This paper deals with the in-plane vibration of circular annular disks under combinations of different boundary conditions at the inner and outer edges. The in-plane free vibration of an elastic and isotropic disk is studied on the basis of the two-dimensional linear plane stress theory of elasticity. The exact solution of the in-plane equation of equilibrium of annular disk is attainable, in terms of Bessel functions, for uniform boundary conditions. The frequency equations for different modes can be obtained from the general solutions by applying the appropriate boundary conditions at the inner and outer edges. The presented frequency equations provide the frequency parameters for the required number of modes for a wide range of radius ratios and Poisson's ratios of annular disks under clamped, free, or flexible boundary conditions. Simplified forms of frequency equations are presented for solid disks and axisymmetric modes of annular disks. Frequency parameters are computed and compared with those available in literature. The frequency equations can be used as a reference to assess the accuracy of approximate methods.

The out-of-plane vibration properties of circular disks subjected to a variety of boundary conditions have been extensively investigated (e.g., [

The in-plane vibration of circular disks was first attempted by Love [

The in-plane vibration analyses in the above reported studies were limited to solid disks with either free or clamped outer edge. The in-plane free vibration of annular disks with different boundary conditions has also been addressed in a few studies. The variations in the in-plane vibration frequency parameters of annular disks with free edges were investigated as function of the size of the opening by Ambati et al. [

The above reported studies on in-plane vibration of solid and annular disks have employed different methods of analyses. The finite-element technique has also been used to examine the validity of analytical methods (e.g., [

The equations of the in-plane vibration of a circular disk are formulated for an annular disk shown in Figure _{r}_{θ}

Geometry and coordinate system used for in-plane vibration analysis of an annular disk.

Following Love’s theory [

Assuming harmonic oscillations corresponding to a natural frequency

Equations (

The radial and circumferential displacements can then be expressed in terms of the Bessel functions by substituting for

Equations (

In a similar manner, the solution must satisfy the following for the clamped inner edge (_{r}

For the clamped inner and outer edges, the equations for the boundary conditions can be obtained directly from (

The in-plane vibration analysis of a solid disk can be shown as a special case of the above generalized formulations. Upon eliminating the coefficients associated with Bessel function of the second kind, Equations (

Frequency equations for the solid disks corresponding to free and clamped edge conditions.

Boundary conditions at | Clamped | Free | |

Radial | |||

Circumferential | |||

Frequency equations of axisymmetric modes for annular disks.

Boundary conditions | radial | Circumferential | |

inner | outer | ||

Clamped | Clamped | ||

Free | Free | ||

Clamped | Free | ||

Free | Clamped |

In the above analysis, the boundary conditions considered are either clamped or free. However, Flexible boundary conditions may be considered more representative of many practical situations. The proposed formulations can be further employed to study the in-plane vibration of solid as well as annular disks with flexible boundary conditions. Artificial springs may be applied to describe the flexible boundary conditions at the inner or the outer edge of an annular disk. A number of studies on the analysis of out-of-plane vibration characteristics of circular plates and cylindrical shells have employed uniformly distributed artificial springs around the edge to represent a flexible boundary conditions or a flexible joint [

The effects of flexible boundary conditions on the in-plane free vibration of circular disks have been considered in a recent study by the authors [

The frequency parameters (

Exact frequency parameters of in-plane vibration of a solid disk with free edge (

Mode | ||||
---|---|---|---|---|

1 | 1.6176 | 1.3877 | 2.1304 | 2.7740 |

2 | 3.5291 | 2.5112 | 3.4517 | 4.4008 |

3 | 4.0474 | 4.5208 | 5.3492 | 6.1396 |

4 | 5.8861 | 5.2029 | 6.3695 | 7.4633 |

5 | 6.9113 | 6.7549 | 7.6186 | 8.5007 |

6 | 7.7980 | 8.2639 | 9.3470 | 10.2350 |

7 | 9.6594 | 8.7342 | 9.8366 | 11.0551 |

Exact frequency parameters of in-plane vibration of a solid disk with clamped edge (

Mode | ||||
---|---|---|---|---|

1 | 1.9441 | 3.0185 | 3.0185 | 4.7021 |

2 | 3.1126 | 4.0127 | 4.0127 | 5.8985 |

3 | 4.9104 | 5.7398 | 5.7398 | 7.3648 |

4 | 5.3570 | 6.7079 | 6.7079 | 8.9816 |

5 | 6.7763 | 7.6442 | 7.6442 | 9.5296 |

6 | 8.4938 | 9.4356 | 9.4356 | 11.1087 |

7 | 8.6458 | 9.9894 | 9.9894 | 12.5940 |

The exact frequency parameters for the annular disks were subsequently obtained under different combinations of boundary conditions at the inner and outer edges. The edge conditions are presented for the inner followed by that of the outer edge. For instance, a “Free-Clamped” condition refers to free inner edge and clamped outer edge. The solutions obtained for conditions involving free and clamped edges (“Free-Free”, “Free-Clamped”, “Clamped-Clamped”, and “Clamped-Free”) are compared with those reported by Irie et al. [

Frequency parameters of in-plane vibration of an annular disk with “Free-Free” conditions.

Radius ratio ( | ||||||||

Reference [ | present study | Reference [ | Present study | Reference [ | Present study | Reference [ | Present study | |

0.2 | 1.652 | 1.651 | 1.110 | 1.110 | 2.071 | 2.071 | 2.767 | 2.766 |

3.842 | 3.842 | 2.403 | 2.402 | 3.401 | 3.400 | 4.389 | 4.388 | |

0.4 | 1.683 | 1.682 | 0.721 | 0.721 | 1.618 | 1.619 | 2.482 | 2.482 |

4.044 | 4.044 | 2.451 | 2.450 | 3.346 | 3.346 | 4.227 | 4.226 |

Frequency parameters of in-plane vibration of an annular disk with “Free-Clamped” conditions.

