Micromechanics - Neural Networks and Micromechanics - page 144

8.3 The Analysis of Micromachine Tool Errors

8.3.1 Thermal Expansion

Let us consider machine tool A and machine tool B , which have the same design and
are made of the same materials. Let each component size of machine tool B be S
times smaller than the corresponding component size of machine tool A . The
thermal expansion of a component having length L can be presented as:

D L ¼ a L D T ; (8.8)

where D L is the thermal deformation, a is the thermal expansion coefficient, and D T

is the difference in temperature. The term is the same in machine tools A and B

because it depends only on the component material. The length of a component B is
S -times smaller than the length of a component A :

a

L A ¼ S L B : (8.9)

The temperature difference, D T , consists of two parts: D T i , the internal tempera-

T e , the external temperature difference. Machine tool B
demands smaller room volume than machine tool A . As it is easier to maintain a
constant temperature in the smaller volume than in the larger one, we can write:

D

ture difference, and

D T e A D T e B : (8.10)

To compare the internal temperature difference, let us consider a simplified
scheme of the origin of temperature difference shown in Fig. 8.2 . Let the tempera-
ture T 2 of the heat sink (HS) be constant and the energy produced by the heat
generator be W . The temperature difference can be obtained from:

W L HT

D T ¼ T 1 T 2 ¼

S HT ;

(8.11)

l

where L HT is the length of the heat transfer (HT) element, S HT is the square area of
the HT element, and l is the heat transfer coefficient. The term l is the same in

machine tools A and B . For L HT and S HT we have:

Heat
Generator
HG
(T 1 )

Heat Sink
HS

Heat Transfer Element
HT

(T 2 )

L

Fig. 8.2 Scheme of the origin
of temperature difference