Glossary

3.1 Undamped free vibrations

4 min read•july 30, 2024

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 . 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 and phase. This knowledge is crucial for analyzing real-world mechanical systems.

Equation of Motion for Undamped Vibrations

Derivation Fundamentals

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  • 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 mx¨+kx=0mẍ + 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=kmωn = \sqrt{\frac{k}{m}}

Solution and Characteristics

  • General solution to the equation of motion takes the form x(t)=Acos⁡(ωnt)+Bsin⁡(ωnt)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)=Ceλtx(t) = Ce^{λt}
    • C and λ are constants to be determined
  • Substituting assumed solution into equation of motion yields characteristic equation mλ2+k=0mλ² + k = 0
  • Roots of characteristic equation are purely imaginary, given by λ=±iωnλ = ±iωn
    • i represents imaginary unit
  • General solution expressed in complex form as x(t)=C1eiωnt+C2e−iωntx(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 rate without external forces
  • of oscillation (T) related to natural frequency by T=2πωnT = \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
  • Sinusoidal function with amplitude and phase determined by initial conditions
  • Amplitude remains constant throughout motion due to absence of damping
  • (φ) calculated as φ=tan⁡−1(BA)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 A2+B2\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
    • Follows 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=kmωn = \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, )
    • 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)

Key Terms to Review (17)

Amplitude: Amplitude refers to the maximum extent of a vibration or oscillation, measured from the position of equilibrium. In mechanical systems, it represents the peak displacement from the resting position during free vibrations and plays a crucial role in determining the energy and intensity of the motion. Understanding amplitude is essential for interpreting vibration data, as it directly affects the behavior and response of mechanical systems under various conditions.
Conservation of Energy: Conservation of energy is a fundamental principle stating that energy cannot be created or destroyed, only transformed from one form to another. In the context of undamped free vibrations, this means that the total mechanical energy of a system, which includes kinetic and potential energy, remains constant over time as the system oscillates. This principle is crucial for understanding the behavior of vibrating systems, as it allows for the prediction of motion and energy distribution throughout the cycle of oscillation.
Differential Equation: A differential equation is a mathematical equation that relates a function with its derivatives, expressing how a quantity changes in relation to another variable. In the study of mechanical systems, these equations help describe the dynamic behavior of systems under various conditions, providing insight into aspects such as motion, stability, and response characteristics.
Displacement-time graph: A displacement-time graph is a visual representation that shows how the position of an object changes over time. In the context of undamped free vibrations, this type of graph illustrates the periodic motion of a vibrating system, depicting how displacement oscillates back and forth around an equilibrium position without any loss of energy. The slope of the graph at any point indicates the velocity of the object, while the overall shape reveals important characteristics such as amplitude and frequency of the vibrations.
Free Vibration: Free vibration occurs when a mechanical system oscillates without any external force acting on it after an initial disturbance. This type of vibration relies on the system's inherent properties, such as stiffness and mass, allowing it to oscillate at its natural frequency until energy is dissipated through damping or other means.
Hooke's Law: Hooke's Law states that the force exerted by a spring is directly proportional to the displacement of the spring from its equilibrium position, represented mathematically as $$F = -kx$$, where $$F$$ is the restoring force, $$k$$ is the spring constant, and $$x$$ is the displacement. This principle underlies many mechanical systems and can be applied to analyze various types of vibrations, as it describes how materials return to their original shape after deformation.
Kinetic Energy: Kinetic energy is the energy possessed by an object due to its motion, defined mathematically as $$KE = \frac{1}{2} mv^2$$, where 'm' is mass and 'v' is velocity. This energy plays a critical role in various mechanical systems, particularly during oscillations and vibrations, where it alternates with potential energy. Understanding how kinetic energy behaves in different contexts helps analyze the dynamics of vibrating systems and their responses to forces.
Natural Frequency: Natural frequency is the frequency at which a system tends to oscillate in the absence of any external forces. It is a fundamental characteristic of a mechanical system that describes how it responds to disturbances, and it plays a crucial role in the behavior of vibrating systems under various conditions.
Oscillation: Oscillation refers to the repetitive variation or movement of an object around a central point or equilibrium position. This phenomenon is crucial in understanding how systems behave over time, particularly in mechanical contexts where forces act upon them. It encompasses various types of motion, including free vibrations that occur without any energy loss and transient responses that follow an impulse or external force application.
Pendulum: A pendulum is a weight suspended from a fixed point that swings back and forth under the influence of gravity. It is a classic example of undamped free vibrations, where the motion occurs without any external forces or energy losses, resulting in a periodic oscillation. The behavior of a pendulum is governed by its length, mass, and the acceleration due to gravity, making it an ideal model for studying harmonic motion.
Period: The period is the duration of one complete cycle of a periodic motion, usually measured in seconds. It is inversely related to frequency, which indicates how many cycles occur in one second. Understanding the period is crucial for analyzing oscillatory systems, as it provides insights into the system's behavior, energy, and stability during undamped free vibrations.
Phase Angle: Phase angle is a measure of the position of a point in time on a waveform cycle, typically expressed in degrees or radians. It is crucial in understanding the relationship between sinusoidal functions, especially when analyzing oscillatory motion and energy transfer in mechanical systems. The phase angle indicates how far along in its cycle an oscillating system is at a given moment, allowing for comparisons between multiple oscillations and insights into the system's dynamic behavior.
Potential Energy: Potential energy is the stored energy in a system that has the potential to do work due to its position or configuration. It is crucial for understanding how mechanical systems behave, especially when analyzing motion and vibrations. This concept is directly related to how energy is exchanged between kinetic and potential forms, and plays a key role in scenarios like undamped free vibrations, resonance, and energy methods for analyzing vibrations in single degree-of-freedom and coupled systems.
Simple harmonic motion: Simple harmonic motion (SHM) is a type of periodic motion where an object oscillates back and forth around an equilibrium position, following a specific sinusoidal pattern. In this motion, the restoring force acting on the object is directly proportional to its displacement from the equilibrium position and acts in the opposite direction, leading to consistent oscillations. This behavior is foundational in understanding various mechanical systems and their vibrations, as it establishes a basis for concepts such as undamped free vibrations and natural frequency.
Spring-mass system: A spring-mass system consists of a mass attached to a spring, which provides a restoring force when the mass is displaced from its equilibrium position. This setup is fundamental in understanding how mechanical vibrations and oscillatory motion work, where the mass and spring interact to produce periodic motion. The behavior of a spring-mass system is crucial in analyzing undamped free vibrations, where the system oscillates indefinitely without energy loss, and also serves as a classic example of a single degree-of-freedom system.
Tuning Fork: A tuning fork is a metal device that vibrates at a specific frequency when struck, producing a pure musical tone. It is often used as a standard for tuning musical instruments and in various applications in science and medicine, demonstrating the principles of undamped free vibrations due to its ability to resonate at a constant frequency without energy loss.
Vibration isolation: Vibration isolation is a technique used to reduce the transmission of vibrations from one object to another, thereby protecting sensitive equipment or structures from potentially damaging oscillations. This concept is important for minimizing the effects of vibrations generated by machinery, traffic, or environmental sources on adjacent structures and systems.
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