597 592 Chapter 10 ...

Chapter 10
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CHAPTER 10 Miscellaneous
10.1 What is an Eigenvector ?
If a structure is released from rest in a position of an Eigenvector, the structure is
vibrating only in the pattern of that Eigenvector with the associated frequency.
This was already shown with the software SPRI-MOVE in chapter 4. Here we verify
it with the numerical procedure RUNGE_KUTTA ; the code is written in VISUAL
BASIC. The system of two differential equations second order are rewritten as a system
of four differential equations first order.
Figure 10.1 SPRI_MOVE Software , Two Mass , Two Spring , Two Damper System
Coded in Visual Basic using Runge_Kutta Method
The demonstration in chapter 4 , was based on a code , written in C ++ .
Chapter 10
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The code reads : part 1, part 2 , part 3 , part 4 , part 5 , part 6
Print y(i) within Sub RKSFX lists the output values : y(i) = velocity mass 1
y(i+1) = displacement mass 1 ; y(i+2) = velocity mass 2 ; y(i+3) = displacement mass 2
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M(i,j) is the mass matrix ; C(i,j) the damping matrix ; and K(i,j) the stiffness matrix ;
TMAX = 3.5 sec ; DELT = 0.002336 (time increment) . The RUNGE_KUTTA Method
is very sensitive to the selected time increment. For larger time increments the Method
leads to wrong results. (See special literature for estimated time increments)
The two Eigenvectors for the system are : {y} 1 = { 1.0 , 1.62 } ; {y} 2 = { 1.0 , -0.618 }
Figure 10.1a Oscillations of the Two Mass System , Initial Value Mass 1 : 1.0
Initial Value Mass 2 : 1.62
Figure 10.1b Oscillations of the Two Mass System , Initial Value Mass 1 : 1.0
Initial Value Mass 2 : -0.618
It can be observed , that the pattern of the second Eigenvector due to damping is nearly
zero after Tmax = 3.5 sec . ( Modal Analysis is so effective , because the higher modes
are canceling out) . If initial displacement conditions are selected , not identical of an
Eigenvector, is shown in the following graph :
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Figure 10.1c Oscillations of the Two Mass System , Initial Values Mass 1 : 1.0
Initial Values Mass 2 : 1.0
It can be seen that at first arbitrary oscillations of the two masses are performed , but
then due to the fact that the initial conditions given , are close to the first Eigenvector ,
the pattern of oscillation is nearly identical with that of the first Eigenvector .
10.2 Adaptive Meshing
Following [31], and [32] the Energy Error Norm Results can be used, to change a regular
mesh in several steps to an Adaptive Mesh. The results set stores a scalar error value for
each element. This data is useful for determining the quality of the analysis results based
on the existing mesh. It can help you to decide whether the existing mesh is acceptable ,
or whether it should be refined. The larger the error, the more the mesh can be refined.
Adaptive meshing calculates the new mesh size as :
New mesh size = old mesh size * (error norm / actual error ) ** (1 / element order )
The adaptive meshing software lets you set the desirable error percentage .
The results set stores a scalar value ξ i for each element i .
ξ i = ‫ ׀׀‬e ‫ ׀׀‬i / ‫ ׀׀‬U ‫ ( ׀׀‬Ώ / Ώ i ) ½ , with the element error strain energy norm ‫ ׀׀‬e ‫ ׀׀‬i .
‫ ׀׀‬e ‫ ׀׀‬i = √ ∫ Ώ i ( σ * - σ ) T ( ε * - ε ) dΏ , with the element total strain energy norm ‫ ׀׀‬U ‫ ׀׀‬i
‫ ׀׀‬U ‫ ׀׀‬i = √ ∫ Ώ i ( σ *) T ( ε * ) dΏ , and ‫ ׀׀‬U ‫ ׀׀‬2 = ∑ ‫ ׀׀‬U ‫׀׀‬2i , Ώ = ∑ Ώ i ;
Ώ i is the element volume , σ * , ε * are averaged stress , strain , σ , ε are the finite element
calculated stress , strain . The mesh is optimal if the specific error (error strain energy per
unit volume) is the same over the entire model.
