To use the viscoelastic capability, the following conditions are necessary:
1. Assume the [K1dd] matrix will be restricted only to the viscoelastic elements. This restriction implies that elastic elements will have a blank or zero entry for geon their associated MATi Bulk Data entries. Conversely, all viscoelastic materials must have representative reference values of ge, and GREF entered on their associated MATi Bulk Data entries.
Then, by definition,
2. The TABLEDi tabular functions TR (f) and TI (f) are defined to represent the complex moduli of all viscoelastic materials.
These two conditions may be combined inEq. 5-38to provide the following expression:
Equation 5-41.
A comparison ofEq. 5-40andEq. 5-41yields the form of the tabular functions TR (f) and TI(f):
Equation 5-42.
Equation 5-43.
Note that the direct input matrix, [K2dd], fromEq. 5-38is still available but not involved in the definition of viscoelasticity.
Direct frequency response analyses that involve viscoelastic materials require some special input data relative to analyses that involve only elastic materials. These special input requirements are given below:
1. Executive Control Section:
None
2. Case Control Section:
SDAMPING = n reference TABLEDi Bulk Data entry that defines the alternate tabular form of TR(f)
3. Bulk Data Section:
a. MATi Bulk Data entry
• G = GREF, the reference modulus
• NU = Poisson’s ratio for the viscoelastic material
• GE, the reference element damping
• All other entries on the MATi Bulk Data entry are utilized in the standard manner.
b. TABLEDi Bulk Data entries:
• A TABLEDi Bulk Data entry with an ID = n is used to define the function TR(f) of Eq. 5-42.
• A TABLEDi Bulk Data entry with an ID = n + 1 is used to define the function TI(f) of Eq. 5-43.
All other input requirements to the NX Nastran Bulk Data entry are typical of direct frequency response analysis. Note that the overall structural damping, g, entered through the PARAM Bulk Data entry (PARAM,G,XX) applies to all elastic materials.
Compare With Theory
The functional form of Eq. 5-42andEq. 5-43requires the user to perform some modest
calculations that involve GREFand gREF. You must also input these two terms through the MATi Bulk Data entry for the viscoelastic materials. In general, you should use representative values for these parameters. However, in those cases where you do not use the NX Nastran OMIT feature, the calculation of TR(f) and TI(f) can be simplified. In this case, define
Advanced Dynamic Analysis User’s Guide 5-41
and select gREFso that
ThenEq. 5-42andEq. 5-43reduce to the following form:
Equation 5-44.
Equation 5-45.
The above simplifications should not be used if OMIT calculations are involved in the analysis in order to avoid possible matrix ill-conditioning.
Note that stress and force data recovery calculations are performed with the reference moduli irrespective of frequency.
Example
To illustrate the representation of viscoelastic material properties in NX Nastran, consider the following structure that may undergo both axial extension along the z-axis and torsion about the z-axis:
where:
flywheel mass, Mz = 2.0 flywheel inertia, Iz = 10.0
axial stiffness, Kz = torsional stiffness, Kqz =
length, l = 2.0
area, A = .9
area moment, J = 2.0
The symbols E(f) and G(f) imply that the extensional and torsional moduli are functions of frequency, i.e., viscoelastic. For simplicity, it is assumed that E(f) = G(f) and that these quantities have the following frequency-dependent characteristics:
f, hz G ´(f) G²²²(f)
.8 1800. 180.
1.1 1850. 185.
1.4 1910. 191.
1.7 1970. 197.
2.0 2030. 203.
2.3 2070. 207.
2.6 2140. 214.
2.9 2210. 221.
A model for the system is shown in the following schematic:
You can generate this model with the following Bulk Data entries:
Advanced Dynamic Analysis User’s Guide 5-43
1 2 3 4 5 6 7 8 9 10
GRID ID CP X1 X2 X3 CD PS
GRID 2 2. 1245
GRID 3 123456
CMASS2 EID M G1 C1 G2 C2
CMASS2 21 2. 2 3 3 3
CMASS2 24 10. 2 6 3 6
CROD EID PID G1 G2
CROD 23 1 2 3
PROD PID MID A J
PROD 1 1 .9 2.
