Frequency Equations for the In-Plane Vibration of Circular Annular Disks

Table 2

Frequency equations of axisymmetric modes for annular disks.

Boundary conditions		radial	Circumferential
inner	outer
Clamped	Clamped	𝐽1(𝜆1)𝑌1(𝜆1𝛽)−𝐽1(𝜆1𝛽)𝑌1(𝜆1)=0	𝐽1(𝜆2)𝑌1(𝜆2𝛽)−𝐽1(𝜆2𝛽)𝑌1(𝜆2)=0
Free	Free	[(−𝜆1/(1−𝑣))𝐽0(𝜆1)+𝐽1(𝜆1)][(−𝜆1/(1−𝑣))𝑌0(𝜆1𝛽)+(1/𝛽)𝑌1(𝜆1𝛽)]−𝜆1/(1−𝑣)𝑌0(𝜆1)+𝑌1(𝜆1)[(−𝜆1/(1−𝑣))𝐽0(𝜆1𝛽)+(1/𝛽)𝐽1(𝜆1𝛽)]=0	[−2𝐽1(𝜆2)+𝜆2𝐽0(𝜆2)][−(2/𝛽)𝑌1(𝜆2𝛽)+𝜆2𝑌0(𝜆2𝛽)]−[−(2/𝛽)𝐽1(𝜆2𝛽)+𝜆2𝐽0(𝜆2𝛽)][−2𝑌1(𝜆2)+𝜆2𝑌0(𝜆20)]=
Clamped	Free	[(𝜆1/(1−𝑣))𝐽0(𝜆1)−𝐽1(𝜆1)]𝑌1(𝜆1𝛽)−𝐽1(𝜆1𝛽)[(𝜆1/(1−𝑣))𝑌0(𝜆1)−𝑌1(𝜆1)]=0	[2𝐽1(𝜆2)−𝜆2𝐽0(𝜆2)]𝑌1(𝜆2𝛽)−𝐽1(𝜆2𝛽)[2𝑌1(𝜆2)−𝜆2𝑌0(𝜆2)]=0
Free	Clamped	𝐽1(𝜆1)[(−𝜆1/(1−𝑣))𝑌0(𝜆1𝛽)+(1/𝛽)𝑌1(𝜆1𝛽)]+[(−𝜆1/(1−𝑣))𝐽0(𝜆1𝛽)+(1/𝛽)𝐽1(𝜆1𝛽)]𝑌1(𝜆1)=0	𝐽1(𝜆2)[−(2/𝛽)𝑌1(𝜆2𝛽)+𝜆2𝑌0(𝜆2𝛽)]−[−(2/𝛽)𝐽1(𝜆2𝛽)+𝜆2𝐽0(𝜆2𝛽)]𝑌1(𝜆2)=0