Forced Oscillations - Differential Equation

Forced Oscillations - Differential Equation

Forced oscillations occur when a system is subjected to an external periodic force. Let's consider a simple harmonic oscillator under the influence of an external force:

\[m\frac{d^2x}{dt^2} + b\frac{dx}{dt} + kx = F_0\cos(\omega t)\]

where:

• \(m\) is the mass of the oscillator,
• \(b\) is the damping coefficient,
• \(k\) is the spring constant,
• \(F_0\) is the amplitude of the external force,
• \(\omega\) is the angular frequency of the external force.

This equation represents the motion of the oscillator, with the first term on the left side representing the inertia, the second term representing damping, the third term representing the spring force, and the right side representing the external force.

Step 1: Homogeneous Solution

First, consider the homogeneous part by setting \(F(t) = 0\):

\[m\frac{d^2x}{dt^2} + b\frac{dx}{dt} + kx = 0\]

The solution to this homogeneous equation is a linear combination of sine and cosine functions:

\[x_h(t) = A\cos(\omega_0 t) + B\sin(\omega_0 t)\]

where \(\omega_0 = \sqrt{\frac{k}{m}}\) is the natural angular frequency of the system. This represents the motion of the oscillator in the absence of external forces.

Step 2: Particular Solution

Now, consider the particular solution for the forced part. We assume that \(x_p(t)\) has the same form as the external force:

\[x_p(t) = X\cos(\omega t - \phi)\]

where:

• \(X\) is the amplitude of the forced oscillations,
• \(\phi\) is the phase difference between the external force and the displacement.

This represents the motion induced by the external force.

Step 3: Combine Solutions

Combine the homogeneous and particular solutions to get the complete solution:

\[x(t) = x_h(t) + x_p(t)\] \[x(t) = A\cos(\omega_0 t) + B\sin(\omega_0 t) + X\cos(\omega t - \phi)\]

This equation describes the motion of the forced oscillator, combining the natural motion with the motion induced by the external force.

Step 4: Find Coefficients

Use the initial conditions and boundary conditions to find the coefficients \(A\), \(B\), \(X\), and \(\phi\). These conditions depend on the specific situation and may include the initial position, initial velocity, or other constraints.

Once these coefficients are determined, you will have the complete solution for the forced oscillations.

This is a general outline of the derivation process for forced oscillations. The specific values of \(m\), \(b\), \(k\), \(F_0\), and \(\omega\) will influence the exact form of the solution.