Intrinsic critical speed of a steel circular shaft

[Mechatroni]

Homework Statement

Determine the intrinsic first critical speed of a circular steel shaft supported by two bearings
E=207GPa

How can i determine it I'm very confused about this question , you can see the question at the attachment. Thanks for the help

 

[SteamKing]

Specifically, what are you confused about? Do you understand how to calculate the critical speed of a shaft? Have you been exposed to the Rayleigh Method? Dunkerley's equation?

 

[Mechatroni]

i'm not so good at this lecture. The part I'm confused is there is 3 different regions in the shaft so do i have to calculate for each of them and take the lowest one? i know the Rayleigh Method and the Dunkerley's Equation but i need to use m ( mass ) in those equations , the problem is only E is given to me. Where do i have to start to solve this

 

[SteamKing]

The OP says it is a steel shaft, so you can look up the density of steel and figure the mass of the shaft.

I think your first order of business is to analyze the shaft statically and determine its deflection between the bearings.

 

[rezeew]

I would approach this question by first understanding the concept of critical speed and its significance in structural engineering. Critical speed is the rotational speed at which a structure experiences resonance, leading to excessive vibration and potential failure. In the case of a circular steel shaft, it is important to determine the intrinsic first critical speed to ensure the structural integrity and safe operation of the shaft.

To determine the intrinsic first critical speed of the steel circular shaft, several factors need to be taken into consideration. These include the material properties of the steel, the geometry of the shaft, and the boundary conditions of the shaft (i.e. supported by two bearings).

The material property of the steel, represented by the elastic modulus E, is given as 207 GPa in the homework statement. This is a crucial factor in calculating the intrinsic first critical speed as it determines the stiffness of the shaft.

The geometry of the shaft, specifically its diameter and length, also plays a significant role in determining the critical speed. The larger the diameter and length of the shaft, the lower the critical speed will be.

Lastly, the boundary conditions of the shaft, in this case, being supported by two bearings, will affect the critical speed. The type and placement of the bearings will determine the support and stability of the shaft.

To determine the intrinsic first critical speed, a mathematical model, such as the Euler-Bernoulli beam equation, can be used. This equation takes into account the material properties, geometry, and boundary conditions of the shaft to calculate the critical speed.

In summary, to determine the intrinsic first critical speed of a circular steel shaft, a thorough understanding of the concept of critical speed and its influencing factors is required. A mathematical model can then be used to calculate the critical speed, ensuring the safe operation of the shaft.