Machine Design

UET, Taxila

“Power Screw & Gears”

What are Power Screws A power screw is a drive used in

machinery to convert a rotary motion into a linear motion for power transmission.

They find use in machines such as lead screw for lathes & machine tools, automotive jacks, vices, micrometers, Presses and C-clamps.

The mechanical advantage in the screw is to produce large axial forces in response to small torques.

Two principal categories of power screws are machine screws and re-circulating ball screws.

The screw threads are typically formed by thread rolling, which results in high surface hardness, high strength, and superior surface finish. Since high thread friction can cause self-locking when the applied torque is removed, protective brakes or stops to hold the load are usually not required.

Three thread forms that are often used are the:

Acme thread(Perfection or highest point)

Square thread, Buttress thread.

Acme Thread

Acme threads have a 29° thread angle, which is easier to machine than square threads. The Acme thread is stronger than the square thread because of the larger thread width at the root or minor diameter.

They are not as efficient as square threads because there is an increased friction induced by the thread angle.

Square Thread

Square threads are named after their square geometry. They are the most efficient power screw, but also the most difficult to machine, thus the most expensive.

As shown in last Figures, the Acme thread and the square thread exhibit symmetric leading and trailing flank angles, and consequently equal strength in raising and lowering.

Buttress ThreadButtress threads are of a triangular shape. It combines the advantages of the square and acme thread forms with only one difference, it only works in one direction.

Power screw requirements

• The thread forms used in power screws.

• Torque required to raise and lower a load in a power screw.

• Efficiency of a power screw and condition for self locking.

There are different series of this thread form and some nominal diameters, corresponding pitch and dimensions a and b are shown in special tables.

Dimensions of three different series of square thread form.

The square thread is designated by its nominal diameter and pitch, as for example, SQ 10 x 2 designates a thread form of nominal diameter 10 mm and pitch 2 mm.

Some details for screw thread

The proportions in terms of pitch are: h1= 0.5 p; h2 = (0.5 p – b); H = (0.5 p + a); e

= 0.5 p a and b are different for different series of

threads.

Some details of an Acme (Trapezoidal) thread form.

A metric Acme (trapezoidal) thread form is shown in last Figure and different proportions of the thread form in terms of the pitch are as follows: Included angle = 29-30o ; H1= 0.5 p

z = 0.25 p + (H1)/2 ; H3 = h3 = H1+ ac = 0.5 p + ac

ac is different for different pitch, for example ac = 0.15 mm for p = 1.5 mm ; ac = 0.25 mm for p = 2 to 5 mm; ac = 0.5 mm for p = 6 to 12 mm ; ac = 1 mm for p = 14 to 44 mm.

Dimensions of an Acme trapezoidal thread form:

According to the design standard, trapezoidal threads may be designated as, for example, Tr 50 x 8 which indicates a nominal diameter of 50 mm and a pitch of 8 mm.

Multiple power drives

When a large linear motion of a power screw is required two or more parallel threads are used. These are called multiple start power drives.

Efficiency of a power screw

A square thread power screw with a single start is shown in next figure. Here p is the pitch, α the helix angle, dm the mean diameter of thread and F is the axial load.

A square thread Development of power screw a single thread

A developed single thread is also shown in last figure, where L = n p for a multi-start drive, n being the number of starts.

In order to analyze the mechanics of the power screw we need to consider two cases:

(a) Raising the load (b) Lowering the load.

Raising the load

This requires an axial force P as shown in the figure below. Here N is the normal reaction and μN is the frictional force. For equilibrium

Forces at the contact surface for raising the load.

Divide the numerator and the denominator by cos

Lowering the load

Divide the numerator and the denominator by cos

Comparison:

The above analysis is for square thread and for trapezoidal thread some modification is required.

However, for sake of simplicity, the above equations could be used in both cases.

Condition for self locking

The load would lower itself without any external force if μπdm < L

and some external force is required to lower the load if

μπdm ≥ L

This is therefore the condition for self locking.

Efficiency of the power screw is given by

Bursting effect on the nut

Bursting effect on the nut is caused by the horizontal component of the axial load F on the screw and this is given by ( as shown in next figure)

Fx = F tan φ For an ISO metric nut 2φ = 60o and Fx = 0.5777 F.

