As for the motion of the balance of the actual watch, the balance wheel receives the impulse from the anchor at the close to the neutral point. As a result, after the balance wheel swings over to the amplitude, the balance wheel will return back to the neutral point by the reactive force of the hairspring.
In the general watch, the following series of the procedures will be repeated that utilizing the rotational energy of the balance wheel (by reducing the velocity of the balance wheel) at a little bit before returning back to the neutral point, in order to unlock the escape wheel so that the main spring torque can be again transferred via the gear train for accelerating the anchor to be able to make the impulse to the balance wheel with this.
At this point above, if the impulse is precisely made right at the neutral point only, no unlocking the escape wheel at all and furthermore, disregarding the viscous damping and the solid friction, the motion of the balance wheel is represented as in clause 2-1, and the period, ''$T$'' can be shown in the equation (1).
\begin{eqnarray}
T=\frac{2\pi}{\omega_n}=2\pi \sqrt{\frac{I}{k}}
\end{eqnarray}
Definition /Condition:
$\;\omega_n\;$: Natural frequency of the balance
$\;I\;$: Moment of inertia of the balance wheel
$\;k\;$: Spring constant of the hairspring
Note that the equation (1) does not include the component of the amplitude, ''$A$''.
This is because of the fact that even though the maximum displacement varies, the period of vibration remains constant and unchanged. This situation specifically represents that the isochronism is maintained.
However, in fact, the impulse to the balance wheel is a little bit off from the neutral point, which means that the energy loss to unlock the escape wheel inevitably occurs short of the neutral point (called as the escapement error). In addition, in fact, the spring constant of the hairspring is not always constant, but varies according to the displacement.
Therefore, strictly speaking, the period of the balance, ''T'' is not exactly the same as shown in the equation (1). In other words, it is obvious that the isochronism will be disturbed mainly due to the conditions above.
As some other factors to cause the period error of the balance actually remain in existence,
it would be common practice to define these as the disturbance torque, ''$f (\theta)$'', and to indicate it in the equation as shown in the following.
When the disturbance torque, ''$f (\theta)$'' applied to the balance, the equation of motion of the balance can be shown in the equation (2), assuming that the damping term of the equation disregarded.
\begin{eqnarray}
I\frac{d^2 \theta}{dt^2}=f(\theta)-k\theta
\end{eqnarray}
Dividing the both sides of the equation by ''$I$'', and sorting it out, it becomes the equation (3).
\begin{eqnarray}
\frac{d^2 \theta}{dt^2}+\frac{k}{I}\theta=\frac{f(\theta)}{I}
\end{eqnarray}
Based on the equation (4) below obtaind in clause 2-1, the equation (3) can be converted to the equation (5).
\begin{eqnarray}
\omega_n=\sqrt{\frac{k}{I}}
\end{eqnarray}
\begin{eqnarray}
\frac{d^2 \theta}{dt^2}+\omega_n^2\theta=\frac{f(\theta)}{I}
\end{eqnarray}
As the equation (5) includes the disturbance torque, ''$f (\theta)$'', the motion of the balance wheel is no longer the complete sine motion. But, it can be regarded to be approximately sine motion, assuming that the disturbance torque, ''$f (\theta)$'' is very small.
Therefore, the equation (6) can be represented.
\begin{eqnarray}
\theta=Asin\omega t
\end{eqnarray}
''$\omega$'' & ''$\omega_n$'' are the physical constant, and ''$\omega$'' is slightly different from ''$\omega_n$'', and the purpose of this clause is to identify the degree of the effect on the period of the vibration due to the disturbance torque.
Substituting the equation (6) to the equation (5), the equation (7) can be obtained.
\begin{eqnarray}
-A\omega^2sin\omega t +\omega_n^2Asin \omega t=\frac{f(\theta)}{I}
\end{eqnarray}
Sorting out the equation (7), the equation (8) can be obtained.
\begin{eqnarray}
A(\omega_n^2-\omega^2)sin\omega t=\frac{f(\theta)}{I}
\end{eqnarray}
Then, multiplying the both sides of the equation by ''$A sin \omega t$'', and integrating from ''$t=0$'' to
''$t=T$'' all over the one period, the equation (9) can be obtained.
\begin{eqnarray}
A^2(\omega_n^2-\omega^2)\int_0^T sin^2\omega t dt=\frac{A}{I}\int_0^Tf(\theta)sin\omega t dt
\end{eqnarray}
Here, from the double angle formula of the trigonometric function (Section 5 Mathematical Formulas 5-1-5), the equation (10) can be indicated.
\begin{eqnarray}
\int_0^Tsin^2\omega t dt=\int_0^T(\frac{1-cos2\omega t}{2})=\frac{T}{2}
\end{eqnarray}
In addition, as for the right side of the equation (9), the equation (6) shows that
''$A sin \omega t =\theta$''. Accordingly, the equation (11) can be obtained from the equation (9).
\begin{eqnarray}
\frac{A^2T}{2}(\omega_n^2-\omega^2)=\frac{1}{I}\int_0^T\theta f(\theta)dt
\end{eqnarray}
Here, based on the equation (12) and definitions of ''$\omega$'' & ''$\omega_n$'' as shown in the equation (13), the equation (14) can be obtained.
\begin{eqnarray}
\omega=\omega_n(1+\delta)
\end{eqnarray}
\begin{eqnarray}
\omega=\frac{2\pi}{T}\;\;\;\;\;\;,\;\;\;\;\;\;\omega_n=\frac{2\pi}{T_0}
\end{eqnarray}
\begin{eqnarray}
\delta=\frac{T_0-T}{T}
\end{eqnarray}
''$\delta$'' is defined as the isochronous error since it means the rate of the period deviation from that of the vibration system, [$f (\theta) = 0$] which maintains the "isochronism".
Substituting the equation (12) to the equation (11), and disregarding the term of ''$\delta^2$'' since the condition is ''$\delta \ll 1$'' ,the equation (15) can be obtained.
\begin{eqnarray}
-A^2\omega_n^2\delta T=\frac{1}{I}\int_0^T\theta f(\theta)dt
\end{eqnarray}
On the assumption of ''$\omega_n^2=k/I$'', sorting out ''$\delta$'' of the equation (15), the equation (16) can be obtained.
\begin{eqnarray}
\delta=-\frac{1}{A^2kT}\int_0^T\theta f(\theta) dt
\end{eqnarray}
This is the general equation of the isochronous error when the disturbance torque, ''$f (\theta)$'' exists. Accordingly, the isochronous error turns out to be inversely proportionate to the square of the amplitude, ''$A$''.