4. Cosserat plate dynamic field equations

The Cosserat plate field equations are obtained by substituting the relations Eqs. (74)–(83) into the system of Eqs. (65)–(70) similar to [10]:

<sup>1</sup> In the following formulas a subindex β = 1 if α = 2 and β = 2 if α = 1.

Distinctive Characteristics of Cosserat Plate Free Vibrations DOI: http://dx.doi.org/10.5772/intechopen.87044

$$L^2 \mathcal{U} = K \frac{\partial^2 \mathcal{U}}{\partial t^2} + \mathcal{F}(\eta),\tag{85}$$

where

L ¼ L<sup>11</sup> L<sup>12</sup> L<sup>13</sup> L<sup>14</sup> 0 L<sup>16</sup> kL<sup>13</sup> 0 L<sup>16</sup> L<sup>12</sup> L<sup>22</sup> L<sup>23</sup> L<sup>24</sup> L<sup>16</sup> 0 kL<sup>23</sup> L<sup>16</sup> 0 �L<sup>13</sup> �L<sup>23</sup> L<sup>33</sup> 0 L<sup>35</sup> L<sup>36</sup> L<sup>77</sup> L<sup>38</sup> L<sup>39</sup> L<sup>41</sup> L<sup>42</sup> 0 L<sup>44</sup> 00 0 0 0 �L<sup>16</sup> �L<sup>38</sup> 0 L<sup>55</sup> L<sup>56</sup> �kL<sup>35</sup> L<sup>58</sup> 0 L<sup>16</sup> 0 �L<sup>39</sup> 0 L<sup>56</sup> L<sup>66</sup> �kL<sup>36</sup> 0 L<sup>58</sup> �L<sup>13</sup> �L<sup>14</sup> L<sup>73</sup> 0 L<sup>35</sup> L<sup>36</sup> L<sup>77</sup> L<sup>78</sup> L<sup>79</sup> �L<sup>16</sup> �L<sup>78</sup> 0 L<sup>85</sup> L<sup>56</sup> �kL<sup>35</sup> L<sup>88</sup> kL<sup>56</sup> L<sup>16</sup> 0 �L<sup>79</sup> 0 L<sup>56</sup> L<sup>55</sup> �kL<sup>36</sup> kL<sup>56</sup> L<sup>99</sup> , K ¼ h3 <sup>ρ</sup> 00 0 0 0 0 0 0 <sup>h</sup><sup>3</sup> <sup>ρ</sup> 00 0 000 0 0 0 h <sup>ρ</sup> 0 0 000 0 <sup>h</sup><sup>2</sup> <sup>J</sup><sup>33</sup> 0 000 0 000 0 <sup>5</sup><sup>h</sup> <sup>J</sup><sup>11</sup> h <sup>J</sup><sup>12</sup> 00 0 000 0 <sup>5</sup><sup>h</sup> <sup>J</sup><sup>12</sup> h <sup>J</sup><sup>22</sup> 00 0 000 0 0 0 h <sup>ρ</sup> 0 0 000 0 0 0 0 <sup>2</sup><sup>h</sup> <sup>J</sup><sup>11</sup> h <sup>J</sup><sup>12</sup> 000 0 0 0 0 h <sup>J</sup><sup>12</sup> h <sup>J</sup><sup>22</sup> , <sup>U</sup> <sup>¼</sup> <sup>Ψ</sup>1; <sup>Ψ</sup>2; <sup>W</sup>; <sup>Ω</sup>3; <sup>Ω</sup><sup>0</sup> ; Ω<sup>0</sup> ; W <sup>∗</sup> ; Ω<sup>0</sup> ; Ω<sup>0</sup> � �<sup>T</sup> , <sup>F</sup>ð Þ¼ <sup>η</sup> �<sup>3</sup>h<sup>2</sup> <sup>λ</sup> <sup>3</sup>p1,1þ5<sup>p</sup> ð Þ <sup>2</sup>,<sup>1</sup> ð Þ <sup>λ</sup>þ2<sup>μ</sup> , �<sup>3</sup>h<sup>2</sup> <sup>λ</sup> <sup>3</sup>p1,2þ5<sup>p</sup> ð Þ <sup>2</sup>,<sup>2</sup> ð Þ <sup>λ</sup>þ2<sup>μ</sup> , �p1, <sup>0</sup>, <sup>0</sup>, <sup>0</sup>, <sup>h</sup><sup>2</sup> <sup>3</sup>p1þ4<sup>p</sup> ð Þ<sup>2</sup> , 0, 0 h i<sup>T</sup> <sup>p</sup><sup>1</sup> <sup>¼</sup> <sup>η</sup>p, p<sup>2</sup> <sup>¼</sup> ð Þ <sup>1</sup> � <sup>η</sup> <sup>p</sup>

