topic14-designofmachinestructures
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ME EN 7960 Precision Machine Design Design of Machine Structures 14-1
Design of Machine Structures
ME EN 7960 Precision Machine Design
Topic 14
ME EN 7960 Precision Machine Design Design of Machine Structures 14-2
Topics Overall design approach for the structure
Stiffness requirements
Damping requirements
Structural configurations for machine tools
Other structural system considerations
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ME EN 7960 Precision Machine Design Design of Machine Structures 14-5
Design Strategies (contd.)
Active temperature control: Air showers Circulating temperature controlled fluid
Thermoelectric coolers to cool hot spots
Use proportional control
Structural configurations:
Where are the center of mass, friction and stiffness located?
What does the structural loop look like?
Open frames (G type)
Closed frames (Portal type)
Spherical (NIST's M3)
Tetrahedral (Lindsey's Tetraform)
Hexapods (Stewart platforms)
Compensating curvatures Counterweights
ME EN 7960 Precision Machine Design Design of Machine Structures 14-6
Design Strategies (contd.) Damping:
Passive:
Material and joint-slip damping
Constrained layers, tuned mass dampers
Active:
Servo-controlled dampers (counter masses)
Active constrained layer dampers
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ME EN 7960 Precision Machine Design Design of Machine Structures 14-7
Summary of Strategies for Accuracy
Accuracy obtained from component accuracy: Inexpensive once the process is perfected
Accuracy is strongly coupled to thermal and mechanical loads onthe machine
Accuracy obtained by error mapping:
Inexpensive once the process is perfected
Accuracy is moderately coupled to thermal and mechanical
loads on the machine
Accuracy obtained from a metrology frame:
Expensive, but sometimes the only choice
Accuracy is uncoupled to thermal and mechanical loads on the
machine
ME EN 7960 Precision Machine Design Design of Machine Structures 14-8
Stiffness Requirements Engineers commonly ask "how stiff should it be?"
A minimum specified static stiffness is a useful but notsufficient specification
Static stiffness and damping must be specified
Static stiffness requirements can be predicted
Damping can be specified and designed into a machine
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ME EN 7960 Precision Machine Design Design of Machine Structures 14-9
Minimum Static Stiffness
For heavily loaded machine tools, the required stiffnessmay be a function of cutting force
For lightly loaded machines and quasi-staticallypositioning, use the following:
First make an estimate of the system's time constant:
The control system loop time loop must be at least twice as fastto avoid aliasing
Faster servo times create an averaging effect by thefactor (mechanical/2loop)
k
mmech 2=
ME EN 7960 Precision Machine Design Design of Machine Structures 14-10
Minimum Static Stiffness (contd.) For a controller withNbits of digital to analog resolution,
the incremental force input is:
The minimum axial stiffness is thus:
servo
mechN
FF
22
max=
34
41
21
2
1
2
max
K
N
servo
mFk
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ME EN 7960 Precision Machine Design Design of Machine Structures 14-11
Minimum Static Stiffness (contd.)
While the controller is calculating the next value to sendto the DAC, the power signal equals the last value in theDAC
The motor is receiving an old signal and is thereforerunning open loop
Assume that there is no damping in the system
The error M due to the mass being accelerated by theforce resolution of the system for a time increment servois
2
2
1
servoM M
F
=
ME EN 7960 Precision Machine Design Design of Machine Structures 14-12
Minimum Static Stiffness (contd.) The maximum allowable servo-loop time is thus
The minimum axial stiffness is thus:
It must also be greater than the stiffness to resist cuttingloads or static loads not compensated for by the servos:
F
MMservo
=
2
( ) 4521
41
25.0
max
2 KN
MFK
maxFK
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ME EN 7960 Precision Machine Design Design of Machine Structures 14-13
Minimum Static Stiffness (contd.)
