# Undamped oscillation examples

In classical mechanicsa harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force F proportional to the displacement x :. If F is the only force acting on the system, the system is called a simple harmonic oscillatorand 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:. The boundary solution between an underdamped oscillator and an overdamped oscillator occurs at a particular value of the friction coefficient and is called critically damped. If an external time-dependent force is present, the harmonic oscillator is described as a driven oscillator.

Mechanical examples include pendulums with small angles of displacementmasses connected to springsand 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. A simple harmonic oscillator is an oscillator that is neither driven nor damped. Balance of forces Newton's second law for the system is. Solving this differential equationwe find that the motion is described by the function.

The motion is periodicrepeating itself in a sinusoidal fashion with constant amplitude A. The period and frequency are determined by the size of the mass m and the force constant kwhile the amplitude and phase are determined by the starting position and velocity.

The velocity and acceleration of a simple harmonic oscillator oscillate with the same frequency as the position, but with shifted phases. The velocity is maximal for zero displacement, while the acceleration is in the direction opposite to the displacement. In real oscillators, friction, or damping, slows the motion of the system.

Due to frictional force, the velocity decreases in proportion to the acting frictional force. While in a simple undriven harmonic oscillator the only force acting on the mass is the restoring force, in a damped harmonic oscillator there is in addition a frictional force which is always in a direction to oppose the motion.

The balance of forces Newton's second law for damped harmonic oscillators is then. A damped harmonic oscillator can be:.

Driven harmonic oscillators are damped oscillators further affected by an externally applied force F t. Newton's second law takes the form.

This equation can be solved exactly for any driving force, using the solutions z t that satisfy the unforced equation. The term overshoot refers to the extent the response maximum exceeds final value, and undershoot refers to the extent the response falls below final value for times following the response maximum.Every object, every particle and every system oscillates in its own natural frequency or set of frequencies.

The natural frequency of an object is the frequency at which the object tends to vibrate or oscillate without any external force applied. All these objects and particles require a source of energy at a specific frequency ranging from few Hz to several MHz.

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This requirement can be fulfilled by an electronic device called an oscillator. It simply produces periodic oscillations in the form of electrical or mechanical energy. An oscillator can produce sinusoidal or non-sinusoidal waveforms. Basically oscillators are categorized into two main types â€” sinusoidal and non-sinusoidal oscillators. An oscillator which produces a sine-wave output is a sinusoidal oscillator.

They are classified according to their frequency-determining components. The oscillations generated by sinusoidal oscillators may be categorized as damped and undamped oscillations.

The friction in an oscillating system is referred to as damping. The electronic oscillations whose amplitude goes on decreasing with time due to the losses inherent in the electrical system in which oscillations are generated are called damped oscillations. It refers to an oscillation that fades away with time. An oscillator is always subject to forces, which dissipate a part of the oscillator energy as heat, or in other forms.

As the energy is proportional to the square of the amplitude, the amplitude decreases gradually until the oscillator returns to equilibrium. The oscillator circuits then produce the damped oscillations. However, the frequency of oscillation remains unchanged because it depends on the circuit parameters. The best example of a damped oscillation is a swinging pendulum, in which the vibration slows down and stops over time.

If the losses incurred in the electrical system could be compensated, the amplitude of oscillation would remain constant and as such the oscillation would continue indefinitely against both external disturbances and changes in the initial conditions. This type of oscillation is called undamped oscillation. So, simply put, the oscillations whose amplitude remains constant with time are called undamped oscillations.Next post Previous post.

You can ignore the free body diagram in the right half for the moment. There are lots of oscillating systems that have the same behavior as the spring-mass system. The pendulum is an important historical example, because it was studied by Galileo and Huygens and was an early means for accurate timekeeping.

Even within mechanical systems, we can change the way our spring-mass system looks by using different types of springs. We could put the mass at the end of a light cantilever beams, for example, or in the center of a stretched string.

These are all analogous systems because the equations that govern their behavior take the same mathematical form.

