# Forced Oscillations

1. Nov 19, 2005

### Aki

I'm doing this problem from Mastering Physics, and I'm really stuck on this problem

Assume that, when we walk, in addition to a fluctuating vertical force, we exert a periodic lateral force of amplitude 25 N at a frequency of about 1 Hz. Given that the mass of the bridge is about 2000 kg per linear meter, how many people were walking along the 144 m-long central span of the bridge at one time, when an oscillation amplitude of 75 mm was observed in that section of the bridge? Take the damping constant to be such that the amplitude of the undriven oscillations would decay to 1/e of its original value in a time t=6T, where T is the period of the undriven, undamped system.

2. Nov 19, 2005

### Physics Monkey

You can calculate the response of a driven damped oscillator in terms of the amplitude of the driving force, right? Since you know the amplitude of the response you want, try to work backwards and find the amplitude of the driving force you need.

3. Nov 15, 2007

### HeLLz aNgeL

did anyone get this problem ?? i've been trying to solve this and im really confused :S

4. Nov 15, 2007

### fubag

this question took me forever

but some things I got out of it

that in order to solve for the number of people you need to find Fmax, and then use the relationship Fmax/ Favg = number of people. In this case u know Favg = 25N..

To find Fmax you need to know that it is Ares(amplitude resonance)*b*w_d...

to find d you need to use the hints, lol its too complicated to explain...

otherwise took me forever but worked

5. Nov 16, 2007

### HeLLz aNgeL

i cant figure out how to find b ??? the damping constant

6. Nov 16, 2007

### fubag

the damping constant is equal to (m*omega)/ (6pi) if u use the information they tell you about decaying in t = 6T.

7. Nov 17, 2007

### HeLLz aNgeL

thanks !! finally figured it out...

8. Dec 15, 2007

### defne

can someone tell me how to calculate the force constant(do we need it?) and the Fmax? do the frequency of pedestrians have a close value to the natural frequency of bridge when resonance occurs?so can we omit the (w_0)^2 - (w_d)^2 in the denominator of the formula? and finally, what is the max value of the driving force when resonance occurs?please heelllllppp

9. Dec 15, 2007

### Staff: Mentor

Pedestrians on a foot bridge do not normally walk with the natural frequency of a well designed bridge. Pedestrians do not normally walk in phase as a group.

10. Dec 16, 2007

### defne

wobbling bridge

given information:
Consider an oscillating system of mass m and natural angular frequency omega_n. When the system is subjected to a periodic external (driving) force, whose maximum value is F_max and angular frequency is omega_d, the amplitude of the driven oscillations is

A= F_max / (sqrt((k-mw_d^2)^2+(bw_d)^2))
or
A= (F_max/m) / (sqrt((w_n^2-w_d^2)^2+(b(w_d)/m)^2))
where k is the force constant of the system and b is the damping constant.

question:
(Assume that, when we walk, in addition to a fluctuating vertical force, we exert a periodic lateral force of amplitude 25 N at a frequency of about 1 Hz. Given that the mass of the bridge is about 2000 kg per linear meter, how many people were walking along the 144 m-long central span of the bridge at one time, when an oscillation amplitude of 75 mm was observed in that section of the bridge? Take the damping constant to be such that the amplitude of the undriven oscillations would decay to 1/e of its original value in a time t=6T, where T is the period of the undriven, undamped system.)

Last edited: Dec 16, 2007
11. Dec 17, 2007

### defne

i have solved the problem, thanks...