The maximum servo-loop time is thus:
Typically, one would set K = M = servo Usually, servo actual = servo /L, where L is the number of
past values used in a recursive digital control algorithm
Example: Required static stiffness for a machine with800 N max. axial force, 250 kg system mass, and 14 bitDAC:
( )
max
75.0 41
43
21
2
F
m KMN
servo
+
ME EN 7960 Precision Machine Design Design of Machine Structures 14-14
Example
1 .10 10 1 .10 9 1 .10 8 1 .10 71 .10 4
1 .103
0.01
0.1
1 .105
1 .106
1 .107
1 .108
1 .109
servo update time
minimum static stiffness
Minimum Servo Update Time and Static Stiffness
Total allowable servor error [m]
M
inimumservoupdatetime[s]
M
inimumstaticstiffness[N/m]
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ME EN 7960 Precision Machine Design Design of Machine Structures 14-15
Servo System Force Output
Lower force drive system for a given servo error Increases the servo update time Lowers the static stiffness requirement
1 .109
1 .108
1 .107
1 .106
1 .103
0.01
0.1
1
1 .104
1 .105
1 .106
1 .107
1 .108
1 .109
servo update time (F_max = 400N)
servo update time (F_max = 800N)
servo update time (F_max = 1600N)
static stiffness (F_max = 400N)
static stiffness (F_max = 800N)
static stiffness (F_max = 1600N)
Minimum Servo Update Time and Static Stiffness
Total allowable servor error [m]
Minimumservoupdatetime[s]
Minimumstaticstiffness[N/m]
ME EN 7960 Precision Machine Design Design of Machine Structures 14-16
System Mass
Lower mass Decreases the servo update time Does not affect static stiffness requirement
1 .109
1 .108
1 .107
1 .106
1 .103
0.01
0.1
1
1 .104
1 .105
1 .106
1 .107
1 .108
servo update time (m = 125kg)
servo update time (m = 250kg)
servo update time (m = 500kg)
static stiffness (m = 125kg)
static stiffness (m = 250kg)
static stiffness (m = 500kg)
Minimum Servo Update Time and Static Stiffness
Total allowable servor error [m]
Minimumservoupdatetime[s]
Minimumstaticstiffness[N/m]
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ME EN 7960 Precision Machine Design Design of Machine Structures 14-17
Servo DAC Resolution
Lower DAC resolution Decreases the servo update time Increases static stiffness requirement
1 .109
1 .108
1 .107
1 .106
1 .103
0.01
0.1
1
1 .104
1 .105
1 .106
1 .107
1 .108
servo update time (N = 12bit)
servo update time (N = 14bit)
servo update time (N = 16bit)
static stiffness (N = 12bit)
static stiffness (N = 14bit)
static stiffness (N = 16bit)
Minimum Servo Update Time and Static Stiffness
Total allowable servor error [m]
Minimumservoupdatetime[s]
Minimumstaticstiffness[N/m]
ME EN 7960 Precision Machine Design Design of Machine Structures 14-18
Dynamic Stiffness Dynamic stiffness is a necessary and sufficient
specification
Dynamic stiffness:
Stiffness of the system measured using an excitation force with a
frequency equal to the damped natural frequency of the structure
Dynamic stiffness can also be said to be equal to thestatic stiffness divided by the amplification (Q) atresonance
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ME EN 7960 Precision Machine Design Design of Machine Structures 14-19
Dynamic Stiffness (contd.)
It takes a lot of
damping to
reduce theamplification
factorto a low level.
Amplificationfactor(output/input)
Source: Alexander Slocum,Precision Machine Design
ME EN 7960 Precision Machine Design Design of Machine Structures 14-20
Dynamic Stiffness (contd.) Material and joint damping factors are difficult to predict
and are too low anyway.
For high speed or high accuracy machines:
Damping mechanisms must be designed into the structure in
order to meet realistic damping levels.
The damped natural frequency and the frequency atwhich maximum amplification occurs are
2
2
21
1
=
=peakd
d
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ME EN 7960 Precision Machine Design Design of Machine Structures 14-21
Dynamic Stiffness (contd.)
