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Frequency Manipulation via SDP
Brian W. Anthony Robert M. Freund
Truss Structures - Natural
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Background : Mechanical Systems
Simple Mechanical System
m1
u1
k1
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System wants to vibrate when forced.
0 2 4 6 8 10 12 14 16 18 20 0
0.5
1
1.5
2
2.5
3
Frequency (Hz or Cycles/Sec)
Am
plitu
de
1 DOF System
(k/m).5
Natural Frequencies – How a Mechanical
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.
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Beam Vibration Narrow Beam
How would we change the frequency of vibration?
Beam Vibration Narrow Beam
Narrow Beam - Natural Frequency = 5.6 Hz Narrow and Short Beam - Natural Frequency = 11.7 Hz Wide Beam - Natural Frequency = 7.8 Hz
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Another Mechanical System
F(t)
m2 m3m1
u1
k1 k2 k3
u2 u3
Write Newton’s Law for Each Mass
F = ma
Another Mechanical System
F(t)
m2 m3m1
u1
k1 k2 k3
u2 u3
m1(d2u1 /dt2) + k1u1+ k2(u1 2) = 0 m2(d2u2 /dt2) + k2(u2 1) + k3(u2 3) = 0 m3(d2u3 /dt2) + k3(u3 2) = F(t)
M(d2U /dt2) + KU = F(t)
The Dynamics Model (The Equations of motion).
In Matrix Form
-u-u -u
-u
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Equations of Motion
• M K are the natural frequencies of vibration (squared).
• The Eigenvectors of M K are the mode shapes (the relative displacement of each degree of freedom)
M(d2U /dt2) + KU = F(t)
The Equations of motion
2
2
KU dt
M =+
The Eigenvalues of -1
-1
And Eigenvalue Analysis
) ( t F U d
Mechanical System wants to vibrate
1 2 3 4 5 6 7 8 9 10 0
0.5
1
1.5
2
2.5
3
Frequency (Hz or Cycles/Sec)
Am
plitu
de
3 DOF System
Natural Frequencies – The Rate a
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Are the shape of the Vibration
Movies…
1 2 3 4 5 6 7 8 9 10 0
0.5
1
1.5
2
2.5
3
Frequency (Hz or Cycles/Sec)
Am
plitu
de
3 DOF System
m2 m3m1
m2 m3m1
m2 m3m1
Mode 1
Mode 2
Mode 3
Natural Modes – (The Eigen-Modes)
Modification?
1 2 3 4 5 6 7 8 9 10 0
0.5
1
1.5
2
2.5
3
Frequency (Hz or Cycles/Sec)
Am
plitu
de
3 DOF System
m2 m3m1
m2 m3m1
m2 m3m1
Mode 1
Mode 2
Mode 3 It is no longer obvious what we have to do in order to change the lowest natural frequency.
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Truss Structure
Truss Structures
• Rigid beams – Axial forces only
• – Concentric joints – Welded or bolted
• Bridges, towers, exoskeletons
Pin-connected
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A truss has large number of natural frequencies and complex motion.
Half of the mass of Each Link is assumed to sit at a node. (A function of Areas)
Link Stiffness is a function of Length, Area, density
Truss : Model as Lumped Masses with Connecting Springs
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1 2 3 5
7
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9
4
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Truss : Modeled as Lumped Masses with Connecting Springs
1 2
4
6
3 5
7
8
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Step 6. Determine System of Equations by applying Newton’s Law at each node (on each ball of mass)
M(d2U /dt2) + KU = F(t)
x
y
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-1 -0.5 0 0.5 1 1.5 2 2.5 3 0
2
4
6
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Each Node (Mass) has, in general, 3 Degrees of Freedom
Equations of Motion
• M K are the natural frequencies of vibration (squared).
• The Eigenvectors of M K are the mode shapes (the relative displacement of each degree of freedom)
M(d2U /dt2) + KU = F(t)
The Equations of motion
The Eigenvalues of -1
-1
And Eigenvalue Analysis
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A proposed design of a Cell Phone Tower
-1 -0.5 0 0.5 1 1.5 2 2.5 3 0
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50 Bars 40 Degrees of Freedom
Planar Approximation:
All beams have an Area of 1 square centimeter
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A proposed design of a Cell Phone Tower
-1 -0.5 0 0.5 1 1.5 2 2.5 3 0
2
4
6
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10
12
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20
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Natural Frequency = 159 Hz M K )( smallest eigenvalue of -1
Mechanical System wants to vibrate
0 0.5 1 1.5 2 2.5
x 104
10-10
10-5
100
Frequency (Hz or Cycles/Sec)
Am
plitu
de
N DOF System
Natural Frequencies – The Rate a
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Natural Modes (Eigenvectors or The Shape of the Vibration
0 0.5 1 1.5 2 2.5
x 104
10-10
10-5
100
Frequency (Hz or Cycles/Sec)
Am
plitu
de
N DOF System
159.7156 Hz 920.3246 Hz 1618.3 Hz 2333.4 Hz
Movies…
Eigen-modes) -
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.
-1 -0.5 0 0.5 1 1.5 2 2.5 3 0
2
4
6
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12
14
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20
22
Movies…
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Optimized Tower
-1 -0.5 0 0.5 1 1.5 2 2.5 3 0
2
4
6
8
10
12
14
16
18
20
22 towershrt250
Natural Frequency = 250 Hz
New Areas:
1.0353
1.5332
2.0785
2.6420
3.2065
3.7667
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.
Movies… -1 -0.5 0 0.5 1 1.5 2 2.5 3 0
2
4
6
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10
12
14
16
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20
22 towershrt250
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-1 -0.5 0 0.5 1 1.5 2 2.5 3 0
2
4
6
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10
12
14
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20
22 towershrt500
Optimized Tower Natural Frequency = 500 Hz
New Areas:
2.0561
4.0845
7.3261
11.8377
17.2199
22.8216
28.2383
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Truss Approximation
Step 0. Complex (Continuous) Mechanical Structure
Continuous Mechanical System –
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Step 1. Apply a Mesh (Truss) to the Mechanical Structure
Step 2. Form Local Regions around ‘Nodes’ of the Mesh
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Step 3. Use Local Regions around ‘Nodes’ to approximate Node Mass (mi) ij)
All of this mass is applied to the contained node.
Step 4. The Finite Element (Truss) Approximation
mi
mj
kij
and Link Stiffness (k
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1 2
4
6
3 5
7
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Step 5. Label the Nodes
1 2
4
6
3 5
7
8
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Step 6. Determine System of Equations by applying Newton’s Law … m6(d2x6 /dt2) + k61x(x6 1) + k62x(x6 2) + … = F6x(t) m6(d2y6 /dt2) + k61y(y6 1) + k62y(y6 2) + … = F6y(t) …
x
y
-x -x-y -y
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1 2
4
6
3 5
7
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Step 6. Determine System of Equations by applying Newton’s Law
M(d2U /dt2) + KU = F(t)
x
y
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If Time Remains
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