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Formally, the small-amplitude resonance can be computed as follows: First, recall Newton's law for angular acceleration:
T = µ A
Where:
T -- torque applied to rotor
µ -- moment of inertia of rotor and load
A -- angular acceleration, in radians per second per second
We assume that, for small amplitudes, the torque on the rotor can be approximated as a linear function of the displacement from the equilibrium position. Therefore, Hooke's law applies:
T = -k
where:
k -- the "spring constant" of the system, in torque units per radian
-- angular position of rotor, in radians
We can equate the two formulas for the torque to get:
µ A = -k
Note that acceleration is the second derivitive of position with respect to time:
A = d2 /dt2
so we can rewrite this the above in differential equation form:
d2 /dt2 = -(k/µ)
To solve this, recall that, for:
f( t ) = a sin bt
The derivitives are:
df( t )/dt = ab cos bt
d2f( t )/dt2 = -ab2 sin bt = -b2 f(t)
Note that, throughout this discussion, we assumed that the rotor is resonating. Therefore, it has an equation of motion something like:
= a sin (2 f t)
a = angular amplitude of resonance
f = resonant frequency
This is an admissable solution to the above differential equation if we agree that:
b = 2 f
b2 = k/µ
Solving for the resonant frequency f as a function of k and µ, we get:
f = ( k/µ )0.5 / 2
It is crucial to note that it is the moment of inertia of the rotor plus any coupled load that matters. The moment of the rotor, in isolation, is irrelevant! Some motor data sheets include information on resonance, but if any load is coupled to the rotor, the resonant frequency will change!
In practice, this oscillation can cause significant problems when the stepping rate is anywhere near a resonant frequency of the system; the result frequently appears as random and uncontrollable motion.
Resonance and the Ideal Motor
Up to this point, we have dealt only with the small-angle spring constant k for the system. This can be measured experimentally, but if the motor's torque versus position curve is sinusoidal, it is also a simple function of the motor's holding torque. Recall that:
T = -h sin( (( /2)/S) )
The small angle spring constant k is the negative derivitive of T at the origin.
k = -dT / d = - (- h (( /2)/S) cos( 0 ) ) = ( /2)(h / S)
Substituting this into the formula for frequency, we get:
f = ( ( /2)(h / S) / µ )0.5 / 2 = ( h / ( 8 µ S ) )0.5
Given that the holding torque and resonant frequency of the system are easily measured, the easiest way to determine the moment of inertia of the moving parts in a system driven by a stepping motor is indirectly from the above relationship!
µ = h / ( 8 f2 S )
For practical purposes, it is usually not the torque or the moment of inertia that matters, but rather, the maximum sustainable acceleration that matters! Conveniently, this is a simple function of the resonant frequency! Starting with the Newton's law for angular acceleration:
A = T / µ
We can substitute the above formula for the moment of inertia as a function of resonant frequency, and then substitute the maximum sustainable running torque as a function of the holding torque to get:
A = ( h / ( 20.5 ) ) / ( h / ( 8 f2 S ) ) = 8 S f2 / (20.5)
Measuring acceleration in steps per second squared instead of in radians per second squared, this simplifies to:
Asteps = A / S = 8 f2 / (20.5)
Thus, for an ideal motor with a sinusoidal torque versus rotor position function, the maximum acceleration in steps per second squared is a trivial function of the resonant frequency of the motor and rigidly coupled load!