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The whirling frequency of a symmetric cross section of a given length between two points is given by: N = 94.251 E I m L 3 RPM {\displaystyle N=94.251{\sqrt {EI \over mL^{3}}}\ {\text{RPM}}} where: E = Young's modulus, I = second moment of area , m = mass of the shaft, L = length of the shaft between points.
Total mass of shaft and attached parts; Unbalance of the mass with respect to the axis of rotation; The amount of damping in the system; In general, it is necessary to calculate the critical speed of a rotating shaft, such as a fan shaft, in order to avoid issues with noise and vibration.
The graphs below show the angle domain equations for a constant rod length (6.0") and various values of half stroke (1.8", 2.0", 2.2"). Note in the graphs that L is rod length l {\displaystyle l} and R is half stroke r {\displaystyle r} .
r is the shaft radius c is the radial clearance μ is the absolute viscosity of the lubricant N is the speed of the rotating shaft in rev/s P is the load per unit of projected bearing area. The second part of the equation is seen to be the Hersey number. However, an alternative definition for S is used in some texts based on angular velocity: [2]
axial stress, a normal stress parallel to the axis of cylindrical symmetry. radial stress , a normal stress in directions coplanar with but perpendicular to the symmetry axis. These three principal stresses- hoop, longitudinal, and radial can be calculated analytically using a mutually perpendicular tri-axial stress system.
The curve () describes the deflection of the beam in the direction at some position (recall that the beam is modeled as a one-dimensional object). is a distributed load, in other words a force per unit length (analogous to pressure being a force per area); it may be a function of , , or other variables.
Strength depends upon material properties. The strength of a material depends on its capacity to withstand axial stress, shear stress, bending, and torsion.The strength of a material is measured in force per unit area (newtons per square millimetre or N/mm², or the equivalent megapascals or MPa in the SI system and often pounds per square inch psi in the United States Customary Units system).
The general solution to the above equation involves complex eigenvectors which are spin speed dependent. Engineering specialists in this field rely on the Campbell Diagram to explore these solutions. An interesting feature of the rotordynamic system of equations are the off-diagonal terms of stiffness, damping, and mass.