Afterword

Thus, for a complete investigation of dynamical systems, we require not only a computer and the direct integration methods. These provide no more than an ideal computer laboratory in which an arbitrary number of experiments can be performed, yielding an immense data flow. We require, in addition, certain principles according to which the data may be evaluated and displayed, thus giving an insight into the astonishing variety of responses of dynamical systems.

J. Argyris and H-P. Mlejnek [4)

Nonlinear problems present special difficulties for a book in that they pose a great challenge to have sufficient representative results. A book example by its nature is a single realization - a single geometry, material, or load case; multiple test cases and examples are just not feasible if the book is to remain reasonable in size. But engineers, being introduced to something new, need to see other examples as well as variations on the given examples. Consequently, a new way must be found to present the results of exploring nonlinear problems.

As computer modeling becomes easier to use and faster to run, this opens up the possibility of understanding problems by visualizing the results. Whereas computer programs once sufficed to provide numbers - discrete solutions of "stress at a point" and the like - now complete simulations, which present global behavior, trends, and patterns, can be explored and sensitivity studies analyzing the relative importance of geometric and material parameters can be evaluated. Being able to observe phenomena and zoom in on significant parameters, making judgments concerning those that are significant and those that are not, will greatly enhance the depth of understanding.

The situation becomes one of how to present and interpret information, what trends to look for, what conclusions can be drawn. The purpose of this discussion is to explore (in very broad terms) the make-up of such a computer laboratory, to share some experiences of a prototype version called QED, and to place the content of the current book within this context.

QED: A Computer Laboratory

As envisioned, the simulation program runs simultaneously with reading the text and becomes a resource to be interacted with and tested against as the examples

502 Afterword

are explored. For instance, in the stress analysis of a plate with a hole introduced in Chapter 2, some logical questions to ask are:

• What if the hole is bigger or smaller?

• What if there are two holes instead of one?

• What if the hole it; ill a tlhell'(

• What if the loading is dynamic?

• What if the clamped boundary has some elasticity?

These questions are too cumbersome and expensive to answer in a single text but are very appropriate for a simulation program.

frames

cylinders

channels

holes

notches

QED: a computer laboratory

geometry bcs loads mesh linear static

linear vibration

linear transient

buckling

nonlinear incremental

nonlinear transient

contours

shapes

tractions

time traces

Nonlinear ODE

sym K

anti-sym K

pendulum

van der pol

frame

parametric

Figure A_I: Overview of a simulation program called QED.

The intent of QED, then, is to provide an interactive simulation environment for studying and understanding a variety of problems in nonlinear structural mechanics - a laboratory for exploration and accumulation of experiences. Fig­ure A.I shows a possible schematic of the functional parts of the program. Its design is such that it isolates the user from having to cope with the underlying enabling programs (indicated in dashed boxes) and presents each problem in terms of a limited (but richly adaptable) number of choices and combinations.

The process of finite element analysis can be broken down into three separate stages. These are presented as independent modules in QED. The pre-processing stage allows the model geometry to be defined, the boundary conditions imposed, the loads applied, and the mesh generated. In the second stage, the analytical solution is obtained. Choices as to the type of solution required and the param­eters best suited to guide the procedure are made. In the post-processing stage, results are displayed in a variety of ways. Contour plots of nodal results, the

Afterword 503

deformed shape, free body diagrams, and time history traces are available. The essence of many nonlinear phenomena is captured in single differential equations. A gallery of standard some nonlinear equations are also presented.

I: Creating Model Geometries

The design philosophy for the model building is to make each module very specific but flexible through parameterization. This is important as it helps to fix focus on the significant aspects of behavior. Each problem has a template of properties, therefore the user need only focus on what they want to change.

The model geometries presented by QED are chosen to encompass a range of interesting shapes, loading patterns, and boundary conditions. The models are separated into categories based upon their geometric configurations. Within each module parameters can be changed to "morph" the shape from one form to another. For instance, the "Frames" module includes the option to define a multispan beam, a plane frame, and a space frame as shown in Figure A.2. These share common geometric characteristics and are thus classified in a single module. There are corresponding variations of the loads .

..-------"- - ,,

(a) (b) (c)

pm;

Figure A.2: Progression of associated geometries. (a) Beam. (b) Plane frame. (c) Space frame.

