Question

a shaft 180mm diameter is supported in two bearing 2.5 metre apart. it carries discs of mass 250kg, 500kg,and 200kg at 0.6m, 1.5m and 2m from left hand. assuming the mass of the shaft 190kg/m, determine critical speed of shaft. Young's modulus of the shaft material is 211 GN/m²​

Answer 1

Critical speed  = 117363.8 $ms^{-1}$.

• The critical speed is the hypothesised angular velocity that stimulates a rotating object's inherent frequency, such as a shaft. Systemic vibration dramatically increases as the rotational speed approaches the object's inherent frequency and the object starts to resonate. No matter how the resonance is oriented, it happens. Critical speed is the rotational speed at which the natural vibration has a numerical value equal to it.
• There is a speed for a spinning shaft at which the centrifugal force and elastic restoring force are equivalent for any modest initial deflection. At this moment, the shaft is termed to "spin" as the deflection significantly rises. Above and below this speed, this effect is drastically diminished. The shaft's dimensions, composition, and loads all affect the critical speed at which it revolves.

Here, according to the given information, we are given that,

A shaft 180mm diameter is supported in two bearing 2.5 metre apart. it carries discs of mass 250kg, 500kg,and 200kg at 0.6m, 1.5m and 2m from left hand.

Now, $N_{1}$ is given as, 2210410$d^{2} \sqrt{\frac{1}{Wl^{3} } }$.

This gives, $N_{1}$

= 2210410$(0.18)^{2} \sqrt{\frac{1}{250}(2.5)^{3} }$ +  2210410$(0.18)^{2} \sqrt{\frac{1}{500}(2.5)^{3} }$ +  2210410$(0.18)^{2} \sqrt{\frac{1}{200}(2.5)^{3} }$

= 2210410$(0.18)^{2} \sqrt{(2.5)^{3} }$ ($\sqrt{\frac{1}{250} +\frac{1}{500} +\frac{1}{200}}$)

=117363.8

Now, N = (1, 4, 9, 16, ...). 117363.8.

Hence, critical speed  = 117363.8 $ms^{-1}$.

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Answer 2

Answer:

Explanation:

From the above question the critical speed is the hypothesised angular velocity that stimulates a rotating object's inherent frequency, such as a shaft. Systemic vibration dramatically increases as the rotational speed approaches the object's inherent frequency and the object starts to resonate. No matter how the resonance is oriented, it happens. Critical speed is the rotational speed at which the natural vibration has a numerical value equal to it.

There is a speed for a spinning shaft at which the centrifugal force and elastic restoring force are equivalent for any modest initial deflection. At this moment, the shaft is termed to "spin" as the deflection significantly rises. Above and below this speed, this effect is drastically diminished. The shaft's dimensions, composition, and loads all affect the critical speed at which it revolves.

Here, according to the given information, we are given that,

A shaft 180mm diameter is supported in two bearing 2.5 metre apart. it carries discs of mass 250kg, 500kg,and 200kg at 0.6m, 1.5m and 2m from left hand.

Now,  is given as, 2210410.

This gives,

         = 2210410 +  2210410 +  2210410

         = 2210410 ()

         =117363.8

Now, N = (1, 4, 9, 16, ...). 117363.8.

Hence, critical speed  = 117363.8 .

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