Encyclopedia
In
classical mechanics, a
Harmonic oscillator is a system which, when displaced from its equilibrium position, experiences a restoring force proportional to the displacement according to
Hooke's law:
where is a positive constant.
If 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 .
If a frictional force proportional to the velocity is also present, the harmonic oscillator is described as a
damped oscillator. In such situation, the
frequency of the oscillations is smaller than in the non-damped case, and the
amplitude of the oscillations decreases with time.
If an external time-dependent force is present, the harmonic oscillator is described as a
driven oscillator.
Mechanical examples include
pendula , masses connected to springs, and acoustical systems. Other analogous systems include electrical harmonic oscillators .
Simple harmonic oscillator
The simple harmonic oscillator has no driving force, and no friction , so the net force is just
Using Newton's Second Law
The acceleration, is equal to the second derivative of .
If we define , then the equation can be written as follows,
and has the general solution
where the
amplitude and the phase are determined by the initial conditions.
Alternatively, the general solution can be written as
where the value of is shifted by relative to the previous form;
or as
where and are the constants which are determined by the initial conditions, instead of and in the previous forms.
The
frequency of the oscillations is given by
The kinetic energy is
.
and the potential energy is
so the total energy of the system has the constant value
Driven harmonic oscillator
A driven harmonic oscillator satisfies the nonhomogeneous second order linear differential equation
where is the driving amplitude and is the driving frequency for a sinusoidal driving mechanism. This type of system appears in AC LC circuits and idealized spring systems lacking internal mechanical resistance or external air resistance.
Damped harmonic oscillator
A damped harmonic oscillator satisfies the second order differential equation
where is an experimentally determined damping constant satisfying the relationship .
An example of a system obeying this equation would be a weighted spring underwater if the damping force exerted by the water is assumed to be linearly proportional to .
Damped, driven harmonic oscillator
This satisfies the equation
The general solution is a sum of a transient that depends on initial conditions, and a steady state that is independent of initial conditions and depends only on driving frequency, driving force, restoring force, damping force, and inertial moment of the oscillator .
The steady-state solution is
where
is the absolute value of the impedance
and
is the phase of the oscillation relative to the driving force.
One might see that for a certain driving frequency, , the amplitude is maximal. This occurs for the frequency
and is called
resonance of displacement.
In summary: at a steady state the frequency of the oscillation is the same as that of the driving force, but the oscillation is phase-offset and scaled by amounts that depend on the frequency of the driving force in relation to the preferred frequency of the oscillating system.
Example:
RLC circuit.
Full mathematical definition
Most harmonic oscillators, at least approximately, solve the differential equation:
where
t is time,
b is the damping constant, ?
o is the characteristic
angular frequency, and
Aocos represents something driving the system with amplitude
Ao and angular frequency ?.
x is the measurement that is oscillating; it can be position, current, or nearly anything else. The
angular frequency is related to the frequency,
f, by
Important terms
- Amplitude: maximal displacement from the equilibrium.
- Period: the time it takes the system to complete an oscillation cycle. Opposite of frequency.
- Frequency: the number of cycles the system performs per unit time .
- Angular frequency
- Pulsation [i]
...
:
- Phase: how much of a cycle the system completed .
- Initial conditions: the state of the system at t = 0, the beginning of oscillations.
Simple harmonic oscillator
A simple harmonic oscillator is simply an oscillator that is neither damped nor driven. So the equation to describe one is:
Physically, the above never actually exists, since there will always be friction or some other resistance, but two approximate examples are a mass on a spring and an
LC circuit.
In the case of a mass attached to a spring, Newton's Laws, combined with Hooke's law for the behavior of a spring, states that:
- where k is the spring constant
- m is the mass
- x is the position of the mass
- a is its acceleration.
Because acceleration
a is the second derivative of position
x, we can rewrite the equation as follows:
The most simple solution to the above
differential equation is
and the second derivative of that is
- where A is the amplitude, d is the phase shift, and ? is the angular frequency
...
.
Plugging these back into the original differential equation, we have:
Then, after dividing both sides by
we get:
or, as it is more commonly written:
The above formula reveals that the
angular frequency ? of the solution is only dependent upon the physical characteristics of the system, and not the initial conditions . We will label this ? as ?
o from now on. This will become important later.
Universal oscillator equation
The equation
is known as the
universal oscillator equation since all second order linear oscillatory systems can be reduced to this form. This is done through
nondimensionalization.
If the forcing function is
f = cos = cos = cos, where ? =
?tc, the equation becomes
The solution to this differential equation contains two parts, the "transient" and the "steady state".
