Simple Harmonic Motion.....
Free Natural Vibrations.....
Underdamped oscillations..... Overdamped oscillations..... Logarithmic Decrement..... Angular Vibrations.....
Simple Harmonic Motion
Generally free natural vibrations occur in elastic system when a body moves away from its
rest position. The internal forces tend to move the body back to its rest
position. The restoring forces are in proportion to the displacement.
The acceleration of the body which is directly related to the force on the body is
therefore always towards the rest position and is proportional to the displacement of the body from its rest position. The body moves with
simple harmonic motion...
Free Natural Vibrations
Damped Vibrations ..(Viscous Damping)
The resisting force of the damper is directly proportional to the velocity of the mass
m.dx2 /dt 2 = - k.x - c.dx /dt
This equates to the general expression of the equation of motion of a ocillating system with damping
m.dx2 /dt 2 + c.dx /dt + k.x = 0
Now by making ωn 2 = k / m
dx2 /dt 2 + 2ζ ωn dx /dt + ωn2 x = 0
Note: ζ is called the damping ratio and is described a little better under heading critical damping ratio below.
The equation is solved by assuming a solution of the form x = e s.t (s = constant)
( s 2 + (c/m)s + (k/m) ) e st = 0 = ( s 2 + 2ζ ωns + ωn2) e st
s 2 + s (c/m) + k/m = 0 = s 2 + 2ζ ωn s + ωn2
This equation has two roots
The solution of the equation in general forms
A and B are constants which can be evaluated from initial values of x and dx/dt..
The term e - ( c / 2 m) t represents an exponentially decaying factor...The expression inside the brackets can have three general results which significantly affect the solution.
Critical damping ...( (c /2 m) 2 = (k / m) and ζ= 1 (see below) )
The damping value c which results in [ (c/2m) 2 - (k/m) = 0 ] is called the critical damping coefficient cc
The damping can be expressed in terms of the damping ratio
This results in the values..
Underdamped Oscillations.. ( ζ <1 or (c /2 m) 2 < (k / m))
In the underdamped case the square root of a negative number is an imaginary number..resulting in roots of
This results in a oscillatory solution of the form..
The frequency of damped oscillation (rads/s) is equivalent to
The equation of motion for the underdamped case ( ζ < 1 ) is rewritten as
This equation can be transformed to
This equation can be further transform into an equation with one trigonometric term as follows
Overdamped Oscillations ....( ζ > 1 and (c /2 m) 2 > (k / m))
The equation of motion for the overdamped case ( ζ > 1 )is rewritten as
The equation of motion for the critally damped case ( ζ = 1 )is written as
The rate at which the amplitude of vibrations decays over time provides a very useful method of identifying the degree of damping. See the figure below for the plot of a typical underdamped vibration.
The equation for of motion for underdamped oscillations arrived at above can be used to establish the amplitude of any of the cycles. i.e.
Considering the amplitudes of successive cycles when cos(ω dt - φ) = 1
The logarithmic decrement δ is ln( xn / xn+1 ).. Normally n = 1 and n+1 is therefore 2.
The damping ratio ζ can be expressed in terms of the logarithmic decrement δ as follows
It is clearly possible to determine the damping ratio experimentally for a mechanical system be initiating vibrations and measuring the amplitude of the vibrations.
Consider a disc with a moment of Inertia I suspended on a vertical bar with a torsional stiffness q with a viscous damper attached with a viscous damping coefficient c
The equation of motion of this system is expressed by the equation ..
This equation can be expressed using the natural circular frequencey ωn and the damping ratio ζ..
It is clear that this equation is the same as the equation of motion for linear motion as reviewed above and can be analysed using the same principles
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