Possible cases:
The nature of this solution based on the term in the square root. There are three following possible cases:
(1) (c/m) 2 > 4 (k /m) - Overdamped case
(2) (c/m)2 = 4 (k/m) - Critically damped case
(3) (c/m)2 < 4 (k/m) - Underdamped case
Let the critical damping coefficient be Cc, Hence,
![1071_Possible cases.png](https://www.expertsmind.com/CMSImages/1071_Possible%20cases.png)
Almost all of the systems are underdamped.
Here,
![1050_Possible cases1.png](https://www.expertsmind.com/CMSImages/1050_Possible%20cases1.png)
The ratio of damping coefficient (c) to critical damping coefficient is known as damping factor 'J'.
J = C /Cc . . . (17)
.....(18)
here ωd is natural frequency of the damped free vibrations.
Hence, underdamped case
x = e - (c /2m) ( X1 ei ωd t + X e- i ωd t ) . . . (19)
For critically damped system
x = ( X1 + X 2 t ) e - ( c/2m) t
For overdamped system
. . . (20)
. . . (21)
![2123_Possible cases5.png](https://www.expertsmind.com/CMSImages/2123_Possible%20cases5.png)
Figure
The Equation (19) may also be written as
x = X e-ζ ωn t cos (ωd t + φ) . . . (23)
Here X and φ are constants. X represents amplitude and φ phase angle.
Assume at t = t, x = x0.
∴ x0 = X e- ζ ωn t cos (ωd t + φ) . . . (24)
After one time period
t = t + t p and x = x1
∴ x = X e- ζ ωn (t + t p ) cos {ωd (t + tp ) + φ} . . (25)
Dividing Eq. (24) by Eq. (25)
![2466_Possible cases6.png](https://www.expertsmind.com/CMSImages/2466_Possible%20cases6.png)
As
t p = 1/ f p = 2π/ ωd
or ωd t p = 2π
![1000_Possible cases7.png](https://www.expertsmind.com/CMSImages/1000_Possible%20cases7.png)
Since cos θ= cos (2π+ θ)
∴ cos (ωd t + φ) = cos {ωd t + 2π + φ}
∴ x0 /x1 = e ζ ωn t p
or L = x0 /x1= ζ ωn tp = ζ ωn (2π/ ωd )=
![1620_Possible cases8.png](https://www.expertsmind.com/CMSImages/1620_Possible%20cases8.png)
or
-------- (26)
is known as logarithmic decrement.
If at t = t + n t p
It might be proved that
--------(27)
If
![1651_Possible cases12.png](https://www.expertsmind.com/CMSImages/1651_Possible%20cases12.png)
Figure 8 shows displacement time diagram for the above indicated three cases. For critically damped & overdamped system mass returns to its original place (indisplaced) gradually and there is no vibration. Vibration is possible just in the underdamped system as the roots of Eq. (14) are hard and solution consists of periodic functions (Eq. (22)).