II: Analyzing the Problem

Irrespective of how the model was created, the same collection of analysis pro­cedures can be applied and all loads are considered to be applied dynamically. Each has a set of parameters associated with the solution algorithm. Thus the "nonlinear incremental" option includes parameters for the time step and conver­gence tolerance for Newton-Raphson iterations. Generally, the algorithmic issues addressed in the text are presented as parameters. Any load can be considered to be applied dynamically or statically.

III: Viewing the Results

In the post-processing stage, a variety of schemes are provided for viewing the results. The same collection of post-processed information can be interrogated

504 Afterword

for all models, but knowing which is best suited to a particular problem is very valuable to a deep understanding of a problem. For example, for buckling and wave propagation problems, each mode or snapshot can be displayed and investigated separately as if an instance of a static problem. But time traces (of velocity or stress at a point) is additionally needed for the wave propagation problem but not the buckling problem.

IV: Gallery of Nonlinear Equations

The collection could range from the symmetric return force to Van der Pol equation to parametric excitation. The typical equation is of the form

Mil + C(u)u + K(u, t) = P(t)

with C(u) and K(u, t) containing many parameters that are adjustable. The loads can be a combination of ramp, forced frequency, multiple pings, and uni­form.

The results are presented as trajectories, component histories, phase-plane plots, Poincare plots, energy and isocline contours, and Fourier transforms. At any stage, an immediate comparison can be made with the linearized model.

V: Enabling Programs

In terms of philosophy, any FEM program could be used for the underlying computations and any graphically oriented 4GL can be used for programming the user interface. Reference [81] is a nice illustration of using MatLab as the host to Ansys for the computations.

NonStaD is the finite element analysis program used as a test bed implementing the major features discussed in this book. Its operation under QED is by way of driver or script files. That is, QED creates the script files to execute NonStaD in batch mode.

The program GenMesh is used to create structure datafiles for use by NonStaD.

Its main capabilities involve generating meshes and performing executive func­tions such as merging meshes and adding material properties and boundary conditions. Here also, QED creates the script files which run GenMesh in batch mode.

The Role of the Book

The role of the book is to provide the principles by which dynamical systems are explored, it is a guide book; the role of the computer is to make many examples readily available. Having a flexible sophisticated simulation running on the computer to augment/counterpoint the text examples would profoundly affect the engineers' perception of both theory and computational methods. The interplay of both would establish an interesting and exciting dynamic.

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Index

Adjustable parameter 96, 138 Airy stress function 100, 106, 166 Arc-length method 418, 422 Assemblage 82, 88, 135, 141, 150, 156,

201, 212, 268, 270, 348 Attractor 312, 333, 334, 338, 362, 485 Autonomous system 332, 450, 456, 487

Beam 1, 102, 125, 230, 259, 377, 387, 458,474

Bifurcation 335, 336, 338, 362, 366, 369, 374, 414, 417, 419, 421, 432, 495

Boundary condition 52, 54, 76, 80, 93, 95, 97, 100, 104, 112, 115, 121, 126, 141, 160, 352, 376, 393, 439

Buckling 161, 229, 361, 366, 380, 385, 389, 391, 403, 408, 421, 426, 430, 431, 436, 446

Catastrophe 417,479 Cauchy stress 39, 41, 47, 70, 71, 202, 205,

206,207 Center of shear 164, 466 Cohesive strength 71, 348, 352 Compatibility 48, 99, 148 Complex notation 113, 246, 279, 293, 296,

302, 448 Conservative system 10, 48, 55, 64, 154,

238, 239, 435, 456, 484 Constant strain triangle 135, 270 Convergence test 143, 151, 281, 356, 405,

409 Coordinates 14, 16, 51, 55, 69, 84, 104,

118, 119, 128, 129, 146, 154, 182, 208, 263, 274, 289, 327, 332, 407, 426, 432

Coriolis acceleration 275, 372 Corotational scheme 208, 214, 219, 231,

401, 404, 406, 411

D'Alembert's principle 50, 267 Damping 8, 58, 93, 242, 245, 248, 251,

254, 268, 289, 290, 293, 294, 317, 323, 339, 347, 351, 356, 444, 467, 475, 484, 495

Degree of freedom 3, 48, 54, 77, 98, 136, 139, 145, 148, 155, 160, 207, 208, 214, 230, 231, 232, 290, 333, 404, 411, 431, 465, 476

Duhamel's integral 257, 290, 307, 314

Eigenanalysis 18, 266, 279, 283, 286, 294, 302, 345, 371, 374, 381, 388, 401, 403, 423, 424, 447, 451, 460,465,485

