HELICAL TORSION SPRINGS As depicted in Figure 14.14, helical torsion springs are wound in a way similar to exten- sion or compression springs but with the ends shaped to transmit torque
Trang 1Comment: If this corresponds to operating speeds (for equipment mounted on this
_spring), it may be necessary to redesign the spring
: @ “Check for buckling for extreme case of deflection (5 = 4;):
extension spring with both ends fixed against axial deflection is the same as that for a heli-
In extension springs, however, the coils are usually close wound so that there is an ini- tial tension ot so-termed preload P No deflection therefore occurs until the initial tension built into the spring is overcome; that is, the applied load P becomes larger than initial ten- sion (P > P,) It is recommended that [1] the preload be built so that the resulting stress Tipitias Calculated from Eq (14.6) equals about 0.68,/C Here, Sy, and C present ultimate strength and spring index, respectively
P = P —~ P,, is given as follows:
8M — P)
dG:
The reduced coil diameter results in a lower stress because of the shorter moment arm
Hence, hook stresses can be reduced by winding the last few coils with a decreasing diameter D No stress concentration factor is needed for the axial component of the load
Active coils refers to all coils in the spring, not counting the end coils, which are bent
to form a hook (Figure 14.3c) Depending on the details of the design, each end hook adds the equivalent of 0.1 to 0.5 helical coil For an extension spring with two end hooks, the total number of coils is then
N, = Na +2(0.1 to 0.5) (14.31)
As before, N, represents the number of active coils
The spring rate is expressed, by the application of Eq (14.30), in the form
The spring load is therefore
P=P.+kô (14,33)
The quantities & and ổ are given by Eqs (14.32) and (14.30), respectively
Critical stresses occur in the end hooks or end loops of extension springs The hooks
must be designed so that the stress concentration effects produced by the presence of bends
are decreased as much as possible It is obvious that sharp bends should be avoided, since the stress concentration factor is higher for sharp bends Maximum bending stress at section A (Figure 14.13a) and maximum torsional stress at section B (Figure 14,13b) in the bend of the end coil may be approximated respectively, by the formulas,
where 7, is the mean radius and +; represents the inside radius
The stresses in coils are obtained from the same formulas as used in compression springs In extension springs, a mechanical stop is desirable to limit deflection to an allowable value; while in compression springs, deflection is restricted by the solid deflec-
tion Maximum stress values may be 70% of those used for extension springs of the
identical compression springs
581
Trang 2The yield strength S, for torsion springs can be estimated from Table 14.3 Based on
the energy of distortion criterion, we divide the Sy, in each part in Table 14.3 by the quantity 0.577 The endurance limit S, for torsion springs can be found in a like manner:
the S’, in each part in Table 14.3 is divided by 0.577 The process of designing of torsion springs is very similar to that of the helical compression springs
HELICAL TORSION SPRINGS
As depicted in Figure 14.14, helical torsion springs are wound in a way similar to exten- sion or compression springs but with the ends shaped to transmit torque These coil ends can have a variety of forms to suit the application The coils are usually close wound like
an extension spring but have no initial tension We note that forces (P) should always be applied to arms of helical torsion springs to close the coil, as shown in the figure, rather than open it The spring is usually placed over a supporting rod The rod diameter is
about 90% smaller than the inside diameter of the spring Square or rectangular wire is
in widespread use in torsion springs However, round wire is often used in ordinary ap- plications, since it costs less The torque about the axis of the helix acts as a bending mo- ment on-each section of the wire The material is therefore stressed in flexure The bend- ing stress can be obtained from curved beam theory It is convenient to write the flexure formula in the form
c = distance from the neutral axis to the extreme fiber
J = moment of inertia about the neutral axis
K = stress concentration factor
_ 3C?+C—0.8
° 3C(C + D where C = D/h The quantity h represents the depth of the rectangular cross section We see from these expressions that K; > K,,, as expected
; The maximum compressive bending stress at the inner fiber of the helical torsion spring is therefore
i=
4.37)
Mec
đi = Kis (14.38)
Carrying the bending moment M = Pa and the section modulus I /c of round and rectan-
gular wires into Eq (14.38) gives the bending stress In so doing, stress on the inner fiber
of the coil is:
The quantity } is the width of rectangular cross section
For commonly employed values of the spring index, k == M/®r.y, the curvature has no effect on the angular deflection Through the use of Eqs (4.15) and (4.14), we have
Trang 3For springs of round wire, to account for the friction between coils, based on experience
[1], Eq (4.41) is multiplied by the factor of 1.06 Interestingly, angle @ in some cases can
be many complete turns, as in Example 14.4
FATIGUE LOADING
A dynamically loaded torsion spring operates between two moment levels Mmax and Moin:
‘The tensile stress components occurring at the outside coil diameter of a round wire heli- cal torsion is then
32M max 32Mrmin
ao max Km Øa mịn = Ấo xa?