The amplification at the damped natural frequency and the peakfrequency can thus be shown to be
For unity gain or less, must be greater than 0.707
Cast iron can have a damping factor of 0.0015
Epoxy granite can have a damping factor of 0.01-0.05
All the components bolted to the structure (e.g., slides on bearings)
help to damp the system To achieve more damping, a tuned mass damper or a shear damper
should be used
2
1
12
1
34
1
2
42
=
=
==
dynamic
staticpeak
k
k
Input
OutputQ
Input
OutputQ
ME EN 7960 Precision Machine Design Design of Machine Structures 14-22
Material Damping
0.000 0.001 0.002 0.003 0.004 0.005 0.006
loss factor
granite
polymer concrete
lead
6063 aluminum
alumina
mild steel
cast iron
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ME EN 7960 Precision Machine Design Design of Machine Structures 14-23
Effects of Changing System Mass
Adding mass: Adding sand or lead shot
increases mass and dampingvia the particles rubbing oneach other
Higher mass slows the servoresponse, but helps attenuateshigh frequency noise
Decreasing mass Faster respond to command
signals Increases a higher natural
frequency Higher speed controller signals
must be used
Improved damping, a result ofthe increased loss factor (theloss factor = c/(2m)
However, low mass systemsshow less noise rejection athigher frequencies
This suggests that themachine will be less able toattenuate noise and vibration
m=2.0
m=1.0
m=0.5
c=0.2
k=1.0
2
4
6
8
10
H( )
0.5 1 1.5 2 2.5 3
Source: Alexander Slocum, Precision Machine Design
ME EN 7960 Precision Machine Design Design of Machine Structures 14-24
Effects of Adding Stiffness to the Machine System Higher stiffness gives a flatter
response at low frequenciesand give smallerdisplacements for a givenforce input
The compromise of decreasednoise attenuation is not asdramatic as is the case withlowering the system mass This is shown by the similar
shapes in the three responsecurves at high frequencies
This suggests that raising thestiffness of a system is alwaysa desirable course of action
However, acoustical noisemay be worsened by addingstiffness (frequency ofvibration is moved to theaudible region of the humanear)
Source: Alexander Slocum, Precision Machine Design
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ME EN 7960 Precision Machine Design Design of Machine Structures 14-25
Effects of Adding Damping to the Machine System
Increasing the systemdamping can make adramatic improvement inthe system response
The trend is fordecreasing amplificationof the output at resonancewith increasing damping
The plot shows thedramatic improvementavailable by doubling thesystem damping:
Although a dampingcoefficient of 0.4 may be
difficult to obtain inpractice
Source: Alexander Slocum, Precision Machine Design
ME EN 7960 Precision Machine Design Design of Machine Structures 14-26
Summary For a servo controlled machine:
The stiffness of the machine structure should be maximized to
improve positioning accuracy
The mass should be minimized to reduce controller effort and
improve the frequency response and loss factor ()
Damping, however, must be present to attenuate vibration in themachine system
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ME EN 7960 Precision Machine Design Design of Machine Structures 14-27
Spindle
Tool
WorkpieceFixtureX-Z table
Structural loop
YX
Z
Open Frame Structures
Easyaccess towork zone
Structuralloop prone
to Abbeerrors (likecalipers!)
Source: Alexander Slocum,
Precision Machine Design
ME EN 7960 Precision Machine Design Design of Machine Structures 14-28
Open Frame Structures (contd.)Spindlehousing
Faceplate
Base
Carriage
Y2C
YR
XR
ZR
X2C
Z2C
Z2S
X2S
Y2S
Source: Alexander Slocum,Precision Machine Design
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ME EN 7960 Precision Machine Design Design of Machine Structures 14-29
Structural loop
X carriage
Y spindle
Tool
Workpiece
Fixture
Z table
Closed Frame Structures
Moderately easyaccess to work zone
Moderately strongstructural loop (like a
micrometer!)
