8.2 - Steady-State Response of Undamped SDOF Systems
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Introduction to Steady-State Response
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Today, we're delving into the steady-state response of undamped single-degree-of-freedom systems. To begin, could someone remind us what is meant by 'undamped'?
Does that mean there’s no damping coefficient affecting the system?
Exactly! In an undamped system, we can simplify our analysis. The response can be expressed as a sinusoidal function, something like this: `x(t) = X sin(ωt − ϕ)`. Student_2, can you tell us what `X` represents?
Isn't `X` the amplitude of the response?
That's correct! Now, the amplitude is influenced by the frequency ratio, which we will explore in more depth.
Understanding the Frequency Ratio
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In the formula for amplitude, we see that `r`, the frequency ratio, plays a significant role. Who can remind us the formula for `r`?
It's `r = ω / ω_n`, right?
Exactly! Now tell me, what happens when `r` equals 1?
That’s resonance! The amplitude goes to infinity, making the system unstable.
Great! Resonance is a critical condition to understand in vibratory systems.
Implications of Phase Angle
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Now, let's discuss the phase angle `ϕ`. How does it relate to our frequency ratio?
I think it depends on whether `r` is less than or greater than 1? Like if `r` is less than 1, `ϕ` is 0?
Correct! And if `r` is greater than 1?
Then `ϕ` is π?
Exactly! This phase relationship is key as it determines the timing of the system's response.
Introduction & Overview
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Quick Overview
Standard
The steady-state response of undamped single-degree-of-freedom (SDOF) systems to harmonic forces is characterized by a sinusoidal displacement function, with the amplitude affected by the frequency ratio. The section highlights the implications of resonance when the forcing frequency matches the natural frequency of the system.
Detailed
Steady-State Response of Undamped SDOF Systems
In this section, we explore the steady-state response of undamped single-degree-of-freedom (SDOF) systems subjected to harmonic forces. The governing equation of motion, when damping is absent (i.e., damping coefficient c = 0), leads to a solution of the form:
Equation of Steady-State Response
$$ x(t) = X imes ext{sin}( ext{ω}t - ϕ) $$
Where:
- X is the amplitude of the response,
- ϕ represents the phase angle.
Amplitude and Frequency Ratio
The amplitude X is determined by the formula:
$$ X = \frac{F_0}{k(1 - r^2)} $$
with the frequency ratio r defined as:
$$ r = \frac{ ext{ω}}{ω_n} $$
Here, ω_n is the natural frequency of the system. The analysis reveals crucial observations regarding resonance—when the frequency ratio r equals 1, the amplitude tends to infinity, indicating an unstable condition. The phase angle ϕ further depends on whether r is less than or greater than 1, which determines the relationship between the excitation and response of the system. This understanding is critical in the design of structures to avoid catastrophic failures due to resonance.
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Steady-State Solution
Chapter 1 of 3
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Chapter Content
In the absence of damping (c=0), the steady-state solution is:
x(t)=Xsin(ωt−ϕ)
Detailed Explanation
The steady-state solution describes how the system behaves after all transient effects have died out due to sustained harmonic input. When there is no damping (which means c=0), the response of the system is primarily sinusoidal and characterized by the formula x(t)=Xsin(ωt−ϕ). Here, X represents the amplitude of the steady-state response, and ϕ is the phase angle. This signifies that the system will oscillate indefinitely at a frequency determined by the forcing frequency ω.
Examples & Analogies
Imagine a swing being pushed in a regular rhythm by a person. Initially, the swing may wobble for a moment until it steadily moves back and forth as the push continues. In this context, the steady-state response is akin to the smooth swings the child experiences after the initial chaos settles down when the pushing is consistent.
Amplitude Calculation
Chapter 2 of 3
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Chapter Content
Where the amplitude X is given by:
X= F/(k(1−r²))
with r = ω/ω_n.
Detailed Explanation
The amplitude X can be determined mathematically using the formula X = F / (k(1 - r²)). Here, F is the amplitude of the harmonic force, k is the system's stiffness, and r is the ratio of the forcing frequency ω to the system's natural frequency ω_n. This formula indicates how much the system will respond to the input force at different frequencies. The relationship suggests that as the frequency approaches the system's natural frequency (where r approaches 1), the amplitude can increase significantly.
Examples & Analogies
Think of a person pushing a swing. If they push the swing slowly and out of sync with the swing’s natural rhythm, the swing moves just a little. However, if they synchronously push just as the swing reaches the peak of its motion, the swing goes much higher, demonstrating resonance. Here, the formula helps predict how high the swing will go based on how hard and at what frequency it's pushed.
Resonance Condition
Chapter 3 of 3
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Chapter Content
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.
Detailed Explanation
When r equals 1 (r=1), it means that the forcing frequency matches the system's natural frequency. This results in an unstable situation where the amplitude of oscillation can theoretically become infinite, a condition known as resonance. On the other hand, the phase angle ϕ represents the relationship in time between the input force and the system's response. If the ratio r is less than 1, the phase angle is typically 0; if r is greater than 1, the phase angle shifts to π, indicating a lag in response.
Examples & Analogies
Consider a child swinging on a playground swing. If someone pushes the swing at just the right time – when it’s about to swing back to them – it goes higher and higher. But if they push it too fast or slow, it doesn’t achieve that height. When they push right at the ideal moment (resonance), it’s like everything around the swing forces aligns perfectly to maximize the result.
Key Concepts
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Undamped SDOF Systems: Systems without damping that oscillate freely under harmonic excitation.
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Steady-State Response: The long-term behavior of systems after transient effects have diminished.
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Resonance: The condition resulting in maximum amplitude when forcing frequency matches the system's natural frequency.
Examples & Applications
An undamped SDOF system like a simple pendulum subjected to periodic forces can resonate under specific conditions.
Engineers must carefully assess the frequency of machinery to avoid resonating with structures.
Memory Aids
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Rhymes
In a system that’s undamped, when forces are jammed, resonance will rise, making things high-stressed and crammed.
Stories
Imagine a child on a swing; when they push at the right rhythm, they go higher and higher. That is resonance—a joyful yet unstable moment.
Memory Tools
Remember 'RAP' for resonance: Ratio matches amplitude peaks.
Acronyms
Use 'RAP' to recall Resonance (R), Amplitude (A), and Phase (P).
Flash Cards
Glossary
- Amplitude
The maximum extent of a vibration or oscillation, measured from the position of equilibrium.
- Frequency Ratio (r)
The ratio of the forcing frequency to the natural frequency of the system, defined as
r = ω / ω_n.
- Resonance
A condition in which a system experiences increased amplitude at specific frequencies, often leading to instability.
- Phase Angle (ϕ)
The angle that represents the phase difference between the excitation and response signals in a vibrating system.
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