Single Degree Rotating Unbalance

One may find many rotating systems in industrial applications. The unbalanced force in such a system can be represented by a mass m with eccentricity e, which is rotating with angular velocity as shown in Figure.

 
Vibrating system with rotating unbalance

 
      Free body diagram of the system

Let x be the displacement of the nonrotating mass (M-m) from the static equilibrium position, then the displacement of the rotating mass m is  

From the free body diagram of the system shown in figure, the equation of motion is
                                              
                                               

This equation is same as equation (1) where F is replaced by . So from the force polygon as shown in figure
                                               
      

                                               
                      Force polygon

                                                                 

                                                                       

So the complete solution becomes plot for system with rotating unbalance

Phase angle ~ frequency ratio plot for system with rotating unbalance

Following observations may be made for a rotating unbalanced system.

• For very low value of frequency ratio (say ), the response of the system is very small.

• For frequency ratio between 0.5 and 1, there is a sharp increase in system response with increase in frequency of excitation of the system.

• At frequency ratio equal to 1, the phase angle is 90º.

• Maximum response amplitude occurs at a frequency slightly greater than .

• With increase in damping, the response of the system decreases.

• For higher value of  (say >2), the response amplitude approaches and phase angle approaches 180º

Whirling of shaft :

Whirling is defined as the rotation of the plane made by the bent shaft and the line of the centre of the bearing. It occurs due to a number of factors, some of which may include

(i) eccentricity,

(ii) Unbalanced mass,

(iii) Gyroscopic forces,

(iv) Fluid friction in bearing, viscous damping.
 


The acceleration of point G can be given by

Assuming a viscous damping acting at S. The equation of motion in radial direction

                                                        
Considering the synchronous whirl case, i.e. 

                                                                  
Where ø is the phase angle between e and r .

Taking ,

 
Hence,

 
as and

Thus we obtain,

 
Substituting value of cos yields

 
Or,

Services: - Single Degree Rotating Unbalance Homework | Single Degree Rotating Unbalance Homework Help | Single Degree Rotating Unbalance Homework Help Services | Live Single Degree Rotating Unbalance Homework Help | Single Degree Rotating Unbalance Homework Tutors | Online Single Degree Rotating Unbalance Homework Help | Single Degree Rotating Unbalance Tutors | Online Single Degree Rotating Unbalance Tutors | Single Degree Rotating Unbalance Homework Services | Single Degree Rotating Unbalance