Primary/follower
actuator oftenrequired for the bridge
Easier to obtain
common centers ofmass, stiffness,
friction
Source: Alexander Slocum,Precision Machine Design
ME EN 7960 Precision Machine Design Design of Machine Structures 14-30
Closed Frame Structures
Source: Precision Design Lab
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ME EN 7960 Precision Machine Design Design of Machine Structures 14-31
Tetrahedral Structures
Composed of six legsjoined at sphericalnodes
Work zone in center oftetrahedron
Bearing ways bolted tolegs
High thermal stability High stiffness Viscous shear damping
mechanisms built intothe legs
Damping obtained at theleg joints by means ofsliding bearing materialapplied to the selfcentering spherical joint
Inherently stable shape
Source: Alexander Slocum,Precision Machine Design
ME EN 7960 Precision Machine Design Design of Machine Structures 14-32
Tetrahedral Structures Like the tetrahedron, the
octahedron is a stable truss-type geometry (comprised oftriangles)
As the work volumeincreases, the structuregrows less fast than atetrahedron
The hexapod (Stewartplatform concept originallydeveloped for flightsimulators) gives six limiteddegrees of freedom
The tool angle is limited toabout 20 degrees from thevertical
Advanced controllerarchitecture and algorithmsmake programming possible
Source: Alexander Slocum,Precision Machine Design
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ME EN 7960 Precision Machine Design Design of Machine Structures 14-33
Cast Iron Structures
Widely used Stable with thermal anneal,
aging, or vibration stressrelieve
Good damping and heattransfer
Modest cost for modest sizes
Integral ways can be cast inplace
Design rules are wellestablished (see text)
Economical in medium to
large quantities
Source: Alexander Slocum,Precision Machine Design
ME EN 7960 Precision Machine Design Design of Machine Structures 14-34
Welded Structures Often used for larger
structures or small-lotsizes
Stable with thermalanneal
Low damping, improvedwith shear dampers
Modest cost
Integral ways can bewelded in place
Structures can be madefrom tubes and plates
Source: Alexander Slocum,Precision Machine Design
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ME EN 7960 Precision Machine Design Design of Machine Structures 14-35
Epoxy Granite Structures
Can be cast with intricate passages and inserts Eepoxy granite/Ecast iron = 5/20
Exterior surface can be smooth and is ready to paint
Cast iron or steel weldments can be cast in place, butbeware of differential thermal expansion effects
Epoxy granites lower modulus, and use of foam coresmeans that local plate modes require special care whendesigning inserts to which other structures are bolted
Sliding contact bearing surfaces can be replicated ontothe epoxy granite
ME EN 7960 Precision Machine Design Design of Machine Structures 14-36
Epoxy Granite Structures Foam cores reduce weight:
For some large one-of-a-kind machines
A mold is made from thin welded steel plate that remains an integral
part of the machine after the material is cast Remember to use symmetry to avoid thermal warping
Consider the effects of differential thermal expansion whendesigning the steel shell
The steel shell should be fully annealed after it is welded together
Source: Alexander Slocum, Precision Machine Design
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ME EN 7960 Precision Machine Design Design of Machine Structures 14-37
Epoxy Granite Structures
Instead of ribs, polymer concrete structures usually useinternal foam cores to maximize the stiffness to weightratio
Polymer concrete castings can accommodate cast inplace components (Courtesy of Fritz Studer AG.)
