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Introduction to Base Excitation
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Today, we'll talk about how buildings respond not just to external forces, but to ground motion during earthquakes. This is known as base excitation.
So, is the force from the earthquake different from other forces we study?
Exactly! In base excitation, the building moves in response to the ground shaking. The equation of motion changes to account for that effect. Can anyone state the modified equation?
Is it \( mx¨(t) + cx˙(t) + kx(t) = -mu¨_g(t) \)?
Correct! This reflects how the structure experiences acceleration due to ground motion. Let’s remember, we are interested in the relative displacement of the structure!
Using Duhamel's Integral
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Now, we can apply Duhamel's Integral to this base-excited system. Can anyone recall what we express the impulse response as?
Is it \( x(t) = -\int_0^t h(t-τ) mu¨_g(τ) dτ \)?
Correct! This integral helps us understand how past ground motions affect the current state of the structure. Why do you think we sum these historical effects?
To get the total response of the structure over time from all previous inputs?
Exactly! Each historical ground motion influences the current response of the system.
Mathematical Representation
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Let’s simplify the integral a bit more. When we express it, we often factor in the damping as well, which gives us an expression like \( x(t) = -\int_0^t e^{-ζω_n(t-τ)}sin[ω_d(t-τ)]u¨_g(τ)dτ \). What does each term represent?
The exponential represents the damping effect, and the sine function relates to the oscillatory nature of the response?
Absolutely! This expression provides a clearer picture of how damping and the oscillatory response are influenced by the ground motion. Remember this equation for solving earthquake-related problems!
Introduction & Overview
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Quick Overview
Standard
In this section, the dynamics of a base-excited SDOF system are analyzed through the application of Duhamel's Integral, allowing for the computation of relative displacement response to ground acceleration caused by earthquakes. The mathematical derivation and implications for structural engineering are highlighted.
Detailed
Detailed Summary
The section on Application to Base Excitation elaborates on the use of Duhamel's Integral to evaluate the response of structures during an earthquake, where the system is subjected to ground acceleration instead of an external force. The governing equation of motion for a base-excited single-degree-of-freedom (SDOF) system is introduced, given by:
$$ mx¨(t) + cx˙(t) + kx(t) = -mu¨_g(t) $$
This formulation accounts for the relative motion induced by the ground acceleration, where \( u¨_g(t) \) represents the ground motion input. By expressing the external force as \( F(t) = -mu¨_g(t) \), Duhamel's Integral can then be utilized to obtain:
$$ x(t) = -\int_0^t h(t-τ) mu¨_g(τ) dτ $$
The section further simplifies this integral to:
$$ x(t) = -\int_0^t e^{-ζω_n(t-τ)}sin[ω_d(t-τ)]u¨_g(τ)dτ $$
where the impulse response is based on damping. This leads to a direct method to calculate the relative displacement of structures in response to dynamic ground motion. The practical implications of these calculations are significant for earthquake engineering, assisting in the design and assessment of structures to withstand seismic activity.
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Ground Acceleration Input
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In earthquake engineering, instead of an external force F(t), the system is subjected to ground acceleration \( u¨ g(t) \), which induces relative motion in the structure.
Detailed Explanation
In the context of earthquake engineering, the primary concern is how structures respond to the motion of the ground caused by an earthquake. Unlike situations where we apply forces directly to a structure, during an earthquake, the ground itself moves. This movement is represented as ground acceleration, noted mathematically as \( u¨ g(t) \). This acceleration affects how the structure reacts, making it crucial to consider this change in force application.
Examples & Analogies
Imagine standing in a car that suddenly accelerates forward. You feel pushed back against your seat, not because something is pushing you directly, but because the car (the ground in a structural context) is moving. When an earthquake occurs, the ground shifts beneath buildings and bridges, causing them to sway or even collapse, much like the sudden motion of that car would affect your stance.
Equation of Motion for Base-Excited System
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The equation of motion in this case becomes: \[ mx¨(t) + cx˙(t) + kx(t) = -mu¨ g(t) \]
Detailed Explanation
This equation describes how the system behaves when subjected to base excitation. Here, \( m \) represents the mass of the structure, \( c \) is the damping coefficient which affects the energy dissipation of the structure, and \( k \) symbolizes the stiffness of the structure. The term on the right, \( -mu¨ g(t) \), indicates that the relative displacement of the structure is affected by the negative of the ground acceleration. This means that as the ground shakes, it induces movement that needs to be accounted for in the dynamics of the system.
