Vibration theory
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CHAPTER 2
BASIC VIBRATION THEORY
Ralph E. Blake
INTRODUCTION
This chapter presents the theory of free and forced steady-state vibration of single degree-of-freedom systems. Undamped systems and systems having viscous damping and structural damping are included. Multiple degree-of-freedom systems are discussed, including the normal-mode theory of linear elastic structures and
Lagrange’s equations.
ELEMENTARY PARTS OF VIBRATORY SYSTEMS
Vibratory systems comprise means for storing potential energy (spring), means for storing kinetic energy (mass or inertia), and means by which the energy is gradually lost (damper). The vibration of a system involves the alternating transfer of energy between its potential and kinetic forms. In a damped system, some energy is dissipated at each cycle of vibration and must be replaced from an external source if a steady vibration is to be maintained. Although a single physical structure may store both kinetic and potential energy, and may dissipate energy, this chapter considers only lumped parameter systems composed of ideal springs, masses, and dampers wherein each element has only a single function. In translational motion, displacements are defined as linear distances; in rotational motion, displacements are defined as angular motions.
TRANSLATIONAL MOTION
Spring. In the linear spring shown in Fig. 2.1, the change in the length of the spring is proportional to the force acting along its length:
F = k(x − u)
(2.1)
FIGURE 2.1 Linear spring.
The ideal spring is considered to have no mass; thus, the force acting on one end is equal and
2.1
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CHAPTER TWO
opposite to the force acting on the other end.The constant of proportionality k is the spring constant or stiffness.
Mass. A mass is a rigid body (Fig. 2.2) whose acceleration x according to Newton’s second law is
¨
CHAPTER 2
BASIC VIBRATION THEORY
Ralph E. Blake
INTRODUCTION
This chapter presents the theory of free and forced steady-state vibration of single degree-of-freedom systems. Undamped systems and systems having viscous damping and structural damping are included. Multiple degree-of-freedom systems are discussed, including the normal-mode theory of linear elastic structures and
Lagrange’s equations.
ELEMENTARY PARTS OF VIBRATORY SYSTEMS
Vibratory systems comprise means for storing potential energy (spring), means for storing kinetic energy (mass or inertia), and means by which the energy is gradually lost (damper). The vibration of a system involves the alternating transfer of energy between its potential and kinetic forms. In a damped system, some energy is dissipated at each cycle of vibration and must be replaced from an external source if a steady vibration is to be maintained. Although a single physical structure may store both kinetic and potential energy, and may dissipate energy, this chapter considers only lumped parameter systems composed of ideal springs, masses, and dampers wherein each element has only a single function. In translational motion, displacements are defined as linear distances; in rotational motion, displacements are defined as angular motions.
TRANSLATIONAL MOTION
Spring. In the linear spring shown in Fig. 2.1, the change in the length of the spring is proportional to the force acting along its length:
F = k(x − u)
(2.1)
FIGURE 2.1 Linear spring.
The ideal spring is considered to have no mass; thus, the force acting on one end is equal and
2.1
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2.2
CHAPTER TWO
opposite to the force acting on the other end.The constant of proportionality k is the spring constant or stiffness.
Mass. A mass is a rigid body (Fig. 2.2) whose acceleration x according to Newton’s second law is
¨