Chapter 10
INDEX
A
ABAQUS 585
Acceleration 395 485
Accuracy of calculations 335 351
Adaptive meshing
594
Analysis, structural 42 134 338
365
Animations 246 520
524
ANSYS
585
Assembly of element matrices 70
135 219 328
Automatic meshing 334 506 507
Axi-symmetric shell elements 357
B
Back-substitution 73 74
Bandwidth of Matrix 9
Bar elements
574
Bending
93
Beam elements 96
Bevel Gear with curved teeth 522
Blue Print 519
Boolean Operations :
unite, subtract, Intersect 500 504
511
Boundary conditions in analysis :
Displacement 48
Force 55
56
Boundary value problem 330
BOUSSINESQ equation 336
B-Spline- , curve , surface I.2
Buckling problems I.11
C
CAMSTUDIO software 520
Cards , NASTRAN
568
GRID,CROD, PROD, MAT1, FORCE,
SPC,CBAR,PBAR,CQUAD4,PSHELL
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596
596
595
634
Characteristic
polynomial
243 277
Characteristic
polynomial
Cholesky
Method
9
Cholesky Method 9
Column of matrix
49
Compatibility in elastic theory 330
Completeness of a polynomial 298
Computer programs I.2 I.3
Computer- aided design (CAD) ,
CAD/CAM
I.2
I.3
51 336
Concentrated loads
Condensation
373
Condition number 15
Conical dome 523
Connectivity – or Rigidity matrix) 309
Consistent-, load vector , mass matrix
107
108
314
Constant strain of a triangle element
319
Constraint equation 374
Convergence 298
Coordinate system of element 56
Curvature 83
85
Cutting edge
31
D
D’Alembert’s principle 311
Damping -, matrix , vibrations 458
Deformation 105 334 349 523
Degree of freedom 18 to 23
Design I.3
Determinant 243
Diagonal matrix 248
Digital simulation I.3
Dimension 519
Displacement method of analysis 293
Distortion of elements
525
Distributed loads 164
Divergence 298 (see convergence)
Documentation 544
Double precision arithmetic 591
Drilling holes 522
Dynamic response
279
DYTRAN 1.3
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631
596
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Chapter 10
E
Efficiency
244
Eigenvalue problem 3 243
Eigenvector 244 368 524
Elbow 514
Element-, matrix , stiffness matrix 185
321 325
Energy norm
594 (see adaptive
meshing)
Engine system 388
Equations of finite elements 134 195
Equilibrium equation 331
Error in solution 335 351 358 364
370 371
F
INDEX
Electrodynamics
631
I.3
I
IDEAS I.2
Idealization 341 347
Ill-conditioning 15
Initial conditions 250
Instability 15
Integration methods I.9 (chapter 13)
Interpolation functions 296 316 322
Intersecting cylinders
499
Inverse of matrix 2
Iteration 244
FEBEAM 58
204
Fillet
499
Finite element method 42 81 204
Finite volume method I.9
Fluid flow analysis I.9
Force
51
207
Four –bar mechanism 474
Frame element 207
Frequency , natural 242
J
G
L
Galerkin method (chapter 18)
Gauss elimination 3
Graphics, computer
235
(chapter 3 , 4 , 5 , 6 , 7 , 8)
GTSTRUDL 561
LANCZOS algorithm 244
Landing gear
448
Least square procedure I.9
(chapter 18)
Line loads 107
108
Loads in analysis
135
Local coordinates
74
Lofting
513
Lumped mass matrix 248
H
Harmonic motion 250
Heat transfer analysis I.9 (chapter 18)
Higher order elements (chapter 13)
Hinge 400 402
Hooke’s law 27
Joints of element
Joint loads 170
183
328
K
Kinematic conditions
Kinetic energy 240
374
M
Man-hole 503 507
Mapped meshing 507
Mass matrix 242
314
Master degrees of freedom
373
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631