Radius ratio ( | ||||||||

Reference [ | Present study | Reference [ | Present study | Reference [ | Present study | Reference [ | Present study | |

0.2 | 2.104 | 2.103 | 2.553 | 2.553 | 3.688 | 3.688 | 4.712 | 4.711 |

3.303 | 3.302 | 3.948 | 3.948 | 4.859 | 4.858 | 5.894 | 5.893 | |

0.4 | 2.517 | 2.517 | 2.721 | 2.721 | 3.214 | 3.214 | 3.955 | 3.956 |

3.508 | 3.508 | 4.147 | 4.147 | 4.998 | 4.998 | 5.874 | 5.873 |

Frequency parameters of in-plane vibration of an annular disk with “Clamped-Clamped” conditions.

Radius ratio ( | ||||||||

Reference [ | present study | Reference [ | present study | Reference [ | present study | Reference [ | present study | |

0.2 | 2.783 | 2.783 | 3.378 | 3.378 | 4.066 | 4.065 | 4.802 | 4.800 |

4.060 | 4.060 | 4.360 | 4.359 | 5.104 | 5.103 | 6.003 | 6.001 | |

0.4 | 3.429 | 3.429 | 4.023 | 4.022 | 4.707 | 4.707 | 5.287 | 5.286 |

5.306 | 5.306 | 5.311 | 5.311 | 5.619 | 5.619 | 6.289 | 6.288 |

Frequency parameters of in-plane vibration of an annular disk with “Clamped-Free” conditions.

Radius ratio ( | ||||||||

Reference [ | present study | Reference [ | present study | Reference [ | present study | Reference [ | present study | |

0.2 | 0.919 | 0.919 | 1.542 | 1.541 | 2.157 | 2.157 | 2.778 | 2.777 |

2.121 | 2.121 | 2.605 | 2.604 | 3.473 | 3.472 | 4.408 | 4.406 | |

0.4 | 1.281 | 1.281 | 1.965 | 1.964 | 2.445 | 2.445 | 2.911 | 2.911 |

2.691 | 2.691 | 2.908 | 2.907 | 3.604 | 3.603 | 4.492 | 4.491 |

Frequency parameters of in-plane vibration of an annular disk with flexible boundary conditions.

Boundary conditions | ||||||||

Reference [ | present study | Reference [ | present study | Reference [ | present study | Reference [ | present study | |

Elastic-Free | 0.771 | 0.771 | 1.408 | 1.408 | 2.121 | 2.121 | 2.772 | 2.772 |

1.906 | 1.906 | 2.524 | 2.524 | 3.444 | 3.444 | 4.397 | 4.397 | |

Elastic-Clamped | 2.590 | 2.590 | 3.117 | 3.116 | 3.928 | 3.926 | 4.759 | 4.760 |

3.625 | 3.625 | 4.134 | 4.136 | 5.000 | 5.001 | 5.957 | 5.957 | |

Elastic-Elastic | 1.494 | 1.492 | 1.815 | 1.813 | 2.474 | 2.472 | 3.123 | 3.120 |

2.603 | 2.601 | 3.209 | 3.207 | 4.004 | 4.002 | 4.859 | 4.857 | |

Free-Elastic | 1.040 | 1.046 | 1.505 | 1.504 | 2.414 | 2.416 | 3.114 | 3.116 |

2.432 | 2.432 | 3.018 | 3.018 | 3.895 | 3.896 | 4.833 | 4.834 | |

Clamped-Elastic | 1.686 | 1.686 | 1.960 | 1.959 | 2.514 | 2.512 | 3.129 | 3.127 |

2.764 | 2.765 | 3.319 | 3.317 | 4.061 | 4.060 | 4.879 | 4.877 |

The exact frequency parameters of the annular disk are further investigated for boundary conditions involving different combinations of free, clamped, and elastic edges. The solutions corresponding to selected modes are obtained for (“Elastic-Free”, “Elastic-Clamped”, “Elastic-Elastic”, “Free-Elastic”, and “Clamped-Elastic”) conditions are presented in Table

The characteristics of in-plane vibration for circular disks are investigated under different combinations of boundary conditions. The governing equations are solved to obtain the exact frequency equation of solid and annular disks. Frequency equations are presented for different combinations of boundary conditions, including flexible boundaries, at the inner and outer edges. The nondimensional frequency parameters obtained by the present approach compare very well with those available in literature, irrespective of the boundary condition and radius ratio considered. The exact frequency parameters can serve as a reference to assess the accuracy of approximate methods. The presented frequency equations can be numerically evaluated to obtain the in-plane modal characteristics of circular disk for a wide range of constraints conditions and geometric parameters.

Deflection coefficients of the exact solution

Inner radius of the annular disk

Outer radius of the annular disk

Young’s modulus of disk

Thickness of the annular disk

Bessel function of the first kind of order

Radial and circumferential stiffness coefficients

Nondimensional radial and circumferential stiffness parameters

Nodal diameter number

Radial and circumferential in-plane forces

Radial coordinate

Radial, circumferential, and normal displacements of the disk

Bessel function of the second kind of order

Normal coordinate

Radius ratio

Circumferential coordinate

Nondimensional frequency parameters

Poisson’s ratio

Nondimensional radial coordinate of the disk

Mass density of the disk

Lamé Potentials

Radial variations of Lamé Potentials

Radian natural frequency.