,

The operators Lij are given as follows

L<sup>11</sup> ¼ c<sup>1</sup> ∂2 ∂x<sup>2</sup> 1 þ c<sup>2</sup> ∂2 ∂x<sup>2</sup> 2 � c3, L<sup>12</sup> ¼ ð Þ c<sup>1</sup> � c<sup>2</sup> ∂2 ∂x1x<sup>2</sup> , L<sup>13</sup> ¼ c<sup>11</sup> ∂ ∂x<sup>1</sup> , L<sup>14</sup> ¼ c<sup>12</sup> ∂ ∂x<sup>2</sup> , L<sup>16</sup> ¼ c13, L<sup>17</sup> ¼ k1c<sup>11</sup> ∂ ∂x<sup>1</sup> , L<sup>22</sup> ¼ c<sup>2</sup> ∂2 ∂x<sup>2</sup> 1 þ c<sup>1</sup> ∂2 ∂x<sup>2</sup> 2 � c3, L<sup>23</sup> ¼ c<sup>11</sup> ∂ ∂x<sup>2</sup> , L<sup>24</sup> ¼ �c<sup>12</sup> ∂ ∂x<sup>1</sup> , L<sup>33</sup> ¼ c<sup>3</sup> ∂2 ∂x<sup>2</sup> 1 þ ∂2 ∂x<sup>2</sup> 2 , L<sup>35</sup> ¼ �c<sup>13</sup> ∂ ∂x<sup>2</sup> , L<sup>36</sup> ¼ c<sup>13</sup> ∂ ∂x<sup>1</sup> , L<sup>38</sup> ¼ �c<sup>10</sup> ∂ ∂x<sup>2</sup> , L<sup>39</sup> ¼ c<sup>10</sup> ∂ ∂x<sup>1</sup> , L<sup>41</sup> ¼ �c<sup>12</sup> ∂ ∂x<sup>2</sup> , L<sup>42</sup> ¼ c<sup>12</sup> ∂ ∂x<sup>1</sup> , L<sup>44</sup> ¼ c<sup>6</sup> ∂2 ∂x<sup>2</sup> 1 þ ∂2 ∂x<sup>2</sup> 2 � 2c12, L<sup>55</sup> ¼ c<sup>7</sup> ∂2 ∂x<sup>2</sup> 1 þ c<sup>8</sup> ∂2 ∂x<sup>2</sup> 2 � 2c13, L<sup>56</sup> ¼ ð Þ c<sup>7</sup> � c<sup>8</sup> ∂2 ∂x1x<sup>2</sup> , L<sup>58</sup> ¼ �c9, L<sup>66</sup> ¼ c<sup>8</sup> ∂2 ∂x<sup>2</sup> 1 þ c<sup>7</sup> ∂2 ∂x<sup>2</sup> 2 � 2c13, L<sup>73</sup> ¼ c<sup>5</sup> ∂2 ∂x<sup>2</sup> 1 þ ∂2 ∂x<sup>2</sup> 2 , L<sup>77</sup> <sup>¼</sup> <sup>c</sup><sup>4</sup> ∂2 ∂x<sup>2</sup> 1 þ ∂2 ∂x<sup>2</sup> 2 , L<sup>78</sup> ¼ �c<sup>14</sup> ∂ ∂x<sup>2</sup> , L<sup>79</sup> ¼ c<sup>14</sup> ∂ ∂x<sup>1</sup> , L<sup>85</sup> ¼ c<sup>7</sup> ∂2 ∂x<sup>2</sup> 1 þ c<sup>8</sup> ∂2 ∂x<sup>2</sup> 2 � 2c13, L<sup>88</sup> ¼ c<sup>7</sup> ∂2 ∂x<sup>2</sup> 1 þ c<sup>8</sup> ∂2 ∂x<sup>2</sup> 2 � c15, L<sup>99</sup> ¼ c<sup>8</sup> ∂2 ∂x<sup>2</sup> 1 þ c<sup>7</sup> ∂2 ∂x<sup>2</sup> 2 � c15:

