A Suggested Analytical Solution for Vibration of Honeycombs Sandwich Combined Plate Structure

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2 International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:16 No:04 10 and mode shapes were investigated. A numerical technique was employed using ANSYS package In this research the natural frequencies of honeycombs sandwich plate with different Regular Hexagonal Honeycomb dimensions effect were evaluated and compared with those obtained numerically adopting finite element method using ANSYS Ver THE SUGGESTED ANALYTICAL SOLUTION The suggested analytical solution of vibration analysis of honeycomb plate included the determination of the mechanical properties of honeycomb structure and then evaluation the equivalent stresses and the natural frequency of with effect of honeycomb structural parameters. 2.1 Mechanical Properties of Honeycomb Materials The cells in a honeycomb-structure are usually hexagonal but they can also be triangular or square or rhombic. The honeycombs can be made of different types of materials such as metals wood polymer and ceramics or a combination of two or more of these materials. Investigating the in-plane properties ( transverse directions) demonstrate the failure mechanism and deformations of the cellular solid structures i.e defining the plane properties such as those shown in Fig. 1.The stiffness and strength in x-y plane. If the hexagonal honeycomb is regular (all angles are 30 o and wall thicknesses are equal as (Fig. 1.) then the in-plane properties are isotropic. This means that the in-plane properties of the honeycomb can be described by only two independent elastic modules for instance by Young's modulus Fig. 2 shows a hexagonal honeycomb regular structure in which the angles are equal ( 30 0 ) with constant wall thickness. This results in using in-plane properties of the honeycomb structures such as modulus of elasticity E and modulus of rigidiy G 1] as two independent properties. 1 1 Fig. 1. Honeycomb Structure Loaded in x or y-direction 1]. If the honeycomb cellular structure is not regular the in-plane properties are used with four modules (e.g. E h1 E h2 G h12 and h12 ). If the hexagonal honeycomb has a low relative density ( is small) get 1] h ( ) ( ) Fig. 2. A Regular Hexagonal Honeycomb 1]. The deflected shape of the honeycomb when loaded in x- or y- direction ( Fig. 1) is in bending direction for all cell walls (see Figure 3.a) and is described by five module: Young's module E h1 and E h2 a shear modulus G h12 and two Poisson's ratio h12 and h21. Using the reciprocal theorem gives (2) Reduce the five independent modules to four independent modules. When loading in x or y directions the four independent module described as 1] l t 2 2 y x (1) ( ) ( ) (3) Fig. 3b shows the honeycomb structure under shear loading. It is possible to show that the shear module G h12 to be 1]

3 International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:16 No:04 11 ( ) ( ) Where E is modulus of elasticity of honeycomb material. 2 (4) And are mechanical properties of plate materials parts in x and y directions. and w is the deflection of plate in z-direction. Then with subjecting eq. 5 to plate structure as shown in fig. 4 to get the assuming bending moments acting on the honeycomb sandwich plate as (6) 1 a. Normal Stress Loaded in x and y b. Shear Stress Fig. 3. Honeycomb Deformed a. Deformed by Normal Stress Loaded in x and y b. Deformed by Shear Stress 1]. 2.2 Vibration Analysis of Honeycombs Sandwich Plate The analysis of honeycomb sandwich plate covered the determination of the general equation of motion of honeycomb sandwich plate structure where the plate is orthotropic properties in x and y direction. Also solving the general equation of motion of plate is included and then evaluating the natural frequency of honeycomb sandwich plate with various parameters for upper and lower plate parts and honeycomb dimensions and properties part. To Derive the general equation of motion of honeycomb sandwich plate using the properties and density of honeycomb sandwich materials equations (1 to 4) into the general equation of motion of plate 9] (10) Where are the bending moments and twisting moment per unit length of the orthotropic plate. The bending and twisting moments per unit length acting on the plate element are as follows 9] h M x x z dz h h M y y z dz h h M xy xy z dz h h h dz (5) h Where And (7) where are density of lower and upper plate parts respectively. For honeycomb sandwich plate shown in Fig. 4 assuming the structure combine of isotropic materials properties of upper and lower plate parts and orthotropic honeycomb sandwich part then stresses formulation giving in equation (6) become 1. Upper plate part since the plate part is isotropic propeties then (8) And Where are mechanical properties of uper plate part. Then by subsutution eq. 8 into eq. 6 givs 2. Orthotropic honeycomb sandwich part since the plate part is orthotropic propeties then (10) Then by subsutution eq. 10 into eq. 6 gives (9) (11)

