Modeling of 3D Beams

Transcription

1 Modeling of 3D Beams Friday, July 11 th, (1 Hour) Goran Golo Goran Golo

2 Preliminaries Goran Golo

3 Preliminaries Limitations of liner beam theory Large deviations Equilibrium is not a straight line 3

4 Preliminaries Non-linear beam theory Geometrically exact beam theory (Reissner 72 ; Simo vu-quoc 86 ) Virtual work Port-Hamiltonian framework Modeling and simulation Analysis Control 4

5 Contents Model derivation Notation Interconnection structure Elastic domain Kinetic domain Discretization (network) Examples Open problem 5

6 Goran Golo

7 Notation Centroid curve Cross-sectional sectional area Arc-length coordinate Centroid curve length 7

8 Spatial manifold If then the spatial manifold is 8

9 Remark If then the flexible body can be described by using either shell theory of general flexible theory. 9

10 10

11 Position of a slice of the beam is completely determined by an element of. Thus, the position of the whole beam is a curve laying in. 11

12 Mathematical intermezzo group is six dimensional matrix group. Elements of this group are represented by where (i) (ii) 12

13 Choice of initial body frame 13

14 Elastic energy Kinetic energy Interconnection structure 14

15 Interconnection structure Internal relations Boundary relations 15

16 There exists such that 16

17 Mathematical intermezzo is Lie algebra corresponding to. It is a six dimensional vector space. Elements of this space are represented by where (i) (ii) 17

18 Equivalently, Therefore If where 18

19 then where 19

20 Remark The position of the beam is completely determined by a point and. 20

21 Meaning of Since is orthogonal is skew-symmetric and thus is -valued 1-form on. 21

22 Example (straight beam) 22

23 Example (straight beam) 23

24 Expression for in non-stressed position 24

25 Strain measure Strain measure is intrinsically defined quantity i.e. it is invariant under a superposed rigid-body motion. Indeed, let then 25

26 Beam in motion The position of the beam in motion is specified by 26

27 Velocity of a point of the beam is defined as is -valued 0-form on. 27

28 Kinematic relation describes the relation between the strain measure and velocity. In the sequel we show that Where Here stands for the space of -valued -forms. 28

29 Based on relation we have that Since, it follows that The term can be rewritten as 29

30 Hence Using the equality The following is obtained 30

31 Consequently, Since the previous equality becomes 31

32 Matrix representation Vector representation 32

33 The matrix equality has the following vector representation Obviously is a linear transformation. 33

34 Elastic forces,, and the strain rate,,are power-conjugate variables. is the dual space of. One-forms are dual to zero-forms. is -valued 0-form on. 34

35 Duality product (elastic domain power) 35

36 Velocity,, and the momenta rate,,are powerconjugate variables. is -valued 1-form on. 36

37 Duality product (kinetic domain power) 37

38 Relation between forces 38

39 Boundary relations 39

40 Interconnection structure 40

41 Remark Expression can be rewritten as 41

42 42

43 Elastic energy Constitutive relations for elastic domain 43

44 Torsion in direction Energy Constitutive relation 44

45 Bending in direction Energy Constitutive relation 45

46 Bending in direction Energy Constitutive relation 46

47 Longitudinal elongation in direction Energy Constitutive relation 47

48 Shear deflection in direction Energy Constitutive relation 48

49 49

50 Total elastic energy Constitutive relation 50

51 Kinetic energy Constitutive relations for kinetic domain 51

52 Total kinetic energy where is ij th element of the matrix 52

53 The constitutive relation is or in vector notation 53

54 54

55 Discretization Goran Golo

56 Discretization 56

57 Discretization 57

58 Example Goran Golo

59 Examples We consider the beam whose motion takes place in XY plane only. This means that Elastic energy 59

60 Examples Kinetic energy where 60

61 Examples Beam parameters 61

62 Examples 62

63 Examples 63

64 Open problem Goran Golo

65 Open problem If then there exists such that Find expression for as a function of 65