E-StoreResonance
About 70 years ago a bridge was constructed near Washington. Just after 4 months of opening the bridge, it broke down.
Actually that was Tacoma Narrows Bridge and the year was 1940.
The reason , almost all textbooks ( including HCV ) say , was resonance.
Wikipedia however tells :
The collapse of the Old Tacoma Narrows Bridge, nicknamed Galloping Gertie, in 1940 is characterized in physics textbooks as a classical example of resonance. This description is misleading, however. It would be more correct to say that the bridge failed due to the action of self-excited forces upon it, largely through a phenomenon known as aeroelastic flutter. Robert H. Scanlan, father of the field of bridge aerodynamics, wrote an article about this misunderstanding.
To begin at the beginning, we can start with SHM. It is a motion that repeats itself after regular time interval such that the force acting on it is directed towards a point over the linen (which it executes motion) and the force is also proportional to displacement of particle from that fixed point. The equation is
F = -kx
where k = mw0 2
k = force constant
w02 = a positive constant , called natural frequency.
m = mass of particle
x = displacement
But in real life all to and fro motion stops after a certain time becoz of air resistance, friction etc. We call these forces damping force, which are speed depended.
The equn then becomes
F = - kx - bv
where b is another constant and v is velocity.
Now solving this we get x = A0 e-bt/2m sin(w12 + @)
where w1 = Square root [ w02 - ( b /2m )2 ]
Obviously for small b,
w1 = w0
Thus the system oscillates with almost natural frequency ( with which the system will oscillate if there is no damping ) and with amplitude decreasing with time as per equn
x = A0 e-bt/2m
The amplitude decreases with time and finally becomes zero.
So far, so good.
Now, in certain situations, apart from restoring force, damping forces, there may be yet another force on the body which itself changes periodically with time.
As a simple case, let this externally applied force be
F = F0 sin wt
Look another new frequency ( angular ) w is there.
the equn of motion then becomes
F = - kx - bv + F0 sin wt
=> m ( dv / dt ) = - kx - bv + F0 sin wt
the soln comes out as
x = A sin ( wt + # )
This type of oscillation is called forced oscillation.
A = amplitude = (F0/m ) /[ squre root { (w2 - w02 ) + ( bw /m )2 }
We once again remember that
w0 = square root (k / m ) = natural frequency
w = frequency of force applied.
Now when frequency of force applied force w becomes = w1 = square root [ w02 - ( b /2m )2 ]
the amplitude is infinite.
For small damping, we again remember
w1 = w0
i,e, when frequency of force applied force w becomes almost, I repeat, almost equal to natural frequency of the system, the amplitude becomes very large.
The bottom line is that in resonance the amplitude becomes very large.
Comments (1)
Preparing for IIT-JEE ?
Arihant Revision Package for IIT JEE - Books, Practice Tests + Rank Predictor
@ INR 1,995/-
For Quick Info
|
|
|
|
|
| 35,229,755 Engineering Questions Attempted |
6,69,530 Visitors Monthly |
79 Engg. Exam Covered |
2174 Students Currently Online |
Connect with Us  |
 |
