S6-1 NAS122, Section 6, August 2005 Copyright 2005 MSC.Software Corporation SECTION 6 RIGID BODY MODES. - презентация

1 S6-1 NAS122, Section 6, August 2005 Copyright 2005 MSC.Software Corporation SECTION 6 RIGID BODY MODES

2 S6-2 NAS122, Section 6, August 2005 Copyright 2005 MSC.Software Corporation

3 S6-3 NAS122, Section 6, August 2005 Copyright 2005 MSC.Software Corporation RIGID BODY MODES n A structure has the ability to displace without developing internal loads or stresses if it is not sufficiently constrained. Examples of this are: u For Cases (a) and (b), the structure can displace as a rigid body. P P P (a) (b) (c) 3D structure, 6 rigid body modes 2D structure, 3 rigid body modes 3D structure, 3 rigid body modes 2D structure, 1 rigid body mode (rbm) 3D structure, 1 rbm, 2 zero freq. modes mechanism 2D structure, 1 zero freq. mode

4 S6-4 NAS122, Section 6, August 2005 Copyright 2005 MSC.Software Corporation RIGID BODY MODES (Cont.) n Rigid body modes - similar to mechanisms as strain free motion occurs, no relative displacement between the grid points. n In statics, rigid body modes result in a singularity in the stiffness matrix u This can be prevented either by applying constraints (if no rigid body motion should occur) or using inertia relief (if the structure is truly unconstrained). n In dynamics, rigid body modes are a common occurrence. u Examples of this are aircraft in flight and spacecraft in orbit. u In these cases, the possible rigid body motion is a part of the solution and may even be important. u Constraining a model to remove the rigid body modes changes the dynamic properties and response of the model.

5 S6-5 NAS122, Section 6, August 2005 Copyright 2005 MSC.Software Corporation RIGID BODY MODES (Cont.) n We will not see clean orthogonal modes, describing rigid body motion, in most cases, i.e. the first one being motion of the structure in the x-direction, followed by the second one being in the y-direction, and so on. n Rigid body modes are an example of a feature of dynamics known as repeated roots, that is, they all occur at the same frequency (0.0 Hz). n In dynamics, when repeated roots occur, the eigenvectors calculated are a linear combination of all eigenvectors that occur at that frequency. n Normally the rigid body modes that are calculated are an arbitrary combination of the clean rigid body modes. They will be orthogonal, but not intuitively so. n Since the rigid body modes calculated by the computer are rarely cleanmodes, it is a good idea to verify the modes. u Look for numeric zero eigenvalues u Animate the eigenvectors u Plot element strain energy to ensure element strain energy is zero. n If a SUPORT entry is used then the rigid body modes will be in the directions defined and will appear as a clean set. The choice is mainly aesthetic. n WARNING – the number of DOFs n defined on the SUPORT entry will be used by NASTRAN to overwrite the first n modes in the assumption these are true rigid body modes.

6 S6-6 NAS122, Section 6, August 2005 Copyright 2005 MSC.Software Corporation n The presence of rigid body and/or mechanism modes is evidenced by zero frequency values in the solution of the eigenvalue problem. n On the assumption that the mass matrix [M] is positive definite, zero eigenvalues result from a positive semi- definite stiffness, i.e., RIGID BODY MODES (Cont.)

7 S6-7 NAS122, Section 6, August 2005 Copyright 2005 MSC.Software Corporation RIGID BODY MODES (Cont.) n SUPORT does not constrain the structure, it simply defines the r-set components. In normal modes analysis, rigid body modes are calculated using the r-set as reference degrees-of-freedom.

8 S6-8 NAS122, Section 6, August 2005 Copyright 2005 MSC.Software Corporation CALCULATION OF RIGID BODY MODES n If the r-set is has been defined, rigid body modes are calculated in MSC.Nastran by the following method. u a-set partitioning. u Solve for {u l } in terms of {u r }. No loads are applied to the DOFs of the l-set (left-over (remaining) DOFs after the a-set is partitioned into the l-set and r-set), so {P l } = {0}.

9 S6-9 NAS122, Section 6, August 2005 Copyright 2005 MSC.Software Corporation CALCULATION OF RIGID BODY MODES (Cont.) u {P r } is not actually applied. u Construct a set of rigid body vectors. u Create the mass matrix corresponding to the r-set. l The mass matrix [M rr ] is not diagonal, in general.