Examples & Analogies
Think of a tall building on a shaky foundation during an earthquake. The ground motion is like a child shaking a toy building back and forth. The building's response (how much it sways) depends on its weight, how well it can absorb the shaking (damping), and its sturdiness (stiffness). This equation helps engineers predict how much the building will sway when the ground shakes, ensuring that it doesn't collapse.
Application of Duhamel's Integral
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Then using Duhamel’s integral: \[ x(t) = -\int_{0}^{t} h(t−τ) mu¨ g(τ) dτ \]
Detailed Explanation
This formulation uses Duhamel's integral to calculate the relative displacement of the structure over time due to the ground acceleration. Here, \( x(t) \) is the displacement response at a given time, while the integral sums the effects of the ground acceleration at previous times. By evaluating this integral, engineers can understand how the structure will behave (sway, deform) during an earthquake, based on the specific way the ground has moved up to that point.
Examples & Analogies
Imagine you're at a concert where each song triggers waves of movement through the crowd. Each wave creates a ripple effect; those at the front feel the movement first, and it travels backward. Similarly, Duhamel’s integral captures how past ground movements affect the current response of a structure. Each previous movement contributes to the overall sway of the building, helping us predict how it will move in response to ongoing shaking.
Mathematical Formulation of Relative Displacement
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Or: \[ x(t) = -\int_{0}^{t} \frac{1}{\omega_d} e^{-ζω_n(t−τ)} \sin[ω_d(t−τ)] u¨ g(τ) dτ \]
Detailed Explanation
This alternative formulation of the relative displacement response again uses Duhamel's Integral but expresses the impulse response in a sinusoidal form, taking damping into account. Here, \( ω_d \) represents the damped natural frequency, and \( ζ \) is the damping ratio, which represents how fast the oscillations decay. This equation is particularly insightful because it encapsulates how the structural response is influenced not only by sheer acceleration but also by how the system naturally reacts over time, considering its intrinsic properties.
Examples & Analogies
Consider a swing: when you push it, it oscillates back and forth. If you push it too hard, it might swing wildly (high response), but if you do it gently, the motion feels smoother and eventually dampens. This equation mathematically describes how a building sways in response to shaking—high-frequency ground movements can cause it to sway significantly, but if the building is well-designed (damping), it will settle down faster, just like the swing returning to a calm state after being pushed.
Definitions & Key Concepts
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Key Concepts
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Base Excitation: The concept of analyzing a structure's response due to ground acceleration during an earthquake.
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Duhamel Integral: The tool for calculating system response from dynamic loads over time.
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Impulse Response Function: Essential for characterizing how structures react to inputs.
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Acceleration Input: The specific type of loading that structures experience during seismic events.
Examples & Real-Life Applications
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Examples
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In an earthquake scenario, if the ground accelerates at 0.3g, a structure's displacement can be calculated using Duhamel's Integral to assess how much it moves.
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For designing a building in a seismic zone, engineers utilize the impulse response function to predict how the structure will behave when faced with ground motion.
Memory Aids
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🎵 Rhymes Time
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Ground shakes and buildings sway, Duhamel helps in every way!
📖 Fascinating Stories
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Imagine a tall building standing firm in the face of a rattling earthquake. The ground rumbles beneath its feet. To predict its sway, the engineers pull out the Duhamel Integral, forecasting its every move — just as a dancer anticipates their steps.
🧠 Other Memory Gems
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Remember S-D-I: S for SDOF systems, D for Duhamel's Integral, and I for Impulse Response Function.
🎯 Super Acronyms
BASE
- B: for Base Excitation
- A: for Acceleration input
- S: for Structural response
- E: for Earthquake engineering.
Flash Cards
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Glossary of Terms
Review the Definitions for terms.
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Term: Base Excitation
Definition:
The phenomenon where a structure responds to ground motion, particularly during an earthquake, leading to relative displacement.
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Term: Duhamel Integral
Definition:
A mathematical formulation used to determine the response of a linear time-invariant system to arbitrary dynamic loading.
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Term: Impulse Response Function
Definition:
The output of a system when subjected to a unit impulse input. It characterizes the system’s dynamic behavior.
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Term: SingleDegreeofFreedom (SDOF)
Definition:
A system characterized by a single mass simplified down to one dimension, generally used in structural analysis.
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Term: Ground Acceleration
Definition:
The rate of change of velocity of the ground due to seismic activity.