Truss Structures - Natural
Frequency Manipulation via SDP
Brian W. Anthony
Robert M. Freund
1
7
Background : Mechanical Systems
8
Simple Mechanical System
k1
m1
u1
9
10
Natural Frequencies – How a Mechanical
System wants to vibrate when forced.
1 DOF System
3
2.5
Amplitude
2
1.5
1
0.5
(k/m).5
0
0
2
4
6
8
10
12
14
Frequency (Hz or Cycles/Sec)
16
18
20
Examples
• Ball on a string
• Beam(s)
• For a simple mechanical systems it is
relatively easy to systematically effect a
change in the natural frequency.
11
Beam Vibration
Narrow Beam
How would we change the frequency of vibration?
Beam Vibration
Narrow Beam
Narrow Beam
Narrow and Short Beam
Wide Beam
-
Natural Frequency = 5.6 Hz
Natural Frequency = 11.7 Hz
Natural Frequency = 7.8 Hz
12
Another Mechanical System
k1
k2
m1
k3
m3
m2
F(t)
u1
u2
u3
Write Newton’s Law for Each Mass
F = ma
Another Mechanical System
k1
k2
m1
k3
m3
m2
F(t)
u1
u2
u3
The Dynamics Model (The Equations of motion).
m1(d2u1 /dt2) + k1u1+ k2(u1-u2)
=0
2
2
k2(u2-u1) + k3(u2-u3) = 0
m2(d u2 /dt ) +
2
2
m3(d u3 /dt ) +
k3(u3-u2) = F(t)
In Matrix Form
M(d2U /dt2) + KU = F(t)
13
Equations of Motion
And Eigenvalue Analysis
The Equations of motion
M(d2U /dt 2) + KU = F(t)
M
d 2U
+ KU = F (t )
dt 2
• The Eigenvalues of M-1K are the natural
frequencies of vibration (squared).
• The Eigenvectors of M-1K are the mode shapes
(the relative displacement of each degree of
freedom)
Natural Frequencies – The Rate a
Mechanical System wants to vibrate
3 DOF System
3
2.5
Amplitude
2
1.5
1
0.5
0
1
2
3
4
5
6
7
Frequency (Hz or Cycles/Sec)
8
9
10
14
Natural Modes – (The Eigen-Modes)
Are the shape of the Vibration
3 DOF System
3
2.5
Mode 1
m2
m3
Amplitude
2
m1
1.5
1
Mode 2
0.5
0
m1
m2
m3
m2
m3
1
2
3
4
5
6
7
Frequency (Hz or Cycles/Sec)
8
9
10
Mode 3
m1
Movies…
Modification?
3 DOF System
3
2.5
Mode 1
m2
m3
Amplitude
2
m1
1.5
1
Mode 2
0.5
0
m1
m2
m3
Mode 3
m1
m2
m3
1
2
3
4
5
6
7
Frequency (Hz or Cycles/Sec)
8
9
10
It is no longer obvious what we
have to do in order to change
the lowest natural frequency.
15
Truss Structure
Truss Structures
• Rigid beams
– Axial forces only
• Pin-connected
– Concentric joints
– Welded or bolted
• Bridges, towers, exoskeletons
16
A truss has large number of natural
frequencies and complex motion.
Truss : Model as Lumped Masses
with Connecting Springs
Link Stiffness is a function
of Length, Area, density
Half of the mass of Each
Link is assumed to sit at a
node. (A function of Areas)
17
Truss : Modeled as Lumped Masses
with Connecting Springs
8
4
9
6
7
1
2
3
5
8
4
9
6
7
y
1
2
3
5
x
Step 6. Determine System of Equations by applying Newton’s
Law at each node (on each ball of mass)
M(d2U /dt 2) + KU = F(t)
18
22
20
18
16
14
12
10
8
6
4
2
0
-1
-0.5
0
0.5
1
1.5
2
2.5
3
19
Each Node (Mass) has, in general, 3 Degrees of Freedom
20
Equations of Motion
And Eigenvalue Analysis
The Equations of motion
M(d2U /dt 2) + KU = F(t)
• The Eigenvalues of M-1K are the natural
frequencies of vibration (squared).