Collar friction

If collar friction μc is considered then another term μFdc/2 must be added to torque expression. Here dc is the effective friction diameter of the collar. Therefore we may write the torque required to raise the load as

Example (1)

The C-clamp shown in next figure uses a 10 mm Acme (Trapezoidal) screw with a pitch of 2 mm. The frictional coefficient is 0.15 for both the threads and the collar. The collar has a frictional diameter of 16 mm. The handle is made of steel with allowable bending stress of 165 MPa. The capacity of the clamp is 700 N. Consider ac = 0.25 mm.

Required:

(a) Find the torque required to tighten the clamp to full capacity.

(b) Specify the length and diameter of the handle such that it will not bend unless the rated capacity of the clamp is exceeded. Use 15 N as the handle force.

C- clamp

Remember that root diameter d3 = (d nominal – 2h3), Pitch dia, d2= (dn-2z), h3= (0.5p+ac ), z = ( 0.5 pc),

Given: ac=0.25 mm.

Example (2)

Answer (2)

Gears

Spur Gears

Hypoid Gear in car differential

Spur gears

Spur have teeth parallel to the axis of rotation and are used to transmit motion from one shaft to another parallel shaft.

Of all types, the spur gear is the simplest and, for this reason, will be used to develop the primary kinematic relationships of the tooth form.

Spur Gear Schematic Representation

Helical gears

Helical Hears shown in next Figure have teeth inclined to the axis of rotation. Helical gears can be used for the same applications as spur gears and, when so used, are not asnoisy, because of the gradual engagement of the teeth during meshing.

The inclined tooth also develops thrust loads and bending couples, which are not present with spur gearing.

Sometimes helical gears are used to transmit motion between nonparallel shafts.

Helical Gears

Bevel gears

Bevel gears, shown in next Figure, have teeth formed on conical surfaces and are used mostly for transmitting motion between intersecting shafts.

The figure actually illustrates straight-tooth bevel gears.

Spiral bevel gears are cut so the tooth is no longer straight, but forms a circular arc.

Hypoid gears are quite similar to spiral bevel gears except that the shafts are offset and nonintersecting.

Bevel Gears

Worms and worm gears

Worms and worm gears, shown in next Fig. represent the fourth basic gear type.

As shown, the worm resembles a screw. The direction of rotation of the worm gear, also called the worm wheel, depends upon the direction of rotation of the worm and upon whether the worm teeth are cut right-hand or left-hand.

Worm-gear sets are mostly used when the speed ratios of the two shafts are quite high, say, 3 or more.

Spur Gears Nomenclature

The terminology of spur-gear teeth is illustrated in Next Figure. The pitch circle is a theoretical circle upon which all calculations are usually based; its diameter is the pitch diameter.

Two gears meshing together The pitch circles of a pair of mating gears

are tangent to each other.

A pinion is the smaller of two mating gears. The larger is often called the gear.

The circular pitch (p) is the distance, measured on the pitch circle, for similar point in two successive teeth. Thus the circular pitch is equal to the sum of the tooth thickness and the width of space.

The module (m) is the ratio of the pitch diameter to the number of teeth.

The customary unit of length used is the millimeter.

The module is the index of tooth size in SI.

The diametral pitch P is the ratio of the number of teeth on the gear to the pitch

Diameter. Thus, it is the reciprocal of the module. Since diametral pitch is used only with U.S. units, it is expressed as teeth per inch.

The addendum a is the radial distance between the top land and the pitch circle.

The dedendum b is the radial distance from the bottom land to the pitch circle.

The whole depth ht is the sum of the addendum and the dedendum.

The clearance circle is a circle that is tangent to the addendum circle of the mating gear. The clearance c is the amount by which the dedendum in a given gear exceeds the addendum of its mating gear.

The backlash is the amount by which the width of a tooth space exceeds the thickness of the engaging tooth measured on the pitch circles.

P = Diametral Pitch (teeth/inch),

= t/D

m = module (mm)

m = D/t

Note that P is the reciprocal of m

p = circular pitch (mm)

= D/t = m

Involute curve :Line de is moving tangent to the circle without sliding

Fundamentals

When two gears are in mesh, their pitch circles roll on one another without slipping. The pitch radii as r1 and r2 and the angular velocities as ω1 and ω2, respectively.

