16.8 Forced Oscillations and Resonance
3 min read•june 18, 2024
occur when an external force drives an object to vibrate. happens when the driving frequency matches the object's , causing maximum . This phenomenon is crucial in many systems, from swings to skyscrapers.
affects oscillation amplitude by dissipating energy. It reduces peak amplitudes and prevents indefinite increase at . Understanding forced oscillations and resonance is vital in engineering, from designing safe structures to creating efficient machines and musical instruments.
Forced Oscillations and Resonance
Resonance and natural frequency
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- Resonance phenomenon where external causes object to oscillate with maximum amplitude
- Frequency of driving force matches of object
- Natural frequency is frequency at which object tends to oscillate when disturbed from equilibrium position
- Energy transfer most efficient at resonance
- Driving force continuously adds energy to system
- Amplitude of oscillation builds up over time
- Examples of resonance:
- Pushing child on swing at right moment (matching swing's natural frequency)
- Acoustic resonance in musical instruments (guitar, violin)
- Resonance in electrical circuits ()
- : A system that experiences a restoring force proportional to its displacement from equilibrium
Damping effects on oscillation amplitude
- Damping dissipates energy in oscillating system, causing amplitude to decrease over time
- Caused by friction, air resistance, or other dissipative forces
- Relationship between damping and amplitude of forced oscillations:
- Increasing damping decreases amplitude of oscillation
- Heavier damping results in smaller peak amplitudes at resonance
- Without damping, amplitude at resonance would theoretically increase indefinitely
- Types of damping:
- : Oscillations gradually decay over time (pendulum in air)
- : System returns to equilibrium in shortest possible time without oscillating (door closer)
- : System slowly returns to equilibrium without oscillating (heavy object in viscous fluid)
- : A dimensionless parameter that describes how an oscillator is, with higher values indicating lower energy loss per cycle
Forced oscillations characteristics
- : The frequency at which an external force is applied to an oscillating system
- : The difference in phase between the driving force and the resulting oscillation of the system
- : The ratio of the applied force to the resulting velocity in a mechanical system, analogous to electrical impedance
Applications of forced oscillations
- Mechanical systems:
- Engines and motors: Balancing and damping minimize vibrations at resonant frequencies
- Bridges and buildings: Designing avoids resonance with wind or seismic forces ( skyscraper)
- Shock absorbers in vehicles: Damping minimizes oscillations from road irregularities
- Biological systems:
- Human hearing: in inner ear resonates at different frequencies, allowing perception of various sounds
- Cardiovascular system: Resonance can occur in aorta, potentially leading to aneurysms
- Trees and plants: Resonance with wind forces can cause damage or uprooting
- Resonance disaster examples:
- collapse (1940): Wind-induced resonance led to structural failure
- in London (2000): Lateral vibrations caused by pedestrian footsteps at resonant frequency
- Mitigating resonance disasters:
- Proper design and material selection (using damping materials)
- counteract resonant oscillations (Taipei 101, )
- Active control systems monitor and adjust for resonance ()
Key Terms to Review (30)
Amplitude: Amplitude refers to the maximum extent of a vibration or oscillation, measured from the position of equilibrium. It plays a crucial role in understanding how energy is transferred in oscillatory systems, impacting the characteristics of waves and sounds.
Burj Khalifa: The Burj Khalifa is the tallest building in the world, located in Dubai, United Arab Emirates. It is a prominent architectural marvel that has become a symbol of modern engineering and construction achievements. The Burj Khalifa's height and unique design make it a significant consideration in the context of forced oscillations and resonance.
Cochlea: The cochlea is a spiral-shaped, fluid-filled structure located in the inner ear that plays a crucial role in the process of hearing. It is responsible for converting sound waves into electrical signals that can be interpreted by the brain.
Critical damping: Critical damping occurs when a damping force is applied to an oscillating system, bringing it to rest in the shortest possible time without oscillation. It represents the threshold between overdamping and underdamping.
Critically Damped: Critically damped refers to a specific type of damping in oscillatory systems where the system returns to equilibrium as quickly as possible without overshooting or oscillating. This concept is crucial in understanding the behavior of damped harmonic motion and forced oscillations.
Damping: Damping refers to the process of reducing or controlling the amplitude of an oscillating or vibrating system over time. It involves the dissipation of energy, which causes the system to gradually come to rest or a steady-state condition.
Driving Force: The driving force is the external influence or stimulus that causes a system to undergo oscillations or vibrations. It is the primary factor that drives the motion or behavior of a system, particularly in the context of forced oscillations and resonance.
Forced Frequency: Forced frequency refers to the frequency at which an external force or driving force causes an oscillating system to vibrate. It is a key concept in the study of forced oscillations and resonance, as the forced frequency can significantly impact the behavior and response of the system.
