16.7 Damped Harmonic Motion
3 min read•june 18, 2024
is all about how oscillations change over time due to resistance. It's like pushing a kid on a swing - eventually, they slow down because of air resistance and friction.
There are three types of damping: (slow decay), (no oscillation), and (fastest return to equilibrium). Damping affects amplitude but not frequency, and causes energy loss through like friction.
Damped Harmonic Motion
Types of damped oscillations
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- systems
- is less than the causes the system to oscillate with gradually decreasing amplitude over time (lightly damped )
- systems
- Damping force is greater than the critical damping force prevents oscillation and returns the system to without overshooting (heavily damped door closer)
- Critically damped systems
- Damping force equals the critical damping force allowing the system to return to equilibrium position in the shortest possible time without oscillating
- Represents the boundary between underdamped and overdamped systems (carefully designed car suspension system)
Effects of damping on motion
- Period and frequency
- Damping forces do not significantly affect the period or frequency of oscillation which are primarily determined by the system's mass and
- Amplitude
- Damping forces cause the amplitude of oscillation to decrease over time at a rate dependent on the strength of the damping force
- Stronger damping leads to faster
- In underdamped systems, the amplitude decreases exponentially with each oscillation
- In overdamped systems, the amplitude decreases rapidly without oscillation
- In critically damped systems, the amplitude decreases to zero in the shortest possible time without oscillation
- Damping forces cause the amplitude of oscillation to decrease over time at a rate dependent on the strength of the damping force
- The is a measure of how quickly oscillations decay in a system
Energy loss in damped systems
- Non-conservative forces (friction or air resistance) dissipate energy from the system by converting into other forms (heat or sound)
- The work done by non-conservative forces is path-dependent and not recoverable
- In damped harmonic systems, non-conservative forces cause the total energy of the system to decrease over time at a rate dependent on the strength of the damping force
- As energy is removed from the system, the amplitude of oscillation decreases
- In underdamped systems, the energy is gradually dissipated over multiple oscillations
- In overdamped systems, the energy is rapidly dissipated without oscillation
- In critically damped systems, the energy is dissipated in the shortest possible time without oscillation
Forced and Free Oscillations
- occur when a system is displaced from equilibrium and allowed to oscillate without external forces
- happen when an external periodic force is applied to the system
- The system's response depends on the frequency of the applied force
- When the driving frequency matches the system's , resonance occurs
- The is a dimensionless parameter that describes how under-damped an oscillator is
- The is the at which a system oscillates with maximum amplitude
Key Terms to Review (24)
Amplitude Decay: Amplitude decay refers to the gradual reduction in the maximum displacement or peak value of an oscillating or vibrating system over time. This phenomenon is observed in damped harmonic motion, where the energy of the system is dissipated, causing the amplitude of the oscillations to diminish with each successive cycle.
Critical Damping Force: Critical damping force is the minimum amount of damping required to prevent an oscillating system from exhibiting any overshooting or oscillations. It represents the boundary between underdamped and overdamped behavior, ensuring the system returns to equilibrium as quickly as possible without oscillations.
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.
Damped Harmonic Motion: Damped harmonic motion is a type of oscillatory motion where the amplitude of the oscillations decreases over time due to the presence of a damping force. This motion is characterized by a gradual reduction in the size of the oscillations until the system eventually comes to rest.
Damping Force: Damping force is a force that opposes the motion of an oscillating system, gradually reducing the amplitude of the oscillations over time. It is a crucial concept in the study of damped harmonic motion, where it plays a central role in determining the behavior and characteristics of the system.
Damping Ratio: The damping ratio is a dimensionless quantity that describes the oscillatory behavior of a system. It determines the rate at which the amplitude of a vibrating system decreases over time, indicating the degree of damping present in the system.
Equilibrium Position: The equilibrium position refers to the stable or balanced state of a system where the net force acting on the system is zero, and the system remains at rest or in a state of constant motion. This concept is particularly important in the study of oscillations, simple harmonic motion, energy of oscillators, and damped motion.
Exponential Decay: Exponential decay is a mathematical model that describes the gradual reduction of a quantity over time. It is characterized by an initial value that decreases at a rate proportional to its current value, resulting in a smooth, continuous decline.
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.
Free Oscillations: Free oscillations refer to the natural, unforced vibrations of a system that occur when the system is displaced from its equilibrium position and then released. These oscillations continue indefinitely, without the need for external driving forces, as long as there is no energy dissipation in the system.
Mass-Spring System: A mass-spring system is a simple model used to describe the motion of an object attached to a spring and subjected to a force. It is a fundamental concept in the study of oscillatory motion and is central to the understanding of topics such as simple harmonic motion and damped harmonic motion.
Mechanical Energy: Mechanical energy is the sum of kinetic energy and potential energy in a system, representing the total energy available for performing work. This concept encompasses various forms of energy related to motion and position, and is crucial for understanding how objects interact under the influence of forces.
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.
Non-Conservative Forces: Non-conservative forces are forces that do not satisfy the work-energy theorem, meaning the work done by these forces depends on the path taken between two points rather than just the initial and final positions. These forces cannot be expressed as the gradient of a potential function.
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.
Pendulum: A pendulum is a weight suspended from a fixed point that swings back and forth under the influence of gravity. Its motion is periodic, characterized by a constant period and frequency when displaced from its equilibrium position, making it an important example of oscillatory motion.
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 Frequency: Resonance frequency is the natural or characteristic frequency at which a system or object tends to oscillate or vibrate with the greatest amplitude when subjected to an external force or stimulus. This concept is fundamental in understanding the behavior of various physical systems, including mechanical and electrical systems.
Spring Constant: The spring constant, also known as the force constant, is a measure of the stiffness of a spring. It represents the force required to stretch or compress a spring by a unit distance and is a fundamental property of the spring that determines its behavior in various physical contexts.
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.