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2.2 Dynamic Characteristics of a Cantilever Beam under Free Vibration
Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium
point. Free vibration occurs when a mechanical system is set off with an initial input and then
allowed to vibrate freely. The mechanical system will then vibrate at one or more of its "natural
frequency" and damp down to zero. Forced vibration is when an alternating force or motion is
applied to a mechanical system.
A normal mode of an oscillating system is a pattern of motion in which all parts of the
system move sinusoidally with the same frequency and with a fixed phase relation. The motion
described by the normal modes is called resonance. The frequencies of the normal modes of a
system are known as its natural frequencies or resonant frequencies. Each physical object has a
set of normal modes that depend on its structure, materials and boundary conditions.
A mode of vibration is characterized by a modal frequency and a mode shape, and is
numbered according to the number of half waves in the vibration. In a system with two or more
dimensions, such as the pictured disk, each dimension is given a mode number. Each mode is
entirely independent of all other modes. Thus all modes have different frequencies (with lower
modes having lower frequencies) and different mode shapes.
2.2.1 Natural Frequencies of a Cantilever Beam under Free Vibration
For an Euler-Bernoulli beam under free vibration, the Euler-Lagrange equation is
Since deflection is a function of time and distance, we have
This makes eq.15:
Volterra, E. (01/01/1966). "Dynamics of Vibrations". Journal of applied mechanics(0021-8936), 33(4),
p.956.
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