Elastic material 3, 59, 65, 67, 70, 92, 134 Elastic stiffness 84, 135, 137, 150, 193,

211, 216, 226, 230, 402, 411 Elastica 170, 218, 388 Energy, kinetic 50, 53, 56, 57, 92, 97, 121,

335, 372, 461, 466, 483, 492 Energy, potential 48, 49, 76, 78, 82, 335,

366, 412, 419, 433 Energy, strain 48, 49, 53, 56, 57, 64, 65,

74, 92, 94, 97, 120, 126, 141, 365, 372, 375, 387, 392, 404, 406, 410, 431, 460, 465, 483

Finite element method 76, 80, 81, 115, 141, 161, 165, 237, 282, 352, 356, 409, 420, 422, 460

Flexure 87, 90, 94, 96, 98, 121, 127, 145, 147, 150, 163, 241, 266, 273, 281, 300, 375, 391, 431

Floquet's theory 301, 309, 470 Forced frequency 242, 245, 248, 253, 279,

446 Frame 1, 84, 157, 165, 182, 189, 229, 269,

294, 348, 389, 404, 475, 498 Fundamental path 336, 420, 424

Geometric stiffness 193, 201, 212, 217, 226, 227, 228, 230, 402, 404, 406,411

Grid 2,160

Hamilton's principle 50, 51, 52, 55, 93, 95, 97, 120, 376, 393

Harmonic motion 242,244,248, 251, 279, 283, 313, 321, 388, 471

Hooke's law 67, 92, 134

510

Imperfection 373, 374, 380, 419, 429, 475, 497

Impulse 256, 297, 330 Inertia 50, 93, 95, 242, 267, 268, 272, 276,

300, 327, 328, 444, 446 Initial condition 244, 252, 257, 292, 305,

312, 316,325, 331, 341, 347, 351,455,485,491,495

Instability 9, 71, 178, 213, 316, 325, 335, 361, 369, 371, 372, 380, 381. 403, 412, 416, 421, 422, 428, 430, 442, 446, 448, 452, 458, 463, 468, 470, 474, 475, 480, 485, 487, 489, 490, 492, 495, 496, 498

Jacobian 21, 24, 42, 130

Kirchhoff shear 97, 115, 122 Kirchhoff stress 40, 41, 43, 47, 71, 202,

205 Kirsch solution 109, 116, 144

Lagrange's equation 54, 55, 57, 336, 372 Lagrangian strain 29, 32, 33, 34, 47, 66,

69, 70, 71, 202, 205, 375, 391, 410

Lame constants 67, 205 Lame solution 107, 122 Limit cycle 317, 333, 442, 449, 460, 484,

490 Limit point 71, 174, 213, 368, 403, 414,

417, 420, 421, 497 Load interaction 74, 359, 378 Loading equation 171, 194, 413, 418

Mass matrix 268, 269, 270, 272, 277, 281, 461, 465, 466

Mathieu functions 303, 452, 471, 480 Membrane action 2, 90, 99, 120, 122, 133,

137, 224, 230, 232, 235, 266, 270, 391, 399,406, 407, 412, 438, 476, 478

Modal matrix 283, 286, 289, 357 Mode jump 409, 430, 432, 435, 437 Mode shape 241, 251, 260, 262, 264, 267,

279, 283, 284, 294, 296, 383, 386, 387, 394, 397, 400, 401, 403, 408, 424, 425, 437, 476

Natural frequency 244, 252, 286, 313, 315, 320, 321, 334, 446, 470, 471,474

Index

Newton-Raphson iteration 195, 197, 206, 212, 233, 354, 403, 413, 423

Numerical stability 343, 345, 475

Orthogonality 14, 15, 18, 191, 284, 285, 2R7, 41~

Orthotropic material 62, 67, 69, 73, 199

Parametric excitation 300, 338, 385, 471 Pascal's triangle 131, 407 Pendulum 6, 56, 332, 335, 355, 454, 489 Perturbation analysis 320, 359, 370, 429,

450,472 Phase plane 244, 252, 312, 337, 427, 442,

460, 485, 492 Ping load 359, 360 Plane stress 69, 92, 134 Plate 2, 24, 34, 69, 90, 92, 95, 96, 107,