The coefficients ci are given as

$$\begin{aligned} c\_{1} &= \frac{h^{3}\mu(\lambda+\mu)}{3(\lambda+2\mu)}, & c\_{2} &= \frac{h^{3}(a+\mu)}{12}, & c\_{3} &= \frac{5h(a+\mu)}{6}, & c\_{4} &= \frac{5h(a-\mu)^{2}}{6(a+\mu)},\\ c\_{5} &= \frac{h(5a^{2}+6a\mu+5\mu^{2})}{6(a+\mu)}, & c\_{6} &= \frac{h^{3}\chi\epsilon}{3(\chi+\epsilon)}, & c\_{7} &= \frac{10h\chi(\beta+\gamma)}{3(\beta+2\gamma)}, & c\_{8} &= \frac{5h(\gamma+\epsilon)}{6},\\ c\_{9} &= \frac{10ha^{2}}{3(a+\mu)}, & c\_{10} &= \frac{5ha(a-\mu)}{3(a+\mu)}, & c\_{11} &= \frac{5h(a-\mu)}{6}, & c\_{12} &= \frac{h^{3}\alpha}{6},\\ c\_{13} &= \frac{5ha}{3}, & c\_{14} &= \frac{ha(5a+3\mu)}{3(a+\mu)}, & c\_{15} &= \frac{2ha(5a+4\mu)}{3(a+\mu)}.\end{aligned}$$

## 5. Numerical validation

For the validation purposes we provide the algorithm and computation results for the three-dimensional Cosserat elastodynamics. We also present the analysis of the numerical results based on the plate theory for the microelements of different shapes and orientations incorporated into the Cosserat plate.

#### 5.1 Analysis of Cosserat plate vibrations based on the three-dimensional theory

In our computations we consider the plates made of polyurethane foam—a material reported in the literature to behave Cosserat like—and the values of the technical elastic parameters presented in [15]: E ¼ 299:5MPa, ν ¼ 0:44, lt <sup>¼</sup> <sup>0</sup>:62mm, lb <sup>¼</sup> <sup>0</sup>:327 mm, <sup>N</sup><sup>2</sup> <sup>¼</sup> <sup>0</sup>:04. Taking into account that the ratio <sup>β</sup>=<sup>γ</sup> is equal to 1 for bending [15], these values of the technical constants correspond

Distinctive Characteristics of Cosserat Plate Free Vibrations DOI: http://dx.doi.org/10.5772/intechopen.87044

to the following values of Lamé and Cosserat parameters: λ ¼ 762:616MPa, μ ¼ 103:993MPa, α ¼ 4:333MPa, β ¼ 39:975MPa, γ ¼ 39:975MPa, ε ¼ 4:505MPa. We consider a low-density rigid foam usually characterized by the densities of 24–50 kg/m<sup>3</sup> [16]. In all further numerical computations we used the density value <sup>ρ</sup> <sup>¼</sup> 34 kg/m3 and different values the rotatory inertia <sup>J</sup>.

