Harmonic oscillators occur widely in nature and are exploited in many manmade devices, such as clocks and radio 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. Other analogous systems include electrical harmonic oscillators such as RLC circuits. Mechanical examples include pendula, masses connected to springs, and acoustical systems. If an external time dependent force is present, the harmonic oscillator is described as a driven oscillator. It is common to set 2 k/m, so that (1.1) becomes d2u dt2 2u. (1.1) Here u represents the displacement from equilibrium of some oscillator, and (1.1) issimplyNewton’sformula ma F withtheforcebeinggivenbyHooke’sformula F ku. 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." Review: The simple harmonic oscillator Recall the simple harmonic oscillator m d2u dt2 ku. ⁕Decay to the equilibrium position, without oscillations. ⁕Oscillate with a frequency smaller than in the non-damped case, and an amplitude decreasing with time. The motion equation for simple harmonic motion contains a complete description of the motion, and other parameters of the motion can be calculated from it.
Depending on the friction coefficient, the system can: If a frictional force proportional to the velocity is also present, the harmonic oscillator is described as a damped oscillator. 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. In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force, F, proportional to the displacement, x: In particular, they can be thought of as an extension which allows us to take the square root of a negative number.Freebase (5.00 / 1 vote) Rate this definition: So I will go quickly over the basics if you haven't seen this recently, make sure to go carefully through the assigned reading in Boas.Ĭomplex numbers are an important and useful extension of the real numbers. This is one of those topics that I suspect many of you have seen before, but you might be rusty on the details. But to understand it, we need to take a brief math detour into the complex numbers. We're not quite done yet: there is actually yet another way to write the general solution, which will just look like a rearrangement for now but will be extremely useful when we add extra forces to the problem. U(y) = U(y_0) + \frackA^2 \) always - I won't go through the derivation, but it's in Taylor.
#Simple harmonic oscillator series
If \( y_0 \) is an equilibrium point of \( U(y) \), then series expanding around that point gives
In fact, we've already seen why it shows up everywhere: expansion around equilibrium points. The simple harmonic oscillator is an extremely important physical system study, because it appears almost everywhere in physics. This happens to be the equation of motion for a spring, assuming we've put our equilibrium point at \( x=0 \): This sort of motion is given by the solution of the simple harmonic oscillator (SHO) equation, Our next important topic is something we've already run into a few times: oscillatory motion, which also goes by the name simple harmonic motion.