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

Table 1

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

Boundary conditions at 𝜉=1		Clamped	Free
𝑛=0	Radial	𝐽1(𝜆1)=0	𝜆1𝐽0(𝜆1)=(1−𝑣)𝐽1(𝜆1)
	Circumferential	𝐽1(𝜆2)=0	𝜆2𝐽0(𝜆2)=2𝐽1(𝜆2)
𝑛=1		𝑋𝑛(𝜆1)𝑋𝑛(𝜆2) = 𝑛^2𝐽𝑛(𝜆1)𝐽𝑛(𝜆2)	[𝑋1(𝜆2)𝐽1(𝜆1)]+[𝑋1(𝜆1)𝐽1(𝜆2)]=(2−(𝜆^22/2))𝐽1(𝜆2)𝐽1(𝜆1)
𝑛>1			[𝑋𝑛(𝜆2)−𝑄𝐽𝑛(𝜆2)][𝑋𝑛(𝜆1)−𝑄𝐽𝑛(𝜆1)]=𝑛^2(𝑄−1)^2 where 𝑄=𝜆^22/2(𝑛^2−1)