Finite Element Analysis of Vibration modes in Notched

Journal of Mechanical Engineering Research and Developments

ISSN: 1024-1752 CODEN: JERDFO

Vol. 44, No. 10, pp. 343-353

Published Year 2021

343

Finite Element Analysis of Vibration modes in Notched

Aluminum Plate

Mohammed Y. Abdellah†,‡*, Hamzah Alharthi‡, Esraa Husein†,‡†, Abdalla Abdal-hay†,‡‡ and

G.T. Abdel-Jaber†

† Mechanical Engineering Department, Faculty of Engineering, South Valley University, Qena, 83523, Egypt ‡Mechanical Engineering Department, College of Engineering and Islamic Architecture, Umm Al-Qura

University Makkah, KSA. ‡†El-yousr station for water desalination, Hurghada, Red Sea, Egypt

‡‡School of Dentistry, The University of Queensland, Herston Campus, 4072, Australia

*Corresponding Author Email: [email protected]; [email protected]

ABSTRACT

Aluminum and its alloy have a competitive and attractive role in many applications such as aerospace field,

shipbuilding, and offshore equipment, due to their excellent mechanical properties concerning their weight.

Natural vibration is considering a characteristic property of a material. Any change in the material contracture

influences the vibration response. The stress raiser due to cracks or and discontinuity shape can affect the values

of the natural frequency of aluminum plate. Therefore, a matrix of the aluminum specimen of 3 mm thickness

with different holes of diameter (2, 4, 6, 8, and 10) mm with constant width of 50 mm is used in a vibration modal

analysis to measure the neutral frequency. The test is carried out using an impact hummer pulse analyzer system.

The vibration modal analysis is used in a cantilever boundary condition case. The finite element Modeling FEM

was built to measure and simulates the vibration response of the aluminum cantilever plates with the holes and

without holes. The finite element model is carried out using the tensile test properties of standard specimens of

aluminum. The results show the natural frequency and damping ratios at 3 shape modes of notch and un notch

aluminum plate, it is showed an increase in the value of damping ratio with increasing hole diameter. The Finite

element modeling gives a good prediction for shape modes. The cantilever beam is considered a suitable problem

to states the effect of stress raiser in the aluminum plate.

KEYWORDS

Natural frequency, damping ratio, aluminum plate,

INTRODUCTION

Aluminum sheets have numerous applications, for example, in pressure vessels, aircraft, shipbuilding, bridges,

and ground vehicles. The plane strain fracture toughness of such thin sheets is known to be higher than that of

thick sheets; therefore, thin sheets have gained importance in many applications [1]. Many applications have been

found for thin metal sheets such as in pressure vessels, aircraft, shipbuilding, bridges, ground vehicles, etc. The

plane strain fracture toughness of such a thin sheet is known to be higher than that of a thick one; therefore, it

takes more importance in a lot of applications [1-3]. Vibration responses were recently used as a non-destructive

test to measure the nominal strength of open-hole specimens and fracture properties of composite material [4].

The study was used vibration modal analysis and basic linear elastic fracture mechanics combined by finite

element modeling; the results were in good agreement with the experimental data. The effect of specimen size of

smart composite material on natural frequency was investigated [5].

In another study the vibration modal analysis was carried out on thin Copper Films Bonded to FR4 Composite

[6], this study also study the effect of holes and scaled on the neutral frequency. A simple finite element model

was derived to calculate the non-dimensional frequency which was considered a design factor [5]. Natural

frequency and mode shapes were influenced by boundary conditions and material types, therefore a study by [7]

was carried out to study these effects using a cantilever beam. Another study used a single rectangular cantilever

Finite Element Analysis of Vibration modes in Notched Aluminum Plate

344

beam using finite element modeling [8]. The vibration modal analysis of a cantilever beam sample of composite

laminates was compared by the natural frequency of aluminum and tined steel plate. The analysis is extended to

involve circular holes for or even elliptical ones The finite element results give a little difference between the

natural vibration of the un-notch plate and others with circular or elliptical ones, moreover, the eigenvalue and

mode contours are explained.

