THUNDER BIRD PRESENTS

DAMPING HARMONIC OSCILLATOR DAMPING HARMONIC OSCILLATOR AND ITS APPLICATIONAND ITS APPLICATION

HARMONIC OSCILLATOR

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The harmonic oscillator is a common model used in physics because of the wide range of problems it can be applied to. For example atoms in a lattice (crystalline structure of a solid) can be thought of as an in¯nite string of masses connected together by springs, whose equation of motion is oscillatory. In fact, the solutions can be generalized to many systems undergoing oscillations, of which the mass spring system is just one example. Since the mass-spring system is easy to visualize it will serve as the primary example as we develop a more complete general theory describing harmonic motion.

In classical mechanics, a harmonic oscillator is a system which, when displaced from its equilibrium position, experiences a restoring force, F, proportional to the displacement, x according to Hooke's law:

where k is a positive constant.

If F is the only force acting on the system, the system is called a simple harmonic oscillator, and it undergoes simple harmonic motion sinusoidal oscillations about the equilibrium point, with a constant amplitude and a constant frequency (which does not depend on the amplitude).

If a frictional force damping proportional to the velocity is also present, the harmonic oscillator is described as a damped oscillator. Depending on the friction coefficient, the system can:

Oscillate with a frequency smaller than in the non-damped case, and an amplitude decreasing with time (underdamped oscillator).

Decay exponentially to the equilibrium position, without oscillations (overdamped oscillator).

If an external time dependent force is present, the harmonic oscillator is described as a driven oscillator.

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Mechanical examples include pendula (with small angles of displacement), masses connected to springs, and acoustical systems. Other analogous systems include electrical harmonic oscillators such as RLC circuits. The harmonic oscillator model is very important in physics, because any mass subject to a force in stable equilibrium acts as a harmonic oscillator for small vibrations. Harmonic oscillators occur widely in nature and are exploited in many manmade devices, such as clocks and radio circuits. They are the source of virtually all sinusoidal vibrations and waves.

SIMPLE HARMONIC MOTION

The simple harmonic oscillator is one of the central problems in physics. It is useful in understanding springs, small amplitude pendulums, electronic circuits, quantum mechanics, and even cars that shake at 53 MPH. Furthermore, many problems can be considered the sum of a large number, or infinite number, of harmonic oscillators.

In physics, simple harmonic motion (SHM) is the motion of a simple harmonic oscillator, a periodic motion that is neither driven nor damped. A body in simple harmonic motion experiences a single force which is given by Hooke's law; that is, the force is directly proportional to the displacement x and points in the opposite direction.

The motion is periodic: the body oscillates about an equilibrium position in a sinusoidal pattern. Each oscillation is identical, and thus the period, frequency, and amplitude of the motion are constant. If the equilibrium position is taken to be zero, the displacement x of the body at any time t is given by

where A is the amplitude, f is the frequency, and φ is the phase.

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The frequency of the motion is determined by the intrinsic properties of the system (often the mass of the body and a force constant), while the amplitude and phase are determined by the initial conditions (displacement and velocity) of the system. The kinetic and potential energies of the system are also determined by these properties and conditions.

Simple harmonic motion. In this graph, the vertical axis represents the coordinate of the particle (x in the equation), and the horizontal axis represents time (t).

Simple harmonic motion can serve as a mathematical model of a variety of motions, such as the oscillation of a spring. Other phenomena can be approximated by simple harmonic motion, including the motion of a pendulum and molecular vibration.

Simple harmonic motion provides the basis of the characterization of more complicated motions through the techniques of Fourier analysis.

EXAMPLE

A typical example of a system that undergoes simple harmonic motion is an idealized spring–mass system, which is a mass attached to a spring. If the spring is unstretched, there is no net force on the mass (that is, the system is in mechanical equilibrium). However, if the mass is displaced from equilibrium, the spring will exert a restoring force, which is a force that tends to restore the mass to the equilibrium position. In the case of the spring–mass system, this force is the elastic force, which is given by Hooke's Law,

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F = − kx,

where F is the restoring force, x is the displacement, and k is the spring constant.