Newton used itso it has a good pedigree. The solution is. You may remember from differential equations class that second order differential equations have two constants of integrationwhich are determined by either the boundary conditions or initial conditions of the system.

## Types of Oscillations

Another form of the solution is. First, what are its units? Circular frequency is mathematically convenient, but not easy to measure. Measuring the period of a spring-mass system is fairly easy.

Imagine we had such a system oscillating in front of us. We could start a stopwatch when the mass was at its extreme right and stop it the next time it reached that position. Neither the spring stiffness nor the mass matter as much as the ratio between the two. One last thing. Consider again our experiment for measuring the period. In the meantime, you might want to take a look at John D.

This will become more clear as I move into damped and forced vibrations.Let us consider to the example of a mass on a spring. We now examine the case of forced oscillations, which we did not yet handle.

That is, we consider the equation. What we are interested in is periodic forcing, such as noncentered rotating parts, or perhaps loud sounds, or other sources of periodic force. We have the equation. Note that we need not have sine in our trial solution as on the left hand side we will only get cosines anyway. If you include a sine it is fine; you will find that its coefficient will be zero. Let us compute. The general solution is.

First use the trigonometric identity. We write the equation. The important term is the last one the particular solution we found. In Figure 2. By forcing the system in just the right frequency we produce very wild oscillations. This kind of behavior is called resonance or perhaps pure resonance. Sometimes resonance is desired.

You were trying to achieve resonance. The force of each one of your moves was small, but after a while it produced large swings. On the other hand resonance can be destructive. In an earthquake some buildings collapse while others may be relatively undamaged. This is due to different buildings having different resonance frequencies.

So figuring out the resonance frequency can be very important. A common but wrong example of destructive force of resonance is the Tacoma Narrows bridge failure. It turns out there was a different phenomenon at play 1. In real life things are not as simple as they were above. There is, of course, some damping. Our equation becomes. That is. We get the tedious details are left to reader.

The exact formula is not as important as the idea. Do not memorize the above formula, you should instead remember the ideas involved.

So there is no point in memorizing this specific formula. You can always recompute it later or look it up if you really need it. The general solution to our problem is. Hence the name transient.

### Simple harmonic motion

This means that the effect of the initial conditions will be negligible after some period of time.Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium point. The word comes from Latin vibrationem "shaking, brandishing".

The oscillations may be periodicsuch as the motion of a pendulumâ€”or randomsuch as the movement of a tire on a gravel road. Vibration can be desirable: for example, the motion of a tuning forkthe reed in a woodwind instrument or harmonicaa mobile phoneor the cone of a loudspeaker. In many cases, however, vibration is undesirable, wasting energy and creating unwanted sound. For example, the vibrational motions of engineselectric motorsor any mechanical device in operation are typically unwanted.

Such vibrations could be caused by imbalances in the rotating parts, uneven frictionor the meshing of gear teeth. Careful designs usually minimize unwanted vibrations.

### 2.6: Forced Oscillations and Resonance

The studies of sound and vibration are closely related. Sound, or pressure wavesare generated by vibrating structures e. Hence, attempts to reduce noise are often related to issues of vibration. Free vibration occurs when a mechanical system is set in motion with an initial input and allowed to vibrate freely. Examples of this type of vibration are pulling a child back on a swing and letting it go, or hitting a tuning fork and letting it ring.

The mechanical system vibrates at one or more of its natural frequencies and damps down to motionlessness. Forced vibration is when a time-varying disturbance load, displacement or velocity is applied to a mechanical system.

The disturbance can be a periodic and steady-state input, a transient input, or a random input. The periodic input can be a harmonic or a non-harmonic disturbance. Examples of these types of vibration include a washing machine shaking due to an imbalance, transportation vibration caused by an engine or uneven road, or the vibration of a building during an earthquake.

