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7.3. Equation of Motion for Free Vibration
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Today, we're going to discuss the equation of motion for free vibration in a Single Degree of Freedom system. It’s expressed as mx¨(t) + kx(t) = 0. Can anyone tell me what those symbols represent?
Is m the mass of the system?
Exactly! And k represents the stiffness of the spring. Now, what about x(t)?
Is it the displacement as a function of time?
Right! So when we substitute these values, we can analyze how the system behaves under free vibration.
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When we divide the initial equation by m, we get x¨(t) + ω²x(t) = 0. Can anyone tell me why this form is important?
It shows how the motion depends on natural frequency ω!
Correct! ω is defined as √(k/m). Understanding this relationship helps us predict the motion of the system. How would increasing the mass affect the frequency?
If mass increases, the natural frequency decreases, right?
That's right! Lower frequency means the system will vibrate more slowly. Keep that in mind for our discussions on building designs.
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Now let’s discuss natural frequency and time period. Can anyone remind us of the equations for natural frequency ω and time period T?
ω = √(k/m) and T = 2π√(m/k)!
Excellent! These relationships are crucial in determining how quickly a system will oscillate. If the stiffness increases, what happens to the frequency?
The frequency would increase!
Correct! And that's why understanding how these parameters interact is vital in structural dynamics.
Overview
Short Summary
This section introduces the equation of motion for an undamped single degree of freedom (SDOF) system undergoing free vibration.
Medium Summary
The equation of motion for an undamped SDOF system is presented along with its components, including mass, stiffness, and displacement. The significance of natural frequency is also highlighted as a key factor in the behavior of such systems under free vibration.
Detailed Summary
Equation of Motion for Free Vibration
The fundamental equation of motion for an undamped Single Degree of Freedom (SDOF) system undergoing free vibration is expressed as:
mx¨(t) + kx(t) = 0
Where:
- m = mass of the system
- k = stiffness of the spring
- x(t) = displacement as a function of time
- x¨(t) = acceleration (the second derivative of displacement)
By dividing the entire equation by m, we derive the more simplified version:
x¨(t) + ω²x(t) = 0
Where ω (omega) is the natural angular frequency of the system defined by:
ω = √(k/m)
This equation is crucial as it allows us to understand the dynamics of the system, primarily focusing on how the parameters such as mass and stiffness influence the natural frequency and subsequently the oscillatory motion of the system. The understanding of this motion is essential for analyzing and designing structures subjected to dynamic loads, especially in earthquake engineering.
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Create a free accountThe general equation of motion for an undamped SDOF system undergoing free vibration is: mx¨(t)+kx(t)=0 Where:
- m = mass of the system
- k = stiffness of the spring
- x(t) = displacement as a function of time
- x¨(t) = acceleration (second derivative of displacement)
Detailed Explanation
The equation of motion is a mathematical representation of how an undamped single degree of freedom (SDOF) system behaves during free vibration. In this equation, 'm' represents the mass of the system that is moving, 'k' is the stiffness of the spring that holds it in position, 'x(t)' represents the displacement of the mass from its resting position as a function of time, and 'x¨(t)' represents the acceleration of the mass, which is the second derivative of displacement with respect to time. The equation indicates that the force due to the mass's acceleration plus the force from the spring's stiffness must equal zero when no external forces are acting on the system.
Examples & Analogies
Imagine a child on a playground swing. When the child starts swinging back and forth, the mass of the child and the swing acts like the mass (m), while the swing’s ability to move back to the resting position (the spring) acts like the stiffness (k). The motion can be described by the equation of motion, which tells us how the swing moves without any outside pushes—just as the equation shows how the system behaves when only internal forces are at play.
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Create a free accountDividing the equation by m, we get: x¨(t)+ω²x(t)=0 Where ω = √(k/m) is the natural circular frequency of the system.
Detailed Explanation
To isolate the acceleration and displacement terms in the equation of motion, we can divide the entire equation by 'm', the mass of the system. This simplification leads us to a new equation where the term 'ω²' is introduced, representing the natural circular frequency of the system. The natural frequency, ω, tells us how fast the system tends to oscillate naturally when disturbed. It is calculated using the formula ω = √(k/m), where 'k' is the stiffness of the spring and 'm' is the mass.
Examples & Analogies
Think of a swing again. If the swing is built sturdily (high stiffness) or if a heavier child sits on it (high mass), the swing will have a specific frequency at which it naturally swings back and forth. If we divide the forces acting on the swing (taking into account both the weight of the child and the swing's stiffness), we can derive this frequency, which gives us insights into how quickly the swing oscillates without any pushes.
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Key concepts
Core takeaways and short definitions to help you quickly recall the key ideas from this section.
- Equation of Motion:
mx¨(t) + kx(t) = 0 defines the motion of an undamped SDOF system.
- Natural Frequency:
ω = √(k/m), indicates how fast a system vibrates.
- Displacement and Acceleration:
x(t) describes the position, while x¨(t) is the acceleration.
Examples
Memory aids
Imagine a strong spring and a heavy mass. The more you add to the mass, the slower it oscillates, just like a heavy person on a trampoline - they don’t bounce as quickly!
To remember the relationship: Mass = Low Frequency, think MLF, like an acronym for 'Mass Low Frequency'.
Flash Cards
Glossary
Free Vibration
The motion of a mechanical system when it vibrates freely, without the influence of external forces after an initial disturbance.
Natural Frequency
The frequency at which a system tends to oscillate in the absence of any driving force.
Displacement
The distance moved by a mass from its equilibrium position.
Stiffness
A measure of the resistance offered by a spring to deformation.