Glossary

occur when an external periodic force drives an oscillating system, determining its frequency. This contrasts with , which happen without external forces. Understanding forced oscillations is crucial for many real-world applications, from musical instruments to engineering systems.

The for forced oscillators includes terms for mass, damping, and stiffness, along with the . , a key concept, occurs when the driving frequency matches the system's , leading to maximum . Factors like damping and driving frequency significantly influence the system's behavior and .

Forced Oscillations

Forced vs free oscillations

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  • Forced oscillations occur when an external periodic force is applied to an oscillating system drives the oscillation and determines its frequency (pendulum clock, guitar string)
  • Free oscillations occur without any external driving force frequency determined by the system's natural frequency depends on its physical properties (mass, stiffness) (tuning fork, spring-mass system)

Equation of motion for forced oscillators

  • Equation of motion for a forced oscillator: mx¨+bx˙+kx=F0cos⁡(ωt)m\ddot{x} + b\dot{x} + kx = F_0 \cos(\omega t)
    • mm mass of the oscillator
    • bb represents energy dissipation (friction, air resistance)
    • kk measure of the system's stiffness
    • F0F_0 of the driving force
    • ω\omega of the driving force
  • Driving force term F0cos⁡(ωt)F_0 \cos(\omega t) represents the external periodic force applied to the system can be any periodic function (sine, square wave)
  • to this equation: x(t)=Acos⁡(ωt−ϕ)x(t) = A \cos(\omega t - \phi)
    • AA amplitude of the forced oscillation
    • ϕ\phi between the driving force and the oscillation
  • The system's initial response to the driving force is called the , which eventually settles into the steady-state solution

Resonance in Forced Oscillations

Resonance in forced oscillations

  • occurs when the frequency of the driving force matches the natural frequency of the oscillating system ω0=k/m\omega_0 = \sqrt{k/m}
  • At resonance, the amplitude of the oscillation reaches a maximum value depends on the damping of the system
    • amplitude theoretically approaches infinity at resonance (ideal case)
    • amplitude reaches a finite maximum value (real-world systems)
  • Resonance can be beneficial (musical instruments, microwave ovens) or destructive (bridges, buildings)
  • The (Q factor) is a dimensionless parameter that describes how under-damped an oscillator is, with higher Q indicating a sharper

Factors influencing resonance behavior

  • Damping affects the amplitude and sharpness of the resonance peak
    1. Higher damping lower maximum amplitude and broader resonance peak (heavily damped systems)
    2. Lower damping higher maximum amplitude and narrower resonance peak (lightly damped systems)
  • Driving frequency determines whether the system is at, below, or above resonance
    1. Driving frequency below the natural frequency ω<ω0\omega < \omega_0 oscillation with the driving force
    2. Driving frequency above the natural frequency ω>ω0\omega > \omega_0 oscillation with the driving force
    3. At resonance \ω=\ω0\ω = \ω_0 oscillation lags behind the driving force by a phase of 90°

System Response and Power Absorption

  • The shows how the amplitude of oscillation varies with the driving frequency
  • The of the response is the range of frequencies for which the power absorption is at least half the maximum value
  • At resonance, the system exhibits maximum power absorption from the driving force

Key Terms to Review (26)

