The interaction between moving surfaces is a concept which is at the heart
of innumerable phenomena in the nature. Whether in the joints of a human
skeleton, in rolling elements of a jet turbine, or in crucial components on a
space station, no natural events nor human-designed devices are exempt from
the action of friction and wear. The intrinsic multidisciplinary character of
these two physical phenomena aroused the interest of mankind since the ancient times and particularly regained great attention in Renaissance with the
pioneering studies of Leonardo and subsequently during the industrial revolution [1, 2]. Nonetheless, the definition of the concept of tribology in terms of
scientific discipline and economical implications originated first in the second
half of the twentieth century thanks to the work of Jost [3]. By means of a
careful analysis of the state of lubrication research in different industrialized
countries, the Jost Report and many following independent studies estimated
a potential annual saving ranging from 1% to 1.4% of a country’s GDP [4–6].
Furthermore, such saving is deemed to be achievable with a very convenient
return on investment ratio of 1/50, implying a saving of 50 $ for each dollar
spent in research and development in the previous year [4, 7]. Particular emphasis is laid on the role of tribological progress in transportation, industrial
and utilities sectors, where it has been estimated that up to 11% of the used
energy can be saved by the application of the new developments in tribology
1
1 Introduction
[8, 9]. In view of such relevant potential impact, the advance in the field of
tribology is nowadays of utmost importance to address the economical and
environmental challenge and also to cast light on a plenty of phenomena in
nature which characterize our everyday life.
In the last decades, the research on tribology has spread in numerous branches,
ranging over multidisciplinary areas such as lubricants development [10, 11]
surface coating [12, 13], or the optimization of automotive and industrial applications [14, 15]. Among these research fields, the enhancement of the tribological performance through the introduction of surface textures has drawn a
considerable attention in the research community [16]. Surface texture nowadays represents an interesting technology for the reduction of friction and wear
thanks to recent advancements in the laser surface texturing techniques (LST)
which reduced the production costs and increased also the manufacturing precision [17].
However, an unanimous consensus in the research community has not been
reached yet, for what concerns the underlying physical mechanisms and the
possibility to obtain an optimal texture design which proves to be robust under different operating conditions [18]. In order to bridge this gap, a more
and more increasing part of literature combines experimental and numerical
works [16]. Among them, the present work represents the numerical counter.
2. rraj@iitp.ac.in 2
Question 1
Using the parallel-axis theorem, determine the product of inertia of
the area shown with respect to the centroidal x and y axes.
3. rraj@iitp.ac.in 3
Solution
1 3
2
We have, 𝐼𝑥𝑦 = 𝐼𝑥𝑦 1
− 𝐼𝑥𝑦 2
- 𝐼𝑥𝑦 3
, where bar on the top denotes “about the
centroidal axes x and y
Symmetry implies, 𝐼𝑥𝑦 1
= 0
For the triangles, 𝐼𝑥𝑦 2
= 𝐼𝑥𝑦 3
, since the distribution of these masses about the set
of x and y axes is equal
And for each triangle, 𝐼𝑥𝑦 2/3
= 𝐼𝑥′𝑦′ 2/3
+ 𝑥2/3𝑦2/3𝐴2/3, where 𝐼𝑥′𝑦′ 2/3
= −
𝑏2ℎ2
72
(product of inertia about triangle's centroidal axes, solved in class), and 𝑥2/3 or 𝑦2/3
is distance of the centroid axis of the respective triangle (denoted by red and blue
dots) from the current set of x and y axes.
6. rraj@iitp.ac.in 6
Question 2
Using Mohr’s circle, determine the moments of inertia and
the product of inertia of the area with respect to new
centroidal axes obtained by rotating the x and y axes
30°counterclockwise.
X’
Y’
𝐼𝑥 = 68.96 × 106
mm4
𝐼𝑦 = 132.48 × 106
mm4
𝐼𝑥𝑦 = −21.6 × 106
mm4
30°
7. rraj@iitp.ac.in 7
Solution
(68.96 × 106, −21.6 × 106)
(132.48 × 106
, +21.6 × 106
)
X
Y
X’
Y’
60°
30°
30° rotation counterclockwise in
physical coordinate system implies
60° counterclockwise rotation on
Mohr’s circle
𝐼𝑥/𝐼𝑦
𝐼𝑥𝑦
10. rraj@iitp.ac.in 10
Question 3: Solve and Submit
Using Mohr’s circle, determine for the area indicated the
orientation of the principal centroidal axes (w.r.t. x-y shown
below) and the corresponding values of the moments of
inertia.
𝐼𝑥 = 68.96 × 106 mm4
𝐼𝑦 = 132.48 × 106
mm4
𝐼𝑥𝑦 = −21.6 × 106 mm4
11. rraj@iitp.ac.in 11
Solution
(68.96 × 106, −21.6 × 106)
(132.48 × 106
, 21.6 × 106
)
X
Y
X’’
Y’’
Principal centroidal axes (X”,Y”) implies points on the x-axis of the Mohr’s
circle such that 𝐼𝑥𝑦=0.
Since 2𝜃𝑚 = −34.22, the principal axes are obtained by rotating the x-y
axes by 17.11° counterclockwise.
𝐼𝑥/𝐼𝑦
𝐼𝑥𝑦
2𝜃𝑚
3 M
