Thermal elongation-ring on a shaft

• Karol
In summary, the ring must be heated to a temperature of 55.6 degrees Celsius in order for it to be 75.05 mm in diameter.
Karol

Homework Statement

A steel ring has a diameter of 75 mm at 20 deg C. how much must it be heated in order to mount on a brass shaft of dia 75.05
Then both are cooled. how must they be cooled so the ring will fall off?

Homework Equations

Thermal elongation coefficient of steel: 12E-6
Thermal elongation coefficient of brass: 20E-6
Expansion of area under heating/cooling: ##A=A_0(1+2\alpha \Delta t)##

The Attempt at a Solution

Heating:
$$\pi\cdot 75.05^2=\pi\cdot 75^2 (1+2\cdot 12E-6\cdot\Delta t)\rightarrow \Delta t=55.6^0C$$
It's too low, isn't it?
For cooling i consider as if the ring isn' t mounted on the shaft, it's aside of it:
$$\pi\cdot 75.05^2(1-2\cdot 20E-6\cdot\Delta t)=\pi\cdot 75^2 (1-2\cdot 12E-6\cdot\Delta t)\rightarrow \Delta t=83.1^0C$$
Also too small, isn't it?
Is it correct what i have done, taking the ring aside? i didn't know how to solve it with the ring mounted, i don't know if to calculate efforts or not and how to do it. i know the formula for stress due to heating in a straight bar.

How much does the steel ring have to be heated such that it's diameter will be 75.05 mm?

Chet

I wrote, it has to be heated with more 55.6C

Karol said:
I wrote, it has to be heated with more 55.6C
Oh, sorry. The full line didn't show up on my iPhone, so I missed most of your equation.
In the second part, your answer also looks correct, if that 83 C is the amount the temperature is lowered below 20C. I did this part a little differently, by writing:

$$75.05(1-20\times 10^{-6} \Delta t)=75.0(1-12\times 10^{-6} \Delta t)$$

I don't think it's correct because you made linear elongation and i read in a book that i have to calculate area expansion, like i did

Karol said:
I don't think it's correct because you made linear elongation and i read in a book that i have to calculate area expansion, like i did
Either way, you get the same answer (solve my equation and see). I used linear expansion because all line segments and arcs of the material must increase in length by the same fraction. I just wanted to show you that the problem could be done using linear expansion also.

Chet

So there is no difference if the ring is mounted on the shaft or aside of it, the same temperature is needed, right? how can you explain that these conditions are the same?

Karol said:
So there is no difference if the ring is mounted on the shaft or aside of it, the same temperature is needed, right? how can you explain that these conditions are the same?
Why would you think they wouldn't be, especially if, in the end, for the mounted case, the shaft and the ring are not even touching.

Chet

1. What is thermal elongation-ring on a shaft?

Thermal elongation-ring on a shaft is the phenomenon where a metal shaft or rod expands or contracts due to changes in temperature. This can cause the shaft to become longer or shorter, depending on the direction of the temperature change.

2. How does thermal elongation-ring affect the performance of a machine?

Thermal elongation-ring can cause misalignment in machines, which can lead to increased friction, wear and tear, and even failure. It is important to consider thermal elongation-ring when designing and operating machines to ensure optimal performance and longevity.

3. What factors can affect thermal elongation-ring on a shaft?

The material of the shaft, the temperature change, and the length of the shaft are all factors that can affect thermal elongation-ring. Different materials have different coefficients of thermal expansion, meaning they will expand or contract at different rates. Longer shafts will also experience more elongation than shorter ones.

4. How can thermal elongation-ring be managed in a machine?

There are several ways to manage thermal elongation-ring in a machine. One method is to use compensating materials, such as different types of metals with varying coefficients of thermal expansion, in the design of the machine. Another option is to incorporate expansion joints or sliding connections to allow for the movement of the shaft. Additionally, maintaining a constant temperature in the machine's environment can help reduce the effects of thermal elongation-ring.

5. What are the potential consequences of not accounting for thermal elongation-ring in machine design?

If thermal elongation-ring is not properly accounted for in machine design, it can lead to misalignment, increased friction, and premature failure of the machine. This can result in costly repairs and downtime, as well as potential safety hazards. It is important to consider thermal elongation-ring in the design and operation of machines to ensure optimal performance and safety.

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