In this study, the effect of radius of curvature on the natural frequency of isotropic square thin plate is investigated. The models examined are obtained by simply changing the radius of curvature by keeping the length constant from the flat plate to the semicircle model. The free vibration analysis of the plates fixed on two straight edges is performed by the finite element method. As a finite element type, the four-node quadrilateral rectangular element type, which has a total of twenty four generalized coordinates, is used. Out-of-plane theory and in-plane theory are used together to form the curve model. The accuracy and validity of the theory are controlled with the data obtained from the finite element program. The effect of the curvature radius on the first five natural frequencies and the mode shapes of these natural frequencies is given in tables and graphs.

Flat and curved plates with different geometries are used as structural elements in many engineering fields. Different theories of plates for a cylindrical structure are also the subject of different studies. Raffo and Quintana propose a general algorithm to obtain approximate analytical solutions for the study of the free vibrations of a rectangular anisotropic thin plate with an internal curved line hinge and general restraints

It is well-known knowledge that the stiffness of the plates increases with increasing the curvature. This information is used in bridge models, dam structures, and vehicle designs. However, the mode shapes that change with the change of curvature should also be considered as a design parameter. Furthermore, the continued increase in curvature sharply reduces the stiffness of the geometry. In this study, the natural frequency analysis of the fixed-length isotropic thin square plate, starting from the flat plate model to the semicircle model fixed on both edges, is performed. The results are confirmed by the finite element program. In addition, the mode shapes affected by the first five natural frequency changes are shown.

To create flat and curved plate models, the four-node quadrilateral element is selected for using the finite element method. While making this selection it is known that to create the curvilinear design, the plate must be rotatable around any of its axes. The selected model of finite elements must therefore also contain six generalized coordinates in each node. To reach this finite element model, out-of-plane theory and in-plane theory are used together.

The principle of flexural vibration is based on the out-of-plane theory. The finite element model to be used for the form of bending motion must have a minimum of twelve corner displacements corresponding to three generalized coordinates at each corner,

In equation (1), the shape function is given for the flexural vibration of plate [12].

(_{1}

Where

[D]=(Eh^3)/(12(1-ν^2)) [■(1&ν&0@ν&1&0@0&0&(1-ν)/2)]

The mass matrix is also calculated using the equation (5).

Where

The finite element model to be used for the in-plane vibration form must have eight displacements at each corner including _{z}

The in-plane theory shape function is given in equation (6)

The components

Where _{i}

Subsequently, it needs to express the derivatives of the displacement functions, which are in

Where J is the Jacobian matrix.

J=[■(∂x/∂ξ&∂y/∂ξ@∂x/∂η&∂y/∂η)]

Equation (8) now yield:

where A is given by:

A=1/detJ [■(■(〖 J〗_22&-J_12 )&■(0 &0)@■(0 &0)&-■(J_21 &J_11 )@-■(J_21&J_11 )&■(J_22&-J_12 ))]

From the interpolation equations, equation (13) is obtained:

By substituting equation (13) into (11), strain displacement matrix is obtained.

Element stiffness matrix _{2}

Where

[D]=E/((1-ν^2)) [■(1&ν&0@ν&1&0@0&0&(1-ν)/2)]

The element mass matrix is also given in equation (17).

The stiffness and the mass matrices of the finite element model are obtained by combining the out-of-plane theory and the in-plane theory. The relationship is defined in equation (18).

[Out of plane,〖 K〗_1 &〖 M〗_1 ]_(3×3)+[In plane,〖 K〗_2 &〖 M〗_2 ]_(2×2)=[■(3×3&0@0&2×2)]_(5×5)

This model has six degrees of freedom, but it has matrices of stiffness and mass (_{z}

〖K_e,M_e=[■(3×3&0@0&2×2)]〗_(5×5)+θ_z=[■(3×3&0&0@0&2×2&0@0&0&1×1)]_(6×6)

The value of _{z}

To examine the effect of the radius of curvature, firstly the flat plate fixed on both sides. At this point, the radius of curvature is theoretically infinite. For the transition to the curved plate, the radius of curvature is then reduced from infinite to 160 mm, where the structure is exactly a semicircle. The length of the plate is kept constant so that the natural frequency change can be seen during the radius change. Figure-4 shows a linear representation of curved plate models ranging from flat model to semicircle.

