Up till now we have seen that the motion of an object in real space and time is an oscillatory motion, if spacetime is a medium which oscillates, so that the motion of the object can be described by the equations of the (damped) harmonic oscillator.
A more complete equation of motion can be taken if we introduce into the system a driving force. Commonly such a force is treated as sinusoidal, of the form
Adding this term to the equation of the damped harmonic oscillator, we have
whereas if the driving force is not present,
we take back the (homogeneous) equation of the damped unforced harmonic oscillator.
The angular frequency ω of the driving force is different from the natural angular frequency ω0 of the simple harmonic oscillator, and it is different from the angular frequency ω΄ of the damped harmonic oscillator. The angular frequency ω is imposed by an ‘external’ object onto the system of the oscillator, so that finally the whole system will oscillate with the frequency ω of the driving force.
This can be seen assuming a solution of the general form
where the amplitudes C1 and C2 are given by the following expressions,
The first term of the solution refers to the damped (unforced) oscillator, while the second term refers to the driven oscillator. The former term is regarded as the transient term, because of the negative exponential which vanishes with time. The latter term is called the steady state solution, as the system thereafter oscillates with the frequency ω of the driving force.
Thus at the steady state we have
where
The negative sign before the phase φ in the solution is a matter of choice. The constant C2 here is replaced by the amplitude y0(ω), which explicitly expresses its dependence on the angular frequency ω of the driving force.
The previous values for the amplitude y0(ω), and for the phase φ, are taken by substituting the steady state solution into the equation of motion, as follows
we take,
Taking the two sides separately and equating them to zero, we have,
This expression gives the phase φ.
Now, equating the second term to zero, we take
The values of sinφ and cosφ, with respect to the expression for tanφ we previously found, are given as follows,
so that
This expression gives the amplitude y0(ω).
Thus, the steady state solution
is valid as long as
Taking into account the previous solution, for the speed and the acceleration of the oscillator we have,
The values for the amplitude y0(ω) at ω=0, and ω=ω0, are respectively,
The state at which the angular frequency ω of the driving force reaches the natural angular frequency ω0 of the system, ω=ω0, is called resonance.
Although at the steady state the transient term (which is due to damping) has disappeared, the behavior of the system still depends on the damping factor γ, which makes its appearance in the expression for the amplitude y0(ω).
The maximum amplitude y0(ω)max can be taken if we differentiate the amplitude y0(ω) with respect to the frequency ω, and set the differential equal to zero,
From the last expression we see that the maximum amplitude can only be defined if γ≠0, or γ≠2ω0.
Therefore resonance is achieved at equal frequencies, ω≈ω0, only if the damping factor is sufficiently small γ≈0. However, if the damping factor γ is equal to zero, then the amplitude y0(ω) will become infinite.
This is a graph of the amplitude y0(ω), for different values of the damping factor γ,
As the graph shows, the smaller the damping factor γ is, the steeper the slope of the amplitude y0(ω) will be. If the damping factor goes to zero, then the amplitude goes to infinity. If the angular frequency ω goes to infinity, then the amplitude goes to zero. Those marginal conditions, as well as the condition γ=2ω0 (which is the condition of critical damping), will be examined in what follows.
As far as the energy of the driven and damped harmonic oscillator is concerned, the mechanical energy Em can be taken from the steady state solution,
where for simplicity we have set the phase φ equal to zero.
The elastic energy will be
while the kinetic energy will be
so that for the mechanical energy, we have that
Choosing boundary conditions,
and
and using the values for the amplitude at ω=0, or ω=ω0,
we can find an expression for the mechanical energy at the boundaries,
will represent the total energy of the oscillator.
The average value of a function is defined as follows,
The average values of the functions sin2θ and cos2θ, θ=ωt+φ, can be calculated using the following identities,
so that
and
Thus the average value of the mechanical energy will be,
This is a graph of the average mechanical energy <Em(t)>, thus the average energy amplitude <Em(ω)>, of the driven and damped harmonic oscillator, for different values of the damping factor γ,
This graph for the average energy amplitude <Em(ω)> is similar to the one we saw for the amplitude y0(ω), since the mechanical energy depends on the amplitude squared.