Transient solution
The solution based on solving the
ordinary differential equation is for arbitrary constants
c1 and
c2 is
The transient solution is independent of the forcing function. If the system is critically damped, the response is independent of the damping.
Steady-state solution
Apply the "complex variables method" by solving the auxiliary equation below and then finding the real part of its solution:
Supposing the solution is of the form
Its derivatives from zero to 2nd order are
Substituting these quantities into the differential equation gives
Dividing by the exponential term on the left results in
Equating the real and imaginary parts results in two independent equations
Amplitude part
Squaring both equations and adding them together gives
By convention the positive root is taken since amplitude is usually considered a positive quantity. Therefore,
Compare this result with the theory section on resonance, as well as the "magnitude part" of the
RLC circuit. This amplitude function is particularly important in the analysis and understanding of the frequency response of second-order systems.
Note that the variables in these equations ought to be identified before showing the equation.
Phase part
To solve for f, divide both equations to get
This phase function is particularly important in the analysis and understanding of the frequency response of second-order systems.
Full solution
Combining the amplitude and phase portions results in the steady-state solution
The solution of original universal oscillator equation is a superposition of the transient and steady-state solutions
For a more complete description of how to solve the above equation, see
linear ODEs with constant coefficients.
Relationship to RLC circuit
Comparing a mechanical harmonic oscillator with an series
RLC circuit or parallel RLC circuit, the following correspond:
- F V I
- x Q F
- k *v I V
- b R
- a
- m L C
Applications
The problem of the simple harmonic oscillator occurs frequently in physics because of the form of its potential energy function:
Given an arbitrary potential energy function , one can do a
Taylor expansion in terms of around an energy minimum to model the behavior of small perturbations from equilibrium.
Because is a minimum, the first derivative evaluated at must be zero, so the linear term drops out:
The constant term is arbitrary and thus may be dropped, and a coordinate transformation allows the form of the simple harmonic oscillator to be retrieved:
Thus, given an arbitrary potential energy function with a non-vanishing second derivative, one can use the solution to the simple harmonic oscillator to provide an approximate solution for small perturbations around the equilibrium point.
Examples
Simple Pendulum
Assuming no damping and small amplitudes, the differential equation governing a simple pendulum is given by
Solution to this equation is given by:
where is the largest angle attained by the pendulum. Period, the time for one complete oscillation , is given by divided by whatever is multiplying the time in the argument of the cosine
Pendulum swinging over turntable
Simple harmonic motion can in some cases be considered to be the one-dimensional projection of two-dimensional circular motion. Consider a long
pendulum swinging over the turntable of a
record player. On the edge of the turntable there is an object. If the object is viewed from the same level as the turntable, a projection of the motion of the object seems to be moving backwards and forwards on a straight line.
It is possible to change the frequency of rotation of the turntable in order to have a perfect synchronization with the motion of the pendulum.
The angular speed of the turntable is the pulsation of the pendulum.
In general, the
pulsation-also known as angular frequency, of a straight-line simple harmonic motion is the
angular speed of the corresponding circular motion.
Therefore, a motion with period
T and frequency
f=1/
T has pulsation
In general,
pulsation and
angular speed are not synonymous. For instance the pulsation of a pendulum is not the angular speed of the pendulum itself, but it is the angular speed of the corresponding circular motion.
Spring-mass system
When a spring is stretched or compressed by a mass, the spring develops a restoring force. The
Hooke's Law gives the relationship of the force exerted by the spring when the spring is compressed or stretched a certain length.
where Fs is the force, k is the spring constant, and the x is the displacement of the mass with respect to the equilibrium position.
This relationship shows that the distance of the spring is always opposite to the force of the spring.
By using either force balance or an energy method, it can be readily shown that the motion of this system is given by the following differential equation:
If the initial displacement is A, and there is no initial velocity, the solution of this equation is given by:
;Energy variation in the spring-damper system
In terms of energy, all systems have two types of energy, potential energy and kinetic energy. When a spring is stretched or compressed, it stores elastic potential energy, which then is transferred into kinetic energy. The potential energy within a spring is determined by the equation
When the spring is stretched or compressed, kinetic energy of the mass gets converted into potential energy of the spring. By conservation of energy, assuming the datum is defined at the equilibrium position, when the spring reaches its maximum potential energy, the kinetic energy of the mass is zero. When the spring is released, the spring will try to reach back to equilibrium, and all its potential energy is converted into kinetic energy of the mass.
References
See also