To obtain these curve models with flat plate, the flat plate finite element model is rotated about the y axis. Rotation of the coordinate system is given in Figure-5.

The transformation matrix used to perform the rotation mathematically, is given in Table-1.

θ x | θ y | w | u | v | θ z | |

cos(θ) | 0 | 0 | 0 | 0 | sin(θ) | θ x |

0 | 1 | 0 | 0 | 0 | 0 | θ y |

0 | 0 | cos(θ) | -sin(θ) | 0 | 0 | w |

0 | 0 | sin(θ) | cos(θ) | 0 | 0 | u |

0 | 0 | 0 | 0 | 1 | 0 | v |

-sin(θ) | 0 | 0 | 0 | 0 | cos(θ) | θ z |

The stiffness and mass matrices can be obtained through the transformation matrix is given in equation (20).

Where _{e}_{e}_{r}_{r}_{e}_{e}

[M] ({q} ) ̈+[[K]]{q}=0

Natural frequencies can be obtained via equation (22), where 𝜆 includes natural frequency parameter ^{2}

The geometric and material properties are: ^{3},

To verify the reliability and validity of the present model, the natural frequency parameters obtained from this study are compared with the results obtained from ANSYS software. Representative flat and semicircle models with two sides fixed are shown in Figure-6. Table-2 shows a comparison of the results between the present study and ANSYS.

Radius (mm) | λ1 | λ2 | λ3 | λ4 | λ5 | |||||

ANSYS | Present | ANSYS | Present | ANSYS | Present | ANSYS | Present | ANSYS | Present | |

Flat | 3.70 | 3.69 | 4.41 | 4.40 | 7.25 | 7.25 | 10.18 | 10.19 | 11.18 | 11.19 |

5000 | 5.90 | 5.95 | 6.37 | 6.41 | 8.77 | 8.76 | 10.18 | 10.21 | 11.32 | 11.35 |

4500 | 6.30 | 6.35 | 6.74 | 6.79 | 9.08 | 9.08 | 10.18 | 10.21 | 11.36 | 11.39 |

4000 | 6.81 | 6.88 | 7.23 | 7.29 | 9.51 | 9.52 | 10.17 | 10.21 | 11.40 | 11.44 |

3500 | 7.49 | 7.57 | 7.88 | 7.95 | 10.09 | 10.11 | 10.17 | 10.21 | 11.47 | 11.51 |

3000 | 8.42 | 8.51 | 8.77 | 8.86 | 10.17 | 10.21 | 10.93 | 10.96 | 11.58 | 11.62 |

2500 | 9.72 | 9.83 | 10.04 | 10.15 | 10.17 | 10.20 | 11.75 | 11.79 | 12.18 | 12.23 |

2000 | 10.17 | 10.19 | 11.60 | 11.73 | 11.93 | 12.06 | 12.06 | 12.11 | 14.15 | 14.22 |

1500 | 10.15 | 10.18 | 12.68 | 12.76 | 14.28 | 14.42 | 14.72 | 14.87 | 17.49 | 17.57 |

1000 | 10.10 | 10.12 | 14.25 | 14.40 | 16.91 | 17.02 | 17.73 | 17.84 | 21.58 | 21.71 |

500 | 9.80 | 9.82 | 17.83 | 17.91 | 18.76 | 18.85 | 19.69 | 20.00 | 29.10 | 29.23 |

160 | 7.18 | 7.22 | 15.75 | 15.88 | 15.92 | 16.08 | 25.36 | 25.73 | 29.20 | 29.58 |

Table-2 shows that the natural frequency parameter results of the present model and the error rates respect to ANSYS results are quite consistent. The results of the first five natural frequency parameters of the curved plate model starting from 5000 mm radius until the semicircle changes are given in Figure-7.

