Damped Sine Wave
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Damping is an influence within or upon an
oscillatory system Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
that has the effect of reducing or preventing its oscillation. In
physical system A physical system is a collection of physical objects. In physics, it is a portion of the physical universe chosen for analysis. Everything outside the system is known as the environment. The environment is ignored except for its effects on the ...
s, damping is produced by processes that
dissipate In thermodynamics, dissipation is the result of an irreversible process that takes place in homogeneous thermodynamic systems. In a dissipative process, energy (internal, bulk flow kinetic, or system potential) transforms from an initial form to ...
the
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat a ...
stored in the oscillation. Examples include
viscous The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the inter ...
drag (a liquid's viscosity can hinder an oscillatory system, causing it to slow down; see
viscous damping In continuum mechanics, viscous damping is a formulation of the damping phenomena, in which the source of damping force is modeled as a function of the volume, shape, and velocity of an object traversing through a real fluid with viscosity. Typic ...
) in mechanical systems, resistance in
electronic oscillators An electronic oscillator is an electronic circuit that produces a periodic, oscillating electronic signal, often a sine wave or a square wave or a triangle wave. Oscillators convert direct current (DC) from a power supply to an alternating curre ...
, and absorption and scattering of light in
optical oscillator A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word "laser" is an acronym for "light amplification by stimulated emission of radiation". The fir ...
s. Damping not based on energy loss can be important in other oscillating systems such as those that occur in biological systems and
bikes A bicycle, also called a pedal cycle, bike or cycle, is a human-powered or motor-powered assisted, pedal-driven, single-track vehicle, having two wheels attached to a frame, one behind the other. A is called a cyclist, or bicyclist. Bic ...
(ex. Suspension (mechanics)). Not to be confused with
friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative lateral motion of t ...
, which is a dissipative force acting on a system. Friction can cause or be a factor of damping. The damping ratio is a
dimensionless A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
measure describing how oscillations in a system decay after a disturbance. Many systems exhibit oscillatory behavior when they are disturbed from their position of static equilibrium. A mass suspended from a spring, for example, might, if pulled and released, bounce up and down. On each bounce, the system tends to return to its equilibrium position, but overshoots it. Sometimes losses (e.g. frictional) damp the system and can cause the oscillations to gradually decay in amplitude towards zero or
attenuate In physics, attenuation (in some contexts, extinction) is the gradual loss of flux intensity through a medium. For instance, dark glasses attenuate sunlight, lead attenuates X-rays, and water and air attenuate both light and sound at variable a ...
. The damping ratio is a measure describing how rapidly the oscillations decay from one bounce to the next. The damping ratio is a system parameter, denoted by (zeta), that can vary from undamped (), underdamped () through critically damped () to overdamped (). The behaviour of oscillating systems is often of interest in a diverse range of disciplines that include
control engineering Control engineering or control systems engineering is an engineering discipline that deals with control systems, applying control theory to design equipment and systems with desired behaviors in control environments. The discipline of controls o ...
,
chemical engineering Chemical engineering is an engineering field which deals with the study of operation and design of chemical plants as well as methods of improving production. Chemical engineers develop economical commercial processes to convert raw materials int ...
,
mechanical engineering Mechanical engineering is the study of physical machines that may involve force and movement. It is an engineering branch that combines engineering physics and mathematics principles with materials science, to design, analyze, manufacture, and ...
,
structural engineering Structural engineering is a sub-discipline of civil engineering in which structural engineers are trained to design the 'bones and muscles' that create the form and shape of man-made structures. Structural engineers also must understand and cal ...
, and
electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...
. The physical quantity that is oscillating varies greatly, and could be the swaying of a tall building in the wind, or the speed of an
electric motor An electric motor is an Electric machine, electrical machine that converts electrical energy into mechanical energy. Most electric motors operate through the interaction between the motor's magnetic field and electric current in a Electromagneti ...
, but a normalised, or non-dimensionalised approach can be convenient in describing common aspects of behavior.


Oscillation cases

Depending on the amount of damping present, a system exhibits different oscillatory behaviors and speeds. * Where the spring–mass system is completely lossless, the mass would oscillate indefinitely, with each bounce of equal height to the last. This hypothetical case is called ''undamped''. * If the system contained high losses, for example if the spring–mass experiment were conducted in a
viscous The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the inter ...
fluid, the mass could slowly return to its rest position without ever overshooting. This case is called ''overdamped''. * Commonly, the mass tends to overshoot its starting position, and then return, overshooting again. With each overshoot, some energy in the system is dissipated, and the oscillations die towards zero. This case is called ''underdamped.'' * Between the overdamped and underdamped cases, there exists a certain level of damping at which the system will just fail to overshoot and will not make a single oscillation. This case is called ''critical damping''. The key difference between critical damping and overdamping is that, in critical damping, the system returns to equilibrium in the minimum amount of time.