Then the pitch-line velocity is:

Example 1

Suppose now we wish to design a speed reducer such that the input speed is 1800 rev/min and the output speed is 1200 rev/min. This is a ratio of 3:2; the gear pitch diameters would be in the same ratio, for example, a 4-in pinion driving a 6-in gear.

Example 2

The various dimensions found in gearing are always based on the pitch circles.

Suppose we specify that an 18-tooth pinion is to mesh with a 30-tooth gear and that the diametral pitch of the gear set is to be 2 teeth per inch.

Then the pitch diameters of the pinion and gear are:

Diameter of gear (t) = Number of teeth / Diametral pitch

D1 = t1 / P = 18 / 2 = 9 in.

D2 = t2 / P =30 / 2 = 15 in

Gear Train

A gear train is two or more gear working together by meshing their teeth and turning each other in a system to generate power and speed. To create large gear ratio, gears are connected together to form gear trains. They often consist of multiple gears in the train.

The most common of the gear train is the gear pair connecting parallel shafts. The teeth of this type can be spur, helical or herringbone.

The angular velocity is simply the reverse of the tooth ratio.

Simple Gear Trains

(Idler gear)GEAR 'C'GEAR 'B'GEAR 'A'

v

v

CBA

The typical spur gears as shown in diagram. The direction of rotation is reversed from one gear to another. The only function of the idler gear is to change the direction of rotation.

The teeth on the gears must all be the same size so if gear A advances one tooth, so does B and C.

Diameter could be d or DNumber of teeth could be N or t

.

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.r velocity = angula

= m tDand = m tD; = m tD

t

D =

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D =

t

Dm =

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C

C

B

B

A

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CCBBAA

CCBBAA

CC

BB

AA

tNtNtN

revoftermsinor

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DDD

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min/

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Gear Ratio:

valueTraingeardriverofSpeed

geardrivenofSpeed

N

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CA

speedOutput

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Application:

a) to connect gears where a large center distance is required

b) to obtain desired direction of motion of the driven gear ( CW or CCW)

c) to obtain high speed ratio

Torque & Efficiency

The power transmitted by a torque T (N.m) applied to a shaft rotating at N rev/min is given by: 60

2 TNP

In an ideal gear box, the input and output powers are the same so;

GRN

N

T

TTNTN

TNTNP

2

1

1

22211

2211

60

2

60

2

It follows that if the speed is reduced, the torque is increased and vice versa. In a real gear box, power is lost through friction and the power output is smaller than the power input. The efficiency is defined as:

11

22

11

22

602

602

TN

TN

TN

TN

InPower

outPower

Because the torque in and out is different, a gear box has to be clamped in order to stop the case or body rotating. A holding torque T3 must be applied to the body through the clamps.

The total torque must add up to zero. T1 + T2 + T3 = 0 If we use a convention that anti-

clockwise is positive and clockwise is negative we can determine the holding torque. The direction of rotation of the output shaft depends on the design of the gear box.

Problem 1

A gear box has an input speed of 1500 rev/min clockwise and an output speed of 300 rev/min anticlockwise. The input power is 20 kW and the efficiency is 70%. Determine the following.

i. The gear ratio; ii. The input torque.; iii. The output power.; iv. v. The holding torque.

11

11

2

1

2

60

60

2

5300

1500.

:

N

PowerInputT

TNPowerInput

N

N

speedOutput

speedInputVRorRG

Solution

)(

3.12715002

20000601

clockwiseNegative

mNTtorqueInput

iseunticlockwPositive

mNTtorqueOutput

6.445

3002

14000602

kWOutputPower

powerInpu

powerOutput

14207.0

7.0

Clockwise

mNT

T

TTT

3.3186.4453.127

06.4453.127

0

3

3

321

Compound gears

Compound gears are simply a chain of simple gear trains with the input of the second being the output of the first. A chain of two pairs is shown below. Gear B is the output of the first pair and gear C is the input of the second pair. Gears B and C are locked to the same shaft and revolve at the same speed.

For large velocities ratios, compound gear train arrangement is preferred.

GEAR 'A'

GEAR 'B'

GEAR 'C'

GEAR 'D'

Compound Gears

A

C

BD

Output

Input

For large velocities ratios, compound gear train arrangement is preferred.

The velocity of each tooth on A and B are the same so:A tA = B tB -as they are simple gears. Likewise for C and D, C tC = D tD.