Hence, the mean and alternating stresses are
Øo max “F Fo,min On, Fo.max ~~ Fo,min
Having the mean and alternating stresses available, helical torsion springs are designed by following a procedure similar to that of helical compression springs
SPIRAL TORSION SPRINGS
- A’spiral torsion spring (Figure 14.15) can also be analyzed by the foregoing procedure
Therefore, the highest stress occurring on the inner edge of the wire is given by Eqs (14.39) and (14.40) Likewise, Eq (14.41) can be applied directly to ascertain the angular deflection Spiral springs are usually made of thin rectangular wire
EXAMPLE 14.4 | : Spiral Torsion Spring: Design for Static Loading
For 4 torsional window-shade spring (Figure 14.15), determine the maximum operating moment and corresponding angular deflection
Design Decisions: We select a music wire of E = 207 GPa; d = 1.625 mm, D = 25 mm, and
Nj = 350 Asafety factor of 1.5 is used
CHAPTER 14 © SPRINGS Solution: - By Eq, (14.12) and Table 14.2,
Springs in the form of a cantilever are often used as electrical contacts For springs with
uniform sections, we may use the results of Chapters 3 and 4 Recall from Sections 4.4 and
4.10 that, when the width 6 of the cross section is large compared with the depth A, it is necessary to multiply the deflection as given by the formula for a narrow beam section by
(1 ~ v?), where v is the Poisson’s ratio
A cantilever spring of uniform stress o with a constant depth A, in a plan view looks like the triangle depicted in Figure 14.16 (see Section 3.8), However, near the free end, the wedge-shaped profile must be modified to have adequate strength to resist the shear force
as depicted by the dashed lines in the figure From the flexure formula, we have
Trang 4586 PART i} @ APPLICATIONS
As the cross section varies, end deflection § may conveniently be obtained using Castigliano’s theorem (see Section 5.6) It can be shown that
The quantity £ represents modulus of elasticity
MULTILEAF SPRINGS nung 4
Springs of varying width present a space problem Multileaf springs are n wi espread usage, particularly in automotive and iailway service An exact analysis of these serine mathematically complex For small deflections, an approximate solution can be obtaine
by the usual equations of beams, as shown in the following brief discussion ; ;
A multileaf spring, approximating a triangular spring of uniform strength, is shown in Figure 14.17 Note that each half of the spring acts as a cantilever of length L We obser from the figure that a constant strength triangle is cut into a series of leaves of caus wi Ẫ and reatranged in the form of a mulBleaf spring A central bolt or clamp, use to ; a the leaves together, causes a stress concentration The triangular spring and equivatent
Figure 14.17 Multileaf spring: {a) front
view of actual spring; (b) top view of approximation; (c) top view of equivalent
n leaves
Figure 14.18 Example 14.5
Automotive-type leaf spring
multipleaf spring have the identical stress and deflection characteristics, with the exception that the interleaf friction provides damping in the multileaf spring Also, the multileaf spring can resist full load in only one direction; that is, leaves tend to separate when loaded
in opposite direction However, this is partially overcome by clips, as in vehicle suspension
Automotive-Type Multileaf Spring: Design for Fatigue Loading
Asix-leaf spring is subjected toa load at the center that varies between Prax and Prin (Figure 14.18)
Estimate the total length 2 arid width of each leaf
Given: = Psi = 80 Ib, Prag = 400 Ib
Assumptions: © Stress concentration at the center is such'that Ky = 1.2 Use a survival rate of 50%
and Cy = Ces-1
Design Decisions: : We use a sieel alloy spring of S, = 200 ksi, S/ = 78 ksi, E = 30 x 10° psi,
ves 0.3, k= 0:25 in, k= 140 Ib/in The material is shot peened A safety factor of 1.4 is applied
Solution: Brom: Table 8.3, C, = 1.) The ‘modified endurance limit, by Eq (8.6), S =
G)C)G) (/1.2)78 = 65 ksi Each half of a spring acts as a cantilever supporting haif of the total load
The mean and the alternating loads are therefore
400-+ 80 _ 400 — 80
By, tôn “20B, B= = 160 Ib
Inasmuch‘ as: bending stress is directly proportional to the load, we have og /om = Pa/P, = 2/3
The méan stress, using Eq (14.42), is
OP, E 24 :
THẾ GR bQ/259 b
EXAMPLE 14.5
Trang 5588 PARTH & APPLICATIONS
