10. Duhamel Integral

Consider a single-degree-of-freedom (SDOF) system subjected to an external time-varying force F(t). The general equation of motion is given by:

mx¨(t)+cx˙(t)+kx(t)=F(t)

Where:

• m = mass of the system
• c = damping coefficient
• k = stiffness of the system
• x(t) = displacement as a function of time
• F(t) = applied external force

This second-order linear differential equation with constant coefficients governs the motion of the system under arbitrary loading.

The equation of motion for a single-degree-of-freedom (SDOF) system describes how the system behaves when subjected to an external force that varies over time. The equation is a second-order differential equation where:

• m represents the mass of the system, which defines how much inertia the system has.
• c indicates the damping coefficient, which measures how the system dissipates energy (for example, through friction).
• k is the stiffness of the system, measuring how resistant the structure is to displacement due to the applied force.
• x(t) shows the displacement of the system at any time t.
• F(t) is the external force that affects the system's motion.

The equation essentially balances these forces: the inertia (mass times acceleration), the damping (which resists motion), and the stiffness (which resists deformation) all responding to the external force.

Imagine a swing (the SDOF system) being pushed (the external force). The more you push, the more it moves. However, if someone is hanging onto the swing (damping), it will not swing as far or as fast. The swing's ability to go higher also depends on the length of the chains (stiffness). This dynamic interaction where the swing moves back and forth in response to the pushes exemplifies how the equation of motion applies.

Before deriving the Duhamel Integral, we define the impulse response function h(t) as the response of the system to a unit impulse force applied at t=0: F(t)=δ(t) The solution x(t) for such an input is the unit impulse response, which depends on the damping condition of the system.

The impulse response function h(t) is crucial as it characterizes how the system reacts over time when subjected to an instantaneous force (an impulse). An impulse is represented mathematically as δ(t), which means an extremely short and powerful force applied at the very start (t=0). The resulting response x(t) varies based on the system's damping condition:

1. In the underdamped case, the system exhibits oscillatory behavior.
2. In critically damped and overdamped cases, the system’s response varies by how quickly it returns to equilibrium without oscillating.

Imagine someone pushing a child on a swing. If you give a hard push (impulse at t=0), the swing goes up high for a moment before coming down again. That trajectory the swing follows after that push describes the impulse response. If someone sits on the swing and gently resists the push (damping), it will return to a steady position more slowly. The impulse response will describe how high the swing went and how it came back down.

The system's response to a general force F(t) can be obtained using the principle of superposition and convolution. The idea is that any arbitrary force can be broken down into infinitesimally small impulses over time.

Using the superposition of the effects of these impulses, the total response is given by: t x(t)=∫h(t−τ)F(τ)dτ 0 This is the Duhamel Integral, where:

• x(t) is the displacement response at time t
• h(t−τ) is the impulse response function
• F(τ) is the force at time τ
• τ is a dummy time variable.

Duhamel’s Integral is derived from the principle of superposition, which states that the response of a system to multiple stimuli can be calculated by summing the responses to each stimulus. Applying this to any general force F(t), we can consider the force as consisting of many small impulses that occur over time. Each of these impulses produces a reaction in the system. The total response x(t) at any time can then be calculated by integrating the effects of all past impulses—this integration yields the Duhamel Integral:

x(t) = ∫h(t−τ)F(τ)dτ. Here, h(t−τ) indicates that the effect of a force applied at an earlier time τ affects the system's response at the current time t—capturing the delayed response essence that is common in dynamic systems.

Think of throwing pebbles into a pond. Each pebble creates ripples (responses) that spread out from the point of impact. If you throw multiple pebbles at different times, the total effect in the water will be the sum of all the ripples caused by each pebble. Duhamel's Integral allows us to sum up these effects mathematically to predict the water's movement (the system's response) at any time after the last pebble was thrown.

Duhamel’s integral expresses the total system response as the weighted accumulation of the impulse responses over time. Each infinitesimal force F(τ)dτ applied at an earlier time τ causes a delayed response that persists until time t, and the integral sums up these effects.

Duhamel’s Integral provides a deep understanding of how a system accumulates responses to forces over time. It tells us that every small force applied earlier influences the system's behavior later. Each of these tiny forces (F(τ)dτ) has an associated impulse response (h(t−τ)), meaning that the effect lingers and contributes to the total displacement at time t. This concept is particularly useful in contexts like earthquake engineering, where ground motions do not impact structures instantaneously but rather create a series of delayed effects that must be accounted for in analysis.

Consider a tree swaying in the wind. When the wind blows (the force), it doesn't just affect the tree instantly; rather, the movement of the branches takes time to respond to each gust. As each wind gust hits the tree at different times, it creates a cumulative effect on how far the tree bends and sways. Duhamel's Integral helps us capture that compounding influence of all past wind gusts on the tree's position at any moment.

--