Carlos L. Castillo, Ph.D.

Spring
T (t ) θ (t )

Time domain
T (t ) = Kθ (t )

Frequency domain
K
K: spring constant

T (s) = K θ (s)

Impedance

T (s) Z M= = K (s) θ (s)

Viscous damper
T (t ) θ (t )

Time domain dθ (t )
T (t ) = D dt

Frequency domain
D
D: coefficient of Viscous friction

T (s) = D s θ (s)

T (s) Z M= = D s (s) θ (s)

Impedance

Inertia
T(t) θ (t )

Time domain d 2θ (t )
T (t ) = J dt 2

J
J: moment of inertia

Frequency domain
T ( s) = Js 2 θ ( s )

T (s) Z M= = J s 2 (s) θ (s)

Impedance

1.

First, we rotate a body while holding all other points still and place on its free-body diagram all torques due to the body’s own motion Then, holding the body still, we rotate adjacent points of motion one at the time and add the torques due to the adjacent motion to the free-body diagram.

2.

( J1s 2 + D1s + K )θ1 ( s ) − Kθ 2 ( s ) =) T (s

− Kθ1 ( s ) + ( J 2 s 2 + D2 s + K )θ 2 ( s ) = 0

( J1s 2 + D1s + K )θ1 ( s )
2

− Kθ 2 ( s ) =) T (s

− Kθ1 ( s ) + ( J 2 s + D2 s + K )θ 2 ( s ) = 0

Sum of impedances  connected to the  θ (s) − Sum of impedances  θ (s) − Sum of applied   between θ andθ  2  torques at θ    1 1 2 1      motion at θ1   

Sum of impedances  Sum of impedances  Sum of applied  θ1 ( s ) + connected to the  θ 2 ( s) = −  torques at θ    between θ1and θ 2  2      motion at θ 2   

( s + s + 1)θ1 ( s ) − ( s + 1)θ 2 ( s ) = ( s ) T
2

− ( s + 1)θ1 ( s ) + (2 s + 2)θ 2 ( s ) = 0

( J1s 2 + D1s + K )θ1 ( s ) − 0θ1 ( s )

− Kθ 2 ( s )

− 0θ3 ( s ) =) T (s 0 − D2 sθ3 ( s ) =

− Kθ1 ( s ) + ( J 2 s 2 + D2 s + K )θ 2 ( s )

− D2 sθ 2 ( s ) + ( J 3 s 2 + D3 s + D2 s )θ3 = 0



In the system shown in Figure P2.25, the inertia, J, of the radius, r, is constrained to move only about the stationary axis A. A viscous damping force of translation value ���� exists between the bodies J and M. If an external force, ��(��), is applied to the mass, find the transfer function, �� �� = ��(��)���(��)

http://www.monarchbearing.com/index.html

Helical Gears

http://www.coeshow.com/shop/i mages/uploads/CRF150Gears.jpg

Star Gears

http://www.freedigitalphotos.net/images/photos/ingranaggi.jpg

θ 2 r1 N1 = = θ1 r2 N 2
• The ratio of the angular displacement of the gears is inversely proportional to the ratio of the number of teeth

•The distance traveled along each gear’s circumference is the same

r1θ1 = r2θ 2

T1θ1 = T2θ 2
• Thus, the torques are directly proportional to the ratio of the number of teeth •Since the translational energy of force times displacement becomes the rotational energy of torque times angular displacement

T2 θ1 N 2 = = T1 θ 2 N1



Rotational mechanical impedances can be reflected through gear trains by multiplying the mechanical impedance by the ratio
 Number of teeth of gear     on destination shaft   Number of teeth of gear     on source shaft 
2



Find the transfer function, ��2 (��)/��1 (��), for the system below



In order to eliminate gears with large radii, a gear train is used to implement large gear ratios by cascading smaller gear ratios



Problems 30 and 31

 Quiz

02