Source: Alexander Slocum,Precision Machine Design
ME EN 7960 Precision Machine Design Design of Machine Structures 14-38
Epoxy Granite Structures With appropriate section design:
Polymer concrete structures can have the stiffness of cast iron
structures
They can have much greater damping
Highly loaded machine substructures (e.g. carriages) are still
best made from cast iron
Polymer concrete does not diffuse heat as well as castiron
Attention must be paid to the isolation of heat sources to prevent
the formation of hot spots When bolting or grouting non-epoxy granite components
to an epoxy granite bed, consider the bi-material effect
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ME EN 7960 Precision Machine Design Design of Machine Structures 14-39
Granite
Dimensionally very stable Must be sealed to avoid absorption of water
Can be obtained from a large number of vendorsproviding excellent flatness and orthogonality
Cannot be tapped, therefore bolt holes consist of steelplugs that have been potted in place that are drilled and
tapped after the epoxy has cured
Can chip
Provides excellent damping
ME EN 7960 Precision Machine Design Design of Machine Structures 14-40
Granite (contd.)
Source: Standridge Granite
Source: Precision Design Lab
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ME EN 7960 Precision Machine Design Design of Machine Structures 14-41
Constrained Layer DampingHow does it work?
Visco-elastic layer damps motion between structure andconstraining layer (from bending or torsion) by dissipating
kinetic energy into heat
Top Constraining Layer
A3, I3, E3
Bottom Constraining Layer
A2, I
2, E
2
Structure
A1, I
1, E
1
Viscoelastic Damping
Material Gd,
x
y
by2
y3
y1
L
ME EN 7960 Precision Machine Design Design of Machine Structures 14-42
Design parameters to tune damper
10
= EI
EIr
0 10 20 30 40 50 600.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8Stiffness Ratio vs. Dynamic Compliance
Constrained Layer Wall Thickness [mm]
ComplianceQ*10,r
stiffness ratio rdynamic compliance Q*10
+=i
iii yyAEEIEI )( 220
x
y
by2
y3
y1
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ME EN 7960 Precision Machine Design Design of Machine Structures 14-43
How is it implemented?
For round structures, inner tube serves as constraining layer
(ShearDamper)
Constraining layer is wrapped with damping material
Coated inner tube is inserted and gap filled with epoxy
Constraining
Layer Ac,
Ic
Structure
As, I
s
Damping Material,
Thickness td
Epoxy
yc
x
yR
c
tc
td
DS,
Thickness tS
Dc,
Thickness tc
ME EN 7960 Precision Machine Design Design of Machine Structures 14-44
ShearDamper - step by stepStep 1: split tube Step 2: wrap damping sheet around
Step 3: fill gap with epoxy Step 4: done
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ME EN 7960 Precision Machine Design Design of Machine Structures 14-45
Any tradeoffs?
Labor intensive inner tube needs to be split Inner tube and epoxy are expensive
Added weight lowers modal frequencies
Challenging if bottom is not accessible for sealing
Constraining layer performance depends on availablewall thickness
Only works for round or rectangular structures where amatching split tube is available
ME EN 7960 Precision Machine Design Design of Machine Structures 14-46
How can we make it better? Replace steel tube AND epoxy with cheaper material
that has better internal damping
Make the need for splitting the constraining layerobsolete
Make design more flexible in terms of shape and
required stiffness
Remove design constraints
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ME EN 7960 Precision Machine Design Design of Machine Structures 14-49
How is it done part 1?
Step 1: Cut damping sheet Step 2: Make lap joint and turn sheet into tube
Step 3: Use fixture to center assembly Step 4: Seal bottom with cable ties
ME EN 7960 Precision Machine Design Design of Machine Structures 14-50
How is it done part 2?
Step 5: Pour expanding concreteStep 6:After concrete has cured cut
off ends: done
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ME EN 7960 Precision Machine Design Design of Machine Structures 14-51
Dynamic compliance - lower is better
0 10 20 30 40 50 6020
25
30
35
40
45
50
55
60
65Total Mass of Damping Assembly
Constrained Layer Wall Thickness [mm]
Mass[kg]
Split Tube
no rebar
0 10 20 30 40 50 605
10
15
20
25
30
35
40
45
50Dynamic Compliance
Constrained Layer Wall Thickness [mm]
Compliance
Split Tube
act. damping ( td = 0.381mm)
opt. damping ( td = opt.)
no rebar
Steel constraining layer is better because it has a higher
Youngs modulus than concrete an equally stiff layer isthinner and hence further away from system neutral axis increased stiffness ratio r.
