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Verification Problems
This manual presents solved verification models including NAFEMS problems.
NAFEMS Normal Modes Analysis
OS-V: 0400 Pin-ended Double Cross
Test No. FV2 A pin-ended double cross, in-plane vibration problem. OptiStruct is used to investigate the coupling between flexural and extensional behavior, as well as the repeated and closed eigenvalues of the double cross in Normal Modes Analysis.

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Verification Problems
This manual presents solved verification models including NAFEMS problems.
- NAFEMS Linear Elastic
- NAFEMS Thermo-elastic
- NAFEMS Heat Transfer
- NAFEMS Nonlinear Static Analysis
- NAFEMS Dynamic Response Analysis
- NAFEMS Random Response Analysis
- NAFEMS Normal Modes Analysis
  - OS-V: 0400 Pin-ended Double Cross
    Test No. FV2 A pin-ended double cross, in-plane vibration problem. OptiStruct is used to investigate the coupling between flexural and extensional behavior, as well as the repeated and closed eigenvalues of the double cross in Normal Modes Analysis.
  - OS-V: 0410 Cantilever with Off-Center Point Masses
    Test No. FV4 This problem tests the behavior of coupling between torsional and flexural, off-center inertial axis, and close eigenvalues. OptiStruct is used in this normal modes analysis.
  - OS-V: 0415 Deep Simply-Supported Beam
    Test No. FV5 OptiStruct tests the repeated eigenvalues, shear deformation, and rotary inertia. The possibility of missing extensional modes is also investigated.
  - OS-V: 0420 Free Thin Square Plate
    Test No. FV12 A thin square plate model that contains three rigid body modes. OptiStruct investigates the repeated eigenvalues and kinematically incomplete suppressions of the thin plate in Normal Modes Analysis.
  - OS-V: 0425 Clamped Thin Rhombic Plate
    Test No. FV15 The behavior of distorted thin elements in normal modes analysis is examined using OptiStruct.
  - OS-V: 0430 Cantilevered Thin Square Plate
    Test No. FV16 A cantilevered thin square plate of 10 m × 10 m dimension. OptiStruct is used to investigate the frequencies at different modes for different test cases.
  - OS-V: 0435 Clamped Thick Rhombic Plate
    Test No. FV22 The behavior of distorted, thick elements in normal modes analysis is examined using OptiStruct.
  - OS-V: 0440 Cantilevered Tapered Membrane
    Test No. FV32 A tapered membrane problem with irregular mesh. The geometry and mesh symmetry are maintained along the x-direction. The shear behavior of membrane elements in Normal Modes Analysis is examined using OptiStruct.
  - OS-V: 0450 Free Cylinder: Axisymmetric Vibration
    Test No. FV41 A free cylinder with axisymmetric vibration. OptiStruct is used to investigate the rigid body mode, close eigenvalues, and the coupling between axial, radial and circumferential behavior.
  - OS-V: 0455 Simply-Supported Solid Square Plate
    Test No. FV52 A well-established solid square plate, which contains three rigid modes with the given boundary condition. The test examines the performance of OptiStruct Normal Modes Analysis on 3D solid elements.
  - OS-V: 0460 Dynamic Behavior of a Fluid-containing Structure using MFLUID
    A vertical clamped free cylindrical shell with a rigid bottom is chosen to demonstrate the applicability of the structures which are partially filled such as liquid storage tanks.
  - OS-V: 0465 Simply-Supported Thin Square Plate
    Test No. 13OptiStruct is used to investigate the repeated eigenvalues.
  - OS-V: 0470 Simply-Supported Thick Square Plate
    Test 21OptiStruct is used to investigate the repeated eigenvalues and the effect of ‘secondary’ restraints.
- NAFEMS Composites
- Response Spectrum Analysis
- Elements
- Materials
- Fatigue Analysis
- Rotor Dynamics
- Frequency Response Analysis
- Explicit Dynamic Analysis
- Aeroelastic Analysis
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OS-V: 0400 Pin-ended Double Cross

Test No. FV2 A pin-ended double cross, in-plane vibration problem. OptiStruct is used to investigate the coupling between flexural and extensional behavior, as well as the repeated and closed eigenvalues of the double cross in Normal Modes Analysis.

fv2 — Figure 1. FE Model with Boundary Conditions and Loadcases

Model Files

Before you begin, copy the file(s) used in this problem to your working directory.

fv2.fem

Benchmark Model

The 2-node simple beam elements are used to model the double cross. Each arm of the cross consists of four elements. The x- and y-translation displacements are fixed at the end of all arms.

The material properties are:

Property: Value
E: 200 x 10⁹ N/m²
$ρ$: 8000 kg/m³

Modal Analysis Results

The frequency of each targeted mode is normalized with the closed form solution.

f*: Closed form solution


Mode 1		Modes 2 and 3		Modes 4, 5, 6, 7 and 8

`f*` CBAR	11.336 Hz 0.9984	`f*` CBAR	17.709 Hz 0.9958	`f*` CBAR	17.709 Hz 0.9974


Mode 9		Modes 10 and 11		Modes 12, 13, 14, 15 and 16

`f*` CBAR	45.345 Hz 1.0000	`f*` CBAR	57.390 Hz .9944	`f*` CBAR	57.390 Hz 0.9999

Reference

NAFEMS R0015 - Selected benchmarks for natural frequency analysis