Substituting the given numerical values into Eq (8.20), we have
“Because the spring is loaded at the center with 2P, Eq (14.44) becomes k= Ebh?/3L2(1 — 92) la-
troducing the given data results in
Many spring functions may also be acquired by the elastic bending of thin plates and shells
of various shapes and by the blocks of rubber Hence, there are spring washers, clips, constant-force springs, volute springs, rubber springs, and so on A volute spring isa wide, thin strip of steel wound flat so that the coils fit inside one another, as shown in Figure 14.19 These-springs have more lateral stability than helical compression springs, and rub- bing of adjacent turns provides high damping Here, we briefly discuss three commonly encountered types of miscellaneous springs
CONSTANT-FORCE SPRINGS
The constant-force (Neg’ator) spring is a prestressed strip of flat spring stock that coils around a bushing or successive layers of itself (Figure 14.20) Usually, the inner coil is fastened to a flanged drum When the spring is deflected by pulling on the outer end of the
Drum
Figure 14.20 Constant-force spring
Figure 14.19 Volute spring
Figure 14.21 Cross section through Figure 14.22 Belleville springs or washers:
a Belleville spring {a) in parallel stack; (b) in series stack
coil, a nearly constant resisting force develops and there is a tendency for the material to recoil around itself A uniform-force spring is widely employed for counterbalancing loads (such as
in window sash), cable retractors, returning typewriter carriages, and making constant-torque spring motors It provides very large deflection at about a constant pull force [1, 15]
BELLEVILLE SPRINGS
Belleville springs or washers, also known as coned-disk springs (Figure 14.21), patented
by J F Belleville in 1867, are often used for supporting very large loads with small de- flections, Some applications include various bolted connections, clutch plate supports, and gun recoil mechanisms On loading, the disk tends to flatten out, spring action being ob- tained thus The load-deflection characteristics are changed by varying the ratio h/t be- tween cone height # and thickness ¢ Belleville springs are extremely compact and may be used singly or in combination of multiples of identical springs to meet needed characteris-
tics The forces associated with a coned-disk spring can be multiplied by stocking them in
parallel (Figure 14.22a) On the other hand, the deflection corresponding to a given force can be increased by stacking the springs in series as shown in Figure 14.22b
The theory of the Belleville springs is complicated The following formulas are based
on the simplifying assumption that radial cross sections of the spring do not distort during deflection The results are in approximate agreement with available test data [1, 2, 21] As
is the case for a truncated cone shell, the upper edge of the spring is in compression and the lower edge is in tension [22]
The load-deflection relationship can be expressed in the form
Trang 6Load-deflection characteristics are changed by varying the ratio between cone height and.thickness, h/t Figure 14.23 illustrates force-deflection curves for Belleville washers with four different h/t ratios These curves are generated by applying Eqs (14.45), where 1.0 deflection and 1.0 force refer to the deflection at the flat condition and the force at the flat condition, respectively [16, 17} We see from the figure that coned-disk springs have nonlinear P — ổ properties For low values (/t = 0.4), the spring acts almost linearly, and large h/t values result in prominent nonlinear behavior At h/t = V2, the central portion
of the curve approximates a horizontal line; that is, the load is nearly constant over a con- siderable deflection range In the range ⁄2<h/2< /8, a prescribed force corresponds to more than one deflection A phenomenon occurring at h/t > A⁄2 is termed snap-through buckling, at which the spring deflection becomes unstable
Interestingly, in snap-through buckling the spring quickly deflects or snaps to the next stable position It can be shown that [6], if h/t > /8 the spring can snap into a deflection
position for which the calculated force becomes negative Then, a load in the direction op-
posite to the initial load will be required to return the spring to its unloaded configuration