Let us consider the plate <sup>B</sup><sup>0</sup> being a rectangular cuboid 0½ �� ; <sup>a</sup> <sup>0</sup>, a�� � <sup>h</sup> 2 ; h 2 � �. Let the sets <sup>T</sup> and <sup>B</sup> be the top and the bottom surfaces contained in the planes <sup>x</sup><sup>3</sup> <sup>¼</sup> <sup>h</sup> <sup>2</sup> and <sup>x</sup><sup>3</sup> ¼ � <sup>h</sup> <sup>2</sup> respectively, and the curve Γ ¼ Γ<sup>1</sup> ∪ Γ<sup>2</sup> be the lateral part of the boundary:

$$\begin{aligned} \Gamma\_1 &= \left\{ (\mathfrak{x}\_1, \mathfrak{x}\_2, \mathfrak{x}\_3) : \mathfrak{x}\_1 \in \{0, a\}, \mathfrak{x}\_2 \in [0, a], \mathfrak{x}\_3 \in \left[ -\frac{h}{2}, \frac{h}{2} \right] \right\}, \\ \Gamma\_2 &= \left\{ (\mathfrak{x}\_1, \mathfrak{x}\_2, \mathfrak{x}\_3) : \mathfrak{x}\_1 \in [0, a], \mathfrak{x}\_2 \in \{0, a\}, \mathfrak{x}\_3 \in \left[ -\frac{h}{2}, \frac{h}{2} \right] \right\}, \end{aligned}$$

We solve the three-dimensional Cosserat equilibrium Eqs. (1)–(2) accompanied by the constitutive Eqs. (3)–(4) and strain-displacement and torsion-rotation relations Eq. (5) complemented by the following boundary conditions:

$$\Gamma\_1: \mu\_2 = \mathbf{0}, \mu\_3 = \mathbf{0}, \rho\_1 = \mathbf{0}, \sigma\_{11} = \mathbf{0}, \mu\_{12} = \mathbf{0}, \mu\_{13} = \mathbf{0};\tag{86}$$

$$
\Gamma\_2: \mu\_1 = 0, \mu\_3 = 0, \rho\_2 = 0, \sigma\_{22} = 0, \mu\_{21} = 0, \mu\_{23} = 0; \tag{87}
$$

$$T: \sigma\_{33} = p(\varkappa\_1, \varkappa\_2), \mu\_{33} = 0;\tag{88}$$

$$B: \sigma\_{33} = \mathbf{0}, \mu\_{33} = \mathbf{0}.\tag{89}$$

where the initial distribution of the pressure is given as <sup>p</sup> <sup>¼</sup> sin <sup>π</sup>x<sup>1</sup> a � � sin <sup>π</sup>x<sup>2</sup> a � � sinωt and the rotatory inertia tensor J is assumed to have a diagonal form

$$\mathbf{J} = \begin{bmatrix} J\_x & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & J\_y & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & J\_z \end{bmatrix}. \tag{90}$$

Using the method of separation of variables and taking into account the boundary conditions Eqs. (86)–(87), we express the kinematic variables in the form:

$$u\_1 = \cos\left(\frac{\pi\chi\_1}{a}\right)\sin\left(\frac{\pi\chi\_2}{a}\right)z\_1(\chi\_3)\sin a t,\tag{91}$$

$$u\_2 = \sin\left(\frac{\pi\chi\_1}{a}\right)\cos\left(\frac{\pi\chi\_2}{a}\right)z\_2(\chi\_3)\sin a t,\tag{92}$$

$$u\_3 = \sin\left(\frac{\pi\chi\_1}{a}\right)\sin\left(\frac{\pi\chi\_2}{a}\right)z\_3(\chi\_3)\sin\alpha t,\tag{93}$$

$$\phi\_1 = \sin\left(\frac{\pi\varkappa\_1}{a}\right)\cos\left(\frac{\pi\varkappa\_2}{a}\right)z\_4(\varkappa\_3)\sin\alpha t,\tag{94}$$

$$\phi\_2 = \cos\left(\frac{\pi\chi\_1}{a}\right)\sin\left(\frac{\pi\chi\_2}{a}\right)\mathbf{z}\_5(\chi\_3)\sin\alpha t,\tag{95}$$

$$\phi\_3 = \cos\left(\frac{\pi\chi\_1}{a}\right)\cos\left(\frac{\pi\chi\_2}{a}\right)z\_6(\chi\_3)\sin\alpha t,\tag{96}$$

where the functions zið Þ x<sup>3</sup> represent the transverse variations of the kinematic variables.

