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INPUT   DATA

♥ EXAMPLE Of Input/Output

Title

Length of shaft, L		mm
Mass of shaft, ms		kg
Modulus of elasticity, E		109N/m2
Area moment of inertia, I		cm4
Mass of rotor #1, m1		kg
Position of rotor #1, a1		mm
Mass of rotor #2, m2		kg
Position of rotor #2, a2		mm
Mass of rotor #3, m3		kg
Position of rotor #3, a3		mm
Mass of rotor #4, m4		kg
Position of rotor #4, a4		mm
Mass of rotor #5, m5		kg
Position of rotor #5, a5		mm

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OUTPUT   VARIABLES   &   GRAPHS

Variables	Values	Units
♦  Critical speed of shaft, ωs		rpm
♦  Critical speed of rotor #1, ω1		rpm
♦  Critical speed of rotor #2, ω2		rpm
♦  Critical speed of rotor #3, ω3		rpm
♦  Critical speed of rotor #4, ω4		rpm
♦  Critical speed of rotor #5, ω5		rpm
♦♦  Critical speed of whole system, ωc		rpm

Critical Speed Of Rotating Shaft

Solid cylinderical shafts involved in transmission of power by means of gears, flywheel, pulleys are subjected not only to torsion but also bending. All rotating shafts even in the absence of external load, deflect during rotation. The combined weight of the shaft and the load can cause deflection that will create resonant vibration at certain speed known as the Critical Speed.

One of the key methods for calculating critical speed is the Dunkerley Equation. In 1894, Stanley Dunkerley published a study "On the Whirling and Vibration of Shafts," Phil. Trans. R. Soc., London. The first sentence of his paper reads, "It is well known that every shaft, however nearly balanced, when driven at a particular speed, bends, and, unless the amount of deflection be limited, might even break, although at higher speeds the shaft again runs true. This particular speed or 'critical speed' depends on the manner in which the shaft is supported, its size and modulus of elasticity, and the sizes, weights, and positions of any pulleys it carries". Dunkerley equation is an approximation to the first natural frequency of vibration of the system, which is assumed to be nearly equal to the critical speed of rotation.

The method consists of reducing the actual system of multi-rotors to a series of single rotor system. The critical speed of a single rotor system can be calculated by a direct formula (i.e. natural frequency of a pinned-pinned beam with intermediate load). Finally these individual critical speeds are combined to obtain the critical speed of the system as given in the equations below:

where
     L = length of shaft
     a_i = location of rotor i from left end of shaft
     m_i = mass of rotor i
     n = total number of rotors, including shaft; i = 1, ...n
     ω_c = critical speed (cycle/sec[Hz], revolution/minute)
     ω_i = critical speed for rotor i
     δ_i = static deflection at at position of rotor i, due to rotor i only
     g = acceleration due to gravity: 9.81 m/s², 32.2 ft/s²
     E = modulus of elasticity of shaft material
     I = area moment of inertia of cross-sectional area about axis through centroid

The shaft is also considered a rotor with mass m_s concentrated at its mid-point.

Tips

    ◊ Use link EXAMPLE Of Input/Output  to demo data entry expectations and results; you may edit & use it as starting point

BIBLIOGRAPHY

• Blevins RD; Formulas For Natural Frequency And Mode Shape; Krieger Publ. Co., Malabar, Florida, 2001.
• Hicks TG; Mechanical Engineering Formulas, Pocket Guide; McGraw Hill, New York, 2003, Section 4.
• Moulton A, Grosjean J & Owen G; The Moulton Formulae and Methods - directly usable for calculations in mechanical engineering; Professional Engineering Publishing Ltd, London, UK, 2005.
• Kelly SG; Mechanical Vibrations, Schaum's Outline Series; McGraw Hill, New York, 1996.

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