110, 112, 116, 144, 147, 151, 154, 159, 235, 257, 259, 261, 262, 264, 270, 281, 297, 391, 394, 396, 398, 400, 406, 407, 408, 424, 427, 429, 431, 436, 476, 480, 496

Positive definite 284, 339, 414, 487 Principal value 18, 62, 86, 182, 190, 327,

372 Proportional loading 174, 370, 395, 402,

413, 418

Ritz method 76, 77, 78, 80, 81, 280, 387, 464

Rod 1, 52, 54, 74, 81, 250, 270, 273, 276, 280, 297, 350, 352

Rotation matrix 156, 157, 159, 184, 186, 190, 276

Shape function 82, 85, 132, 138, 147, 154, 200, 268, 272, 405, 406, 411

Shear 27, 32, 37, 63, 69, 72, 75, 84, 92, 94, 96, 102, 115, 118, 122, 148, 161, 166, 204, 206, 218, 229, 377, 394, 396, 457

Shell 3, 118, 119, 122, 155, 189, 192, 230, 235, 266, 282

Snap-through 9, 174, 213, 316, 349, 368, 421,499

Spectral analysis 111, 248, 250, 254, 257, 258, 261, 279, 286, 294, 295, 299, 357, 384, 385, 393, 396, 399, 423, 456

Steady-state response 245, 326, 331, 335

Index

Strain 4, 22, 24, 28, 29, 32, 33, 34, 47, 53, 60, 66, 68, 71, 74, 78, 84, 91, 96, 99, 119, 120, 129, 134, 140, 146, 148, 149, 172, 199, 202, 205, 208, 210, 214, 221, 228, 231, 375, 391, 392, 410, 477

Stress 4, 7,36,37,40,44, 46, 47, 62, 67, 71, 73, 92, 98, 100, 102, 106, 107, 110, 116, 122, 123, 125, 138, 141, 144, 151, 161. 164, 199, 201, 202, 204, 205, 209, 230, 235, 348, 360, 375, 379, 387, 395, 396, 398, 411, 438

Structural stiffness 58, 82, 83 Subspace iteration 280, 282, 403, 407,

424,465

Tangent stiffness 9, 73, 193, 198, 209, 212, 213, 218, 227, 233, 235, 347, 349, 352, 359, 371, 374, 403, 414, 423, 434, 451, 460, 480

Thermoelasticity 68, 398 Thin-walled structure 3, 24, 35, 64, 69,

90, 122, 161, 169, 207, 231, 241, 326, 360, 398

Time integration 338,340,341,344,345, 347, 348, 351, 352, 356, 357, 423, 460

Traction 36, 40, 46, 51, 71, 93, 103, 107, 124, 138, 142, 206

Trajectory 332, 338, 455, 483, 485, 487, 492

Triad 16, 181, 186, 188, 190, 191, 238, 274,329

Truss 1, 4, 9, 13, 75, 85, 159, 170, 172, 173, 193, 208, 211, 212, 223, 228, 229, 316, 404, 416, 421, 482, 492, 497

Vander Pol equation 317, 485, 490 Variational principle 47, 52, 54, 76, 411 Vibration 9, 80, 161, 242, 244, 250, 257,

259, 278, 279, 281, 283, 291, 293, 299, 301, 313, 316, 334, 337, 349, 360, 373, 385, 387, 399, 403, 414, 423, 424, 425, 428, 437, 438, 444, 445, 448, 454, 476, 487

Virtual work 46, 47, 50, 55, 85, 134, 138, 140, 153, 201, 203, 209, 215, 222, 233, 267, 276, 460, 465

Wavenumber 250, 259, 261, 396 Waves 73, 94, 250, 297, 346, 350, 351,

356, 411, 496

511

Mechanical Engineering Series (continued from page ji)

R.A Layton, Principles of Analytical System Dynamics

C.V. Madhusudana, Thermal Contact Conductance

D.P. Miannay, Fracture Mechanics

D.K. Miu, Mechatronics: Electromechanics and Contromechanics

D. Post, B. Han, and P. Ifju, High Sensitivity Moire: Experimental Analysis for Mechanics and Materials

F.P. Rimrott, Introductory Attitude Dynamics

S.S. Sadhal, P.S. Ayyaswamy, and J.N. Chung, Transport Phenomena with Drops and Bubbles

AA. Shabana, Theory of Vibration: An Introduction, 2nd ed.

AA Shabana, Theory of Vibration: Discrete and Continuous Systems, 2nd ed.