Ondes et vibrations
Section outline
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Vibratory phenomena deal with the oscillations of mechanical systems, which are well known for their significant importance in the field of physics. For example, the vibratory phenomenon that can affect civil engineering systems and structures is the earthquake. Indeed, physical quantities (displacement, velocity, angular frequency, etc.) are time-dependent variables. They are studied through the behavior of systems with one or multiple degrees of freedom, whether free or forced, and in the presence or absence of damping.
In the “vibrations” section, the systems studied are characterized by equations of motion of the type: linear differential equations. This makes it possible to describe various important characteristics of vibrations.
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CHAPTER 01: Generalities on Vibrations
- Definition of periodic motion
- Definition of an oscillation
- Definition of sinusoidal motion
- Degrees of freedom
- Complex representation of vibratory motion
- Definition of Fourier series
CHAPTER 02: Undamped Free Oscillations with One Degree of Freedom
II.1 Undamped free systems (free oscillators)
II.2 Harmonic oscillator
II.3 Equation of motion
II.4 Kinetic energy and potential energy
II.4.1 Kinetic energyII.4.2 Potential energy
II.4.2.1 Gravitational potential energy
II.4.2.2 Electrical potential energy
II.4.2.3 Elastic potential energy
II.5 Equilibrium conditions
II.6 Equivalent systems
II.6.1 Equivalent springs
II.6.1.1 Springs in series
II.6.2 Springs in parallel and a solid placed between two springs
II.6.3 Case of a spring with non-negligible mass
II.7 Moments of inertia of regular solids
II.8 Solved exercises
II.9 Additional exercises
CHAPTER 03: Damped Free Oscillations with One Degree of Freedom
III.1 Introduction
III.2 Damped oscillator
III.3 Friction and damping coefficient
III.3.1 Viscous friction
III.3.2 Solid friction
III.4 Lagrange’s equation
III.5 Regimes of the damped oscillator
III.5.1 Aperiodic regime
III.5.2 Critical regime
III.5.3 Pseudo-periodic regime
III.6 Logarithmic decrement
III.7 Quality factor
III.8 Mechanical energy
III.9 Solved exercises
III.10 Additional exercises
CHAPTER 04: Forced Oscillations of Systems with One Degree of Freedom
IV.1 Introduction
IV.2 Differential equation of motion
IV.2.1 Example of a damped forced system (mass–spring–damper system)
IV.3 Solution of the differential equation of motion
IV.3.1 Sinusoidal excitation
IV.3.1.1 Calculation of amplitudeIV.3.1.2 Calculation of phase
IV.3.2 Resonance frequency
IV.3.3 Bandwidth
IV.3.4 Quality factor
IV.3.5 Periodic excitation
IV.4 Mechanical impedance
IV.4.1 Mechanical impedances
IV.5 Solved exercises
IV.6 Additional exercises
CHAPTER 05: Free Oscillations of Systems with Multiple Degrees of Freedom
V.1 Introduction
V.2 Two-degree-of-freedom systems
V.2.1 Types of coupling
V.2.1.1 Elastic coupling
V.2.1.2 Inertial coupling
V.2.1.3 Viscous coupling
V.2.2 Differential equations of motion
V.2.3 General method for solving equations of motion
V.2.4 Study of a two-degree-of-freedom mechanical system
V.2.4.1 Complex system (mass–spring systems)
V.2.4.2 Study of normal modes
V.2.4.3 Beat phenomenonV.3 Forced oscillations of two-degree-of-freedom systems
V.3.1 Lagrange equations
V.3.2 Differential equation of a forced two-degree-of-freedom system
V.3.3 Study of sinusoidal steady-state regime (solving differential equations)
V.3.4 Calculation of X2 in the case of weak damping
V.3.4.1 Negligible damping
V.3.5 Variation of amplitudesV.3.6 Application
V.4 Oscillations of mechanical systems with N degrees of freedom
V.4.1 Definition
V.4.2 Lagrange method for formulating equations of N-degree-of-freedom systems
V.4.3 Formulation of equations for N-degree-of-freedom systems
V.4.3.1 General case of N degrees of freedom
V.4.3.2 Normal vibration modes of a three-degree-of-freedom mechanical systemV.5 Solved exercises
V.6 Additional exercises -
CHAPTER 1
Definition of periodic motion
A motion is said to be periodic if it repeats itself identically over equal time intervals.
Examples:
- The Moon’s revolution motion: the Moon completes a full cycle around the Earth in about 29 days.
- Heartbeats: a heartbeat is a sequence of contractions and relaxations of the heart muscles that activate valves and cause blood circulation throughout the body.
Definition of an oscillation
An oscillation is a motion that takes place around an equilibrium position