8.1. Equation of Motion for Harmonic Excitation

The general form of the equation of motion for a single-degree-of-freedom (SDOF) system subjected to a harmonic force is:

mx¨(t)+cx˙(t)+kx(t)=F sin(ωt)

Where:

• m = mass
• c = damping coefficient
• k = stiffness
• x(t) = displacement as a function of time
• F = amplitude of harmonic force
• ω = forcing frequency
• ωₙ = natural frequency of the system.

In this equation, we have three main components which contribute to how a system will respond to external vibrations:

1. Mass (m): This is the resistance of the system to acceleration. A larger mass means the system will be less responsive to the external force.
2. Damping Coefficient (c): Damping is what dissipates energy, reducing oscillations over time. Higher damping means the vibrations will die out faster.
3. Stiffness (k): This is how rigid the system is. A stiffer system will resist deformation more strongly.

The equation shows how these three factors combine with the applied force (F) to determine the motion (x(t)) of the system over time (t).

Think of this equation like a playground swing. The mass of the swing is the seat (the mass), the ability of the swing to stop swinging represents damping (when you push it lightly and it slows down), and how far the swing can move up from its resting position relates to stiffness (the direction of where you push the swing). Each part influences how the swing behaves after being pushed.

The solution of this second-order differential equation consists of two parts:

1. Homogeneous solution (transient response)
2. Particular solution (steady-state response)

When we solve the differential equation of motion, we divide the solution into two components:

1. Homogeneous Solution: This part represents the system's response after an initial disturbance, such as a push. It describes how the system oscillates due to its own characteristics without ongoing external forces.
2. Particular Solution: This portion addresses how the system behaves under the continuous influence of an external force, which, in this case, is harmonic or sinusoidal. Together, these solutions give a complete picture of how the structure will behave over time.

Imagine you're on a swing again. When someone gives you a big push (the external force), your swing moves in a certain way (the particular solution). After that push is gone, your swing will continue to move back and forth on its own because of momentum (the homogeneous solution). Together, they describe the entire experience of swinging.

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