Any system that undergoes simple harmonic motion exhibits two key features.

1. When the system is displaced from equilibrium there must exist a restoring force that tends to restore it to equilibrium.

2. The restoring force must be proportional to the displacement, or approximately so.

The spring-mass system satisfies both.

Once the mass is displaced it experiences a restoring force, accelerating it, causing it to start going back to the equilibrium position. As it gets closer to equilibrium the restoring force decreases; at the equilibrium position the restoring force is 0. However, at x = 0, the mass has some momentum due to the impulse of the force that has acted on it; this causes the mass to shoot past the equilibrium position, in this case, compressing the spring. The restoring force then tends to slow it down, until the velocity reaches 0, whereby it will attempt to reach equilibrium position again.

As long as the system does not lose energy, the mass will continue to oscillate like so; thus, the motion is termed periodic motion. Further analysis will show that in the case of the spring-mass system the motion is simple harmonic.

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SIMPLE HARMONIC OSSCILATOR

A simple harmonic oscillator is an oscillator that is neither driven nor damped. Its motion is periodic repeating itself in a sinusoidal fashion with constant amplitude, A. Simple harmonic motion SHM can serve as a mathematical model of a variety of motions, such as a pendulum with small amplitudes and a mass on a spring. It also provides the basis of the characterization of more complicated motions through the techniques of Fourier analysis.

In addition to its amplitude, the motion of a simple harmonic oscillator is characterized by its period T, the time for a single oscillation, its frequency, f, the reciprocal of the period f = 1⁄T (i.e. the number of cycles per unit time), and its phase, φ, which determines the starting point on the sine wave. The period and frequency are constants determined by the overall system, while the amplitude and phase are determined by the initial conditions (position and velocity) of that system. Overall then, the equation describing simple harmonic motion is

.

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Alternatively a cosine can be used in place of the sine with the phase shifted by π⁄2.

The general differential force equation for an object of mass m experiencing SHM is:

,

where k is the spring constant which relates the displacement of the object to the force applied to the object. The general solution for this equation is given above with the frequency of the oscillations given by:

. The velocity and acceleration oscillate with a quarter and half a period delay

The velocity and acceleration of a simple harmonic oscillator oscillate with the same frequency as the position but with shifted phases. The velocity is maximum for zero displacement, while the acceleration is in the opposite direction as the displacement.

The potential energy of SHM is

.

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MASS SPRING DAMPER SYSTEM

Mass-spring-damper

A mass attached to a spring and damper. The damping coefficient, usually c, is represented by B in this case. The F in the diagram denotes an external force, which this example does not include.

An ideal mass-spring-damper system with mass m (in kilograms), spring constant k (in newtons per meter) and viscous damper of damping coefficient c (in newton-seconds per meter or kilograms per second) is subject to an oscillatory force

and a damping force

Treating the mass as a free body and applying Newton's second law, the total force Ftot on the body is

where a is the acceleration (in meters per second squared) of the mass and x is the displacement (in meters) of the mass relative to a fixed point of reference.

Since Ftot = Fs + Fd,

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This differential equation may be rearranged into

The following parameters are then defined:

The first parameter, ω0, is called the (undamped) natural frequency of the system . The second parameter, ζ, is called the damping ratio. The natural frequency represents an angular frequency, expressed in radians per second. The damping ratio is a dimensionless quantity.

The differential equation now becomes

Continuing, we can solve the equation by assuming a solution x such that:

where the parameter γ is, in general, a complex number.

Substituting this assumed solution back into the differential equation,

Solving for γ,

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MASS SPRING SYSTEM

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DAMPING STRUCTURE

A damped oscillations generic structure is simply a sustained oscillations generic structure with an additional negative feedback loop. In Figure 1, for example, the negative feedback loop between “Stock Two” and “outflow” gradually drains “Stock Two.” As “Stock Two” decreases, “change in stock one” decreases, so “Stock One” does not quite grow as much as it would in a sustained oscillations structure. The damped growth of “Stock One” hinders positive “change in stock two.” A reduced flow limits thegrowth of “Stock Two,” and the effect propagates in the system until the two stocks approach equilibrium. The graph in Figure 2 illustrates the behavior of a damped oscillations generic structure (simulated with arbitrary parameter values and initial values).