ME EN 7960 Precision Machine Design Design of Machine Structures 14-52
Reinforced works even better
Structure
As, I
s
Constraining
Layer Ac, I
c
Damping Material
Thickness td
Reinforcement
Bar Ar, I
r
Ds
Thickness ts
?
Support Tube
AST
, IST
Structure
As, I
s
Constraining
Layer Ac, I
c
Damping Material
Thickness td
Reinforcement Bar
Ar, Ir
?
Ds
Thickness ts
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ME EN 7960 Precision Machine Design Design of Machine Structures 14-53
Concrete cast rocks! (Virtually)
0 10 20 30 40 50 6020
25
30
35
40
45
50
55
60
65Total Mass of Damping Assembly
Constrained Layer Wall Thickness [mm]
Mass[kg]
Split Tube
0.5" rebar (3 pcs.)
0.75" rebar (2 pcs.)
1.0" rebar (1 pcs.)
0 10 20 30 40 50 605
10
15
20
25
30
35
40Dynamic Compliance
Constrained Layer Wall Thickness [mm]
Compliance
Split Tube
act. damping ( td = 0.381mm)
opt. damping ( td = opt.)
0.5" rebar (3 pcs.)
0.75" rebar (2 pcs.)
1.0" rebar (1 pcs.)
ME EN 7960 Precision Machine Design Design of Machine Structures 14-54
Reinforced concrete cast
Cable ties press
damping sleeve againstfixture
Hot glue seals rebars
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ME EN 7960 Precision Machine Design Design of Machine Structures 14-55
Test setup
HP 4-channel frequency analyzer
3-axis accelerometer
28 data points
Free-free setup
ME EN 7960 Precision Machine Design Design of Machine Structures 14-56
Response to impulse
-2 0 2 4 6 8 10
x 10-3
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
Concrete Filled Structure
Time [s]
-2 0 2 4 6 8 10
x 10-3
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2Split Tube Damped Structure
Time [s]-2 0 2 4 6 8 10
x 10-3
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2Concrete Cast Damped Structure
Time [s]
-2 0 2 4 6 8 10
x 10-3
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2Reinforced Concrete Cast Damped Structure
Time [s]
-2 0 2 4 6 8 10
x 10-3
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
Sand Filled Structure
Time [s]-2 0 2 4 6 8 10
x 10-3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25Undamped Structure
Time [s]
E. Bamberg, A.H. Slocum, Concrete-based constrained layer damping,Precision Engineering, 26(4), October 2002, pp. 430-441.
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ME EN 7960 Precision Machine Design Design of Machine Structures 14-57
Transfer Functions
0 1000 2000 3000 4000 5000 6000 700010
-2
10-1
100
101
Concrete Cast Damped Structure (#16)
Frequency [Hz]
Magnitude
0 1000 2000 3000 4000 5000 6000 700010
-2
10-1
100
101
102
Reinforced Concrete Cast Damped Structure (#11)
Frequency [Hz]
Magnitude
0 1000 2000 3000 4000 5000 6000 700010
-3
10-2
10-1
100
101
102
Concrete Filled Structure (#2)
Frequency [Hz]
Magnitude
0 1000 2000 3000 4000 5000 6000 700010
-2
10-1
100
101
Split Tube Damped Structure (#12)
Frequency [Hz]
Magnitude
ME EN 7960 Precision Machine Design Design of Machine Structures 14-58
Measured performance Split Tube
f=1530 Hz and =0.055 (predicted: 0.032)
Material cost 100%
Concrete Cast f=1260 Hz and =0.145 (predicted: 0.035)
Material cost 23.5%
Reinforced Concrete Cast f=1640 Hz and 0.3 (predicted: 0.044) Material cost 28.8%
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