Damped sine wave

A damped sine wave or damped sinusoid is a sinusoidal function whose amplitude approaches zero as time increases. It corresponds to the ''underdamped'' case of damped second-order systems, or underdamped second-order differential equations. Damped sine waves are commonly seen in
science Science is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe. Science may be as old as the human species, and some of the earliest archeological evidence for ...
and
engineering Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...
, wherever a
harmonic oscillator In classical mechanics, a harmonic oscillator is a system that, when displaced from its Mechanical equilibrium, equilibrium position, experiences a restoring force ''F'' Proportionality (mathematics), proportional to the displacement ''x'': \v ...
is losing
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat a ...
faster than it is being supplied. A true sine wave starting at time = 0 begins at the origin (amplitude = 0). A cosine wave begins at its maximum value due to its phase difference from the sine wave. A given sinusoidal waveform may be of intermediate phase, having both sine and cosine components. The term "damped sine wave" describes all such damped waveforms, whatever their initial phase. The most common form of damping, which is usually assumed, is the form found in linear systems. This form is exponential damping, in which the outer envelope of the successive peaks is an exponential decay curve. That is, when you connect the maximum point of each successive curve, the result resembles an exponential decay function. The general equation for an exponentially damped sinusoid may be represented as: y(t) = A e^ \cos(\omega t - \phi) where: *y(t) is the instantaneous amplitude at time ; *A is the initial amplitude of the envelope; *\lambda is the decay rate, in the reciprocal of the time units of the independent variable ; *\phi is the phase angle at ; *\omega is the
angular frequency In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
. Other important parameters include: *
Frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
: f = \omega / (2\pi), the number of cycles per time unit. It is expressed in inverse time units t^, or
hertz The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose expression in terms of SI base units is s−1, meaning that on ...
. *
Time constant In physics and engineering, the time constant, usually denoted by the Greek letter (tau), is the parameter characterizing the response to a step input of a first-order, linear time-invariant (LTI) system.Concretely, a first-order LTI system is a sy ...
: \tau = 1 / \lambda, the time for the amplitude to decrease by the factor of '' e''. *
Half-life Half-life (symbol ) is the time required for a quantity (of substance) to reduce to half of its initial value. The term is commonly used in nuclear physics to describe how quickly unstable atoms undergo radioactive decay or how long stable ato ...
is the time it takes for the exponential amplitude envelope to decrease by a factor of 2. It is equal to \ln(2) / \lambda which is approximately 0.693 / \lambda. * Damping ratio: \zeta is a non-dimensional characterization of the decay rate relative to the frequency, approximately \zeta = \lambda / \omega, or exactly \zeta = \lambda / \sqrt < 1. *
Q factor In physics and engineering, the quality factor or ''Q'' factor is a dimensionless parameter that describes how underdamped an oscillator or resonator is. It is defined as the ratio of the initial energy stored in the resonator to the energy los ...
: Q = 1 / (2 \zeta) is another non-dimensional characterization of the amount of damping; high ''Q'' indicates slow damping relative to the oscillation.


Damping ratio definition

The ''damping ratio'' is a parameter, usually denoted by ''ζ'' (Greek letter zeta), that characterizes the
frequency response In signal processing and electronics, the frequency response of a system is the quantitative measure of the magnitude and phase of the output as a function of input frequency. The frequency response is widely used in the design and analysis of sy ...
of a second-order ordinary differential equation. It is particularly important in the study of
control theory Control theory is a field of mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a ...
. It is also important in the
harmonic oscillator In classical mechanics, a harmonic oscillator is a system that, when displaced from its Mechanical equilibrium, equilibrium position, experiences a restoring force ''F'' Proportionality (mathematics), proportional to the displacement ''x'': \v ...
. In general, systems with higher damping ratios (one or greater) will demonstrate more of a damping effect. Underdamped systems have a value of less than one. Critically damped systems have a damping ratio of exactly 1, or at least very close to it. The damping ratio provides a mathematical means of expressing the level of damping in a system relative to critical damping. For a damped harmonic oscillator with mass ''m'', damping coefficient ''c'', and spring constant ''k'', it can be defined as the ratio of the damping coefficient in the system's differential equation to the critical damping coefficient: : \zeta = \frac = \frac , where the system's equation of motion is : m\frac + c\frac + kx = 0 and the corresponding critical damping coefficient is : c_c = 2 \sqrt or : c_c = 2 m \sqrt = 2m \omega_n where : \omega_n = \sqrt is the
natural frequency Natural frequency, also known as eigenfrequency, is the frequency at which a system tends to oscillate in the absence of any driving force. The motion pattern of a system oscillating at its natural frequency is called the normal mode (if all par ...
of the system. The damping ratio is dimensionless, being the ratio of two coefficients of identical units.


