ENMEC4060 Vibration and Machine Dynamics Assignment

The arrangement of cranks in a six-cylinder in-line engine is shown in the figure. The cylinders are separated by a distance in the
axial direction, and the angular positions of the cranks are given. If the crank length, connecting-rod length, and the reciprocating mass of each cylinder, respectively, find the primary and secondary unbalanced forces and moments with respect to the reference plane

Answer:

As given in question,

It is a case of measurement of single plane balancing, with following data

Unbalance amplitude = 0.165 mm at phase angle degree = 15^o 

Trial weight = 50 gm at phase angle degree 45^o CCW and amplitude 0.225 mm.

First, we will express the above condition in the form of vector, and assumption was taken as counter clockwise will be positive direction

Suppose the vector of original unbalance =

">  = = 0.165*Cos(-15^o) +0.165*Sin(-15^o) = 0.16 - 0.0427j

Measured vector of trial weight,

 = 0.18431+0.130j

Now trial weight vector

 = 35.36 + 35.36j

Now we know that, measured vector can be expressed,

  …………. (a)

  …………. (b)

By subtracting equation (a) from (b), we will get

Since vector  is obtained, now we can calculate , with the following relation

Putting the value in above equation, 

= -6.15-2.0812j

This is the required unbalance vector is calculated, to make it stable we have to place the mass in opposite side,

i.e.

or  

The balance vector in polar coordinate

CCW.

As per above calculation, we must place 6.49 gm of weight at 18.67^o CCW. 

As given in question, the weight 1 kg, 3 kg and 2 kg, are in the radius of 50, 75 and 25 mm, in the planes C, D, and E, the support bearing is at the place B, and F

In this condition we must determine plane A and G. Taking G as the reference plane, the axial position of each weight and unbalance weight 

The weight vector for each position C, D, E can be given as  

As can see that, the weight vector is positioned at different radius position, we must convert reference radius position.

Then new weight vector , at reference position of 50 mm.

Then new weight vector , at reference position of 50 mm.

Then new weight vector , at reference position of 50 mm.

Now I must convert this unbalance weight into rectangular form of vector

The weight vector , then rectangular form of vector will be 0+ 0.77j at radius position of 50 mm.

The weight vector , then rectangular form of vector will be -1.59 - 1.34j at radius position of 50 mm.

The weight vector , then rectangular form of vector will be 0.2 -0.12j at radius position of 50 mm.

The total unbalance weight in rectangular form = -1.39-0.69j.

The required weight at position A in rectangular form = -1.39-0.69j or 1.552<206.39^o  Ans

Similarly, we must calculate unbalance weight in plane G

  Then new weight vector , or (0, 0.23j) at reference position of 50 mm.

Then new weight vector , or (-1.86, -1.56j) at reference position of 50 mm.

Then new weight vector , or (0.67, -.39j) at reference position of 50 mm.

Total unbalance vector =

-1.19-1.72j or 2.09<23532 at radius 50 mm

Therefore, required weight at A = 1.552 kg and at G = 2.09 kg at a radius of 50 mm

The required weight at position A in rectangular form = -1.39-0.69j or 1.552<206.39^o  Ans

Similarly, we must calculate unbalance weight in plane G

  Then new weight vector , or (0, 0.23j) at reference position of 50 mm.

Then new weight vector , or (-1.86, -1.56j) at reference position of 50 mm.

Then new weight vector , or (0.67, -.39j) at reference position of 50 mm.

Total unbalance vector =

-1.19-1.72j or 2.09<23532 at radius 50 mm

Therefore, required weight at A = 1.552 kg and at G = 2.09 kg at a radius of 50 mm

The other format for equation is  

Where m, c, k is the mass, damping and stiffness matrix respectively.  

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