8. Response to Harmonic Excitation

In earthquake engineering and structural dynamics, understanding how structures respond to external forces is vital. Among all dynamic forces, harmonic excitation—a type of periodic force that varies sinusoidally with time—is foundational for studying the dynamic behavior of single and multi-degree-of-freedom systems. This chapter focuses on the theoretical and mathematical aspects of the response of linear systems to harmonic excitation. These concepts are essential for analyzing the vibratory behavior of structures subjected to dynamic loads such as machinery vibrations, seismic waves, or wind-induced forces.

The introduction explains the importance of how structures react to external forces, emphasizing harmonic excitation, which is a type of force that varies in a smooth, repeating pattern over time. Understanding this is crucial in fields like earthquake engineering because it helps engineers design structures that can withstand different dynamic loads. Examples of such dynamic loads include vibrations from machinery or seismic waves during an earthquake.

Imagine a swing at a park. When pushed at certain intervals (like the harmonic excitation), it goes higher and higher. If you push at just the right time (the natural frequency of the swing), it can swing to a much larger height with less effort, much like how resonance can amplify vibrations in buildings.

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

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

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

This equation captures how a system with mass, damping, and stiffness responds to an external harmonic force. Each term in the equation represents a key component of the system's behavior: mass resists motion, damping opposes oscillations, and stiffness tends to return the system to its original position. The solution is broken into two parts: the homogeneous solution reflects how the system would respond if disturbed and then left alone (its natural behavior), while the particular solution shows how the system behaves under the continuous influence of the forcing function.

Think of a rubber band tied to a fixed point. If you pull it (apply force), the band stretches (displacement). If you suddenly let go, it will oscillate back and forth (transient response). If you keep pulling and releasing it steadily (harmonic force), it will eventually reach a steady movement pattern (steady-state response).

In the absence of damping (c=0), the steady-state solution is:

x(t)=Xsin(ωt−ϕ)

Where the amplitude X is given by: X=F/(k(1−r^2)) with r=ω/ωn

Observations:

• At r=1, the amplitude tends to infinity — a condition called resonance.
• The phase angle ϕ=0 or π depending on r<1 or r>1.

This chunk discusses the behavior of a system that has no damping. The equation shows how the displacement of the system varies with time and how its amplitude relates to the frequency ratio (r) compared to its natural frequency. When the driving frequency matches the natural frequency (r=1), the system can experience resonance, leading to potentially dangerous amplitudes. The phase angle indicates how much the response lags or leads behind the external force, which can affect how the system interacts with dynamic loads.

Imagine pushing a child on a swing. If you push in rhythm with their natural swinging motion (r=1), their swings get higher and higher (resonance). But if you push out of sync, the swings won't reach as high. The way you time your pushes is similar to how the phase angle affects the system's response to external forces.

For a damped system, the steady-state amplitude becomes:

X=F/(c√(1−r^2))

Where:

• ξ= damping ratio
• r=ω/ωn

8.3.1 Phase Angle

2ξr tanϕ=1−r^2

The phase angle determines the lag between the excitation and the response.

In this chunk, we learn that when damping is introduced into the system, the formula for amplitude changes. The damping ratio (ξ) describes how much energy is lost in the system due to damping. The phase angle formula shows how damping and frequency ratio interact to influence how the system's response lags or leads relative to the excitation force. This is crucial for tuning systems to minimize destructive responses during events like earthquakes.

Think of a shock absorber in a car. It dampens (reduces) the bounciness, allowing the car to settle after hitting a bump smoothly. Similarly, in structures, damping controls how much vibration is felt and keeps the structure stable, preventing excessive motion which can lead to failure.

Maximum response does not occur at r=1 but at: r=√(1−2ξ^2)

• Damping reduces the peak response and shifts the resonance frequency leftward.

This section explains that in a damped system, the condition for maximum response (or resonance) is modified due to the presence of damping. Instead of reaching peak response when the driving frequency is equal to the natural frequency, it occurs at a lower frequency determined by the damping ratio. This highlights the importance of understanding damping in design to predict and control system behavior effectively.

Imagine tuning a guitar. If the strings (the system's natural frequencies) are dampened slightly but are still playable, you will find that you can play notes in a slightly different harmonic range. This is similar to how engineers have to adjust their designs to account for damping when anticipating how structures will react to forces.

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