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Chapter 10
Material properties 345
MATH-TYPE Software 520
Matrix multiplication 2 188
MBB – Company
I.1
Mechanism 388
Pressure 348
Principle of virtual work
chapter 13)
Publication 520
Meshing
Meshing 506
507
Minimization of bandwidth
9
Minimum of potential energy 294
Modeling 496 (chapter 7)
Mode shape 352 368 369 524
N
Moment
Modulusofofinertia
elasticity101 27
Mold
495
Moment of inertia 84 93
N
Q
NAVIER-STOKES equation
I.3 (chapter 17)
Nodal loads (single joint loads) 170
Node, (see joint) 183 328
Nonlinear analysis I.10
Normal modes 352 368 369 524
Norms 15
Numerical integration I.15
NURBS (non-uniform rational
b-splines) I.2
NX-, IDEAS , NASTRAN , NX-6 I.2
O
Optimization , structural I.11
(chapter 22)
Orthogonality of eigenvectors
I.9 (chapter 15)
P
Partitioning of matrices 219
Plane stress element ,( plate stretching)
330
338
Plate bending element 343
Poisson’s ratio 309
Positive definiteness 15
Post-processing 528 (chapter 8)
Potential energy 240 307
Pre-processing
495 (chapter 7)
I.9
636
632
312
Quasi- uniform sequence of meshes
506
507
R
Rayleigh-Ritz analysis 294
Reaction 55
Rectangular element matrix 325
Reduced linear system 50
Reduction procedure I.4
(Reduktions-Verfahren )
Regular mesh 506
507
Resonance 279
Response
279 (chapter 15)
Results 56 334 369 524
Reverse engineering 179
Roller boundary condition 437
Rigidity Matrix 309
Rigid element
373
RITZ-, procedure , condition 294
Rotation 135
Row of matrix 135
RUNGE-KUTTA code
591
S
Scanning, 3D I.1 (ME 595)
(Rapid Product Development)
Section 249
Shape function 301 316 322
Shear- , deformation , modulus 80 195
Shell element 23 24
Simulation 338 343
365
Singularity 15
Slave nodes 373
Solid elements 365
Solid modeling I.3 495 (chapter 7)
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637
633
Chapter 10
Solutions of equations 54 56
Spherical-, pressure vessel , dome ,
roof
357
SPRI-MOVE code 245
591
Spring 239
Stability 15
Static condensation
373
Stiffness matrix 207 321 325
Strain-displacement matrix 319
323
Stress-, calculation , distribution 320
Stress-,lines
calculation
, distribution
,
Contour
324 342
354 566
Stress – strain relation
309
STRUDL
STRUDL I.1 591
562
580
Substructure
29)19)
Substructure I.9
I.9 (chapter
(chapter
Support
50
Surface
509
Symmetry of a matrix 2 49 328
Systems of ordinary differential
Equations (solutions)
93
T
Thermal coefficient 324
Thickness of element 321
325
Torsion
80
Torus
495
Transcendental equation 249 to 259
399
479
493
Transformation of a matrix 46 181
Transmission structures 465 468
Transpose of a matrix 2
Triangle plane stress element
316
Truss- , element , structure 37 to 56
U
Un-damped vibrations 239 to 265
Unit matrix 2
Units (conversion-factors) 16 17
Unstable structure 15
INDEX
V
V
V
Variational formulation I.9
(chapter 13)
Vector 1
Velocity 394 482 493
453
Velocity
Vibration analysis 239 to 265
Visualizer
40515)
I.9 (chapter
Virtual work principle I.9
(chapter 25)
Visualizer 349 369 522 524
VON MISES
VON
MISES stress
stress 379
329
W
Weighted residual I. 9 (chapter 18)
Wheel assembling 513
Wing 513
WORD- 2007 software 520
X
X-Y plane
104
Y
Young’s modulus
27
Z
Zero (null) matrix , vector 3
Chapter 10
REFERENCES
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