Problem r = 3 in

β

r = 3 in A

ω ω

Problem 18.154

A thin ring of 3-in. radius is attached by a collar at A to a vertical shaft which rotates at the constant rate ω.

Determine (a) the constant angle β that the plane of the ring forms with the vertical when ω = 12 rad/s, (b) the maximum value of ω for which the ring will remain vertical (β = 0 ).

Solving Problems on Your Own Problem 18.154

β

r = 3 in A

ω

ω A thin ring of 3-in. radius is attached by a collar at A to a vertical shaft which rotates at the constant rate ω. Determine (a) the constant angle β that the plane of the ring forms with the vertical when ω = 12 rad/s, (b) the maximum value of ω for which the ring will remain vertical (β = 0 ).

1. Determine the angular velocity ωω of the body and the angular velocity ΩΩ of the rotating frame: ω is the angular velocity of theω body with respect to a fixed frame of reference. The vector ωω may be resolved into components along the rotating axes. ΩΩ is the angular velocity of the rotating frame. If the rotating frame is rigidly attached to the body, ΩΩ = ω ω .

Solving Problems on Your Own Problem 18.154

β

r = 3 in A

ω

ω A thin ring of 3-in. radius is attached by a collar at A to a vertical shaft which rotates at the constant rate ω. Determine (a) the constant angle β that the plane of the ring forms with the vertical when ω = 12 rad/s, (b) the maximum value of ω for which the ring will remain vertical (β = 0 ).

2. Determine the mass moments and products of inertia of the body: For a three dimensional body these are the quantities I_x, I_y, I_z, I_xy, I_xz, and I_yz, where xyz is the rotating frame. If the rotating frame is centered at G (mass center) and is in the direction of the principal axes of inertia (Gx’y’z’), then the products of inertia are zero and I_x, I_y, and I_z, are the principal centroidal moments of inertia.

Solving Problems on Your Own Problem 18.154

β

r = 3 in A

ω

ω A thin ring of 3-in. radius is attached by a collar at A to a vertical shaft which rotates at the constant rate ω. Determine (a) the constant angle β that the plane of the ring forms with the vertical when ω = 12 rad/s, (b) the maximum value of ω for which the ring will remain vertical (β = 0 ).

3. Determine the angular momentum of the body: The angular momentum H_Gof a rigid body about the principal axes of inertia (Gx’y’z’) is given terms of the components of its angular velocity ω

ω and its principal centroidal moments of inertia.

H_x’= I_x’ω_x’ H_y’= I_y’ωy’ H_z’= I_z’ω_z’

Solving Problems on Your Own Problem 18.154

β

r = 3 in A

ω

ω A thin ring of 3-in. radius is attached by a collar at A to a vertical shaft which rotates at the constant rate ω. Determine (a) the constant angle β that the plane of the ring forms with the vertical when ω = 12 rad/s, (b) the maximum value of ω for which the ring will remain vertical (β = 0 ).

4. Compute the rate of change of angular momentum : The rate of change of H_Gwith respect to a fixed frame is given by

H_G= ( H_G)_Oxyz+ ΩΩ x H_G

where ( H_G)_Oxyzis the rate of change of H_Gwith respect to the rotating frame, and ΩΩ is the angular velocity of the rotating frame.

If the rotating frame is rigidly attached to the body, ΩΩ is equal to ωω, the angular velocity of the body.

. .

Solving Problems on Your Own Problem 18.154

β

r = 3 in A

ω

ω A thin ring of 3-in. radius is attached by a collar at A to a vertical shaft which rotates at the constant rate ω. Determine (a) the constant angle β that the plane of the ring forms with the vertical when ω = 12 rad/s, (b) the maximum value of ω for which the ring will remain vertical (β = 0 ).

5. Draw the free-body-diagram equation: The diagram shows that the system of the external forces exerted on the body is equivalent to the vector ma applied at G and the couple vector H_G.

6. Write equations of motion: Six independent scalar equations can be written from

ΣF = m a, ΣM_G= H. _G

.

Problem 18.154 Solution Determine the angular velocity ωω of the body.

β

r = 3 in A

ω ω

β

r = 3 in x’

y’

z’

G

Use the principal axes Gx’y’z’, with x’ to the plane of the ring.

ω

ω ω_x’= ω sin β

ω_y’= ω cos β ω_z’= 0

Determine the angular velocity ΩΩ of the rotating frame.

Determine the mass moments of inertia.

ω

ω = ω sinβ i’ + ω cosβ j’

Ω Ω = ωω

I_x’= m r² I_y’= m r² I_z’ = m r²

1 12 2

Problem 18.154 Solution

β

r = 3 in A

ω ω

β

r = 3 in x’

y’

z’

G ω ω

Determine the angular momentum of the body.

H_G= I_x’ ω_x’i’ + I_y’ω_y’j’ + I_z’ω_z’k’

H_G= m r²ω ( sin β i’ + cosβ j’ )¹₂ Compute the rate of change of angular momentum.

Ω

Ω = ω sinβ i’ + ω cosβ j’

Ω

Ω x H_G= (mr²ω ) i’ j’ k’

ω sinβ ω cosβ 0 sin β ¹₂cosβ 0 H

.

2

H

.

.

( H

.

Problem 18.154 Solution

Draw the free-body-diagram equation.

β

r = 3 in A

ω ω

A

β

r sinβ W

A_x A_y

A

G

m a = m ω² (r sinβ ) H

.

r cosβ

=

Problem 18.154 Solution A

β

r sinβ W

A_x A_y

A

G

m a = m ω² (r sinβ ) H

.

r cosβ

=

Write equations of motion.

Moments about A : + ΣM_A= Σ(M_A)_eff :

W = mg r = 3 in ω = 12 rad/s H

.

2

- mg r sinβ = - (r cos β ) m ω² (r sinβ ) - mr¹₂ ²ω sinβ cosβ β = 53.4^o

cos β = =2 g 3 r ω²

2 (32.2) 3 (0.25)(12)²

(a) The constant angle β the ring forms with the vertical.

Problem 18.154 Solution A

β

r sinβ W

A_x A_y

A

G

m a = m ω² (r sinβ ) H

.

r cosβ

=

cos β = 2 g 3 r ω²

(b) The maximum value of β for which the ring remains vertical.

From part (a) : For β = 0^o

cos β = 1 = ^{2 (32.2)}

3 (0.25) ω² ω = 9.27 rad/s