• The Eigenvectors of M-1K are the mode shapes
(the relative displacement of each degree of
freedom)
21
22
23
24
25
A proposed design of a Cell
Phone Tower
Planar Approximation:
All beams have an
Area of 1 square centimeter
50 Bars
40 Degrees of Freedom
22
20
18
16
14
12
10
8
6
4
2
0
-1
-0.5
0
0.5
1
1.5
2
2.5
3
26
A proposed design of a Cell
Phone Tower
Natural Frequency = 159 Hz
( smallest eigenvalue of M-1K )
22
20
18
16
14
12
10
8
6
4
2
0
-1
-0.5
0
0.5
1
1.5
2
2.5
3
Natural Frequencies – The Rate a
Mechanical System wants to vibrate
N DOF System
Amplitude
10
10
10
0
-5
-10
0
0.5
1
1.5
Frequency (Hz or Cycles/Sec)
2
2.5
x 10
4
27
Natural Modes (Eigenvectors or
Eigen-modes) - The Shape of the Vibration
N DOF System
0
Amplitude
10
-5
10
-10
10
0
159.7156 Hz
0.5
1
1.5
Frequency (Hz or Cycles/Sec)
920.3246 Hz
2
Movies…
2.5
x 10
4
1618.3 Hz
2333.4 Hz
Cell Phone Tower
• The initial design of the cell tower has a
natural frequency of 159 Hz.
• We expect ground vibration induced by a
nearby railroad near 100 Hz and 159 Hz.
22
20
18
16
14
12
10
8
6
4
Movies…
2
0
-1
-0.5
0
0.5
1
1.5
2
2.5
3
28
Optimized Tower
Natural Frequency = 250 Hz
towershrt250
22
20
New Areas:
18
16
1.0353
1.5332
2.0785
14
12
10
8
6
2.6420
3.2065
4
2
0
3.7667
-1
-0.5
0
0.5
1
1.5
2
2.5
3
Cell Phone Tower
• The optimized design of the cell tower has a
natural frequency of 250 Hz.
• This vibration of 100 Hz and 159 Hz will
now have minimal effect.
towershrt250
22
20
18
16
14
12
10
8
6
4
2
Movies…
0
-1
-0.5
0
0.5
1
1.5
2
2.5
3
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Optimized Tower
Natural Frequency = 500 Hz
towershrt500
22
New Areas:
2.0561
4.0845
20
18
16
14
12
7.3261
11.8377
10
8
6
17.2199
22.8216
4
2
0
28.2383
-1
-0.5
0
0.5
1
1.5
2
2.5
3
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Continuous Mechanical System – Truss
Approximation
Step 0. Complex (Continuous) Mechanical Structure
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Step 1. Apply a Mesh (Truss) to the Mechanical Structure
Step 2. Form Local Regions around ‘Nodes’ of the Mesh
32
All of this mass is applied to the contained node.
Step 3. Use Local Regions around ‘Nodes’ to approximate
Node Mass (mi) and Link Stiffness (kij)
mj
kij
mi
Step 4. The Finite Element (Truss) Approximation
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8
4
9
6
7
1
Step 5. Label the Nodes
2
3
5
8
4
9
6
7
y
1
2
3
5
x
Step 6. Determine System of Equations by applying Newton’s Law
…
m6(d2 x6 /dt2) + k61x(x6-x1) + k62x(x6-x2) + … = F6x(t)
m6(d2 y6 /dt2) + k61y(y6-y1) + k62y(y6-y2) + … = F6y(t)
…
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8
4
9
6
7
y
1
2
3
5
x
Step 6. Determine System of Equations by applying Newton’s Law
M(d2U /dt 2) + KU = F(t)
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If Time Remains
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