Undamped free vibrations are the simplest form of mechanical oscillations. They occur when a system moves without external forces or energy loss, swinging back and forth at its natural frequency. This fundamental concept sets the stage for understanding more complex vibration scenarios.

In this part of the chapter, we'll explore how mass and stiffness affect a system's natural frequency. We'll also look at the equation of motion, its solution, and how initial conditions determine the vibration's amplitude and phase. This knowledge is crucial for analyzing real-world mechanical systems.

Equation of Motion for Undamped Vibrations

Derivation Fundamentals

• Newton's Second Law of Motion serves as the fundamental principle for deriving the equation of motion for mechanical vibration systems
• Single degree-of-freedom (SDOF) system characterized by one independent coordinate describing its motion, typically displacement x(t)
• Free body diagram of an SDOF system includes mass (m), spring force (kx), and external forces acting on the system
• Equation of motion for an undamped SDOF system expressed as $mẍ + kx = 0$
  • m represents mass
  • k represents spring constant
  • ẍ represents second derivative of displacement with respect to time
• Natural frequency of the system (ωn) defined as $ωn = \sqrt{\frac{k}{m}}$

Solution and Characteristics

• General solution to the equation of motion takes the form $x(t) = A \cos(ωnt) + B \sin(ωnt)$
  • A and B are constants determined by initial conditions
• Characteristic equation derived by assuming solution of form $x(t) = Ce^{λt}$
  • C and λ are constants to be determined
• Substituting assumed solution into equation of motion yields characteristic equation $mλ² + k = 0$
• Roots of characteristic equation are purely imaginary, given by $λ = ±iωn$
  • i represents imaginary unit
• General solution expressed in complex form as $x(t) = C₁e^{iωnt} + C₂e^{-iωnt}$
  • C₁ and C₂ are complex constants

Natural Frequency and Mode Shape

Frequency Analysis

• Natural frequency (ωn) represents the system's inherent oscillation rate without external forces
• Period of oscillation (T) related to natural frequency by $T = \frac{2π}{ωn}$
  • Represents time required for one complete vibration cycle
• Frequency analysis crucial for understanding system behavior (structural vibrations, acoustic resonance)
• Natural frequency affected by system parameters
  • Increasing mass decreases natural frequency
  • Increasing stiffness increases natural frequency

Mode Shape Characteristics

• Mode shape for undamped SDOF system described by simple harmonic motion
• Sinusoidal function with amplitude and phase determined by initial conditions
• Amplitude remains constant throughout motion due to absence of damping
• Phase angle (φ) calculated as $φ = \tan^{-1}\left(\frac{B}{A}\right)$
  • Represents initial angular position of oscillation
• Mode shape visualization aids in understanding system behavior (nodal points, maximum displacement locations)

Response to Initial Conditions

Initial Condition Analysis

• Initial conditions for vibrating system typically include
  • Initial displacement x(0)
  • Initial velocity ẋ(0) at time t = 0
• Constants A and B in general solution determined by applying initial conditions and solving resulting system of equations
• Amplitude of vibration given by $\sqrt{A² + B²}$
  • Remains constant throughout motion due to absence of damping
• Total energy of system remains constant in undamped free vibrations
  • Consists of kinetic and potential energy
  • Follows conservation of energy principle

Response Visualization

• System response visualized using phase plane plots
  • Show relationship between displacement and velocity over time
• Phase plane analysis reveals important system characteristics (stable equilibrium points, limit cycles)
• Time-domain response plots illustrate displacement, velocity, and acceleration variations
• Frequency-domain analysis (Fourier transform) reveals dominant frequency components of response

Resonance in Undamped Vibrations

Resonance Phenomenon

• Resonance occurs when frequency of external force matches system's natural frequency
• Results in maximum amplitude of vibration
• In undamped systems, resonance theoretically leads to infinite amplitude
  • Real systems always have some damping limiting growth
• Resonance frequency for undamped SDOF system equals its natural frequency $ωn = \sqrt{\frac{k}{m}}$
• Beat frequency phenomenon occurs when two vibrations with slightly different frequencies interfere
  • Results in periodic variations in amplitude (acoustic beats, optical interference)

Implications and Applications

• Resonance has both beneficial and detrimental effects
  • Beneficial applications include musical instruments (guitar strings, piano soundboards)
  • Detrimental effects include structural failure in buildings or bridges (Tacoma Narrows Bridge collapse)
• Crucial for designing structures and machines to avoid or utilize resonant frequencies
  • Avoiding resonance in structural design (earthquake-resistant buildings, vibration isolation)
  • Utilizing resonance in energy harvesting devices (piezoelectric energy harvesters)
• Modal analysis extends resonance understanding to multi-degree-of-freedom systems
  • Multiple natural frequencies and mode shapes exist
  • Important for complex structure analysis (aircraft, spacecraft, large buildings)