Derivation

Using the natural frequency of a
harmonic oscillator In classical mechanics, a harmonic oscillator is a system that, when displaced from its Mechanical equilibrium, equilibrium position, experiences a restoring force ''F'' Proportionality (mathematics), proportional to the displacement ''x'': \v ...
\omega_n = \sqrt and the definition of the damping ratio above, we can rewrite this as: : \frac + 2\zeta\omega_n\frac + \omega_n^2 x = 0. This equation is more general than just the mass–spring system, and also applies to electrical circuits and to other domains. It can be solved with the approach. : x(t) = C e^, where ''C'' and ''s'' are both
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
constants, with ''s'' satisfying : s = -\omega_n \left(\zeta \pm i \sqrt\right). Two such solutions, for the two values of ''s'' satisfying the equation, can be combined to make the general real solutions, with oscillatory and decaying properties in several regimes: ; Undamped: Is the case where \zeta = 0 corresponds to the undamped simple harmonic oscillator, and in that case the solution looks like \exp(i\omega_nt), as expected. This case is extremely rare in the natural world with the closest examples being cases where friction was purposefully reduced to minimal values. ; Underdamped: If ''s'' is a pair of complex values, then each complex solution term is a decaying exponential combined with an oscillatory portion that looks like \exp\left(i \omega_n \sqrtt\right). This case occurs for \ 0 \le \zeta < 1 , and is referred to as ''underdamped'' (e.g., bungee cable). ; Overdamped: If ''s'' is a pair of real values, then the solution is simply a sum of two decaying exponentials with no oscillation. This case occurs for \zeta > 1 , and is referred to as ''overdamped''. Situations where overdamping is practical tend to have tragic outcomes if overshooting occurs, usually electrical rather than mechanical. For example, landing a plane in autopilot: if the system overshoots and releases landing gear too late, the outcome would be a disaster. ; Critically damped: The case where \zeta = 1 is the border between the overdamped and underdamped cases, and is referred to as ''critically damped''. This turns out to be a desirable outcome in many cases where engineering design of a damped oscillator is required (e.g., a door closing mechanism).


''Q'' factor and decay rate

The ''Q'' factor, damping ratio ''ζ'', and exponential decay rate α are related such that : \zeta = \frac = . When a second-order system has \zeta < 1 (that is, when the system is underdamped), it has two
complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
poles that each have a
real part In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
of -\alpha; that is, the decay rate parameter \alpha represents the rate of
exponential decay A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where is the quantity and (lambda) is a positive rate ...
of the oscillations. A lower damping ratio implies a lower decay rate, and so very underdamped systems oscillate for long times. For example, a high quality
tuning fork A tuning fork is an acoustic resonator in the form of a two-pronged fork with the prongs (tines) formed from a U-shaped bar of elastic metal (usually steel). It resonates at a specific constant pitch when set vibrating by striking it against ...
, which has a very low damping ratio, has an oscillation that lasts a long time, decaying very slowly after being struck by a hammer.


Logarithmic decrement

For underdamped vibrations, the damping ratio is also related to the
logarithmic decrement Logarithmic decrement, \delta , is used to find the damping ratio of an underdamped system in the time domain. The method of logarithmic decrement becomes less and less precise as the damping ratio increases past about 0.5; it does not apply a ...
\delta. The damping ratio can be found for any two peaks, even if they are not adjacent. For adjacent peaks: : \zeta = \frac where \delta = \ln\frac where ''x''0 and ''x''1 are amplitudes of any two successive peaks. As shown in the right figure: : \delta = \ln\frac=\ln\frac=\ln\frac where x_1, x_3 are amplitudes of two successive positive peaks and x_2, x_4 are amplitudes of two successive negative peaks.


Percentage overshoot

In
control theory Control theory is a field of mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a ...
, overshoot refers to an output exceeding its final, steady-state value. For a step input, the percentage overshoot (PO) is the maximum value minus the step value divided by the step value. In the case of the unit step, the ''overshoot'' is just the maximum value of the step response minus one. The percentage overshoot (PO) is related to damping ratio (''ζ'') by: : \mathrm = 100 \exp \left(\right) Conversely, the damping ratio (''ζ'') that yields a given percentage overshoot is given by: : \zeta = \frac


Examples and Applications


Viscous Drag

When an object is falling through the air, the only force opposing its freefall is air resistance. An object falling through water or oil would slow down at a greater rate, until eventually reaching a steady-state velocity as the drag force comes into equilibrium with the force from gravity. This is the concept of viscous drag, which for example is applied in automatic doors or anti-slam doors.


Damping in Electrical Systems / Resistance

Electrical systems that operate with
alternating current Alternating current (AC) is an electric current which periodically reverses direction and changes its magnitude continuously with time in contrast to direct current (DC) which flows only in one direction. Alternating current is the form in whic ...
(AC) use resistors to damp the electrical current, since they are periodic. Dimmer switches or volume knobs are examples of damping in an electrical system.


Magnetic Damping

Kinetic energy that causes oscillations is dissipated as heat by electric eddy currents which are induced by passing through a magnet's poles, either by a coil or aluminum plate. In other words, the resistance caused by magnetic forces slows a system down. An example of this concept being applied is the brakes on roller coasters.


References

{{reflist11. Britannica, Encyclopædia. “Damping.” ''Encyclopædia Britannica'', Encyclopædia Britannica, Inc., www.britannica.com/science/damping. 12. OpenStax, College. “Physics.” ''Lumen'', courses.lumenlearning.com/physics/chapter/23-4-eddy-currents-and-magnetic-damping/. Dimensionless numbers of mechanics Engineering ratios Ordinary differential equations Mathematical analysis Classical mechanics