Vibration Analysis.ppt

8/14/2019 Vibration Analysis.ppt

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PREDICTION OF MAXIMUM UNBALANCE

RESPONSES OF A GEAR COUPLED TWO-SHAFT

ROTOR-BEARING SYSTEM

An Sung Leea, Jin Woong HabRotor Dynamics Group, Korea Institute of Machinery and Materials, Yoosung, P.O.

Box 101, Daejeon 305-600,Republic of Korea

R&D Department, Century Corporation, Asansi, Chungnam 336-842, Republic ofKorea

A TECHNICAL PAPER REVIEW

HARSHAL AVINASH MUNGIKARGraduate Student

Mechanical Engineering DepartmentUniversity of Houston, Texas

(PS. ID.: 0930631)


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DERIVATION OF EQUATIONS OF MOTION

1. The gear pair has ten degrees of freedom.

2. A displacement vector of a gear pair from a coordinate system

lying on the pressure line of the two gears is:

u1,2, v1,2, x,yare the translational and rotary degrees of freedom.

z is the torsional degree of freedom


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The equations of motion for the gear pair can be written as follows:

1. M1*(d2u1/dt2) = Q1

2. M1 *(d2v1/dt

2) + cm *(dv1/dt)cm*(dv2/dt)+ cmr1 *(d z1/dt) + cmr2 *(d z2/dt)

+ kmv1kmv2+ kmr1z1+ kmr2z2= Q2

3. It1 *(d2x1/dt

2) + Ip11 *(d y1/dt) = Q3

4. It1*(d2y1/dt2)Ip11*(d x1/dt) = Q4

5. M2*(d2u2/dt2) = Q5

6. M2*(d2v2/dt

2) - cm *(dv1/dt) + cm*(dv2/dt)- cmr1 *(d z1/dt) - cmr2 *(d z2/dt)

- kmv1+ kmv2- kmr1z1- kmr2z2= Q6

7. It2*(d2x2/dt

2) + Ip22*(d y2/dt) = Q7

8. It2*(d2y2/dt

2) Ip22*(d x2/dt) = Q8


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The equations of motion for the gear pair can be written as follows:

9. Ip1*(d2z1/dt

2) + cmr1*(dv1/dt)cmr1*(dv2/dt) + cmr12*(d z1/dt) +

cmr1r2*(d z2/dt) + kmr1v1kmr1v2+ kmr12z1+ kmr1r2z2= Q9

10. Ip2*(d2z2/dt

2) + cmr2*(dv1/dt) cmr2*(dv2/dt) + cmr22*(d z2/dt) +

cmr1r2*(d z1/dt) + kmr2v1kmr2v2 + kmr22z2 + kmr1r2z1= Q10


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For the whole finite element model, all the finite elements

matrices need to be assembled together.

This involves, assembling the coupled vibration FE model

of the gear pair and the general vibration FE models of the

other elements like shafts and bearings.

This is obtained by placing the pure lateral and torsional

vibration matrices diagonally and the coupled ones off

diagonally.

The final equations of motion can be represented as:


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UNBALANCE RESPONSE FORMULATION


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In the rotor case the unbalance exciting force can be

represented as:

Unbalance response solution can be expressed as:

Substituting these into the generalized equation of motion,

we get:

[C](-{a}w1sinw1t + {b}w1cosw1t{c}w2sinw2t +{d}w2cosw2t) + [M](-{a}w12cosw1t{b}w1

2sinw1t

{c}w22cosw2t{d}w2

2sinw2t) + [K]({a}cosw1t + {b}sinw1t

+ {c}cosw2t + {d}sinw2t) = {Uc}1w12cosw1t +

{Us}1w12sinw1t + {Uc}2w2

2cosw2t + {Us}2w22sinw2t


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Now, equating the coefficients of cosw1,2t and sinw1,2t on LHS and

RHS and expressing them in matrix form, we get:

Solving the above matrices, we get {a}, {b}, {c}, {d} (modal

vectors of the unbalance response)

Now, the horizontal and vertical responses q1 and q2 of the

unbalanced node can be expressed as:


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ANALYSIS OF RESULTS


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CONCLUSIONS BY THE AUTHOR

1. Analytical solutions to the maximum and minimum radii of the

unbalance orbit using finite element modeling had beenproposed.

2. When applied to 600kW turbo chiller rotor bearing system, the

lateral coupling between the driver and driven systems waslimited.

3. Due to the torsional-lateral coupling between the two systems,

bumps in the unbalance lateral responses for driver system were

observed at the first torsional natural frequency.

4. The results were validated with those obtained using a full

numerical approach.

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Vibration Analysis.ppt