If we substitute the expressions Eqs. (91)–(96) into Eqs. (3)–(4) and then into Eqs. (1)–(2), we will obtain the following eigenvalue problem

$$\mathbf{Bz} = \omega^2 \mathbf{A} \mathbf{z} \tag{97}$$

where

B ¼ b1L<sup>2</sup> þ b2L<sup>0</sup> b3L<sup>0</sup> b4L<sup>1</sup> 0 �b5L<sup>1</sup> �b6L<sup>0</sup> b3L<sup>0</sup> b1L<sup>2</sup> þ b2L<sup>0</sup> b4L<sup>1</sup> b5L<sup>1</sup> 0 b6L<sup>0</sup> �b4L<sup>1</sup> b4L<sup>1</sup> b7L<sup>2</sup> b6L<sup>0</sup> �b6L<sup>0</sup> 0 �b5L<sup>1</sup> b6L<sup>0</sup> b9L<sup>2</sup> þ b10L<sup>0</sup> b11L<sup>0</sup> b12L<sup>1</sup> b5L<sup>1</sup> 0 �b6L<sup>0</sup> b11L<sup>0</sup> b9L<sup>2</sup> þ b10L<sup>0</sup> b12L<sup>1</sup> �b6L<sup>0</sup> b6L<sup>0</sup> 0 �b12L<sup>1</sup> �b12L<sup>1</sup> b13L<sup>2</sup> þ b2L<sup>14</sup> , (98) A ¼ �a2<sup>ρ</sup> 00 0 0 0 �a2<sup>ρ</sup> <sup>0000</sup> 0 0 �a2<sup>ρ</sup> <sup>000</sup> �a<sup>2</sup>Jx 0 0 000 0 �a<sup>2</sup>Jy <sup>0</sup> 000 0 0 �a<sup>2</sup>Jz , (99) z ¼ ½ � z1, z2, z3, z4, z5, z<sup>6</sup> T, (100)

and the differential operators Li are defined as

$$L\_0 = I, \quad L\_1 = \frac{d}{d\mathbf{x}\_3}, \quad L\_2 = \frac{d^2}{d\mathbf{x}\_3^2}$$

and the coefficients bi are defined as

$$\begin{aligned} b\_1 &= a^2(\mu + a), & b\_2 &= -\pi^2(a + \lambda + 3\mu), & b\_3 &= -\pi^2(\lambda + \mu - a), \\ b\_4 &= a\pi(\lambda + \mu - a), & b\_5 &= 2a^2a, & b\_6 &= 2a\pi a, \\ b\_7 &= a^2(2\mu + \lambda), & b\_8 &= -2\pi^2(a + \mu), & b\_9 &= a^2(\chi + \varepsilon), \\ b\_{10} &= -\pi^2(\beta + \varepsilon + 3\chi), & b\_{11} &= -\pi^2(\beta + \chi - \varepsilon), & b\_{12} &= -a\pi(\beta + \chi - \varepsilon), \\ b\_{13} &= a^2(\beta + 2\chi), & b\_{14} &= -2\pi^2(\chi + \varepsilon) - 4a^2\alpha \end{aligned}$$

The system of differential Eq. (97) is complemented by the following boundary conditions <sup>D</sup><sup>z</sup> <sup>¼</sup> <sup>D</sup><sup>0</sup> for <sup>x</sup><sup>3</sup> <sup>¼</sup> <sup>h</sup> and <sup>D</sup><sup>z</sup> <sup>¼</sup> 0 for <sup>x</sup><sup>3</sup> ¼ � <sup>h</sup> .

$$\mathbf{D} = \begin{bmatrix} d\_1 L\_1 & 0 & d\_2 L\_0 & 0 & -d\_3 L\_0 & 0 \\ 0 & d\_1 L\_1 & d\_2 L\_0 & d\_3 L\_0 & 0 & 0 \\ d\_4 L\_0 & d\_4 L\_0 & d\_5 L\_1 & 0 & 0 & 0 \\ 0 & 0 & 0 & d\_6 L\_1 & 0 & d\_7 L\_0 \\ 0 & 0 & 0 & 0 & d\_6 L\_1 & d\_7 L\_0 \\ 0 & 0 & 0 & d\_8 L\_0 & d\_8 L\_0 & d\_9 L\_1 \end{bmatrix}, \tag{101}$$
 
