Harmonic and Forced Vibrations: A Comprehensive Overview

Posted on Sep 6, 2024 in Physics

Harmonic Vibration

We call harmonic vibration vibration limited to any movement around a position that is repeated several times. When it reaches all positions are repeated in the mobile regular intervals, we say that the motion is periodic. In this case the temporal function of position, x(t), holds that x(t) = x(t + T) where T is the repeat interval and is called the period.

In nature can be observed oscillatory movements related to any freak. Therefore, the study of the motions that fulfill this condition is of great interest. As will be seen in the development of the subject, its study can be made from a particular movement: simple harmonic motion. This is defined as the movement of the mobile whose position is expressed as x(t) = A cos(ωt + j)0 where A is the range of motion, j is the initial phase, and ω₀ is called angular frequency that relates to the period.

Regarded as the projection of uniform circular motion, A, ω₀ corresponding to the radius and angular velocity respectively. The study of the derivatives of this function allow us to know its velocity and acceleration versus time.

Consider now a mass requested by an elastic force. This force, proportional and opposite to the movement, also cause an acceleration proportional and opposite to the displacement at each given moment allows us to identify both expressions and establish a body that requested by an elastic force describes a simple harmonic vibration angular frequency.

In general, a body mass subjected to a restoring force in the opposite direction to the movement, tends to describe oscillatory movements, although not simple. Recognize the set (action recuperative inert element) with the generic name for mechanical oscillator.

Vibration Shock

Vibration shock in the experience of all is to recognize and remember oscillators of various types, (ie any pendulum), we excited by making them run. What is most difficult is that someone has seen this oscillator function indefinitely, as described earlier. If we settle it “indefinitely” meaning a significant number of oscillations, we can surely give some examples. But the reality is that whenever rigorous excited oscillator and abandon it, stopping just more or less quickly.

The above study is not wrong, although we have accepted conditions idealizadas. De fact, all the oscillators that we are surrounded by other things, the force can do some form. These forces when we can not control, we call friction, and accept that they can reduce the energy of our system. As we did before, let’s give these forces a way which is convenient for calculation. Not analyze all possibilities because most cases we complicate too. But nevertheless the results are again “seem” to real situations enough to learn something.

The case we consider here is that of a force that opposes motion and is proportional to the rate F_R = –bv where b is the proportionality constant. We see that the function of the proposed motion, may, be substituted and the equation works. I really do not need much more. Now we see what happens when we consider the frictional force we have proposed. We see that it multiplies the cosine is a decreasing exponential, ie the range of movement is becoming smaller over time, the faster the higher b. The root of the cosine is the current angular frequency of the movement that is smaller, the greater is b. This movement is called sub-oscillation amortiguada.

Si b > b_c, is large, and the ventral root is imaginary. In this case no oscillations and whether the oscillator is left out of equilibrium, is gradually approaching the equilibrium position without passing it. If b = b_c, we say that the oscillator has the critical damping. In this situation the movement toward the equilibrium point is the fastest possible without exceeding it.

One thing to consider, very important, is that lower range, a loss of energy. A damped oscillator loses energy much more rapidly, the greater is b. If we want to consider the matter as a set of oscillators, it is easy to see that when one of them is excited exert forces on the neighbors so you can make work, and this is one way to lose your energy. You can enter the category of damped oscillator. This idea has a counterpart, the other oscillators neighbors may also do energy work on him… ydarle

Forced Vibration: Resonant

Until now we thought that we were acting on an oscillator and we left. In different areas may be important to know what he will do an oscillator if kept under some force that varies with time: f(t). Whose solutions can be as varied as the ideas we may happen to the function f(t). However, we restrict our considerations to a case of great interest that our strength is periodic and in particular: f(t) = F₀ cos(ωt). Note that ω has no subscript. I mean think about a device that would swing left, but about which we will operate with a periodic force of any frequency.

Here too, we avoid solving the equation, we simply give the solution and study about it, the interesting aspects. Initially, the movement of the oscillator depends on how you’ve found to start acting, but after some time, and regardless of how it was at first, the result is a vibration of the same frequency as the applied force. If we shake fast, moves fast, if you stir slowly, move slowly. This is the response of our oscillator. Try to substitute in the equation a solution of the form: x(t) = A cos(ωt – a) will find the solution is correct and satisfies the equation but with the proviso that A and have the proper values. Sounds complicated, but really we are only interested to make some general considerations.

Consider:

• The scope of the response is proportional to the amplitude of the action.
• The denominator increases with increasing b. The response amplitude is lower if you increase the dissipation.
• For b small, the denominator decreases as the frequency of force, ω, is similar to the natural frequency of the oscillator ω₀. This is really interesting. Suppose that b is so small that we despise, the root can be made as small as desired, so that the amplitude reached a value as high as you want. It is an experience easy to achieve if we act on an oscillating system with a frequency equal to his energy can be communicating at each oscillation, so the result is an extraordinary oscillation amplitude. This phenomenon is called resonance, and is of great importance. The answer may reach significant values, although the performance is weak.
• The energy of this oscillator is stable, then the energy dissipated energy must be received from the acting force f(t). Therefore, the oscillator will take over the closer power is resonance. The power dissipated, equal to that absorbed, it turns out, on average: P = 1/2 A²bω²