Another numerical finite element study [9] based on vibration model analysis was carried out on a thin aluminum

plate fixed over the circumference. The natural frequency and modal analysis were carried out on the biomedical

material of different fiber reinforcement [10]. In a study by Baillargeon [11] The natural frequency of beams was

found using a finite element for the first four modes by each the location of the actuator from the fixed end of the

structure, The free vibration of the mild steel and aluminum beams were carried out by varying the initial

displacement and input voltage to the PZT to find out the settling time and the damping factor of both of the

beams. It was found when The actuator placed close to the beam center performed better than those located at the

other position of Mode shapes and variation of natural frequency have been studied using ANSYS [12]. Moita et

al. [13] have been studied analytically about optimizing the thickness and width of the PZT actuators. They

reached that the width is a better design variable than the thickness. Experimental study on active vibration control

of cantilever beam with PZT actuator is done and simulated using ANSYS. [21]. Several studies have been

performed to examine composite structures with different shapes, sizes and material systems (depending on the

application) under different kinds of loading conditions including tensile, compression, and fatigue testing [14].

THE OBJECTIVE OF THE PRESENT STUDY

The two main goals of the present study are as follows:

1-Experimentally measure the natural frequency and damping ratio of aluminum plate with holes

2- To determine the natural frequency using FEM of the cantilever beam

MATHEMATICAL MODAL ANALYSIS

The damping force can be measured as [15]:

𝐹𝑑𝑎𝑚𝑝𝑖𝑛𝑔 =�̇�

𝑄 (1)

Where is (𝑄 =𝜔𝑡

2𝜋 ) is the equality factor and �̇� is the generalized velocity the generalized equation of motion for

damped simple harmonic motion can be as:

�̈� +�̇�

𝑄+ 𝑞 = 0 (2)

Where is (𝑄 =𝜔𝑡

2𝜋 ) is the equality factor and �̇� is the generalized velocity the generalized equation of motion for

damped simple harmonic motion can be as:

�̈� +�̇�

𝑄+ 𝑞 = 0 (3)

The simple harmonic oscillator Lagrangian with characteristic frequency.

𝜔2 ≡ −𝐷

𝐹 (4)

Where x is the amplitude or deviation of the oscillating motion (m), ẋ is the velocity (m/s), ẍ is the acceleration

(m/s2), F is the excitation force (N), (𝜔) is the natural frequency of the mode (rad/sec), and stiffness constant of

the plate (k) or (D) and damping coefficient or damping constant (c) of the system. Then it is found that (Q)

increases as the spring constant or mass increases relative to the damping constant (c). to understand the physical

meaning of Q, it is should to rewriting equations in dimensional form.

The equation of motion of a mechanical vibrated system subject to a linear damping force is measured as:

𝑚�̈� + 𝑐�̇� + 𝑘𝑥 = 0 (5)


345

By divided this equation by (𝑄), one can obtain the expression for damping force;

𝐹𝑑𝑎𝑚𝑝𝑖𝑛𝑔 = −𝑐�̇� =𝑚𝜔

𝑄�̇� =

√𝑘𝑚

𝑄�̇� (6)

By divided Eqn. 5 by component (𝑚), it is estimated natural frequency else;

�̈� +𝜔

𝑄�̇� + 𝜔2𝑥 = 0

(𝑚𝑥̈ + 𝐹𝑑𝑎𝑚𝑝𝑖𝑛𝑔 + 𝜔2𝑚𝑥 = 0)

(7)

The natural frequency in terms of non-dimensional vibration into more familiar units (Hz) can be measured as

[16] :

𝑓𝑛 =𝜋

2 √

𝐷

𝜌ℎ

𝜆

𝑙2 (8)

Where (h) is plate thickness, (𝑙) plate length, (𝜌) material density, and D is the cantilever flexural stiffness in the

present study:

D =Eℎ3

12(1 − 𝜈) (9)

Where E young’s modulus and 𝜈 is passion young’s modulus, and (𝜆) is a non-dimensional frequency factor which

is for cantilever plate [17]. The problem with open holes will be changed according to D as the open holes

specimen has different strengths which give different young’s modulus. Thus, the effect of holes may be model

using the coefficient of damping (c) and the plate stiffness D [18, 19].

Finite element method

FEM is one of the methods used for solving the natural frequencies and the mode shapes of oscillations. For

solutions of the modal analysis of the thin aluminum plate, ABAQUS software is used with toolbox Modal The

natural frequency of these vibrating components was used to validate the model. In Figs. (1a, b, c, d, e, and f), the

number of mesh elements is as follows (9570,2810, 12396, 11984, 12050, 11766) for specimens with (2, 4, 6, 8,

and 10) mm diameter respectively. An 8-node linear brick, reduced integration, hourglass control (C3D8R) [20-

22]. The model for this solution is the rectangular aluminum plate. The plate is fixed as a cantilever and boundary

conation shown in Fig. 2 with dimensions of the isotropic plates are 50× 200 × 3 mm and material properties are

shown in Table 1.

