1.2 Types of vibration and their characteristics

Vibrations in mechanical systems come in various forms, each with unique characteristics. Free vibrations occur naturally, while forced vibrations result from external forces. Damped vibrations lose energy over time, and undamped vibrations continue indefinitely. Understanding these types is crucial for engineers.

Linear vibrations follow simple rules, making them easier to analyze. Nonlinear vibrations, however, are more complex and common in real-world applications. Transient vibrations are temporary responses, while steady-state vibrations represent long-term behavior. Single and multi-degree-of-freedom systems differ in complexity and analysis methods.

Vibration Types

Free and Forced Vibrations

• Free vibration occurs when a system oscillates without external force after initial disturbance governed by natural frequency and initial conditions
• Forced vibration results from application of external force or motion to system often leading to resonance when forcing frequency matches natural frequency
• Free vibration examples include plucking a guitar string or releasing a pendulum from its equilibrium position
• Forced vibration examples include a washing machine during spin cycle or a building subjected to earthquake ground motion

Damped and Undamped Vibrations

• Undamped vibration assumes no energy dissipation resulting in constant amplitude oscillations that continue indefinitely in idealized system
• Damped vibration accounts for energy dissipation typically through friction or fluid resistance leading to decrease in amplitude over time
• Types of damping (underdamped, critically damped, overdamped) affect system's response and return to equilibrium
• Undamped vibration example frictionless pendulum in vacuum
• Damped vibration examples shock absorbers in vehicles or door closer mechanisms

Combined Vibrations and Equations of Motion

• Combination vibrations (free-damped or forced-damped) exhibit characteristics of multiple vibration types simultaneously
• Equation of motion for each vibration type differs reflecting specific forces and system properties involved
• Free vibration equation: $m\ddot{x} + kx = 0$
• Forced vibration equation: $m\ddot{x} + kx = F(t)$
• Damped vibration equation: $m\ddot{x} + c\dot{x} + kx = 0$
• Combined forced-damped vibration equation: $m\ddot{x} + c\dot{x} + kx = F(t)$

Linear vs Nonlinear Vibrations

Linear Vibrations Characteristics

• Linear vibrations characterized by systems where principle of superposition applies and output proportional to input
• Natural frequency independent of amplitude of vibration allowing for simpler mathematical analysis
• Linear systems exhibit proportional relationship between force and displacement (Hooke's Law)
• Examples of linear vibrations small amplitude oscillations of simple pendulum or mass-spring system

Nonlinear Vibrations Characteristics

• Nonlinear vibrations occur in systems where restoring force not proportional to displacement leading to complex behavior
• Exhibit phenomena such as multiple equilibrium points limit cycles and chaotic behavior not present in linear systems
• Frequency response can be amplitude-dependent phenomenon known as "jump phenomenon" or "frequency pulling"
• Examples of nonlinear vibrations large amplitude pendulum swings or vibrations in structures with material nonlinearities

Analysis and Real-World Applications

• Analytical solutions for nonlinear vibrations often difficult or impossible to obtain necessitating numerical methods or approximation techniques
• Real-world systems inherently nonlinear but linear approximations often used for small amplitude vibrations around equilibrium points
• Techniques for analyzing nonlinear vibrations include perturbation methods Lyapunov stability analysis and numerical integration
• Applications of nonlinear vibration analysis aerospace structures (flutter) automotive suspensions and earthquake engineering

Transient vs Steady-State Vibrations

Transient Vibrations

• Transient vibrations temporary responses occurring when system disturbed from equilibrium typically decaying over time
• Influenced by system's natural frequency damping and nature of initial disturbance
• Characterized by rapid changes in amplitude and frequency often containing multiple frequency components
• Examples of transient vibrations initial bounce of a car after hitting a bump or vibrations in a structure immediately after impact

Steady-State Vibrations

• Steady-state vibrations represent long-term behavior of system after transients died out often in response to continuous forcing
• In linear systems steady-state response to harmonic excitation occurs at same frequency as excitation but with different amplitude and phase
• Resonance steady-state phenomenon where response amplitude maximized when forcing frequency matches system's natural frequency
• Examples of steady-state vibrations constant hum of a motor or vibrations in a building due to consistent wind loading

Transition and Analysis

• Transition from transient to steady-state behavior governed by system's time constant related to damping characteristics
• Time constant measure of how quickly system reaches steady-state typically defined as time for response to decay to 37% of initial value
• Analysis of transient and steady-state vibrations crucial for understanding system stability fatigue life and performance in engineering applications
• Methods for analyzing transient response include Laplace transforms convolution integrals and numerical time-domain simulations

Single vs Multi Degree-of-Freedom Systems

Single Degree-of-Freedom (SDOF) Systems

• SDOF systems characterized by one independent coordinate required to describe system's motion completely
• Possess one natural frequency and one mode shape simplifying analysis and solution methods
• Equation of motion for SDOF system: $m\ddot{x} + c\dot{x} + kx = F(t)$
• Examples of SDOF systems simple pendulum mass-spring system or single-story building (lateral motion only)

Multi Degree-of-Freedom (MDOF) Systems

• MDOF systems require multiple independent coordinates to fully describe motion resulting in coupled equations of motion
• Possess multiple natural frequencies and corresponding mode shapes each representing unique pattern of vibration
• Number of degrees of freedom equal to number of independent coordinates needed to describe configuration
• Examples of MDOF systems multi-story buildings vehicle suspensions or complex machinery with multiple moving parts

Analysis Techniques and Phenomena

• Modal analysis techniques (eigenvalue analysis) used to decouple equations of motion in MDOF systems and identify natural frequencies and mode shapes
• MDOF systems can exhibit phenomena like mode coupling and internal resonance not present in SDOF systems
• Complexity of analysis and computational requirements increase significantly with number of degrees of freedom in system
• Methods for analyzing MDOF systems include matrix methods finite element analysis and substructuring techniques