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Depending on the parameter values and initial values of the system, a damped oscillations generic structure can produce behavior of varying amplitude, degree of damping, and period of oscillations.

Amplitude of Oscillations

The maximum amplitude of the damped system depends on the initial imbalance in the system, that is, the gap between the equilibrium and initial stock values.

The maximum amplitude of the damped system is lower than that of the undamped system.

Altering parameter values other than desired stock values does not change the

equilibrium point.

Degree of Damping

The amplitude of each successive peak in the damped oscillations system is determinedby the strength of the additional negative feedback loop.

Changing the strength of the negative feedback loop produces under-damped, Critically-damped, or over-damped oscillations. A system is critically-damped when the value of its damping factor (“parameter three” in Figure 1) is the square root of four times the product of “parameter one” and “parameter two.”

A system is under-damped when “parameter three” is smaller than the critically damped value and over-damped when “parameter three” is larger than the critically damped value.

Period of Oscillations

The period of oscillations in a damped oscillations system is greater than the period in the corresponding sustained oscillations system.

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DAMPING MOTION

While the undamped system provides a measure of elegant beauty and simplicity in its solution it is to a certain extent boring. Once the initial conditions are set it will continue to oscillate forever, never deviating from its simple sinusoidal pattern. This is also unrealistic as any physical system will eventually come to rest. To create a more accurate model a damping (resistive) force must be added. It does not make sense for this to be a constant force. If a mass-spring system is sitting at rest at its equilibrium point it will not all of a sudden begin moving under the in°uence of some mysterious force. It also does not make sense for the force to depend on the displacement, or position, of the mass. If the entire system were to be translated intuition says the motion will be the same, simply moved to another location. It then makes sense to model the damping force as a function of the objects velocity. This adds an additional force of the form Fd = bv, so that our equation of motion is

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Damped LC Oscillator

In practice there is some resistance in the circuit, which causes the energy to burn away as heat over time. Usually this resistance is undesirable, like friction in general, but if it is significant, the oscillator must be modelled as an LCR oscillator.

It's important to note that the LC frequency formula below does not properly apply to damped circuits, unless the resistance is small.

There are two distinct kinds of damped oscillator, with the borderline being known as "critical damping". Light damping occurs when the resistance is low. The system still oscillates but dies away. High resistance causes heavy damping (or overdamping). In this case there is no oscillation; any energy simply dissipates.

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Mechanical Damped Harmonic Motion

The simple harmonic motion relates to the motion of a body acted on by a special kind of force and in friction-free conditions. The amplitude of SHM is a constant. If the amplitude of the oscillation gradually decreases to zero as a result of friction, the motion is said to be damped harmonic motion. The magnitude of the frictional force usually depends on the speed.

The equation of motion of the damped harmonic motion is given by

                   

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Electromagnetic Damped Oscillation

The resistance in the LC circuit will dissipate the energy. The variation of voltage across the capacitor is shown in the following figure.

Using a square wave

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Apply a square wave to the LC-series combination, at each rising edge of the square wave, the capacitor would be charged. Since the charging current also passes through the inductor, oscillation would occur. At the falling edge, the charged capacitor would be discharged through the inductor. Again, oscillation would occur. 

           In the following figure, the first diagram represents the variation of voltage across the capacitor and the second diagram represents the variation of voltage across the inductor.

APPLICATIONS

Inertial Pendulum Systems

Almost all seismometers are based on damped inertial-pendulum systems of one form or another. Simple vertical and horizontal seismometer designs are illustrated in Figure. The frame of the seismometer is rigidly attached to the ground, and the pendulum is designed so that movement of the internal proof mass, m, is delayed relative to the ground motion by the inertia of the mass. Each pendulum system has an equilibrium position in which the mass is at rest and to which it will return following small transitory disturbances. The orientation of the pendulum further determines which component of ground motion will induce relative pendulum motion.