4 International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:16 No: Lower plate part since the plate part has isotropic propeties then (12) Where are mechanical properties of uper plate part. Then by subsutution eq. 8 in to eq. (6) gives a y z x (13) ( ] ] ( ] ) ] ) Then with integration eq. 14 gives ] ] (14) Upper Plate b ( ) Honeycomb structural h Up ( ) ] hh ( ) Lower Plate h l t h Lp ( ) ] ( ) Dimensions of Regular Hexagonal Honeycomb Fig. 4. Dimensions of Honeycomb Sandwich Combined Plate Structure. Therefore by substation eqs and 13 into eq. 7 gives the bending moments acting on the honeycomb sandwich plate as ( ) ] (15) And then by substitution eq. 15 into the general equation of motion of orthotropic plate eq. (10) gives the suggested general equation of motion of honeycomb sandwich plate as ] ] ( ] )

5 International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:16 No:04 13 ( ) the suggested general equation of natural frequency for honeycomb sandwich plate structure with various mechanical properties and dimensions of honeycomb sandwich plate combined as ( ) ] ( ) ( ) ( ) ] ( ) ] ( ) ( ) ( ) ] ( ) ( ( ) ] ) (16) To solve equation (16) separation method can be used with assuming the function of deflection as 11] (17) To evaluate the behavior of deflection plate as a function of x and y directions the boundary conditions of the plate are required. Then for the equation of plate as a function of x and y direction as 10] (18) Then by substation eq. 18 into eq. 16 the suggested general equation of motion for honeycomb sandwich plate structure is obtained as ( ) ( ) ] ( ) ( ) ] ( ( ) ] (21) By building a computer program with using Fortran Power Station 4.0 can be evaluated the natural frequency of honeycomb sandwich plate with various parameters effect as 1. Thickness of upper and lower plates and honeycomb parte. 2. Hexagonal honeycomb. 3. t h and l of honeycomb parte. 4. Mechanical properties and density of honeycomb sandwich parte. 3. NUMERICAL INVESTIGATION The numerical investigation is carried out to calculate the natural frequency of honeycomb sandwich plate structure by using the finite elements method ANSYS program (Ver.14). Where the shell element (SHELL93) is used to build the model of honeycomb sandwich plate structure figure 5.It is assumed that the shell element is employed with zero thickness or element tapering down to a zero thickness at any corner is not allowed Also the shear deflections are included in this element. The out-of plane stress for this element varies linearly through the thickness..the transverse shear stresses (SYZ and SXZ) are assumed to be constant through the thickness and the transverse shear strains are assumed to be small in a large strain analysis. ) ( ) ( ( ) ] ) (19) With comparison eq. (19) with general equation of motion of single degree of freedom for free vibration structure as follows 12] (20)