$$D\_0 = \begin{bmatrix} 0 & 0 & a & a, & 0, & 0, & 0 \end{bmatrix}^T,\tag{102}$$

and the coefficients di are defined as

$$\begin{aligned} d\_1 &= a(\mu + a), & d\_2 &= -\pi(\mu - a), & d\_3 &= 2a a, \\ d\_4 &= a(\lambda + 2\mu), & d\_5 &= -\pi \lambda, & d\_6 &= a(\gamma + \varepsilon), \\ d\_7 &= a(\gamma - \varepsilon), & d\_8 &= \pi \beta, & d\_9 &= a(\beta + 2\gamma). \end{aligned}$$

Distinctive Characteristics of Cosserat Plate Free Vibrations DOI: http://dx.doi.org/10.5772/intechopen.87044

The idea for the solution of the eigenvalue problem Eq. (97) is based on the following algorithm:

Step 1. Fix certain frequency value.

We fix certain value of the frequency ω and force the Cosserat body to vibrate at this frequency.

Step 2. Solve the three-dimensional Cosserat system of equations.

Mathematically, fixing certain value of ω implies that three-dimensional system of Eq. (97) has a constant right-hand side and therefore can be solved for the kinematic variables as a static system of equations. We solve the system Eq. (97) using the high-precision Runge-Kutta method incorporated in Mathematica software similar to how it was done in [7].

Step 3. Find large amplitudes of the kinematic variables.

We runω through an interval of positive real values and take note where the solution changes its sign and the amplitude of the solutions starts to grow indefinitely. This corresponds to the oscillation of the Cosserat body at its resonant frequency. Thus, when the frequency ω coincides with the natural frequency of the plate the resonance will occur and the large amplitude linear vibrations can be observed (Figure 1).

The comparison of the eigenfrequencies of the Cosserat plate with the eigenfrequencies of the three-dimensional Cosserat elasticity is given in the Table 1. The rotatory inertia principle moments used are Jx ¼ 0:001, Jy ¼ 0:001, Jz ¼ 0:001, which represent a ball-shaped microelement (Figure 2). The relative error of the natural macro frequencies associated with the rotation of the middle plane and the flexural motion is less than 1%.

#### Figure 1.

Large amplitude linear vibrations of the Cosserat body forced to vibrate close to its natural frequency ω1.


#### Table 1.

Comparison of the eigenfrequencies ω<sup>i</sup> (Hz) with the exact values of the 3D Cosserat elasticity.

Figure 2.

Ball-shaped micro-elements: Jx ¼ 0:001, Jy ¼ 0:001, Jz ¼ 0:001 (left) and horizontally stretched ellipsoid micro-elements: Jx ¼ 0:002, Jy ¼ 0:001, Jz ¼ 0:0001 right ð Þ.

#### 5.2 Analysis of Cosserat plate vibrations based on the plate theory

We consider a plate a � a of thickness h with the boundary G ¼ G<sup>1</sup> ∪ G<sup>2</sup>

$$G\_1 = \{ (\mathfrak{x}\_1, \mathfrak{x}\_2) : \mathfrak{x}\_1 \in \{ 0, a \}, \mathfrak{x}\_2 \in [0, a] \}$$

$$G\_2 = \{ (\mathfrak{x}\_1, \mathfrak{x}\_2) : \mathfrak{x}\_2 \in \{ 0, a \}, \mathfrak{x}\_1 \in [0, a] \}$$

and the following hard simply supported boundary conditions [7]:

$$G\_1: W = 0, \, W^\* = 0, \, \Psi\_2 = 0, \, \Omega\_1^0 = 0, \, \hat{\Omega}\_1^0 = 0, \, \Omega\_3 = 0, \, \frac{\partial \Psi\_1}{\partial n} = 0, \, \frac{\partial \Omega\_2^0}{\partial n} = 0, \, \frac{\partial \hat{\Omega}\_2^0}{\partial n} = 0;$$

$$G\_2: W = 0, \, W^\* = 0, \, \Psi\_1 = 0, \, \Omega\_2^0 = 0, \, \hat{\Omega}\_2^0 = 0, \, \Omega\_3 = 0, \, \frac{\partial \Psi\_2}{\partial n} = 0, \, \frac{\partial \Omega\_1^0}{\partial n} = 0, \, \frac{\partial \hat{\Omega}\_1^0}{\partial n} = 0.$$