For linear systems, the frequency of the steady-state vibration response resulting from the application of a periodic, harmonic input is equal to the frequency of the applied force or motion, with the response magnitude being dependent on the actual mechanical system. Damped vibration: When the energy of a vibrating system is gradually dissipated by friction and other resistances, the vibrations are said to be damped. The vibrations gradually reduce or change in frequency or intensity or cease and the system rests in its equilibrium position.

An example of this type of vibration is the vehicular suspension dampened by the shock absorber. Vibration testing is accomplished by introducing a forcing function into a structure, usually with some type of shaker.

Alternately, a DUT device under test is attached to the "table" of a shaker. Vibration testing is performed to examine the response of a device under test DUT to a defined vibration environment. The measured response may be ability to function in the vibration environment, fatigue life, resonant frequencies or squeak and rattle sound output NVH. Squeak and rattle testing is performed with a special type of quiet shaker that produces very low sound levels while under operation.In mechanics and physicssimple harmonic motion is a special type of periodic motion or oscillation where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement.

Simple harmonic motion can serve as a mathematical model for a variety of motions, such as the oscillation of a spring. In addition, other phenomena can be approximated by simple harmonic motion, including the motion of a simple pendulum as well as molecular vibration. Simple harmonic motion is typified by the motion of a mass on a spring when it is subject to the linear elastic restoring force given by Hooke's law. The motion is sinusoidal in time and demonstrates a single resonant frequency.

For simple harmonic motion to be an accurate model for a pendulum, the net force on the object at the end of the pendulum must be proportional to the displacement. This is a good approximation when the angle of the swing is small. Simple harmonic motion provides a basis for the characterization of more complicated motions through the techniques of Fourier analysis. The motion of a particle moving along a straight line with an acceleration whose direction is always towards a fixed point on the line and whose magnitude is proportional to the distance from the fixed point is called simple harmonic motion [SHM].

In the diagram, a simple harmonic oscillatorconsisting of a weight attached to one end of a spring, is shown. The other end of the spring is connected to a rigid support such as a wall. If the system is left at rest at the equilibrium position then there is no net force acting on the mass. However, if the mass is displaced from the equilibrium position, the spring exerts a restoring elastic force that obeys Hooke's law. Once the mass is displaced from its equilibrium position, it experiences a net restoring force.

As a result, it accelerates and starts going back to the equilibrium position. When the mass moves closer to the equilibrium position, the restoring force decreases. At the equilibrium position, the net restoring force vanishes. Therefore, the mass continues past the equilibrium position, compressing the spring. A net restoring force then slows it down until its velocity reaches zero, whereupon it is accelerated back to the equilibrium position again.

As long as the system has no energy loss, the mass continues to oscillate. Thus simple harmonic motion is a type of periodic motion. Note if the real space and phase space diagram are not co-linear, the phase space motion becomes elliptical. The area enclosed depends on the amplitude and the maximum momentum. In Newtonian mechanicsfor one-dimensional simple harmonic motion, the equation of motion, which is a second-order linear ordinary differential equation with constant coefficients, can be obtained by means of Newton's 2nd law and Hooke's law for a mass on a spring.

Undamped and damped oscillations

Solving the differential equation above produces a solution that is a sinusoidal function :. Using the techniques of calculusthe velocity and acceleration as a function of time can be found:.

By definition, if a mass m is under SHM its acceleration is directly proportional to displacement. These equations demonstrate that the simple harmonic motion is isochronous the period and frequency are independent of the amplitude and the initial phase of the motion. In the absence of friction and other energy loss, the total mechanical energy has a constant value.

The following physical systems are some examples of simple harmonic oscillator. A mass m attached to a spring of spring constant k exhibits simple harmonic motion in closed space. The equation for describing the period.This article has two sections: One discusses criteria for selecting an oscillator while other one discusses various types of oscillations. Here are the parameters that are to be noted while selecting an oscillator for a particular application.

Damped oscillations is clearly shown in the figure a given below. In such a case, during each oscillation, some energy is lost due to electrical losses I 2 R. The only parameters that will remain unchanged are the frequency or time period. They will change only according to the circuit parameters.