1.5. Forced Vibration of SDOF Systems

For a harmonic external force F(t)=F_0 sin(ωt), the equation becomes:

mx¨ + cx˙ + kx = F_0 sin(ωt)

This chunk introduces the concept of forced vibration in Single Degree of Freedom (SDOF) systems. Forced vibration happens when an external force continuously acts on the system. The external force is represented as a sine function, indicating that it varies periodically over time. The equation combines the mass (m), damping (c), and stiffness (k) effects to understand how these components react to the imposed force.

Imagine pushing a child on a swing at regular intervals. The force you apply (pushing) alters how high the swing goes. If the pushes are timed perfectly to match the swing's natural rhythm, your pushes will enhance the swing's motion, leading to larger swings—this is akin to forced vibration.

The solution has two parts:

• Transient response: Decays with time
• Steady-state response: Dominates at long times

The response of the system can be broken down into two main components. The transient response occurs immediately after the force is applied and consists of initial oscillations that gradually disappear over time. The steady-state response, however, is what we observe after a long time has passed, where the system oscillates at a consistent frequency and amplitude. This concept helps engineers predict how structures will behave under continuous external forces.

Think of a car radio. When you first turn it on, you might hear some static (transient response) as the system settles on a station. After a moment, the music comes through clearly (steady-state response). Similarly, structures will respond initially to forces but will stabilize to a predictable behavior over time.

Steady-state amplitude is given by:

X = F_0 / (k - (mω²))

This amplitude peaks near the natural frequency, causing resonance when ω ≈ ω_n.

The steady-state amplitude X shows how much the system will vibrate under the influence of the external harmonic force F_0, while taking into account the spring constant (k) and the inertia of the mass (m). It highlights that the amplitude of oscillation will peak when the frequency (ω) of the applied force is close to the system's natural frequency (ω_n). This phenomenon is known as resonance, which can significantly amplify the system's response.

Consider a swing again, but this time, think about timing your pushes. If you push the swing at just the right moments (matching its natural swing frequency), it will go much higher than if you pushed at the wrong times. This is the same idea as resonance—small inputs can lead to much larger outputs if timed correctly!

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