Answer:
The given Problem describes the problem of balancing in a single plane, these are the following data which is tabulated for ease of calculation given below
As given in problem 

Condition 
Amplitude 
Phase 

Vibration displacement 
Angle 

mm 
Degree 
Original Unbalance 
0.165 
15^{o} CW 
Trial Weight 
0.225 
35^{o} 
Weight = 50 at 45^{o} CCW 


The calculation of such kind of problem starts with defining vector form of the data as given in question, we have assumed that, original unbalance vector =
_{ }From, data, this can be defining as, the rectangular form of this vector can be given as
In the same way trial weight can be given as
The trial weight can be represented as
The further equation for measured vector can be given as follows
The resultant magnitude can be obtained from subtracting (iv) from (v)
Since amplitude of resultant vector is calculated, on this basis we can calculated weight of the balanced mass
The placement balance mass will be just opposite of the result calculated,
= 6.148 +2.08123j
The balance vector in polar coordinate CCW.
As per above calculation, we must place 6.491 gm of weight at 18.672^{o} CCW. Ans
To solve the condition further we must decide a reference plane, and in this condition, the reference plane is G, we must calculate all the distance from reference plane G, which is as follows.
We can represent the weight vector as follows
The above radius vector is positioned at different radius on axis, for solving the problem, it is necessary that, we must, take one radius as reference radius. Which is given as follows. The reference radius is taken as R for the given condition which is equal to 50 mm
The converted weight vector , with reference to the standard radius.
, with reference to the standard radius R = 50 mm.
, with reference to the standard radius R = 50 mm.
The further calculation is possible only after converting the rectangular form of weight vector with reference to the standard radius R = 50 mm.
The rectangular form of weight is 0+0.772j with reference to the standard radius R = 50 mm.
The rectangular form of weight is 1.591.34j with reference to the standard radius R = 50 mm.
The rectangular form of weight is 0.20.12j with reference to the standard radius R = 50 mm.
The sum of unbalance vector =
If we add the above rectangular vector, we will get sum of unbalanced rectangular vector.
= 1.390.69j or 1.552 <206.39^{o} Ans
Similarly, for plane G the unbalance vector is given as.
The rectangular form of weight with reference to the standard radius R = 50 mm, the new weight vector , or (0, 0.23j)
The rectangular form of weight with reference to the standard radius R = 50 mm, the new weight vector , or (1.862, 1.561j).
The rectangular form of weight with reference to the standard radius R = 50 mm, the new weight vector , or (.67, 0.39j).
The sum of unbalance vector =
If we add the above rectangular vector, we will get sum of unbalanced rectangular vector.
= 1.191.72j or 2.09 <235.32^{o} with reference to the standard radius R = 50 mm
The calculated weight is A = 1.552 kg, and at G = 2.09 kg, with reference to the standard radius R = 50 mm
The distance of different cylinder is given as follow. The cylinder 1 is taken as reference plane.
The distance of cylinder 2 = d2 = 2a
The distance of cylinder 2 = d3 = 3a
The distance of cylinder 2 = d4 = 4a
The distance of cylinder 2 = d5 = 5a
In the same way, the primary and secondary forces along y direction is given
Similarly, for by putting the value of α and t = 0
Now equation
Simplifying with trigonometrical theories (b)
Same calculation for y and z axis
We must calculate the moment in z axis
References
Callister, W., & Rethwisch, D. (2010). Materials Science and Engineering : An Introduction (8th ed.). New York: John Wiley & Sons.
Haym Benaroya, M. L. (2017). Mechanical Vibration: Analysis, Uncertainties, and Control (1st ed.). Boca Raton: CRC.
Inman, D. J. (2014). Engineering Vibration. New Jersey: Pearson Education.
Palm, W. J. (2010). Mechanical vibration (2nd ed.). New York: Wiley.
Rao, S. (2013). Mechanical Vibration (6th ed.). New Jersey: pearson.
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