Industrial Control Systems

344 Appendix: Mathematical andengineering formulae

c. Accelerations:

a =

−

+

gr

f r

I k

2

2 2

sin ( / )cos

a a

d. Tilting angle:

tan ; tan

min max

a a m= =

+ −

f

r

r k r

k

0

2 2

2

f

The value of k can be the calculated by the following formulas:

Ball Solid cylinder Pipe with low wall thickness

k

r

2

2

2

5

=

k

r

2

2

2

= k

r r

r

i

2

2

0

2

2

2

=

+

≈

where

s is the distance (m)

ν is the velocity (m/s)

a is the acceleration (m/s

2

)

α is the tilting angle (°)

f is the lever arm of rolling resistance (m)

k is the radius of gyration (m)

μ

0

is the coefcient of static friction

g is the acceleration due to gravity (m/s

2

)

Mechanics: Dynamics

Newton’s rst law of motion

Newton’s rst law is called the Law of ...