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Bernhard Bettig

Mechanical Design Research Lab

Mechanical Engineering - Engineering Mechanics Dept.

Michigan Technological University

Web site: http://www.me.mtu.edu/~mdrl

Rotordynamics: Introduction

& Unit 1- Rotating Masses

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Units of this short course1. Rotating masses Coriolis forces and imbalances (1)

2. Timoshenko beam model (1)

3. Modeling coupling misalignments (1)

4. Understanding journal bearing fluid dynamic forcesand coefficients and bearing misalignment (2)

5. Non-linear static calculations (0.5)

6. Steady state response calculations (0.5)

7. Time integration (0.5)

8. Natural frequency and mode shape calculations (1)

9. Finite difference calculation of bearing forces (1)

10. Modeling Seven Sisters G.S. Unit 1 includingestimating imbalances and rotating misalignments (1)

11. Applying concepts to a unit at New York PowerAuthority, Robert Moses Station (1)

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Rotating Masses

In this first unit we will study the

forces and accelerations of a rotatingmass as pictured:

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Important Variables

The coordinates X, Yand Z. These

identify three orthogonal directions.

The displacements uand v.

The rotations , and .

The rotational velocity about Zis .

or .

The mass properties m, Ixand Iz.

td

d= =

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Important Assumptions

The net forces Fand moments M

and accelerations aand .

Assumptions

No displacements in the Zdirection.

and are small compared to .

u, v, and are small.

=

y

x

F

FF

=

y

x

M

MM

=

v

u

a

=

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Lateral Forces and Reactions

First let us consider the lateral forces

and reactions. These are independent

of the rotations. According to

Newtons second law:

or

where Fis the sum of all forces on m.

aF m=

=

v

u

m

m

F

F

y

x

0

0

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Lateral Forces and Reactions

If gravity is acting on the mass in the-Ydirection we need to add one moreterm:

We then need to remember NOT toinclude gravity in Fy.

=

+

v

u

m

m

mgF

F

y

x

0

00

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Rotational Forces ( = 0)

If we assume that = 0, then thesame kinds of equations apply to the

moments and rotational accelerations:

or

where Mis the sum of all moments onthe mass. Note that Iy= Ix.

IM=

=

y

x

y

x

I

I

M

M

0

0

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Rotational Forces ( 0)

If the rotational speed about Zis not

zero ( 0), then we must considerCoriolis affects. These are dependent

on , and :

+

=

0

0

0

0

z

z

y

x

y

x

I

I

I

I

M

M

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Imbalance ForcesImbalance forces (Fu) are proportional to

the mass, eccentricity (e) and squared.

If we are using a non-rotating (inertial)

reference frame, then:

)sin(0

)cos(0

2

2

,

,

tme

tme

F

F

yu

xu

+

=

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Imbalance Forces

If the unbalanced mass, does not happen

to be sitting on the +Xaxis at time t = 0,

then we need to consider the phase angle

of the imbalance ().

The phase angle is the position of the

mass as measured from the +Xaxis

towards the +Yaxis at time t= 0.

)sin()cos()sin()cos(

)sin()cos(

2

2

2

2

,

, tmemet

meme

FF

yu

xu

+

=

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Sample Questions What are the reaction forces if a hollow, cylindrical

solid steel mass (24 in. height x 100 in. outside dia., 90in. inside dia.) is accelerated laterally in the Ydirectionat 2 in./sec2?

What if the mass also has an angular acceleration of0.002 rad/sec2 about the Yaxis?

What if the mass is also spinning about the Zdirectionat 2 Hz and is rotating about the Ydirection at 0.02rad/sec?

What if there is an imbalance of 50 lb-in acting at aphase angle of 20 degrees and we are considering theforce at time t= 20 seconds.

Can we neglect the affect of gravity in the Ydirectionfor these examples?