Moment of Inertia of Stepped Shafts

CHAPTER 15 SHAFTSCHAPTER 15 SHAFTSA shaft is one of the most important components in machines. Shafts are used

to support rotating elements. And transmit rotational motion and power.

15.1 INTRODUCTION TO SHAFTSThe types of shafts

According to the shapes, there are crank shaft, the axis is not at a horizontal line; straight shaft: The axis is at a horizontal line; flexible shaft, convoluted by groups of steel wire ,goodflexibility.

According to the straight shafts, there are plain shaft, no-diameter-change shafts, simple, is prone to machining, inconvenient to install the components on the shafts; stepped shaft: inconvenient to install the components on the shafts; hollow shaft and solid shaft

To machining technics, straight shafts are the best, but difficult to fixed. To the forces on the shafts, equality intensity shafts are good, but difficult to machining and install. Hence, there are all stepped shafts.

The word “shaft” covers numerous variations and according to the loads exerted on the shafts, the shafts can be classified into three groups:(a) Shafts with only bending moments, usually called axle;(b) Shafts with only torsional moments, usually called spindle.(c) Shafts with bending and torsional moments;

rotating axle fixed axle

Spindle Rotary shaft

(a) shaft (b) axle

(c) spindle

The material of shafts① Carbon steel

In common use:30、40、45steel

② Alloy steel

40Cr、40CrNi、20Cr、20Cr2Ni4A、38SiMnMo

③ Cast steel

QT600—3、QT800—2

Selecting the material and heat treatment of the shafts mainly accord to the strength and wearing resistance, but not the bending and tension rigidity.

15.2 CONFIGURATION DESIGN OF SHAFTS1. The factors relative to the configuration of shafts

The location and modality of the shafts installing on machine ;The types, size, amount of elements on the shaft ;The character, magnitude orientation and distributing of load on the shaft; The types, size, collocation of bearing;

The roughcast and machining technics of shafts；

The assembly project of elements on shaft.

2、The conditions the configure of shafts should to satisfy

The elements on the should have exact working location, convenient for to dismantle and install and rectify, Have good machining technics.

3、The names of segments of shaft

4、The ways of shaft design

When select the project, it is possible to reduce the quantity of elements on shaft and deadweight of shaft, have good machining technics and proper forces on it.

Where n—rotate speed of shaft（r／min）

P—the power transmitted by the sh（kW）

[τT]—the permissible shear stress for the shaft（MPa）

A0—coefficient，Table 15—3

To hollow shaft：

When d＞100mm：

single key groove aggrandize3%， double key groove aggrandize7%；

When d≤100mm：

single key groove aggrandize5～7%， double key groove aggrandize10～15%

The location of elements on shaft

Orientation by sleeveUsed for orientation between two elements, inconvenient to locate the elements

with big distance, not feasible for the occasion where rotational speed is high. Orientation by retaining ringsOrientation by nuts

Orientation by bearing coverOrientation by flexible retaining ringsOrientation by fixed screwOrientation by fixed retaining ringsOrientation by cone

Peripheral orientation of elements on shaft, key、spline、interference-fit joint、fixed

screw.

15.3 CONSIDERATION FOR SHAFT GEOMETRYThe location of elements on shaft

In order to ensure the fixed position of elements on the shaft, it is needed to fix each element on the shaft radially and axially Radial location of shafts

Key joints Spline joints

Pin joints interference-fit joints

Axial location

Locknut Shoulder fillet

Locknut Snap ring

End plate Screw

configuration technics of shaftAs long as possible to aggrandize the transition fillet radius, as soon as possible to

unify the fillet radius of shaft. For install elements conveniently, we should wipe off the burr, the ends of the shaft should be 45, there must be grinding wheel outre groove to grind and cut rounding chamfer. To machining screw thread by lathe, there should be a groove for the reamer exit. The groove for keys should be in a horizontal line

15.4 CALCULATION OF SHAFT①Calculate accord to bending-torsion strength.

To ensure horizontal forces and vertical forces

Calculate bending moment MH and MV

Draw torque moment diagram（T）

Calculate bending moment

Checking the strength

② Checking exactly according to fatigue strength

Only normal stress

Only torsional shear stress

Safety factor

When even material, precise calculated load and stress ,S＝1.3～1.5

When uneven material, low precise calculated load and stress, S＝1.5～1.8

When even material, lowest precise load and stress, or the diameter of shaft d＞

200mm, S＝1.8～2.5

The measure when calculating dissatisfy

Lower concentrative stress effect when configuration design;

Aggrandize fillet radius；

Machining with crew thread on unload region；

Loosen combination of shaft and elements.

Intensify mechanical property Minish the roughness of the surfaceSurface preeningRoller compaction, rotoblast the surface

Using the material with high strength, or increase the diameter of shaft ② Checking according to static strengthCondition of strength

SSca— static strength calculated safety factor of the jeopardy section

SS—Calculate design safety factor according to yield strength

SSσ—the safety factor when only consider bending and axial force

SSτ—the safety factor when only consider torque

2、Checking calculate rigidity of shaft

①check by bending rigidity of shaft

li—the length of the i segment of stepped shaft

di—the radius of the i segment of stepped shaft L—the calculated length of stepped

shaftz—the number of segments of shaft in calculated length

[y]—allowable flexivity of shaft，Table 15—5

[θ]—allowable deflection angle of shaft,Table 15—5 ② Check the torsional rigidity of shaft

plain shaft ：

Stepped shaft：

T—the torque on shaft

G—shear elasticity modulus of material of shaft

Steel G＝8.1×104

IP—polar moment of inertia of shaft section

L—the length of the shaft supporting torque

Ti、li、IPi—the torque, length, polar moment of inertia of i segment of shaft

z—the number of segments of stepped shaft[φ]—allowable torsion angle per mile of shaft

MAIN CONTENT OF THIS CHAPTERConfiguration design of shaftThree calculation methods of shaft strengthRigidity calculation of shaft, stability of vibration, etc.

KEY POINTSStepped shaft design , strength and rigidity checking

Example 15.1 A transmission shaft ,given: transmitted power p=10kw,rotational

speed n=120r/min,estimate the diameter of shaft.

Example 15.2 There is a shaft system shown in Fig.15.6, and there are some errors and unsuitable points, mark them by order, and make a brief illustration.