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Dezi et. al (2010)
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Projectile motion
This template is used to calculate the natural frequency of a rod undergoing longitudinal vibration. Such vibrations are a fundamental aspect in the fields of mechanical engineering, structural analysis and materials science.
Calculation
Inputs
ρ
:7,850kg/m^3
L
:1.00m
E
:290.00GPaOutput
v
:6,078m/s
f
:3,039HzWhere:
- is the density, the mass per unit volume of the rod's material, measured in kilograms per cubic meter
- is the length of the rod, measured in meters
- is Young's Modulus, the modulus of elasticity of the material, measured in gigapascals
- is the speed of sound in the rod measured in meters per second. The formula is:
- is the fundamental frequency, which is the frequency at which the rod naturally vibrates, measured in Hertz. The formula is:
Explanation
What is a vibrating rod?
Analogous to a spring system which is often used as an idealised model of how a beam bends, a vibrating rod is an idealised model of longitudinal vibrations of many different objects. A vibrating rod can also be considered as an elongated elastic band or string. It is used to understand longitudinal vibrations in diverse engineering contexts including the shaking behaviour of a building, the sound of instruments and the toughness of materials.
What is longitudinal vibration?
Longitudinal vibration is when particles within our idealised “rod” move back and forth in a direction parallel to the rod's length. Longitudinal vibration is a form of harmonic motion, which means this back and forth movement is a periodic oscillation of particles. This oscillatory motion leads to alternating regions of compression and elongation along the rod, a phenomenon critical for the analysis and design of mechanical systems to prevent resonant frequencies that could cause structural failure.
What is a fundamental frequency?
The fundamental frequency, also known as the natural frequency, is the frequency at which our rod naturally vibrates in. It is the simplest form of vibration where the whole rod stretches or compresses along its length. It is important to predict the natural frequency of your object under consideration in order to avoid external vibrations at the same frequency, which would lead to resonance and therefore potential failure.
The fundamental frequency is given by:
The fundamental frequency has an inverse relationship to the rod’s length, which means an increase in rod length leads to a reduction of the natural frequency.
The fundamental frequency is found in the rod’s first "mode", a mode is a specific pattern of vibration. The second mode shape has an additional node in the centre showing the two halves of the rod moving in opposite direction to each other, leading to more complex stress distributions in the rod.
As a general pattern the frequency of mode
is given by:
As the mode number increases, so does the number of nodes and therefore dividing the rod into more segments that vibrate out of phase with adjacent segments, leading to increasingly complex patterns of vibration.
This template is particularly useful in scenarios such as:
- Predicting and avoiding resonant frequencies in mechanical systems and structures
- Automotive and aerospace industries for assessing the vibrational characteristics of components
- Material science for studying the properties of different materials under vibrational stress.
References
- Rao, S. S. (2017). Mechanical Vibrations (6th ed.). Pearson.
- Inman, D. J. (2013). Engineering Vibration (4th ed.). Pearson.
- Thomson, W. T. (1993). Theory of Vibration with Applications (5th ed.). Prentice Hall.
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