• Overview
• Theoretical Basis for Reduction Methods
• Dynamics of Rotating Structures
• Overview of Aeroelastic Analysis
Advanced Dynamic Analysis User’s Guide 3-1
3.1 Overview
The solutions used in dynamic analysis are more complicated than the solution of a simple matrix equation. Compared to linear static analysis, the running time for a dynamics problem is more costly, the required file space is more excessive, and the larger amounts of output data can be difficult to digest. Therefore, attention should be paid to reducing the number of solution equations to manageable levels.
Other special effects enter into many cases of dynamic analysis. Certain loads, such as those used in rotating structures and in aerodynamics, require special matrix formulations and solution strategies. These problems use conventional matrices to represent structural properties, but they can also require extra degrees-of-freedom and may need to account for fundamental changes such as nonlinear effects. For example, in rotating structures the connections vary with phase angles. In aeroelasticity the effects of the air flow vary with frequency.
This chapter begins with an explanation of the various options available in NX Nastran for reducing the size of the solution matrices in dynamic analysis. Direct methods, modal formulations, and combinations are discussed.
The next section presents methods for the analysis of finite element models in rotating coordinate systems. Applications include turbo machinery, disk drives, and flexible shafts.
The last section is an introduction to the use of aeroelasticity, which can be applied to any problem with a steady fluid flow attached to a flexible structure.
3.2 Theoretical Basis for Reduction Methods
In the following development we will start from the full-size structural matrix equations and derive the equations for Static Condensation, Guyan Reduction, and Component Mode Synthesis.
These operations will apply to both single structures and superelement models. We will also try to explain the physical consequences of the assumptions involved in reducing the systems.
Definition
The basic dynamic equation before reduction is given in the uf set (after SPC and MPC constraints have been applied, but before DMIGs and extra points). The standard matrix equation to be reduced is:
Equation 3-1.
where:
ua, , üaare the displacements, velocities, and accelerations of the analysis (a) set, to be retained.
uo, , üoare the displacements, velocities, and accelerations of the omit (o) set, to be eliminated.
M, B, K are the mass, damping, and stiffness matrices (assumed to be real and symmetric).
a, Poare the applied loads.
Note that all free-body motions must be included in the uapartition. Otherwise, Koowill be singular. The bar quantities ( , etc.) indicate unreduced values.
Statics
For statics problems, we may ignore the mass and damping effects and solve the lower partition ofEq. 3-1for uo:
Equation 3-2.
The two parts ofEq. 3-2become the Guyan matrix Goand the static corrective displacement uoo:
Equation 3-3.
Equation 3-4.
The exact static solution system is obtained by substitutingEq. 3-2throughEq. 3-4 into the upper partition terms ofEq. 3-1, resulting in the reduced equations used in the static solution
Equation 3-5.
and
Equation 3-6.
where:
Equation 3-7.
Equation 3-8.
Advanced Dynamic Analysis User’s Guide 3-3
In actual practice the size of the uaset is usually small compared to uo, but the reduced matrices are dense, resulting in no savings in cost. The savings in solvingEq. 3-5are usually offset by the costs of calculating Goand uoo.
However, for dynamics, we also may approximate the vectors üoand to reduce the order of the system. A good place to start is to use the static properties. FromEq. 3-6, define the transformation
Equation 3-9.
where:
Equation 3-10.
Equation 3-11.
Here uooare the incremental displacements relative to the static shape. The system described inEq. 3-1may be transformed to the new coordinates with no loss of accuracy. The stiffness matrix in the transformed system is
Equation 3-12.
Performing the multiplication and substitutingEq. 3-3 results in
Equation 3-13.
Although the stiffness matrix becomes decoupled, the mass and damping matrices tend to have more coupling than the original system. Since the damping terms have the same form as the mass, we will not include them here. The exact transformed system becomes
Equation 3-14.
where:
Equation 3-15.
Equation 3-16.
Equation 3-17.
The damping matrix terms of B¢ffare similar in form to the mass matrix partitions. An
alternative derivation which does not rely on symmetric transformation is given below. Starting fromEq. 3-1throughEq. 3-8, we may estimate the acceleration effects of the omitted points by the equation
Equation 3-18.
SubstitutingEq. 3-18into the lower partition ofEq. 3-1, and solving for uo, with damping neglected, we obtain the approximation
Equation 3-19.
SubstitutingEq. 3-3for KoaandEq. 3-16for the mass terms intoEq. 3-19, we obtain
Equation 3-20.
SubstitutingEq. 3-18andEq. 3-20into the upper half ofEq. 3-1(ignoring damping), we obtain
Advanced Dynamic Analysis User’s Guide 3-5
Equation 3-21.
Combining the terms, we obtain the same results asEq. 3-14throughEq. 3-17.
The significance of this exercise is to show that the Guyan transformation has very interesting properties, namely:
1. The approximation occurs only on the acceleration terms (Eq. 3-18).
2. The stiffness portion of the reduced system is exact.
3. The interior displacements defined byEq. 3-20andEq. 3-14are nearly identical.
The significant aspects of the partially decoupled system described byEq. 3-9throughEq. 3-21 are that most of the NX Nastran reduction methods are easily developed from this form and the approximations are conveniently explained in these terms. The Guyan reduction and the modal synthesis methods are described below.