Amplitude: Amplitude is the maximum displacement of a point on a wave from its equilibrium position. It is a measure of the energy carried by the wave.
Amplitude: Amplitude is the maximum displacement or extent of a periodic motion, such as a wave or an oscillation, from its equilibrium position. It represents the magnitude or size of the motion and is a fundamental characteristic of various physical phenomena described in the topics of 1.7 Solving Problems in Physics, 8.4 Potential Energy Diagrams and Stability, 15.1 Simple Harmonic Motion, and beyond.
Angular frequency: Angular frequency, denoted by $\omega$, is the rate of change of angular displacement with time. It is commonly measured in radians per second (rad/s).
Angular Frequency: Angular frequency, often represented by the Greek letter $\omega$ (omega), is a fundamental concept that describes the rate of change of the angular position of an object undergoing rotational or oscillatory motion. It is a crucial parameter in understanding various physical phenomena, including simple harmonic motion, wave propagation, and the behavior of oscillating systems.
Bandwidth: Bandwidth refers to the range of frequencies or the maximum amount of data that can be transmitted over a communication channel or a computer network in a given amount of time. It is a fundamental concept in the study of forced oscillations, as it determines the response of a system to an external driving force.
Damped System: A damped system is a dynamic system that experiences a reduction in the amplitude of its oscillations over time due to the presence of some form of damping. Damping refers to the dissipation of energy within the system, which causes the oscillations to gradually diminish in magnitude.
Damping coefficient: The damping coefficient is a parameter that quantifies the extent to which oscillations decrease in amplitude over time due to energy loss from the system. It indicates how much resistance is present in a system, impacting the rate of decay of oscillatory motion and influencing the behavior of both damped and forced oscillations. This coefficient plays a crucial role in determining the system's response to external forces and its tendency to return to equilibrium after being disturbed.
Driving Force: The driving force is the external influence or energy that causes an object or system to move or change in a particular way. It is the primary factor that initiates and sustains the motion or behavior of a physical system.
Equation of Motion: The equation of motion is a fundamental concept in classical mechanics that describes the relationship between the position, velocity, acceleration, and time of an object undergoing motion. It is a mathematical expression that allows for the prediction and analysis of an object's movement under the influence of various forces.
Forced Oscillations: Forced oscillations refer to the oscillatory motion of a system that is driven by an external force or periodic input, rather than by the system's own natural frequency. This type of oscillation occurs when a system is subjected to a continuous, time-varying force that causes it to vibrate at a frequency different from its natural frequency.
Free Oscillations: Free oscillations refer to the natural, unforced vibrations of a system that occur when it is displaced from its equilibrium position and allowed to oscillate without any external driving force. These oscillations are determined by the inherent properties of the system, such as its mass, stiffness, and damping characteristics.
Frequency Response Curve: A frequency response curve is a graphical representation of the magnitude and phase response of a system or device as a function of the frequency of the input signal. It is a fundamental tool used in the analysis and design of various systems, including electronic circuits, mechanical systems, and control systems.
In-Phase: In-phase refers to the state where two or more periodic signals or waves have their maxima and minima aligned, meaning they reach their peak and trough values at the same time. This synchronization of the signals is a crucial concept in the study of forced oscillations.
Natural Frequency: Natural frequency is the inherent frequency at which a system tends to oscillate when it is not affected by external forces. It is a fundamental property of a system that depends on its physical characteristics and determines how the system will respond to various inputs or disturbances.
Out-of-phase: Out-of-phase refers to the condition where two oscillating systems have their peaks and troughs misaligned, meaning that when one system reaches its maximum displacement, the other is at its minimum. This relationship can significantly affect the behavior of forced oscillations, particularly in how energy is transferred and how systems respond to external forces. In forced oscillations, being out-of-phase can lead to destructive interference, impacting the amplitude and stability of the system.
Periodic driving force: A periodic driving force is an external force applied to a system at regular time intervals, causing the system to oscillate with a frequency determined by the force. This can lead to resonance if the driving frequency matches the natural frequency of the system.
Phase Difference: Phase difference refers to the difference in phase between two oscillating systems or waves at a given point in time. It is typically measured in degrees or radians and plays a crucial role in understanding how waves interact with each other, including their constructive and destructive interference, as well as their collective behavior in various physical contexts.
Power Absorption: Power absorption refers to the process by which a system or medium absorbs energy from an external source, typically in the form of electromagnetic radiation or mechanical vibrations. This concept is particularly relevant in the context of forced oscillations, where the system's response to an external driving force can lead to the absorption and dissipation of energy.
Quality Factor: The quality factor, or Q-factor, is a dimensionless parameter that describes the quality or performance of a resonant system. It quantifies the ratio of a system's stored energy to its dissipated energy, and is an important concept in the analysis of damped and forced oscillations.
Resonance: Resonance occurs when a system is driven at its natural frequency, leading to a significant increase in amplitude. It is a crucial concept in oscillations and wave phenomena.
Resonance: Resonance is a phenomenon that occurs when a system is driven by a force that matches the system's natural frequency of oscillation, leading to a significant increase in the amplitude of the system's response. This concept is fundamental across various fields in physics, including mechanics, acoustics, and electromagnetism.
Resonance Peak: A resonance peak is the maximum amplitude or response of a system that is undergoing forced oscillations at a particular frequency. It occurs when the frequency of the driving force matches the natural frequency of the system, causing the system to vibrate with maximum intensity.
Spring Constant: The spring constant, often denoted as 'k', is a measure of the stiffness of a spring. It quantifies the force required to stretch or compress a spring by a unit distance, and it is a fundamental property of a spring that is crucial in understanding its behavior in various physical contexts.
Steady-State Solution: A steady-state solution refers to a condition in a system where the variables of interest, such as position, velocity, or energy, remain constant over time. This means that the system has reached an equilibrium state where the input and output of the system are balanced, and the overall behavior of the system does not change with time.
Transient Response: Transient response refers to the temporary, time-dependent behavior of a system or signal before it reaches a steady-state condition. It describes the initial, dynamic response of a system to a change in input or initial conditions, prior to the system settling into a stable, long-term behavior.
Undamped System: An undamped system is a type of oscillatory system that experiences no energy dissipation or loss over time. In an undamped system, the oscillations continue indefinitely without any decrease in amplitude, maintaining a constant energy level within the system.
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