hw 9 finite element

7/29/2019 Hw 9 Finite element

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Homework Set 9

Problem 1

Find: Determine the first 10 natural frequencies.

Solution:

Element type used:

Beam (B31: A 2-node linear beam in space.)

Material Properties: Steel. Youngs Modulus (E) = 210x109 N/m2, Poissons Ratio = 0.29.

Steel density = 7800 kg/m3

All the dimensions were introduced in meters (m).

Model3D

N of elements 140

N of Nodes 133

Restraints were applied as is shown in Figure 1.1.

Fixed support (translations and rotations are restrained)

Figure 1.1. FE Model 3D

Results:

Mode N Modes

(Eigenvalues)

Frequencies

[cycles/sec]

1 69.3076 1.32498

2 218.55 2.35286

3 451.254 3.38089


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4 696.007 4.19882

5 1.16396E+03 5.42986

6 1.73013E+03 6.62003

7 2.37023E+03 7.74846

8 3.64572E+03 9.60974

9 3.84421E+03 9.86787

10 4.19053E+03 10.3028

Actives DOF = 133*6 4*6 = 774. This result implies that it can be obtained 70 frequencies

with a high degree of accuracy. (Using the ratio 10:1)


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Problem 2

Given: The Aluminum bracket, shown in Figure 2.1, is 5mm thick and is clamped attwo ends

(vertical right edge and horizontal bottom edge).

Find: Compare the first 5 fundamental frequencies of the structure using planar and shell

elements. Comment on any discrepancies between the results of the two models.

Figure 2.1. Aluminum Bracket

Solution:

Element types used:

CPS4: A 4-node bilinear plane stress quadrilateral. (2D representation)

C3D8: An 8-node linear brick. (3D Solid)

S4: A 4-node doubly curved general-purpose shell, finite membrane strains. (3D Shell)

Material Properties: Aluminum. Youngs Modulus (E) = 70x103 N/mm2, = 0.33.

Aluminum density = 2.7x10-6 kg/mm3

All the dimensions were introduced in millimeters (mm).

Restraints were applied as is shown in Figure 2.2.

Fixed support (translations and rotations are restrained)

(a) 2D representation (b) 3D representation

Figure 2.2. Aluminum Bracket restraints


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Number of nodes and element:

Type of element

CPS4 C3D8 S4

N of elements 367 344 384

N of Nodes 427 636 444

Results:

Table 1. Modes and Frequencies

Element CPS4

(Plane Stress)

Element C3D8

(Solid 3D)

Element S4

(Shell 3D)

Mode

N

Modes

(Eigenvalues)

Frequencies

[cycles/sec]

Modes

(Eigenvalues)

Frequencies

[cycles/sec]

Modes

(Eigenvalues)

Frequencies

[cycles/sec]

1 20.5605E+06 721.668 534.474E+03 116.355 507.747E+03 113.408

2 25.0292E+06 796.24 5.01956E+06 356.577 4.79669E+06 348.571

3 41.2062E+06 1.02165E+03 9.02051E+06 478.009 8.98531E+06 477.075

4 96.1187E+06 1.56036E+03 20.9506E+06 728.482 20.2717E+06 716.581

5 157.982E+06 2.00043E+03 22.5963E+06 756.551 21.973E+06 746.045

Modes using CPS4 element. (Plane stress)


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Modes using C3D8 element. (3D Solid)


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Modes using S4 element. (Shell)


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hw 9 finite element

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