Table 1: Mechanical property of Aluminum plates

Specimen 2 mm 4 mm 6 mm 8mm 10mm

Strength, MPa 73 53 46 42.6 30

Fracture strain .0466 .0466 .04 .04 .04

Denisity Kg/m3 2400

Young modulus GPa 71

Poission ratio µ .33


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Figure 1: Mesh element domain a) un-notch, b) 2 mm, c) 4 mm, d) 6mm e) 8mm and f) 10mm

Figure 2: B. C domain of aluminum cantilever

Vibration modal analysis

The free vibration test set consists of various components and tools showed as follows: Delta Tron High-

Temperature Accelerometer (Bruel & Kjaer), type 4526, with a voltage sensitivity of 100mV / g ± 10% used from

160 Hz and 25 kHz combined resonance frequency to measure sample response. The transducer was involved

with the sample using beeswax. A shocker hammer (Bruel & Kjaer), type 8206, with a voltage sensitivity of 22

mV / N and a built-in power transformer with a pressure of a full power range of 220 N, was used to push the

model through with a driving force. The hammerhead was made of hard plastic (ratios, and mode shapes were

determined using the software. The test sample has a rectangular cross-section of 50mm x200 mm long was fixed

to one end of an attachment so that it can be considered as a cantilever. The sample was connected to a steel plate

with bolts and nuts. the samples have central holes of different diameters (2, 4, 6, 8, and 10). The specimens are

made of aluminum with mechanical properties listed in Table 1.


347

This gives the start input frequency and magnitude for the modal setup (see Fig. 3). To obtain the sample I / O

data, a six-channel input unit, LAN-XI (Bruel & Kjaer) type 3050, was used with a frequency range of 51.2 kHz.

Two input channels were used for taking measurements, one for the impact hammer The crash hammer and

accelerometer were attached to the LAN XI using BNC cables and connectors, while the hardware unit was

connected to a computer (Latitude E6400), which had a PULSE LabShop V13.5.0 with a LAN cable (see Fig. 4).

There are 3 different nodes in each sample. To collect data, the transducer is assigned to node 3. In each sample,

a shock hammer test technique is applied to measure the response in all nodes, one at a time, using the program.

The hammer acts as the system entry, while the accelerometer measures the system's response at each node.

Conditional parameters, such as natural frequencies and damping ratios, are extracted from the FRF curve at each

node in different ways obtained using the program. The natural frequencies and damping ratios are taken at

positions 1, 2 and 3. The frequency is calculated for 3 possible modes.

Figure 3: Schematic drawing of measurement system set up

Figure 4: Sample of modal analysis test with holes.

RESULT AND DISCUSSION

Table 2 illustrates the natural frequency and damping ratio of un–notch cantilever aluminum plate, the average

natural frequencies were 137.5, 916, and 16000) and the damping ratio was (0.506, 1.95, and 0.9) for modes 1, 2,

and 3 respectively.

Table 3 shows the natural frequency and damping ratio of 2mm hole the average natural frequencies were

(172.8and 662.5) and the damping ratio was (.34and .89)

Table 4 clarifies the natural frequency and damping ratio of 4mm hole the average natural frequency was

(168.5and 734.4) and the damping ratio were (.29 and .7)

Table 5 explains the natural frequency and damping ratio of 6mm hole the average natural frequency was

(133.5,596.9and 1287.9) and the damping ratio were (.62,1.07 and 2 .87)


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Table 6 states the natural frequency and damping ratio of 8mm hole the average natural frequency were

(146.9,675.2and 1293.5) and the damping ratio were (..34,1.07 and 2 .2.8).

Table 7 presents the natural frequency and damping ratio of 8mm hole the average natural frequency was

(168.8,538.8 and 1203.3) and the damping ratio was (,69,2.04 and 2 .16)

