## Installing a Bearing

Consider an aluminum bearing with a maximum room-temperature outer diameter of 50.15 mm is to be installed inside of an aluminum mounting plate with a minimum room-temperature bore diameter of 50 mm. If the room temperature is 25°C and the required diametral clearance for assembly is 0.1 mm, what is the minimum temperature the mounting plate must be heated to install a room-temperature bearing? Note, aluminum’s coefficient of linear expansion is 23x10^(-6)/°C.

##
__
__**Hint**

**Hint**

$$$\delta_t=\alpha L(\Delta T)$$$

where
$$\delta_t$$
is the deformation caused by a change in temperature,
$$\alpha$$
is the temperature coefficient of expansion,
$$L$$
is the length of the member, and
$$\Delta T$$
is the change in temperature.

##
__
__**Hint 2**

**Hint 2**

Required diameter change:

$$$\delta_{diameter}=d_{bearing}-d_{plate}+required\:assy\:clearance$$$

where
$$d$$
is the diameter.

Required diameter change:

$$$\delta_{diameter}=d_{bearing}-d_{plate}+required\:assy\:clearance$$$

where
$$d$$
is the diameter. Thus,

$$$\delta_{diameter}=50.15mm-50mm+0.1mm=0.25\:mm$$$

Required temperature change for the mounting plate’s bore diameter:

$$$\delta_t=\alpha L(\Delta T)$$$

where
$$\delta_t$$
is the deformation caused by a change in temperature,
$$\alpha$$
is the temperature coefficient of expansion,
$$L$$
is the length of the member, and
$$\Delta T$$
is the change in temperature.

$$$\Delta T=\frac{\delta_{diameter}}{\alpha(L)}=\frac{0.25mm^{\circ}C}{(23\times 10^{-6})(50mm)}=\frac{0.25^{\circ}C}{0.00115}=217.39^{\circ}C$$$

Minimum temperature to which the mounting plate must be heated:

$$$T=T_{room}+\Delta T=25^{\circ}C+217.39^{\circ}C=242.39^{\circ}C$$$

242°C