Stress distribution in the washer is nonuniform The largest stress o4 occurs at the upper inner édge A (convex side) at deflection 5 and is compressive The outside lower
2.0
18 Aft = V8
16 L4
RUBBER SPRINGS
A rubber spring and cushioning device is referred to as a rubber mount (Figure 14.24)
Springs of this type are widely used due to their essentially shock and vibration damping
Trang 7592 PART II: ` 9 APPLICATIONS
qualities and low elastic moduli The foregoing properties help dissipate energy and pre- vent sound transmission Stresses and deformations in the rubber mounts for smail deflections can be derived by the use of appropriate equations of mechanics of materials
A cylindrical rubber spring with direct shear loading is shown in Figure 14.24a The rubber is bonded to a steel ring on the outside and a steel shaft in the center The shear stress
The quantity T represents the torque
Note that rubber does not follow Hooke’s law but becomes increasingly stiff as the de- formation is increased The modulus of elasticity is contingent on the durameter hardness number of the rubber chosen for the mount [12] The results of calculations must therefore
be considered only approximate
REFERENCES
1 Associated Spring-Barnes Group Design Handbook Bristol, CN: Associated Spring-Barnes
Group, 1987
2 Wahl, A M Mechanical Springs, 2nd ed New York: McGraw-Hill, 1963
3 Shigley, J E., and C R Mischke Standard Handbook of Machine Design, 2nd ed New York:
CHAPTER 14 ® SPRINGS
4 Carlson, H C R “Selection and Application of Spring Materials.” Mechanical Engineering 78
(1956), pp 331-34,
5 Carlson, H.C R Spring Designer’s Handbook New York: Marcel Dekker, 1978
6 Burr, A H., and J B Cheatham Mechanical Analysis.and Design, Ind ed Upper Saddle River, NJ: Prentice Hall, 1995
7 Metals Handbook, vol.1, 9th ed Metals Park, OH: ASM, 1978
8 Rothbart, H A., ed Mechanical Design and Systems Handbook, 2nd ed New York: McGraw-
Hill, 1995
9, Avallone, E A., and T Baumeister Il, eds Mark’s Standard Handbook for Mechanical
Engineers, 10th ed New York: McGraw-Hill, 1996
i0 Samonov, C “Computer-Aided Design of Helical Compression Springs.” ASME paper
no 80-DET-69, 1980
li Dietrich, A “Home Computers Aid Spring Design.” Design Engineering (June 1981), pp 31-35
12 Spotts, M F, and T E Shoup Machine Design, 7th ed Upper Saddle River, NJ: Prentice
Hall, 1998
13 Juvinail, R C., and K M Marshak Fundamentals of Machine Component Design, 3rd ed
New York: Wiley, 2000
14 Hamrock, B J., B O., Jacobson, and S R Schmid Fundamentals of Machine Elements
New York: McGraw-Hill, 1999
15 Levinson, I J Machine Design Reston, VA: Reston/Prentice Hall, 1978
16 Mott, R L Machine Elements in Mechanical Design, 2nd ed New York: Macmillan, 1992
17 Norton, R L Machine Design: An Integrated Approach, 2nd ed Upper Saddle River, NJ:
Prentice Hall, 2000
18 Shighley, J E., and C R Mischke Mechanical Engineering Design, 6th ed New York:
McGraw-Hill, 2001
19 Dimarogonas, A D Machine Design: A CAD Approach New York: Wiley, 2001
20 Zimmerli, P F Human Failures in Spring Design Mainspring, Associated Spring Corp., Bristol,
CN, August-September 1957
21 Ortwein, W C Machine Component Design St Paul, MN: West, 1990
22 Ugural, A C Stresses in Plates and Shells, 2nd ed New York: McGraw-Hill, 1999
(b) The change in shear stress
Given: L = 1.25 m, đ = 8mm, G=79Pa 14.2 A steel bar supports a load of2 kN with a moment arm R = 150 mm (Figure 14.24) Calculate (a) The wire diameter
(b) The length for a deflection of 40 mm
Given: n = 1.5, Sys = 350 MPa, G = 79 GPa
Trang 814.3 A helical spring must exert a force of 1 KN after being released 20 mm from its most highly
compressed position Determine the number of active coils
Désign Assumptions: The loading is static, tay = 450 MPa, G = 29GPa, d = 7mm, and
` =5
14.W1 Using the website at www.leesspring.com, rework Example 14.1
14,W2 Check the site at www.acxesspring.com to review the common spring materials presented
List five commonly employed wire spring materials and their mechanical properties
A helical compression spring used for static loading has d = 3 mm, D = 15 mm, Ny = 10, and squared ends Determine
(a) The spring rate and the solid height
(b) The maximum load that can be applied without causing yielding
Design Decision: The spring is made of ASTM A227 hard-drawn steel wire of G = 79 GPa
A helical compression spring is to support a 2 KN load Determine (a) The wire diameter