Table 2: Natural frequency and damping ratio of un-notch aluminum plate

Node Mode1 Mode2 Mode3

# Natural

frequency

Damping

ratio

Natural

frequency

Damping

ratio

Natural

frequency

Damping

ratio

1 125.6 .44 929.5 2.2 -- --

2 125.6 .44 910.3 1.83 -- --

3 160 .64 910.3 1.83 1600 0.9

Average 137.5 .506 916.7 1.95 1600 .9

Table 3: Natural frequency and damping ratio of 2 mm aluminum plate

Node Mode1 mood2 Mood3

# Natural

frequency

Damping

ratio

Natural

frequency

Damping

ratio

Natural

frequency

Damping

ratio

1 172.8 .34 662.5 .89 -- --

2 172.8 .34 662.5 .89 -- --

3 172.8 .34 662.5 .89 -- --

Average 172.8 .34 662.5 .89


Node Mode1 mode2 Mode3

# Natural

frequency

Damping

ratio

Natural

frequency

Damping

ratio

Natural

frequency

Damping

ratio

1 160 .2 630 .52

2 172.8 .34 786.3 .79

3 172.8 .34 786.3 .79

Average 168.5 .29 734.3 .7


Node Mode1 mode2 Mode3

# Natural

frequency

Damping

ratio

Natural

frequency

Damping

ratio

Natural

frequency

Damping

ratio

1 131.5 .412 660.5 .66 - -

2 131.5 .412 550.8 1.41 - -

3 137.4 1.05 579.9 1.14 1287.9 2.87

Average 133.5 .62 596.9 1.07 1287.9 2.87


Node Mode1 Mode2 Mode3

# Natural

frequency

Damping

ratio

Natural

frequency

Damping

ratio

Natural

frequency

Damping

ratio

1 131.5 .412 721.5 1.6 - -

2 149.2 .421 674.3 1.09 1293.5 2.8

3 160 .2 630 .52 - -

Average 146.9 .34 675.2 1.07 1293.5 2.8


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Table 7: Natural frequency and damping ratio of 10mm aluminum plate

Node Mode1 mood2 Mood3

# Natural

frequency

Damping

ratio

Natural

frequency

Damping

ratio

Natural

frequency

Damping

ratio

1 172.8 .523 568.2 1.26 1293.7 2.88

2 172.8 .523 379.3 2.15 674.3 1.09

3 161 1.04 804.1 2.72 1641.9

2.52

Average 168.8 .69 583.8 2.04 1203.3 2.16

Figure 5 shows the relationship between hole diameter and natural frequency at different modes 1,2 and 3

instability is observed at mode 1 after specimen 4mm hole. A large increase occurred at the 6 mm hole and then

began to decrease while at modes 2 and 3 The ratio is close; it may be because the sample is disturbed because of

beating with the hammer.

Figure 6 states the relationship between mode and natural frequency, it was found that natural frequency gives the

maximum value at mode 3.

Figure 7 shows the relationship between mode and damping ratio, it was noticed an increase in the value of

damping ratio with increasing hole diameter.

Figure 5: The relation between hole diameter and natural frequencies

Figure 6: Relation between vibration modes and natural frequencies


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Figure 7: The relation between vibration modes and damping ratio.

Figure 7 shows the vibration response for the un-notch specimen it is observed that there is a focus area for

displacement at the fixed end then dispersion occurs for wave in the blue color region and the medium value in

the green color region. The dispersion of waves around the hole increased with increasing the hole diameter clarify

in Figure 8 (a, b, C) for 3 modes for un notch aluminum Figure 8 (d, e, f) for plate while 2 mm holes. The same

results for specimens with 4-, 6-, 8-, and 10-mm holes are shown in Figure 9 (g, h, i), Figure 9 (j, k, l), Figure 10

(m. n. o), and Figure 10 (p, q, r) respectively.

Figure 8: FEM displacement un-notch specimen a) mode 1, b) mode 2, c) mode3 and for 2 mm hole d) mode 1,

e) mode 2, f) mode3


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Figure 9: FEM displacement of 4 mm holes g) mode 1, h) mode 2, i) mode3 and for 6 mm hole j) mode 1, k)

mode 2, l) mode3

Figure 10. FEM displacement of 8 mm holes m) mode 1, n) mode 2, o) mode3 and for 10 mm hole p) mode 1,

q) mode 2, r) mode3

CONCLUSION

Aluminum and its alloy have a competitive role in a lot of application; therefore, the vibration response is of great

intense to be effectively studied, the free natural frequency of notched and unnotched is measured experimentally

and it is observed that It was found that with the increase in the diameter of the samples, a decrease in the natural

frequency occurs and an increase in the damping ratio and a prediction was made to a great extent by finite element


352

(FEM). Normal finite element modeling is used to simulate the vibration response using simple plate cantilever

theory and related to the spring system of differential equations to measure the damping ratios.

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Finite Element Analysis of Vibration modes in Notched