynamic response of engineering structures, and explains the importance of such a study in its design. The dynamic forces acting on a structure may be the result of different causes such as rotating machinery, wind, blast, earthquake, and etc. The forces can be classified according to the nature of their vibration with time as periodic, non-periodic, or random. To classify the dynamic forces as deterministic or nondeterministic, can be useful in statistical problems. The above classification is also the topic of Chapter 1. Chapter 2 deals with the single-degree-of-freedom systems. In this chapter, the equations of motion are formulated by using the Newton vectorial mechanics and the principle virtual displacements. However, the procedures of Newtonian mechanics are adequate only for simple structures. For more complex single-degree as well as multi-degree-of-freedom systems the derivatives of equations of motion employ also the energy principle or the principles of analytical mechanics. Chapter 2 presents additionally the methods of representing a continuous system or a discrete multi-degree-of-freedom system by an equivalent single-degree-of-freedom system. Chapter 3 describes the formulation of the equations of motion for a multi-degree-of-freedom system through the principles of vectorial mechanics. Ritz’ method is applied here to the modelling of continuous and discrete systems. Chapter 4 introduces the principles of analytical mechanics and their application to the formulation of equations of motion. The author describes here the generalized coordinates, the generalized forces, the important principle of virtual work, and derives the Lagrange equations of motion. Chapter 5 describes vibrations of a single-degree-of-freedom system. Various damped systems, in the presence of viscous damping, structural damping, and Coulomb damping, are considered. The application of the phase plane to the analysis of vibrational response of damped and free systems is described. Chapter 6 deals with the response of single-degree-of-freedom systems under a harmonic excitation. Resonant response of undamped systems is considered. The author also considers the Nyquist plots for a single-degree-of-freedom system subjected to a harmonic force. Unfortunately, the text does not give information how to find the coefficients of energy dissipation, see [Yu. N. Sankin, Tr. Sredn. Mat. Obshch. 8, No. 2, 22–33 (2006; Zbl 1150.74385)].

Chapter 7 presents the response of single-degree-of-freedom systems to general dynamic loading, such as an impulsive force and shock loading. In Chapter 8, the author gives a review of approximate and numerical methods for the analysis of single-degree and multi-degree-of-freedom systems. Frequency determination of such systems is often complex. On the other hand, if a reasonable estimate can be made on the vibration shape, the system can be represented by an equivalent single-degree-of-freedom model whose frequency of vibration can be easily determined by the Rayleigh method. Chapter 9 deals with transform methods for single-degree-of-freedom systems, in particular with Fourier transform and Laplace transform. Unfortunately, the author does not explain how to use initial conditions [loc. cit.].

Chapter 10 considers free vibrations of multi-degree-of-freedom systems and the application of eigenvalue solutions to the determination of free vibration response. Various methods for the solution of eigenvalue problems of structural dynamics are given in Chapter 11. The methods include transformation methods, iteration methods, and the determinant search methods. Chapter 12 considers the force dynamic response based on normal coordinate transformation, called here the mode superposition method. In Chapter 13 the author presents approximate and numerical methods for multi-degree-of-freedom systems. He considers the Rayleigh-Ritz method and the direct numerical integration in frequency domain. Chapter 14 describes the equations of motion for simple continuous systems such as transverse vibration of a beam, axial and torsional vibrations of a rod, and transverse vibrations of a string and shear beams. Chapter 15 deals with free-vibration response of a simple continuous system. Here, the author considers the associated eigenvalue problem and mode superposition method. Chapter 16 examines the forced response of a simple continuous system under the action of time-varying forces. In Chapter 17, one-dimensional wave propagation is considered. The mode superposition method is not applicable to a system of semi-infinite or infinite extent. In contrast to a system of finite extent, a system of infinite extent has a continuous band of frequencies and the term “mode shape” loses its meaning. Thus the author presents new methods to study the propagation of travelling and standing waves, harmonic waves, wave velocity, wave length and frequency, reflection and refraction of waves and wave dispersion. Chapter 18 gives an introductory treatment of finite element method considered as a variational Bubnov-Galerkin method. Chapter 19 presents the component mode synthesis methods. These methods are effective in structural dynamic analysis of large complex structures, such as an aircraft, an aerospace structure, or an automobile. The final Chapter 20 analyzes the nonlinear response of single-and multi-degree-of-freedom systems.

On balance, this is a fundamental textbook on the vibration theory. It provides the classical principles and modern methods of analysis, modelling and simulation of mechanical systems. The book can be very useful for specialists in the theory of vibrations, for students, and for postgraduate students.

Reviewer: Yuri N. Sankin (Ul’yanovsk)

##### MSC:

74-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mechanics of deformable solids |

70-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mechanics of particles and systems |

74H45 | Vibrations in dynamical problems in solid mechanics |

74S05 | Finite element methods applied to problems in solid mechanics |

70Jxx | Linear vibration theory |

70Kxx | Nonlinear dynamics in mechanics |