MAT1 MID E G NU RHO A TREF GE
MAT1 1 2000. 2000. .09
Note that the reference values of 2000. for both E and G are specified on the MAT1 Bulk Data entry. The reference value for structural damping, gREF, is set to .09 under the GE field of the MAT1 Bulk Data entry. Once the reference values GREFand gREFhave been assigned, one can evaluateEq. 5-42andEq. 5-43for the values to be assigned to TR(f) and (TI(f). The values for these functions are entered on TABLEDi Bulk Data entries. As elastic portions of the structure may exist in addition to viscoelastic portions, assume that a value of overall structural damping, g, is to be utilized for these elastic portions of the model. The overall structural damping value of .06 is assigned through the following PARAM Bulk Data entry.
1 2 3 4 5 6 7 8 9 10
PARAM N V1 V2
PARAM G .06
This value of g must be considered inEq. 5-43. The evaluation ofEq. 5-42andEq. 5-43will result in the values shown in the following TABLED1 Bulk Data entries:
1 2 3 4 5 6 7 8 9 10
TABLED1 ID
TABLED1 ID TABR1
X1 Y1 X2 Y2 X3 Y3 X4 Y4
+ABR1 .0 .0 .8 –1.11111 1.1 –.833333 1.4 –.5 TABR2
X5 Y5 X6 Y6 X7 Y7 X8 Y8
+ABR2 1.7 –.166667 2. .1666667 2.3 .3888889 2.6 .7777777 TABR3
X9 Y9
+ABR3 2.9 .5611111 ENDT
TABLED1 ID
TABLED1 11 TABI1
X1 Y1 X2 Y2 X3 Y3 X4 Y4
+ABI1 0. 0. .8 .3333333 1.1 .3611111 1.4 .3944444 TABI2
X5 Y5 X6 Y6 X7 Y7 X8 Y8
+ABI2 1.7 .4277778 2. .4611111 2.3 .4833333 2.6 .5222222 TABI3
X9 Y9
+ABI3 2.9 .5611111 ENDT
To demonstrate that elastic as well as viscoelastic elements may be included in the same analysis, the following single degree-of-freedom is added to the Bulk Data Section:
The following Bulk Data entries are required to represent the foregoing single degree-of-freedom oscillator:
1 2 3 4 5 6 7 8 9 10
CELAS2 EID K G1 C1 G2 C2 GE S
CELAS2 1 1000. 1
CDAMP2 EID B G1 C1 G2 C2
CDAMP2 2 2. 1
Advanced Dynamic Analysis User’s Guide 5-45
CMASS2 EID M G1 C1 G2 C2
CMASS2 3 10. 1
The excitation for both disjoint models is a force (moment for the torsional system) with a magnitude of coswt. This function can be generated with the following Bulk Data entries:
1 2 3 4 5 6 7 8 9 10
To perform a frequency response analysis is necessary to provide a list of frequencies (Hz) at which solutions are desired. The following FREQ1 Bulk Data entry is used for this purpose.
FREQ1 SID F1 DF NDF
FREQ1 1 .5 .3 10
The complete NX Nastran data entry for the two disjoint problems is listed below:
ID TEST,DAMPING
$ INCLUDE ALL BULK DATA ENTRIES FOR BOTH
$ DISJOINT MODELS DISCUSSED IN THE PRECEDING
$ REMARKS ENDDATA
5.7 Free Body Techniques
Free body motion in a structure occurs when a structure may move freely without applied forces.
Although the stiffness matrix will have one or more singularities, the combined dynamic system, with mass and damping, may not be singular. Examples are flying objects such as aircraft or rockets, and structures with kinematic mechanisms such as a gyroscope or a pendulum.
In most cases, the dynamic response solutions in NX Nastran do not require any special attention for free body motions. In general, NX Nastran decomposes matrix combinations that are not singular. For example, an unbalanced load applied to a free body has a unique dynamic solution, namely a constant acceleration. The only exception is that of a frequency response analysis at a frequency of zero. In this case it is recommended that a small frequency be used instead of zero for free bodies.
In some cases, it may be necessary to use the SUPORT Bulk Data to provide a method of defining the free body modes of a structure. The primary use for this process is in inertia relief option for static analysis.
See Also
• “Inertia Relief in Linear Static Analysis” in the NX Nastran User’s Guide
In other cases, the basic geometry is used for the definition of the rigid body displacements. The parameter GRDPNT specifies a grid point or a location for the six rigid body displacement vectors—three translations and three rotations. This method is used in the weight and CG calculations and is used for inertia relief with superelements.