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Bernhard Bettig
Mechanical Design Research LabMechanical Engineering - Engineering Mechanics Dept.
Michigan Technological University
Web site: http://www.me.mtu.edu/~mdrl
Rotordynamics: Unit 2
- Timoshenko Beam Model
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2-Node Beam ModelsThe beam models we will be lookingat have 8 degrees of freedom (2translations and 2 rotations at eachnode):
(u1, v1, 1, 1 )
( u2, v2, 2, 2 )
L, I, J, , E, G
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Important Variables
The displacements and rotations at Node 1:
u1, v1, 1, 1.
The displacements and rotations at Node 2:
u2, v2, 2, 2.
The beam geometric properties L, I, and J.
The beam material properties , Eand G
The beam rotation:
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Important Assumptions
No displacements in the Zdirection.
Linearization: u1, v1, 1, 1, u2, v2, 2, and
2 are small.
Gyroscopics about Z axis only:
and are small compared to .
Cross-section planes remain plane.
The displacements are represented by aHermitian shape function
,,, 211 &&&
2&
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Hermitian Shape Function
The deflected shape is represented by a cubicpolynomial. I.e.:
u(Z) = a0 + a1Z+ a2Z2 + a3Z
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v(Z) = b0 + b1Z+ b2Z2 + b3Z
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As well, the coefficients are determined from thedisplacements and slopes at the ends:
dZ
duu =
2
3212
33
22102
11
01
32)(
)(
)0(
)0(
LaLaaLu
LaLaLaauLu
au
auu
++==+++==
==
==
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Hermitian Shape FunctionSolving for as
in terms of u1,
1, u2, and 2:
( )
( )
( )
( )22334
23
33
3223
32
323
31
1)(
321
)(
21
)(
321
)(
LzLzL
zN
LzzL
zN
zLLzLzL
zN
LLzzL
zN
=
+=
+=
+=
42322111
42322111
)(
)(
NNvNNvzv
NNuNNuzu
+++=
+++=
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Hermitian Shape Function
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Euler-Bernoulli Beam ModelWe start the lecture with a simpler beammodel, the Euler-Bernoulli beam. This
model considers only:
- stresses and strains due to bending
- inertia due to lateral velocity
Z
X
MM
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Euler-Bernoulli Beam Model
Stiffness matrix (kE-B
) for deflections
in X and Y directions:
2
2
2
2
1
1
1
1
2
2
22
22
3
4
612
004
00612
26004
61200612
0026004
0061200612
v
u
v
u
L
LSYM
L
L
LLL
LL
LLL
LL
L
EI
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Timoshenko Beam ModelThis model also considers shear
deformation and rotational inertia.
This model is important for stubby
shafts and higher frequencies.
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Timoshenko Beam Model
Mass matrix (mri) for rotational inertia:
2
2
2
2
1
1
1
1
2
2
22
22
4
336
004
00336
3004
33600336
003004
0033600336
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&&
&&
&&
&&
&&
&&
&&
&&
v
u
v
u
L
LSYM
L
L
LLL
LL
LLL
LL
L
I
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Timoshenko Beam ModelNew stiffness matrix (k
T):
( )
+
+
+
+
+
2
2
2
2
1
1
1
1
2
2
22
22
3
4
612
004
00612
26004
61200612
0026004
0061200612
1
v
u
v
u
aL
LSYM
aL
L
aLLaL
LL
aLLaL
LL
La
EI
2
12
GAL
EIa =
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Timoshenko Beam Model
Timoshenko Beam Equation:
( ) Fukumm =++ TriBE &&
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Accounting for CoriolisCoriolis affects appear as cross-
coupling coefficients in the FE
damping matrix (the same as with the
spinning mass).
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Accounting for Coriolis
Damping matrix (g):
2
2
2
2
1
1
1
1
2
2
22
0
00
430
33600
0030
0033600
300430
3360033600
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&
&
&
&
&
&
&
&
v
u
v
u
SYM
ANTI
LL
L
LL
L
LLLL
LL
L
I
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Accounting for Coriolis
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Accounting for Torque
Torque in the shaft also has a cross-
coupling affect, but in the stiffnessmatrix. The affect is somewhat like
wringing a wet towel.
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Accounting for TorqueAdditional stiffness matrix (k
To):
2
2
2
2
1
1
1
1
005.01005.01
00100010
5.01005.0100
10001000
005.01005.01
00100010
5.01005.0100
10001000
v
u
v
u
LL
LL
LL
LL
L
T
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Accounting for Torque
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Accounting for Axial ForcesAn axial force in the shaft has the
affect of straightening it.
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Accounting for Axial Forces
Additional stiffness matrix (kax
):
2
2
2
2
1
1
1
1
2
2
22
22
4
336
004
00336
3004
33600336
003004
0033600336
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v
u
v
u
L
LSYM
L
L
LLL
LL
LLL
LL
L
Fax
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Accounting for Axial Forces
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Complete Rotordynamic BeamModel
Dynamic Equation for complete beam
model:
( ) ( ) Fukkkugumm =+++++ axToTriBE &&&
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Sample Questions1. What end moments are required to statically bend a
solid steel shaft (3 m length x 300mm dia.) to 0.1 rad
at each end? (Bent into a bow shape.) Use the Euler-Bernoulli beam model.
2. What is the deflection in the middle?
3. Repeat question 1 using the Timoshenko model.
4. What if the shaft is also being bent (bowed) in thesame plane at an angular rate of acceleration of 0.002
rad/sec2 at each end? Use the Euler-Bernoulli model.
5. Repeat question 2 using the Timoshenko model.
6. What if the shaft is also spinning about the +Zdirection at 2 Hz and is bowing in the same plane at0.02 rad/sec at each end?
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Sample Questions
7. What if the shaft is carrying a torque of 1000 N-m(about +Z at the upper end, -Z at lower end)?
8. What if the shaft is carrying an axial force of 1000 N?
9. In the equation:
which matrices couple deflections in the X and Y
directions?
10. In the final equation , which
matrices will end up being symmetric? Whichmatrices will be anti-symmetric? Which will beneither? What if the affects of torque were not
included?
FkuucuM =++ &&&
( ) ( ) Fukkkugumm =+++++ axToTriBE &&&
