machine element chapter 4

1. Chapter 4
Torque Transmitting Joints:
Keys; Spline Joints; Pin Joints; Interference Fit

2. Introduction
• Shafts serve to transmit torque from
an input gear or pulley, through the
shaft, to an output gear or pulley
• Common torque-transfer elements are:
Keys, Splines, Setscrews, Pins, Press or
shrink fits, Tapered fits, etc..
• Many of these devices are designed to
fail, to protecting more expensive
components.

3. Axial Positioning
• Axial positioning: retaining rings, locking collars, shoulders machined into
shaft, etc….
• A retaining ring is frequently used instead of a shaft shoulder or a sleeve to
axially position a component on a shaft or in a housing bore.

4. Splines
• Splines are essentially stubby gear teeth formed on the outside of the shaft
and on the inside of the hub of the load-transmitting component.
• It is more expensive to manufacture than keys, and usually not necessary for
simple torque transmission
• For cases of low torque transmission, various means of transmitting torque are
available. These include pins, setscrews in hubs, tapered fits, and press fits.

5. Splines Design
• Torque capacity is based bearing stress on the sides of the splines
𝑇 = 1000𝑥𝑁𝑥𝑅𝑥ℎ
𝑅 =
1
2
𝐷 + 𝑑
2
=
𝐷 + 𝑑
4
ℎ =
1
2
(𝐷 − 𝑑)
𝑇 = 1000(
𝐷+𝑑
4
) (
𝐷−𝑑
2
)=1000
𝐷2−𝑑2
8
where N = number of splines
R = mean radius of the splines
h = depth of the splines

6. Cont’d…

7. Cont’d….
• 𝑇 = 𝑘𝐷2 𝐿
Where
T = torque capacity
𝑘𝐷2 = torque capacity per unit
length (from Table 11-5)
L = length of spline in inches

8. Keys and pins
• Both are used on shafts to secure rotating elements, such as gears, pulleys,
or other wheels.
• Keys are used to enable the transmission of torque from the shaft to the
shaft-supported element.
• Pins are used for axial positioning and for the transfer of torque or thrust or
both.
• Pins are useful when the principal loading is shear and when both torsion and
thrust are present

9. Cont’d….
Taper pins are sized according to the diameter at the large end. The diameter
at the small end is
𝑑 = 𝐷 − 0.0208𝐿
where d= diameter at small end, in
D= diameter at large end, in
L= length, in

10. Keys:
• A key is the piece inserted in an axial direction
between a shaft and hub of the mounted machine
element such as pulley or gear etc., to prevent
relative rotation….
• a keyway is the groove cut in the shaft
or hub to accommodate a key.

11. Parallel Key Design
• keys are sunk in the shaft and the hub.
• Let D = diameter of the shaft
- width of the key W = D/4
• Rectangular cross-section
nominal thickness H = (2/3)W = (1/6)D
• Square cross-section:
nominal thickness H = W = D/4.
Steps to key design:
Step 1 – Complete Shaft Design
Step 2 – Select key size based on
shaft diameter.(W,H) from the table
Step 3 – Calculate required length, L,
based on torque.

12. Cont’d….
• Direct shear stress
• 𝜏 =
𝐹
𝐴 𝑠
=
𝑇
𝐷/2
.
1
𝑊𝐿
=
2𝑇
𝐷𝑊𝐿
• Using MSS theory: 𝑛 =
0.5𝑆 𝑦
𝜏 𝑚𝑎𝑥
, if n is known
• 𝜏 𝑎𝑙𝑙 =
0.5𝑆 𝑦
𝑛
=
2𝑇
𝐷𝑊𝐿
⇒ 𝐿 =
4𝑛𝑇
𝑆 𝑦 𝐷𝑊
• Bearing Compressive stress
• 𝜎 =
𝐹
𝐴 𝑐
=
𝑇
𝐷/2
.
1
𝐻
2
𝐿
• Using MSS theory
• 𝑛 =
0.5𝑆 𝑦
𝜏 𝑚𝑎𝑥
=
0.5𝑆 𝑦
𝜎
2
⇒ L =
4𝑛𝑇
𝑆 𝑦 𝐷𝐻

