9.7 - Numerical Example
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Understanding the System Parameters
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Today we are analyzing a numerical example related to the impulse response of a single degree of freedom system. Can anyone tell me why the mass, damping coefficient, and stiffness are important?
They determine how the system will react to forces, right?
Exactly! The mass (m) affects the inertia of the system, damping (c) influences how energy is dissipated, while stiffness (k) relates to how much the system resists deformation. Let’s calculate the natural frequency. Who remembers the formula?
Is it ω_n = √(k/m)?
Correct! So what do we get when we plug in our values?
For m=1kg and k=4N/m, ω_n equals 2 rad/s!
Great job! Now we can explore how this affects our impulse response.
Damping Ratio Calculation
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Next, let's calculate the damping ratio ζ. Who can remind us how to find that?
It’s ζ = c / (2√(mk)).
Correct! Now substituting the values, what do we find?
That makes ζ = 0.25.
Right again! Understanding the damping ratio is key, as it tells us whether our system is underdamped or overdamped. What does it mean in our case?
It indicates that our system is underdamped since ζ < 1.
Perfect, let’s keep that in mind as we move on to formulate the impulse response.
Formulating the Impulse Response Function
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Now, let’s write down the impulse response function. Who remembers how we approach this?
It’s x(t) = A * e^{-ζω_nt} * sin(ω_d t).
Exactly! So now, what do we set our values into for A, ω_d, and other variables?
A can be calculated from initial conditions, and ω_d will be √(k/m) adjusted for ζ!
Wonderful! Plugging those values, we get the full expression. What does it look like?
It results in x(t) = 0.5166 * e^{-0.5t} * sin(1.936t).
Very well done! This describes how the system will respond to an impulsive force at t=0.
Interpreting the Impulse Response Output
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Now that we’ve derived the impulse response function, what can we conclude about the system's behavior over time?
It will oscillate while gradually losing amplitude due to damping.
Yes, and the frequency of oscillation is governed by the damped natural frequency!
Exactly! This response is crucial in earthquake engineering to understand how structures behave under seismic activity.
So, it helps in designing structures that can withstand shocks!
You're all correct! This example exemplifies how theoretical concepts apply to real-world structures.
Introduction & Overview
Read summaries of the section's main ideas at different levels of detail.
Quick Overview
Standard
Here, the parameters of a damped single-degree-of-freedom (SDOF) system are specified, leading to the derivation of the impulse response function. The example demonstrates the significant aspects of system response to an impulsive force.
Detailed
Numerical Example (Section 9.7)
In this section, we evaluate a numerical example based on the parameters of a damped single degree of freedom (SDOF) system. We are given the mass (m), damping coefficient (c), and stiffness (k) to generate the impulse response function for the system.
Given parameters:
- Mass (m): 1 kg
- Damping Coefficient (c): 1 N·s/m
- Stiffness (k): 4 N/m
From these values, we compute:
- Natural frequency (C9_n) = 2 rad/s
- Damping ratio (ζ) = 0.25
- Damped natural frequency (C9_d) = 1.936 rad/s
Impulse Response Function
The derived impulse response function (displacement response of the structure to an impulsive force at t=0) is given as:
$$x(t) = 0.5166 imes e^{-0.5t} imes ext{sin}(1.936t)$$
This function crucially encapsulates how the structure responds over time to impulsive forces applied, illustrating both the damped response envelope and oscillatory characteristics inherent to the system.
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Given Parameters
Chapter 1 of 4
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Chapter Content
Given:
• m=1kg, c=1N·s/m, k =4N/m
Detailed Explanation
In this numerical example, we are given three important parameters for a damped system: mass (m), damping coefficient (c), and stiffness (k). Here, the mass of the system is 1 kilogram, the damping coefficient is 1 Newton-second per meter, and the stiffness is 4 Newtons per meter. These parameters are crucial for defining the motion of the system when subjected to external forces, particularly impulsive forces.