The factor of ½ was added because the initial (total) mechanical energy Em(t=0) is twice its average value <Em(ω=0)>,
while the mechanical energy Em(t=T) at the other boundary, t=T, ω=ω0, is the same as the average value <Em(ω=ω0)>,
In fact, the latter energy, at t=T (according to the boundary conditions we chose earlier), is purely kinetic energy, so that if the damping factor γ is comparable to the natural angular frequency ω0, γ≈ω0≈1, then the total energy (the initial elastic energy Eel) of the oscillator will have transformed into the kinetic energy Ek at resonance, ω≈ω0, ω≈ω0≈γ≈1, so that
A more general relationship between the driving angular frequency ω and the damping factor γ will be established later on.
Notes:
A common approximation for the mechanical energy Em(t) is taken if we suppose that the damping factor γ is sufficiently small, γ≈0, so that it will also be ω≈ω0. This can be seen from the formula which gives the maximum amplitude,
However this does not necessarily mean that the driving force has achieved resonance, because, as we shall see, the damping factor γ may depend on the angular frequency ω of the driving force.
In any case, if the angular frequency ω of the driving force is close to the natural angular frequency ω0 of the oscillator, in order to further manipulate the expression for the average mechanical energy,
we may use the following approximations,
so that the expression for the average mechanical energy takes the form
which has a maximum at ω0=ω,
However, if γ≈0, then the maximum amplitude of the average mechanical energy of the system will be infinite,
Thus the mechanical energy of the system cannot be defined if the damping factor is zero, γ≈0.
A more complete equation of motion can be taken if we introduce into the system a driving force. Commonly such a force is treated as sinusoidal, of the form
The angular frequency ω of the driving force is different from the natural angular frequency ω0 of the simple harmonic oscillator, and it is different from the angular frequency ω΄ of the damped harmonic oscillator. The angular frequency ω is imposed by an ‘external’ object onto the system of the oscillator, so that finally the whole system will oscillate with the frequency ω of the driving force.
This can be seen assuming a solution of the general form
Thus at the steady state we have
The previous values for the amplitude y0(ω), and for the phase φ, are taken by substituting the steady state solution into the equation of motion, as follows
Using the identities
so that if both sides of the previous equation are equal to zero, then the previous equation is true.
Taking the two sides separately and equating them to zero, we have,
Now, equating the second term to zero, we take
Thus, the steady state solution
Although at the steady state the transient term (which is due to damping) has disappeared, the behavior of the system still depends on the damping factor γ, which makes its appearance in the expression for the amplitude y0(ω).
The maximum amplitude y0(ω)max can be taken if we differentiate the amplitude y0(ω) with respect to the frequency ω, and set the differential equal to zero,
Inserting this value for the angular frequency ω back into the expression for the amplitude, we have
Therefore resonance is achieved at equal frequencies, ω≈ω0, only if the damping factor is sufficiently small γ≈0. However, if the damping factor γ is equal to zero, then the amplitude y0(ω) will become infinite.
This is a graph of the amplitude y0(ω), for different values of the damping factor γ,
As the graph shows, the smaller the damping factor γ is, the steeper the slope of the amplitude y0(ω) will be. If the damping factor goes to zero, then the amplitude goes to infinity. If the angular frequency ω goes to infinity, then the amplitude goes to zero. Those marginal conditions, as well as the condition γ=2ω0 (which is the condition of critical damping), will be examined in what follows.
As far as the energy of the driven and damped harmonic oscillator is concerned, the mechanical energy Em can be taken from the steady state solution,
The elastic energy will be
where
where the quantity
In order to plot the graph of the mechanical energy Em(t) with respect to the driving angular frequency ω, thus the amplitude of the mechanical energy Em(ω), it is convenient to eliminate the sinusoidal terms by taking the average value of the mechanical energy,
where we have set
This graph for the average energy amplitude <Em(ω)> is similar to the one we saw for the amplitude y0(ω), since the mechanical energy depends on the amplitude squared.
The factor of ½ was added because the initial (total) mechanical energy Em(t=0) is twice its average value <Em(ω=0)>,
Notes:
A common approximation for the mechanical energy Em(t) is taken if we suppose that the damping factor γ is sufficiently small, γ≈0, so that it will also be ω≈ω0. This can be seen from the formula which gives the maximum amplitude,
In any case, if the angular frequency ω of the driving force is close to the natural angular frequency ω0 of the oscillator, in order to further manipulate the expression for the average mechanical energy,
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