13. Interference Fit
• Another common means of coupling a hub to the shaft is to use a press or
shrink fit, also called interference fitting.
• Obtained by machining the hole in the hub to slightly smaller diameter than
the shaft.
• There are two standards for limits and fits in the United States, one based on
inch units and the other based on metric units.
• The metric version is the newer of the two and is well organized,
• In using the standard, capital letters always refer to the hole; lowercase
letters are used for the shaft.
• The definitions illustrated in Fig. 7–20 are explained as follows:
• Basic size is the size to which limits or deviations are assigned and is the same
for both members of the fit.
• Deviation is the algebraic difference between a size and the corresponding
basic size.

14. Cont.
• Upper deviation is the algebraic difference between the maximum limit and
the corresponding basic size.
• Lower deviation is the algebraic difference between the minimum limit and
the corresponding basic size.
• Fundamental deviation is either the upper or the lower deviation, depending
on which is closer to the basic size.
• Tolerance is the difference between the maximum and minimum size limits of
a part.
• International tolerance grade numbers (IT) designate groups of tolerances
such that the tolerances for a particular IT number have the same relative
level of accuracy but vary depending on the basic size.
• Hole basis represents a system of fits corresponding to a basic hole size. The
fundamental deviation is H.
• Shaft basis represents a system of fits corresponding to a basic shaft size.
The fundamental deviation is h. The shaft-basis system is not included here.

16. Cont.
• The fundamental deviations for shafts are given in table A-11 and A-13.
For letter codes c, d, f, g, and h,
Upper devotion = fundamental deviation
Lower deviation = upper deviation – tolerance grade
• For letter k, n, p, s, and u, the deviations for shafts are
Lower deviation = fundamental deviation
Upper deviation = lower deviation + tolerance grade

17. Type of Fit description symbol
clearance Loose running fit: for wide commercial tolerances or allowances on external
members
H11/c11
Free running fit: not for use where accuracy is essential, but good for large
temperature variations, high running speeds, or heavy journal pressures
H9/d9
Close running fit: for running on accurate machines and for accurate location at
moderate speeds and journal pressures
H8/f7
Sliding fit: where parts are not intended to run freely, but must move and turn
freely and locate accurately
H7/g6
Locational clearance fit: provides snug fit for location of stationary parts, but can
be freely assembled and disassembled
H7/h6
Transition Locational transition fit: for accurate location, a compromise between clearance
and interference
H7/k6
Locational transition fit: for more accurate location where greater interference is
permissible
H7/n6
Interference Locational interference fit: for parts requiring rigidity and alignment with prime
accuracy of location but without special bore pressure requirements
H7/p6
Medium drive fit: for ordinary steel parts or shrink fits on light sections, the
tightest fit usable with cast iron
H7/s6
Force fit: suitable for parts that can be highly stressed or for shrink fits where
the heavy pressing forces required are impractical
H7/u6

18. • The lower deviation H (for holes) is zero.
For these, the upper deviation equals the
tolerance grade.
• as shown in figure 7-20, we use the
following notation:
D = basic size of the hole
d = basic size of the shaft
𝛿 𝑢 = upper deviation
𝛿𝑙 = lower deviation
𝛿 𝐹 = fundamental deviation
∆𝐷 = tolerance grade for hole
∆𝑑 = tolerance grade for shaft

19. Cont.
Note that these
quantities are all
deterministic. Thus,
for the hole,
𝐷 𝑚𝑎𝑥 = D + ∆𝐷
𝐷 𝑚𝑖𝑛 = D
For shafts with
clearance fits c, d,
f, g, and h,
𝑑 𝑚𝑎𝑥 = 𝑑 + 𝛿 𝐹
𝑑 𝑚𝑖𝑛 = d + 𝛿 𝐹 - ∆𝑑