Forced Oscillations: Forced oscillations refer to the oscillations of a system that are driven by an external, time-dependent force or input, rather than by the system's own natural frequency. These oscillations occur when a system is subjected to a periodic driving force, causing it to vibrate at the frequency of the driving force, even if this frequency differs from the system's natural frequency.
Harmonic Oscillator: A harmonic oscillator is a system that exhibits oscillations, or repetitive motion, around an equilibrium position. It is a fundamental concept in physics that describes the behavior of various physical systems, including mechanical, electrical, and quantum-mechanical systems.
Magnetorheological Dampers: Magnetorheological dampers are a type of semi-active vibration control device that can rapidly and continuously vary their damping characteristics in response to an applied magnetic field. They are used to control and mitigate unwanted vibrations in various engineering applications, particularly in the context of forced oscillations and resonance phenomena.
Mechanical Impedance: Mechanical impedance is a measure of the opposition to motion in a mechanical system. It is the ratio of the applied force to the resulting velocity, and it determines how much a system will vibrate or oscillate in response to an external force or excitation.
Millennium Bridge: The Millennium Bridge is a steel suspension bridge that crosses the River Thames in London, England. It was designed to allow pedestrians to cross the river, providing a direct link between the City of London and the Tate Modern art gallery on the South Bank.
Natural frequency: Natural frequency is the frequency at which a system oscillates when not subjected to a continuous or repeated external force. It is determined by the system's physical properties such as mass and stiffness.
Natural Frequency: Natural frequency is the intrinsic frequency at which a system naturally vibrates or oscillates when it is not affected by any external forces. It is a fundamental property of a system that depends on its physical characteristics and is independent of any applied forces.
Overdamped: Overdamped describes a system where the damping force is so strong that it prevents oscillations and the system returns to equilibrium without oscillating. This occurs when the damping coefficient is greater than the critical damping coefficient.
Overdamped: Overdamped is a term used to describe a system that exhibits damped harmonic motion where the motion decays without oscillation. In an overdamped system, the damping force is so strong that the system returns to its equilibrium position in a smooth, non-oscillatory manner.
Phase Shift: A phase shift refers to a change in the timing or position of a wave relative to a reference point. It describes the displacement of a wave's position in time or space, often measured in degrees or radians. This concept is crucial in understanding the behavior of waves and their interactions in various physical phenomena.
Quality factor: Quality factor (QF) is a dimensionless factor used in radiological protection to account for the effectiveness of different types of ionizing radiation in causing biological damage. It is used to convert absorbed dose (measured in grays) into equivalent dose (measured in sieverts).
Quality Factor: The quality factor, or Q-factor, is a dimensionless parameter that describes the ratio of a system's stored energy to its dissipated energy. It is a measure of the system's efficiency and is commonly used in the analysis of oscillating systems, electrical circuits, and the biological effects of ionizing radiation.
Resonance: Resonance occurs when a system is driven at its natural frequency, causing it to oscillate with maximum amplitude. It is a phenomenon observed in various physical systems where the input energy is efficiently transferred.
Resonance: Resonance is a phenomenon that occurs when a system is driven by a periodic force at a frequency that matches the system's natural frequency of oscillation, resulting in a significant increase in the amplitude of the system's motion. This concept is fundamental in understanding various physical phenomena, including the behavior of oscillating systems, the propagation of waves, and the operation of electronic circuits.
Resonate: Resonate occurs when a system oscillates at its natural frequency due to an external periodic force. This results in a large amplitude of oscillation.
RLC Circuits: An RLC circuit is an electrical circuit containing a resistor (R), an inductor (L), and a capacitor (C) connected in series or parallel. These circuits are known for their ability to exhibit resonance and forced oscillations, which are important concepts in physics.
Simple harmonic oscillator: A simple harmonic oscillator is a system where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement. It exhibits periodic motion characterized by sinusoidal oscillations.
Tacoma Narrows Bridge: The Tacoma Narrows Bridge was a suspension bridge in the U.S. state of Washington that dramatically collapsed in 1940, just four months after its opening. This catastrophic event provided important insights into the principles of forced oscillations and resonance, which are fundamental concepts in physics.
Taipei 101: Taipei 101 is a landmark skyscraper located in Taipei, Taiwan. It is known for its unique design and innovative engineering features that make it a prominent example of how structural elements can be utilized to mitigate the effects of forced oscillations and resonance in tall buildings.
Tuned Mass Dampers: A tuned mass damper (TMD) is a device that is used to reduce the vibrations of a structure, such as a building or a bridge, by counteracting the motion of the structure. It consists of a mass, a spring, and a damper, which are tuned to the natural frequency of the structure to be controlled.
Underdamped: An underdamped system is one where the damping force is not strong enough to prevent oscillations. The system will oscillate with a gradually decreasing amplitude over time.
Underdamped: Underdamped refers to a system that experiences oscillations that gradually decrease in amplitude over time due to the presence of a relatively small amount of damping. This type of damping allows the system to exhibit a series of oscillations before coming to rest.