6 International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:16 No:04 14 maximum percentage discrepancy between the analytical and numerical results is about (2%). Fig. 5. Shell93-8 Node Element Geometry. 4. RESULTS AND DISCUSSIONS The results evaluated are included the natural frequency of honeycomb sandwich simply supported plate with various angle and thickness of honeycomb part and different upper and lower plates parts thicknesses. The materials are assumed to be distributed to upper and lower plate parts and honeycomb part are aluminium materials with mechanical properties as And the dimensions of plates are (22) the results of natural frequency evaluated are shown in figures 6 to 9 using various parameters. Figs. 6 and 7 show the natural frequency of honeycomb plate structure with different Hexagonal Honeycomb thickness angle various honeycomb parts thickness with upper and lower plates parts are respectively. From figures it is seen that the natural frequency increases with increase the Hexagonal Honeycomb thickness and angle. Figs. 8 and 9 show the natural frequency of honeycomb plate structure with different upper and lower plates parts and various of Hexagonal Honeycomb angle and various honeycomb part thickness for Hexagonal Honeycomb thickness. From figures it is clear that the natural frequency is increasing with increase the Hexagonal Honeycomb angle and honeycomb part thickness. The natural frequency decreases with increasing the upper and lower plates parts thickness. The results indicate that the increasing of Hexagonal Honeycomb thickness and angle and increasing the honeycomb plate part thickness causes an increase in the strength to weight ratio therefore an increase in the natural frequency of plate. The increase of upper and lower plate parts causes a decrease in the strength to weight ratio therefore decreasing the natural frequency of honeycomb sandwich plate. Table I shows the comparison between theoretical and numerical results of fundamental natural frequency of honeycomb plate structure with different honeycomb parts and upper and lower plate parts thickness. It is clear that the suggested analytical solution of honeycomb plate gives a good agreement compared with numerical results evaluated by using finite element technique ANSYS program ver. 14. The Table I Comparison between analytical and numerical natural frequency results for various honeycomb and upper and lower plate thickness. Natural Frequency Case h h =1 cm h Lp =h Up = 1 mm t=0.05 mm h h =2 cm h Lp =h Up = 1 mm t=0.05 mm h h =0.5 cm h Lp =h Up = 1 mm t=0.05 mm h h =0.5 cm h Lp =h Up = 2 mm t=0.05 mm (rad/sec) (Deg Analytica Numerica ) l l a. h h=0.5 cm Discrepancy %

7 International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:16 No:04 15 b. h h=1 cm b. h h=1 cm c. h h=1.5 cm Fig. 6. Natural frequency of honeycomb plate with different dimensions of Regular Hexagonal Honeycomb and various honeycomb thickness for thickness of upper and lower plate hlp=hup=1 mm. c. h h=1.5 cm Fig. 7. Natural frequency of honeycomb plate with different dimensions of Regular Hexagonal Honeycomb and various honeycomb thickness for thickness of upper and lower plate hlp=hup=2 mm. a. h h=0.5 cm a. h Lp=h up=1 mm

8 International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:16 No:04 16 b. h Lp =h up =1.5 mm b. h h=1 cm c. h Lp=h up=2 mm Fig. 8. Natural frequency of honeycomb plate with different angle of Regular Hexagonal Honeycomb and various honeycomb thickness and thickness of upper and lower plate for t=0.025 mm and. a. h h=0.5 cm c. h h=2 cm Fig. 9. Natural frequency of honeycomb plate with different angle of Regular Hexagonal Honeycomb and various thicknesses of upper and lower plate and honeycomb thickness for t=0.025 mm and. 5. CONCLUSIONS Some concluding observations from the investigation are given below: 1. The increase of Hexagonal Honeycomb thickness causes an increase in the strength of honeycomb structure then increasing the natural frequency of honeycomb plate. 2. With increasing the Hexagonal Honeycomb angle increases the strength and stiffness of Honeycomb structure therefore with increase the Hexagonal Honeycomb angle increases the natural frequency of honeycomb sandwich plate structure. 3. The increase of honeycomb thickness causes an increase in the strength of honeycomb structure then increasing the natural frequency of honeycomb plate. 4. The increasing of upper and lower plate parts thickness of honeycomb sandwich plate causes a decrease in the strength to weight ratio of honeycomb plate structure. Therefore with increasing the upper and lower plate parts thickness decreases the natural frequency of honeycomb sandwich plate structure.

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