Similar to [12] we apply the method of separation of variables for the eigenvalue problem Eq. (85) to solve for the kinematic variables Ψα, W, Ω3, Ω<sup>0</sup> <sup>α</sup>, W <sup>∗</sup> and Ω<sup>0</sup> α. The kinematic variables can be further expressed in the following form

Ψnm <sup>1</sup> ¼ A<sup>1</sup> cos nπx<sup>1</sup> a sin <sup>m</sup>πx<sup>2</sup> a sin ð Þþ <sup>ω</sup><sup>t</sup> <sup>B</sup><sup>1</sup> sin <sup>n</sup>πx<sup>1</sup> a cos mπx<sup>2</sup> a sin ð Þ <sup>ω</sup><sup>t</sup> , Ψnm <sup>2</sup> <sup>¼</sup> <sup>A</sup><sup>2</sup> sin <sup>n</sup>πx<sup>1</sup> a cos mπx<sup>2</sup> a sin ð Þþ <sup>ω</sup><sup>t</sup> <sup>B</sup><sup>2</sup> cos nπx<sup>1</sup> a sin <sup>m</sup>πx<sup>2</sup> a sin ð Þ <sup>ω</sup><sup>t</sup> , <sup>W</sup>nm <sup>¼</sup> <sup>A</sup><sup>3</sup> sin <sup>n</sup>πx<sup>1</sup> a sin <sup>m</sup>πx<sup>2</sup> a sin ð Þþ <sup>ω</sup><sup>t</sup> <sup>B</sup><sup>3</sup> cos nπx<sup>1</sup> a cos mπx<sup>2</sup> a sin ð Þ <sup>ω</sup><sup>t</sup> , Ωnm <sup>3</sup> ¼ A<sup>4</sup> cos nπx<sup>1</sup> a cos mπx<sup>2</sup> a sin ð Þþ <sup>ω</sup><sup>t</sup> <sup>B</sup><sup>4</sup> sin <sup>n</sup>πx<sup>1</sup> a sin <sup>m</sup>πx<sup>2</sup> a sin ð Þ <sup>ω</sup><sup>t</sup> , Ω<sup>0</sup>,nm <sup>1</sup> <sup>¼</sup> <sup>A</sup><sup>5</sup> sin <sup>n</sup>πx<sup>1</sup> a cos mπx<sup>2</sup> a sin ð Þþ <sup>ω</sup><sup>t</sup> <sup>B</sup><sup>5</sup> cos nπx<sup>1</sup> a sin <sup>m</sup>πx<sup>2</sup> a sin ð Þ <sup>ω</sup><sup>t</sup> , Ω<sup>0</sup>,nm <sup>2</sup> ¼ A<sup>6</sup> cos nπx<sup>1</sup> a sin <sup>m</sup>πx<sup>2</sup> a sin ð Þþ <sup>ω</sup><sup>t</sup> <sup>B</sup><sup>6</sup> sin <sup>n</sup>πx<sup>1</sup> a cos mπx<sup>2</sup> a sin ð Þ <sup>ω</sup><sup>t</sup> , <sup>W</sup> <sup>∗</sup> ,nm <sup>¼</sup> <sup>A</sup><sup>7</sup> sin <sup>n</sup>πx<sup>1</sup> a sin <sup>m</sup>πx<sup>2</sup> a sin ð Þþ <sup>ω</sup><sup>t</sup> <sup>B</sup><sup>7</sup> cos nπx<sup>1</sup> a cos mπx<sup>2</sup> a sin ð Þ <sup>ω</sup><sup>t</sup> , <sup>Ω</sup>^ <sup>0</sup>,nm <sup>1</sup> <sup>¼</sup> <sup>A</sup><sup>8</sup> sin <sup>n</sup>πx<sup>1</sup> a cos mπx<sup>2</sup> a sin ð Þþ <sup>ω</sup><sup>t</sup> <sup>B</sup><sup>8</sup> cos nπx<sup>1</sup> a sin <sup>m</sup>πx<sup>2</sup> a sin ð Þ <sup>ω</sup><sup>t</sup> , <sup>Ω</sup>^ <sup>0</sup>,nm <sup>2</sup> ¼ A<sup>9</sup> cos nπx<sup>1</sup> a sin <sup>m</sup>πx<sup>2</sup> a sin ð Þþ <sup>ω</sup><sup>t</sup> <sup>B</sup><sup>9</sup> sin <sup>n</sup>πx<sup>1</sup> a cos mπx<sup>2</sup> a sin ð Þ <sup>ω</sup><sup>t</sup> ,