When the results of the natural frequency parameters from the flat plate model to the semicircle plate model are examined, it is seen that from Figure-7, each natural frequency parameter curve changes its behavior at a different radius of curvature. Different mode shapes are observed with these natural frequency parameter curve changes. Moreover, each natural frequency parameter shows a sharp decrease when the radius of curvature is close to the semicircle.

When the mode shapes belong the first five natural frequencies are examined, the constant mode shape occurs up to a radius of curvature of 2400 mm for the first natural frequency parameter, and then a different mode shape occurs. The two different mode shapes are given in the Figure-8. Also, all mode shape graphs are given in the rectangular plate plane.

Figure 8-a represents the mode shape for the radius of curvature value between the flat plate and the 2400 mm, Figure 8-b represents the mode shape in the section where the radius varies between 2400 mm and 160 mm.

Figure 9-a represents the mode shape for the radius of curvature value between the flat plate and the 2500 mm. Figure 9-b represents the mode shape for all radius of curvature in the range of 2500 to 2000 mm, Figure 9-c represents the mode shape for all radius of curvature in the range of 2000 to 600 mm, Figure 9-d represents the mode shape for all radius of curvature in the range of 600 to 160 mm.

Figure 10-a represents the mode shape for the radius of curvature value between the flat plate and the 3500 mm, Figure 10-b represents the mode shape for the radius of curvature in the range of 3500 to 2500 mm, Figure 10-c represents the mode shape for the radius of curvature in the range of 2500 to 2000 mm, Figure 10-d represents the mode shape for the radius of curvature in the range of 2000 to 800 mm, Figure 10-e represents the mode shape for the radius of curvature in the range of 800 to 500 mm, Figure 10-f represents the mode shape for the radius of curvature in the range of 500 to 160 mm

Figure 11-a represents the mode shape for the radius of curvature value between the flat plate and the 3500 mm, Figure 11-b represents the mode shape for the radius of curvature in the range of 3500 to 2700 mm, Figure 11-c represents the mode shape for the radius of curvature in the range of 2700 to 2000 mm, Figure 11-d represents the mode shape for all radius of curvature in the range of 2000 to 1600 mm, Figure 11-e represents the mode shape for the radius of curvature in the range of 1600 to 600 mm, Figure 11-f represents the mode shape for the radius of curvature in the range of 600 to 160 mm.

Figure 12-a represents the fifth mode shape for the radius of curvature value between the flat plate and the 2700 mm, Figure 12-b represents the fifth mode shape for the radius of curvature in the range of 2700 to 1400 mm, Figure 12-c represents the fifth mode shape for the radius of curvature in the range of 1400 to 400 mm, Figure 12-d represents the fifth mode shape for the radius of curvature in the range of 400 to 160 mm.

Figure 8-12 shows the mode shapes resulting from changes in the natural frequency parameter curves given in Figure 7. Consequently, different mode shapes occur at different points where the shape of the natural frequency parameter curvature changes. In addition, it is seen that the natural frequency parameter value shows a sudden decrease for all five natural frequencies where the geometry is closer to the semicircle.

In this study, the effect of the radius of curvature on the natural frequency parameter is investigated by keeping the plate length constant. The first five natural frequencies are examined for all radius of curvature from the flat plate model to the semicircle plate model. Moreover, the mode shape changes due to the radius of curvature are also examined. According to the results obtained:

As the curvature of the plate increases with constant length, the natural frequency values also increase to a certain level for all natural frequency parameters.

By changing the radius of the curvature, more than one mode shape occurs for each natural frequency parameter.

With the change of the radius of curvature, the natural frequency parameter curve behaves differently for each natural frequency parameter. Each change in the curves shows that a different mode shape occurs.

As the radius of curvature moves the plate closer to the semicircle geometry, the increase in all natural frequency parameters stops, and then all values begin to decrease sharply.