Loading the lesson
Loading the lesson
Search AllRounder.ai
Search the courses, subjects, tracks, games and features, or jump straight to a page.
Search the courses, subjects, tracks, games and features, or jump straight to a page.
Learn it your way
A short class with Sarah and her students. The other ways are under More.
Sign up free to listen. Transcripts are free to read.
Today we will be learning about forced vibrations in Single Degree of Freedom systems. Can anyone tell me what we mean by 'forced vibration'?
Isn’t it when an outside force continuously excites the system?
Exactly! The equation we use for forced vibration in SDOF systems is mx¨ + cx˙ + kx = F_0 sin(ωt). Here, what do each of the symbols represent?
m is mass, c is damping, k is stiffness, and F_0 is the external force amplitude.
And ω is the frequency of the external force, right?
Correct! Now, let’s talk about the responses of the system. Can anyone tell me the difference between transient and steady-state responses?
The transient response decays with time, but the steady-state response persists.
Great summary! Remember, the steady-state response dictates the long-term behavior of our structures.
Now that we understand forced vibrations, let’s explore resonance. What happens when the frequency of the external force matches the natural frequency of the system?
That’s when resonance occurs, and it can lead to large oscillation amplitudes.
Exactly! Resonance is crucial in earthquake engineering since it can cause structures to vibrate uncontrollably. How can we mitigate this risk?
We can shift the natural frequency or add damping to the system!
Yes, by introducing sufficient damping, we can reduce the impact of resonance significantly.
Let’s discuss how to quantify the steady-state amplitude. The formula is . Can someone break this down for me?
F_0 represents the applied force, and the denominator calculates the effective stiffness and damping!
Exactly! The relationship indicates that as the external frequency approaches the natural frequency, our amplitude peaks. Why do you think this is important in design?
Because if we design a structure with a natural frequency close to expected external forces, we might get dangerous resonant oscillations!
Great observation! Always remember to assess natural frequencies during structural design to enhance safety.
Continuous external excitation leading to sustained oscillation.
Decays over time after the force is applied.
Long-term behavior dominated by constant forces.
Occurs when excitation frequency matches natural frequency.
The maximum displacement of the system from its equilibrium position.
When forced vibes start to dance, Beware of resonance, take a chance!
Once a bridge swayed gently in the breeze during a storm. But with each gust, it felt the force of the wind pulling it closer to danger—resonance, looming overhead until safety measures were placed.
FRS | Forced vibration, Resonance, Steady-state response — keep these at hand when engineering structures!
TRS | Transient, Resonance, Steady-state — remember the three responses of forced vibrations!
Term
Steady-State Amplitude
Click to flip
Term
Resonance
Click to flip
Term
External Force
Click to flip
A type of vibration that occurs when an external force regularly excites a system.
The part of the system's response that diminishes over time after the external force is applied.
The long-term behavior of a system under continuous external force.
A phenomenon that occurs when the frequency of the external force matches the system's natural frequency, causing large amplitude oscillations.
The frequency at which a system tends to oscillate when not subjected to a continuous external force.
This section explores the dynamics of forced vibration in Single Degree of Freedom (SDOF) systems, focusing on the effects of harmonic external forces.
In this section, we delve into the mathematical modeling of forced vibrations in SDOF systems, characterized by an external harmonic force. The response of these systems includes both transient and steady-state components, with key concentration on resonance phenomena and amplitude of vibration.
In Single Degree of Freedom (SDOF) systems, forced vibrations arise when external forces continuously impact the system. The equation governing this behavior is represented as:
Where:
The solution to this equation can be divided into two main components:
Steady-State Amplitude is quantified using:
The amplitude reaches its maximum when the frequency of the external force is close to the natural frequency of the system, a condition that leads to resonance. This phenomenon can cause dramatic increases in vibration, necessitating careful analysis during design phases to mitigate the risk of structural failure due to excessive oscillation.