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from IPython.display import HTML
HTML(open("00_custom.css", "r").read())









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01 Vibration Isolation

A rotating machine is characterized by

its mass, $m=108000\,{}$kg;
its working frequency, $f_\text{w}=30\,{}$Hz,
the value of the unbalanced load it exerts on its supports, $f_\text{w}={4200}\,{}$N.



In [2]:

    
m = 108000.0
fw = 30.0
pw = 4200.0

Design a suspension system for the machine knowing that it is necessary to reduce the transmitted force to ${1000}\,{}$N and that, to reduce the vibration amplitude during transients, the suspension must have a viscous damping ratio of $7\%$.



In [3]:

    
p = 1000.0
TR = p/pw
print "The required transmissibility ratio is TR = %5.2f%%."%(100*TR)
z = 0.07









    



The required transmissibility ratio is TR = 23.81%.

Solution

We import a few things we need, we define a function to compute the transmissibility ratio as a function of $\beta$ and $\zeta$.



In [4]:

    
from math import pi, sqrt
from scipy.optimize import bisect
def tr(b, z):
    return sqrt((1+(2*z*b)**2)/((1-b*b)**2+(2*z*b)**2))

To solve our problem, we need to pass to bisect a function that

is equal to zero when tr(b,z) equals TR,
is a function of b only.

We fulfill our requirements using the lambda syntax to define on the fly an appropriate anonymous function:



In [5]:

    
b = bisect(lambda b: tr(b,z)-TR, 1.4, 10.0)
print "The frequency ratio fw/fn is %6.4f"%b









    



The frequency ratio fw/fn is 2.3248

We need $\omega_n$ (and we compute also $\omega_D$ while we are at it) so we preliminarily need the circular frequency of the machine, $\omega_w=2\pi f_w$.



In [6]:

    
ww = 2*pi*fw
wn = ww/b
wd = wn*sqrt(1.0-z**2)
print """\
The circular frequency of the machine            %7.3f rad/s,
The natural frequency of the suspension system   %7.3f rad/s,
The damped frequency of the suspension system    %7.3f rad/s."""%(
ww, wn, wd)









    



The circular frequency of the machine            188.496 rad/s,
The natural frequency of the suspension system    81.079 rad/s,
The damped frequency of the suspension system     80.880 rad/s.

To complete our solution, we have that $$k = \omega_n^2m$$ and $$c = 2\zeta\omega_nm$$



In [7]:

    
k  = wn**2*m
c  = 2*z*wn*m
print """\
The stiffness of the suspension system is        %7.3f kN / mm,
The damping of the suspension system is          %7.3f kN s/mm."""%(
k/1E6, c/1E6)









    



The stiffness of the suspension system is        709.973 kN / mm,
The damping of the suspension system is            1.226 kN s/mm.