(b) The free height
(ec) Whether buckling will occur in service
Given: The spring has r, = 10%, C = 5, and k = 90 N/mm
Assumptions: Both ends are squared and ground and constrained by parallel plates
Design Decisions: The spring is made of steel of Sy; = 500 MPa, S,, = 280 MPa, G =
79 GPa Use a safety factor of 1.3
A helical compression spring with ends squared and ground has d = 1.8 mm, D = 15 mm,
rp = 15%, and hy = 21.6 mm Determine, using a safety factor of 2, (a) The free height
(b) Whether the spring will buckle in service, if one end is free to tip
Design Decision: The spring is made of steel having S,, == 900 MPa and G = 79 GPa
Design a helical compression spring with squared and ground ends for a static load of 40 Ib,
C = 8, k = 50 Ibfin,, r, = 20%, and n = 2.5 Also check for possible buckling
Assumption: The ends are constrained by parallel plates
Design Decision: The spring is made of steel of Sy, = 60 ksi and G = 11.5 x 10° psi
A machine that requires a helical compression spring of k = 120 Ib/in., tay = 75 ksi, re = 10%, D = 3 in., and is to support a static load of 400 Ib Determine
(2) The wire diameter
(b) The free height
(c) Whether the spring will buckle in service, if one end is free to tip
Assumption: The ends are squared
Design Decision: The spring is made of steel having G = 11.5 x 10® psi
CHAPTER 14 @ SpRINGS Sections 14.7 and 14.8
14.9 14.10
141
14.12
1413
1414 14.15
Redo Problem14.5 for a load that varies between 2 kN and 4 KN, using the Soderberg relation
Also determine the surge frequency ‘
A helical compression spring for a cam follower supports a load that varies between 30 and
180 N, Determine
(a) The factor of safety, according to the Goodman criterion
(b) The free height `
(c) The surge frequency
(d) Whether the spring will buckle in service
Design Decisions: The spring is made of music wire Both ends are squared and ground; one
end is free to tip
Given: d = 3mm, D= 15mm, Ng = 22, re = 10%, G=79 GPa
A helical compression spring, made of 0.2-in diameter music wire, carries a fluctuating load
The spring index is 8 and the factor of safety is 1.2 If the average load on the spring is 100
1b, determine the allowable values for the maximum and minimum loads Employ the Good- man theory
A helical compression spring made of a music wire has d = 5 mm, D = 24 mm, and G =
79 GPa Determine
(a) The factor of safety, according to the Goodman relation
(b) The number of active coils, Requirements: The height of the spring varies between 65 and 72 mm with corresponding
loads of 400 and 240 N
Asteel helical compression spring is to exert a force of 4 Ib when its height is 3 in and a max- imum load of 18 Ib when compressed to a height of 2.6 in Determine, using the Soderberg criterion with a safety factor of 1.6,
(a) The wire diameter
{b) The solid deflection
(c) The surge frequency
(d) Whether the spring will buckle in service, if ends are constrained by parallel plates
Given: The spring has C = 6, S,; = 80 ksi, Si, = 45 ksi, G = 11.5 x 10° psi, r, = 10%,
Design Assumption: Ends will be squared and ground
Resolve Problem 14.13 for the case in which the helical spring is to exert a force of 2 Ib at Sin height and a maximum load of 10 Ib at 4.2 in height
An engine valve spring must exert a force of 300 N when the valve is closed (as shown in
Figure P14.15) and 500 N when the valve is open Apply the Goodman theory with a safety
595
Trang 9
14.19 Design a window-shade spring similar to that depicted in Figure 14.15, Determine (a) The number of active coils, if a pull on shade of 15 N is exerted after being wound up to
16 revolutions
(b) The maximum bending stress
Assumptions: The spring will be made of 1.2-mm square wire having E = 207 GPa, D =
18 mm, and a roller diameter of 32 mm
14.20 Design a helical torsion spring similar to that shown in Figure 14.14 Calculate (a) The maximum operating moment
(b) The maximum angular rotation
Assumptions: A safety factor of 1.4 is used The spring is made of oil-tempered steel wire
Given: E = 30 x 10®psi, d = 0.08in., D=0.6in., Ng = 6
Figure P14.15
14.21 A multileaf steel spring is to support a center load that varies between 300 and 1100 N (Fig- factor of 1.6 to calculate ure 14.17) Estimate, using the Goodman criterion with a safety factor of 1.2,
(a) The wire diameter (a) The appropriate values of A and b for a spring of proportions b = 40h