where Ai and Bi are constants.


Distinctive Characteristics of Cosserat Plate Free Vibrations DOI: http://dx.doi.org/10.5772/intechopen.87044

Table 2.

Eigenfrequencies ω<sup>11</sup> <sup>i</sup> (Hz) for different shapes of micro-elements.

We solve an eigenvalue problem by substituting these expressions into the system of Eq. (85). The obtained nine sequences of positive eigenfrequencies ωnm i are associated with the rotation of the middle plane (ωnm <sup>1</sup> and ωnm <sup>2</sup> ), flexural motion and its transverse variation (ωnm <sup>3</sup> and ωnm <sup>7</sup> ), micro rotatory inertia (ωnm <sup>4</sup> , ωnm <sup>5</sup> and ωnm <sup>6</sup> ) and its transverse variation (ωnm <sup>8</sup> and ωnm <sup>9</sup> ) [12].

We perform all our numerical simulations for a ¼ 3:0 m and h ¼ 0:1 m. We consider different forms of micro elements: ball-shaped elements, horizontally and vertically stretched ellipsoids (see Figure 2). For simplicity we will use the notation ω<sup>i</sup> for the first elements ω<sup>11</sup> <sup>i</sup> of the sequences ωnm <sup>i</sup> . The results of the computations are given in the Table 2. The shape of the micro-elements does not effect the natural macro frequencies ω<sup>1</sup> and ω<sup>2</sup> associated with the rotation of the middle plane and ω<sup>3</sup> and ω<sup>7</sup> associated with the flexural motion and its transverse variation. The ellipsoid elements have higher micro frequencies associated with the micro rotatory inertia (ω4, ω<sup>5</sup> and ω6) and its transverse variation (ω<sup>8</sup> and ω9), than the ball-shaped elements.

Let Jx, Jy and Jz be the principal moments of inertia of the microelements corresponding to the principal axes of their rotation. We assume that the quantities Jx, Jy and Jz are constant throughout the plate B0. If the microelements are rotated around the z-axis by the angle θ the rotatory inertia tensor J can be expressed as

$$\mathbf{J} = \begin{pmatrix} J\_x \cos^2 \theta + J\_y \sin^2 \theta & \begin{pmatrix} J\_x - J\_y \end{pmatrix} \sin 2\theta & \mathbf{0} \\\ \begin{pmatrix} J\_x - J\_y \end{pmatrix} \sin 2\theta & J\_x \sin^2 \theta + J\_y \cos^2 \theta & \mathbf{0} \\\ \mathbf{0} & \mathbf{0} & J\_z \end{pmatrix} \tag{103}$$


Table 3.

Eigenfrequencies ω<sup>11</sup> <sup>i</sup> (Hz) for different angles of rotation of horizontal ellipsoid micro-elements.

Figure 3. Micro frequencies ω4, ω5, ω8, ω<sup>6</sup> and ω9.

The eigenfrequencies for different angles of microrotation of the microelements are given in the Table 3 and the Figure 3. The rotatory inertia principle moments used are Jx ¼ 0:002, Jy ¼ 0:001, Jz ¼ 0:0001, which represent a horizontally stretched ellipsoid microelement. The case when the microelements are not aligned with the edges of the plate the model predicts some additional natural frequencies related with the microstructure of the material.