10 S6-10 NAS122, Section 6, August 2005 Copyright 2005 MSC.Software Corporation Use Gram-Schmidt orthogonalization (accessible in the READ module) to orthogonalize the matrix [M rr ] using the transformation matrix [ ro ]. u Construct the rigid body modes. l with the properties. CALCULATION OF RIGID BODY MODES (Cont.)

11 S6-11 NAS122, Section 6, August 2005 Copyright 2005 MSC.Software Corporation SPECIFICATION OF SUPORT DEGREES-OF-FREEDOM n Care must be taken when selecting SUPORT DOFs. n SUPORT DOFs must be able to displace independently without developing internal stresses in the structure. The structure must be constrained so it is statically determinate. Bad Selection (The independent displacement of 1 and 4 may produce internal stress.) Good Selection

12 S6-12 NAS122, Section 6, August 2005 Copyright 2005 MSC.Software Corporation n There is no capability to input SUPORT data directly from Patran. However, the Direct Text Input method can be used. Example: SUPORTID1C1ID2C2ID3C3ID4C4 SUPORT SPECIFICATION OF SUPORT DEGREES-OF-FREEDOM (Cont.)

13 S6-13 NAS122, Section 6, August 2005 Copyright 2005 MSC.Software Corporation CHECKING SUPORT DEGREES-OF-FREEDOM n MSC.Nastran calculates (internal) strain-energy (work) associated with each rigid body vector. u where [K rr ] is the stiffness matrix corresponding to the r-set. u [V rr ] RIG is the strain energy matrix; its diagonal terms are output. n If rigid body modes exist, the strain-energy is approximately zero.

14 S6-14 NAS122, Section 6, August 2005 Copyright 2005 MSC.Software Corporation CHECKING SUPORT DEGREES-OF-FREEDOM n Note that [V rr ] RIG is also the transformation of the stiffness matrix [K aa ] to the r-set, which by definition of a rigid body mode (zero frequency mode), it should be null. n MSC.Nastran also calculates the rigid body error ratio u where || || means the Euclidian norm of a matrix, Only one value of is calculated using [V rr ] RIG and [K rr ] based on all SUPORT DOFs.

15 S6-15 NAS122, Section 6, August 2005 Copyright 2005 MSC.Software Corporation n Except for round-off errors, the rigid body error ratio and the strain energy should be zero if a compatible set of statically determinate supports are chosen by the user. These quantities may be nonzero for any of the following reasons. u Round-off error accumulation u The {u r } set is overdetermined leading to redundant supports (high strain energy). u The {u r } set is underspecified leading to a singular reduced stiffness matrix (high rigid body error ratio). u The multipoint constraints are incompatible (high strain energy and high rigid body error ratio). u There are too many single-point constraints (high strain energy and high rigid body error ratio). u [K rr ] is null (unit value for rigid body error but low strain energy). This is an acceptable condition and may occur when generalized dynamic reduction is used. CHECKING SUPORT DEGREES-OF-FREEDOM

16 S6-16 NAS122, Section 6, August 2005 Copyright 2005 MSC.Software Corporation MODAL ANALYSIS WITH RIGID BODY MODES n In MSC.Nastran all modes (nearly rigid body and flexible body) associated with the a-set mass and stiffness matrices are calculated. The first m modes calculated by the eigenanalysis (where m is the number of DOFs in the r-set) are discarded. The m rigid body modes, calculated by the method described above, are substituted in their place. MSC.Nastran does not check that the discarded normal modes are rigid body modes ( =0).

17 S6-17 NAS122, Section 6, August 2005 Copyright 2005 MSC.Software Corporation n The transformation from modal space (i-set) to physical space (a-set) is n When this transformation is applied to the dynamic equations of motion and the normal modes are unit mass normalized, the following is obtained MODAL ANALYSIS WITH RIGID BODY MODES

18 S6-18 NAS122, Section 6, August 2005 Copyright 2005 MSC.Software Corporation MODAL ANALYSIS WITH RIGID BODY MODES

19 S6-19 NAS122, Section 6, August 2005 Copyright 2005 MSC.Software Corporation n As a result of the transformation, the following consequences occur u Constraint forces are not externally active, e.g., u If damping elements are not connected to ground, then u Thus MODAL ANALYSIS WITH RIGID BODY MODES

20 S6-20 NAS122, Section 6, August 2005 Copyright 2005 MSC.Software Corporation u If damping is proportional, then MODAL ANALYSIS WITH RIGID BODY MODES