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Solution using differential equation (Forced oscillation with no friction)

Figure 1. The material point connecting with the spring.

Herein, we consider a material point connecting with the spring. The point of the natural length of this spring is defined as \(x_0\) and the right-hand side is positive. Next, we apply the force expressed as \(F_0\cos \omega t\) to the material point. This situation is called “forced oscillation”. The solution for the forced oscillation changes when frictions between the material point and the ground are involved. Herein, we consider a situation with no friction.

We set the momentum equation for this material point as follows: \begin{eqnarray} m \frac{d^2}{dt^2} x + kx = F_0 \cos \omega t\ \ \ \ \ \ \ \ \ \ (1) \end{eqnarray} This is the inhomogeneous differential equation of the second order. First, we try to derive the general solution by an assumption where the right-hand side of Equation (1) is zero. If \(\omega_0^2=k/m\), the general solution of Equation (1) is: \begin{eqnarray} x = C_1 e^{i \omega_0 t} + C_1 e^{- i \omega_0 t}\ \ \ \ \ \ \ \ \ \ (2) \end{eqnarray} where \(C_1\) and \(C_2\) are constants. Equation (2) can be expanded by Euler’s formula as follows: \begin{eqnarray} x = A \sin \omega_0 t + B \cos \omega_0 t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3) \end{eqnarray} where A and B are also constants. Equation (3) indicates the oscillation component of the forced oscillation. Next, we try to derive the particular solution.

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The general solution is the solution expressing the oscillation. Thus, the particular solution is assumed to also define the oscillation solution. Here, we assume that the particular solution is \(x=a \cos \omega t\) and is substituted to Equation (1). Then: \begin{eqnarray} - \omega^2 a + \omega^2_0 a= \frac{F_0}{m}\ \ \ \ \ \ \ \ \ \ (4) \end{eqnarray} We obtain: \begin{eqnarray} a = \frac{F_0/m}{ \omega^2_0 - \omega^2 }\ \ \ \ \ \ \ \ \ \ (5) \end{eqnarray} As the solution of the inhomogeneous differential equation of the second order is the addition of the general and particular solution, the solution of the forced oscillation is: \begin{eqnarray} x = A\sin \omega_0 t + B\cos \omega_0 t + \frac{F_0/m}{ \omega^2_0 - \omega^2 } \cos \omega t\ \ \ \ \ \ \ \ \ \ (6) \end{eqnarray} If the frequency of the forced oscillation \(\omega\) is greater than the frequency of the spring oscillation \(\omega_0\), the simple harmonic oscillation is damped by the force \(F\). On the other hand, if \(\omega\) is inferior than \(\omega_0\), the simple harmonic oscillation increases.

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