I am simulating an impulse signal of a resonant bearing fault with this formula: $$ y(k)=\sum_r A_r \sin\left(\displaystyle \frac{2\pi f \left(k - rF/f_m\right)}{F}\right)\cdot e^{\displaystyle-\beta(k−rF/f_m)/F} $$
where:
- Amplitude signal $A_r =1.5$
- Resonant frequency $f =2000$
- Sampling frequency $F=10\cdot 10\cdot 3$
- Characteristic fault frequency $f_m =50$
- Decay factor $\beta =500$
What I can't figure out is the interpretation of $r$ in this equation and why is the sum of this term. If someone understands it please help and also what value of this $r$ should I consider.
The signal should look like:
EDIT:
I am adding the Python code for the function, it goes right for value of r=0
also I'm not adding any r
. I cant understand the interpretation of this.
#Parameters:
Ar= 1.5 #Amplitude
f = 2000 #Resonant Frequency in Hz
b=500 #Exponential decay factor
fm=50 #Fault characteristic frequency
F=10*10**3 #Freq. Sampling
Tot_samples=20000
r=0
def y(k):
t=(k-r*F/fm)/F
exp=np.exp(-b*t)
sin=np.sin(2*pi*f*t)
return (Ar*sin*exp)
#Create the summatory of r
_y=[]
y_signal=[]
for j in np.linspace(0,200,200,endpoint=False): #only first 200 points of
the impulse
_y.append(y(j))
plt.plot(_y)
plt.show()