Ground displacements, U(t\ are communicated to the proof mass via the attached springs or lever arms, with favorably oriented motions perturbing the system from its equilibrium position, leading to periodic oscillation of the mass. Friction or viscous damping, represented by the dashpots, is generally proportional to the velocity of the mass and acts to restore the system to its equilibrium position. Small scale fluctuations in the springs and damping elements determine the intrinsic instrument noise level, below which actual ground motions cannot be detected. Although many early seismometers were designed empirically without mathematical analysis, the equation of motion for simple, damped harmonic oscillators provides insight into instrument characteristics.

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Ground displacements, U(t\ are communicated to the proof mass via the attached springs or lever arms, with favorably oriented motions perturbing the system from its equilibrium position, leading to periodic oscillation of the mass. Friction or viscous damping, represented by the dashpots, is generally proportional to the velocity of the mass and acts to restore the system to its equilibrium position. Small scale fluctuations in the springs and damping elements determine the intrinsic instrument noise level, below which actual ground motions cannot be detected. Although many early seismometers were designed empirically without mathematical analysis, the equation of motion for simple, damped harmonic oscillators provides insight into instrument characteristics.

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The motion of the pendulum mass in an inertial reference frame is given by the sum of the ground motion plus the deviation of the mass from its equilibrium state, y(t). For the vertical seismometer in Figure, the forces on the mass must act through the spring and dashpot, with recording-system friction effects included in the dashpot. The force from the spring is –K(t), which is directly proportional to movement of the mass from its equilibrium position and which must involve stretching or contraction of the spring, which has a spring constant K. The damping force, -Dy(t), is directly proportional to the velocity of the mass, with D being a damping coefficient. Newton's law (F = ma) is then.

ROLE OF DAMPING IN EARTHQUAKE STATION

The damping of the oscillator in Figure generally, but not always, reduces the response to earthquake excitation. Because of the change in frequency that the damping introduces, it is possible for very small values of damping to increase the response over the undamped case and for spectral curves for different dampings to cross. In practice, this happens rarely and the amounts are insignificant. If earthquake acceleration were a stationary random process, the average square of steady-state responsewould be inversely proportional to the damping (Crandall and Mark, 1963; Lutes and Sarkani, 1997).

This dependence appears to approximate the effects of damping in most cases of response to earthquakes of significant duration (several natural periods, Tn); that is, response spectra in such cases tend to vary as the inverse square root of the damping value. Damping is less effective in reducing the response to pulse-like excitations. For example, if the oscillator, Figure is subjected to a half sine pulse at its natural frequency, 5% damping only reduces the peak response by about 7.5% from the undamped case. In the limiting case of a very short pulse (of unit area).

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This result shows that small values of damping are relatively ineffective in reducing the maximum response to this type of excitation; for all natural frequencies the maximum response for = 0.05 is only 7.6% less than that for = 0. Doubling the damping from 0.05 to 0.10 decreases the response only by another 7.3%, whereas for steady-state response to stationary random excitation, the reduction would be 29%. Truly viscous damping can appear in the mounting systems for mechanical equipment and in some special structures such as passively damped buildings and bridges, but it is not present in typical buildings. In the case of typical buildings, viscous damping is usually used in analysis and design to approximate the combined effects of such mechanisms as material damping, nonstructural damage, and low levels of yielding and structural damage.

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EQUATIONS TO SATISFY DAMPING IN SEISMOMETER

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DAMPING IN REAL STRUCTURES

It is possible to estimate an “effective” viscous damping ratio directly from laboratory or field tests of structures. One method is to apply a static displacement by attaching a cable to the structure and then suddenly removing the load by cutting the cable. If the structure can be approximated by a single degree of freedom, the displacement response will be of the form shown in Figure, For multi-degree of freedom structural systems, the response will involve the response of more modes and the test and the analysis method required to predict the damping ratios will bemore complex

It should be pointed out that the decay of the typical displacement response only indicates that energy dissipation is taking place. The cause of the energy dissipation may be due to many different effects such as material damping, joint friction and radiation damping at the supports. However, if it is assumed that all energy dissipation is due to linear viscous damping, the free vibration response is given by the following equation.

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Free Vibration Test of Real Structures, Response vs. Time

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