Examples & Analogies
Think of a toy car on a smooth surface. The car’s mass is equivalent to the resistance it has to acceleration. The damping coefficient can be compared to friction between the car and the surface, affecting how quickly it slows down after being pushed. Stiffness is akin to how hard the springs in the car’s suspension work to return it to its position after a bump.
Natural Frequency and Damping Ratio
Chapter 2 of 4
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Chapter Content
Then:
• ω_n=2rad/s, ζ=√1/2=0.25
• ω_d=2√(1−0.25²)=1.936rad/s
Detailed Explanation
From the initial parameters, we can calculate the natural frequency (ω_n) and the damped natural frequency (ω_d) of the system. The natural frequency indicates how the system behaves in a free oscillation without any dampening. The damping ratio (ζ) helps determine how the oscillations reduce over time. In this case, with ζ calculated as 0.25, we note that the system is underdamped, which means it will oscillate before eventually coming to rest.
Examples & Analogies
Imagine a swing at a playground. The natural frequency is like how fast the swing moves back and forth when pushed, while the damping ratio shows how quickly it slows down after being pushed. A lower damping ratio means more oscillations before the swing stops.
Impulse Response Function
Chapter 3 of 4
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Chapter Content
Impulse response:
1
x(t)= e^{-0.25·2·t} sin(1.936·t)
1·1.936
x(t)=0.5166·e^{-0.5·t} sin(1.936·t)
Detailed Explanation
The impulse response function describes how the system will respond to an impulsive force applied at time t=0. The exponential term, e^{-0.5·t}, represents the decay in amplitude of the oscillation over time due to damping. The sine function, sin(1.936·t), shows that the displacement will oscillate at the damped frequency. This response helps predict the motion of the structure over time following the impulse.
Examples & Analogies
Think of a plucked guitar string. When you pluck it (the impulse), it vibrates and produces sound. The sound starts loud (high displacement) and gradually gets quieter (decaying amplitude) until it stops vibrating. Here, we are mathematically modeling that same behavior for a different system.
Physical Interpretation
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Chapter Content
This function describes the displacement response of the structure to an impulsive force at t=0.
Detailed Explanation
The derived function provides a complete description of how the structure behaves immediately after the application of an impulse. It allows engineers to understand the maximum displacement, the frequency of oscillations, and the rate at which the system returns to rest. Such understanding is critical in designing structures that can withstand impulsive forces, like those from earthquakes.
Examples & Analogies
Consider a trampoline. When someone jumps on it (the impulse), it bends and oscillates up and down. The shape and speed of that bounce can be predicted using similar mathematical functions, which allow us to ensure that the trampoline can handle repetitive jumps without breaking.
Key Concepts
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Impulse: A short-term high-magnitude force acting on a structure.
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Impulsive Forces: Critical in understanding real-world applications like earthquakes.
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Natural Frequency: Denotes the inherent ability of a system to oscillate.
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Damping Ratio: Determines how quickly a system returns to rest.
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Impulse Response Function: Reveals the behavior of structures under impulsive loads.
Examples & Applications
An example is using an impact hammer to test structural responses in earthquake engineering.
Simulating an impulsive load can help in predicting responses during seismic activity.
Memory Aids
Interactive tools to help you remember key concepts
Rhymes
In systems damped, impulse reigns, energy fades, while sine remains.
Stories
Imagine a bridge swaying gently after a sudden earthquake — how it oscillates gracefully illustrates our response derived from impulse guidelines!
Memory Tools
DAMP: Damping ratio, Amplitude decay, Motion description, Parameters involved.
Acronyms
FIND
Frequency
Impact
Natural
Damping — key features of impulse responses!
Flash Cards
Glossary
- Impulse
A force of very large magnitude acting over a very short period of time.
- Dirac Delta Function
A mathematical function used to model an impulse, denoted as δ(t).
- Damping Ratio (ζ)
A measure of how oscillations in a system decay after a disturbance.
- Natural Frequency (ω_n)
The frequency at which a system oscillates when not subjected to external forces.
- Impulse Response Function
The output of a system when subjected to a unit impulse input, often used to analyze the system's behavior over time.
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