Question

4. Consider the rotational system shown below. For steel, G- 8.27 x 101° Nt/m2 and p 7800 kg/m', and for the fluid, μ = 0.309 Nt-sec/m2. Given dı = 0.01m, d,-0.40m, Li-0.50m, L2 = 0.30m and h-0.2mm. a). find the torsional stiffiness K, of the shaft; b). find the moment of inertia J of the steel rotor; c). find the torsional damping constant B, ignoring the viscous effects of the oil on the left and right ends of the rotor. d), find the natural frequency, con and damping ratio ζ. Oil pa Steel slee 44). 丁こ 5.ppu-m-Sec2 roe! rad

Answer 1

Given:

For steel-

Modulus of rigidity:

G- 827.1010. n*

Density:

ρ 7800kg

Fluid viscosity:

μ-0.309 m2 n*

Diameter of shaft:

d,0.01m

Diameter of rotor:

d, 0.40m

Length of shaft:

L. 0.50m

Length of rotor:

0.30m2

Thickness of oil layer:

h = 0.2mm

a). Torsional stiffness Ks of the shaft

Torsional stiffness of the shaft is given by the relation

GJ K,우 ..(1)

J is the polar moment of inertia of the shaft given by the relation

zudl( 32

From equation (2) and (1) torsional stiffness

32L (3)

For shaft

On substituting values

K, = π·8232 10150001

b). Moment of inertia of the rotor

Mass Moment of inertia for a cylindrical object is given by the relation

2 -mr ..(4)

is the radius of rotor

is mass rotor

Mass of the rotor can be evaluated by product of volume and density

m=ρ*-*L ..(5) 4

From equation (4) and (5)

32

For rotor

L-L,-0.30m

On substituting values

π * 7800 * 0.30 * 0.404 32

*g -5.88 in

c). Torsional damping constant B

The rotor is rotating in the surrounding environment of viscous oil, due to which damping effect(viscous force) will be generated., and this viscous force will cause resisting viscous torque.Viscous force is given by the following relation

, is the area between rotor and stator

is the viscosity of oil

is a relative velocity between rotor and stator

is the gap between rotor and stator

is resisting torque due to viscous effect

is rotor radius

is rotor angular velocity

On substituting in equation (6)

Torsional damping constant is given by

Resisting torque T Angular velocityé

4h

For rotor

0.30m2

d, 0.40m

On substituting values

π * 0.30 * 0.309 * 0.403 в= 4 2 10-4

N.m. s B- 23.298 raa

d). Natural frequency and damping coefficient

The natural frequency is given by the relation

is shaft stiffness

N.m K, - 162.38n rad

mass moment inertia of the rotor

On substituting values

530 162.38 5.88

Damping coefficient is given by the relation

torsional damping coefficient

B 23.298Nm. rad

is shaft stiffness

mass moment inertia of rotor

23.298 2V162.385.88