Given: The lift is 8 mm | Given: S, = 1400 MPa, S; = 500 MPa, Œ = 207 GPa, and » = 0.3 The total length 27, is tọ Design Decisions: The spring is made of steel having S„; = 720MPa, S,, = 330MPa, be 800 mm
G = 79 GPa, and C = 6 : Assumptions: Use C, = Cy = C, = 1 Stress concentration at the center is such that
Kp = L4
14.16 A helical spring, made of hard-drawn wire having G == 29 GPa, supports a continuous load
Determine
(a) The factor of safety based on the Soderberg criterion
(6) The free height
{e) The surge frequency
(d) Whether the spring will buckle in service
Given: d = 6mm, D = 30mm, r„ = 20%
Design Requirements: The ends are squared and ground; one end is free to tip In the most-
compressed condition, the force is 600 N; after 13 mm of release, the minimum force is
340N
Sections 14.9 through 14.12
14.17 A helical tension spring has d, == 3 mm and D; = 30 mm If a second spring is made of the
same material and the same number of coils with Dz = 240 mm, find the wire diameter dy that would be required to give the same spring rate as the first spring
14/18 An extension coil spring is made of 0.02-in music wire and has a mean diameter of coil of
0.2 in The spring is wound with a pretension of 0.2 Ib, and the load fluctuates from this value
up to 1.0 Ib Determine the factor of safety guarding against a fatigue failure Use the Good- man criterion
Trang 10
This chapter is devoted to the analysis and design of power screws, threaded fasteners,
bolted joints in shear, and permanent connectors such as rivets and weldments Adhesive
bonding, brazing, and soldering are also discussed briefly Power screws are threaded de- vices used mainly to move loads or accurately position objects They are employed in ma- chines for obtaining motion of translation and also for exerting forces The kinematics of power screws is the same as that for nuts and screws, the only difference being the geometry
of the threads, Power screws find applications as motion devices
The success or failure of a design can depend on proper selection and use of its
fasteners A fastener is a device to connect or join two or more members Many varieties of fasteners are available commercially The threaded fasteners are used to fasten the various parts of an assembly together We limit our consideration to detachable threaded fasteners such as bolts, nuts, and screws (Figure 15.1) General information for threaded fasteners as well as for other methods of joining is presented in some references listed at the end of this
chapter and at the websites www.americanfastener.com and www.machinedesign.com
Listings of a variety of nuts, bolts, and washers are found at www.nutty.com For bolted joint technology, see the website at www.boltscience.com
Figure 15.1 An assortment of threaded fasteners {Courtesy of Clark Craft Fasteners.)
Trang 11and relies on available experimental results As with the threaded fasteners, rivets exist in
great variety Note that, while welding has replaced riveting and bonding to a considerable extent, rivets are customarily employed for certain types of joints Welding speeds the man- ufacturing of parts, assembly of these components into structures, and reduces the cost com- pared to casting and forging Soldering, brazing, cementing, and adhesives are all means of bonding parts together
15.2 STANDARD THREAD FORMS
Threads may be external on the screw or bolt and internal on the nut or threaded hole The thread causes a screw to proceed into the nut when rotated The basic arrangement of a helical thread cut around a cylinder or a hole, used as screw-type fasteners, power screws, and worms, is as shown in Figure 15.2 Note that the length of unthreaded and threaded
portions of shank is called the shank or bolt length Also observe the washer face, the fillet
under the bolt head, and the start of the threads Referring to the figure, some terms from geometry that relate to screw threads are defined as follows ; Pitch p is the axial distance measured from a point on one thread to the corresponding point on the adjacent thread Lead L represents the axial distance that a nut moves, or advances, for one revolution of the screw Helix angle, 2, also called the lead angle, may be cut either right-handed (as in Figure 15.2) or left-banded All threads are assumed to be right-handed, unless otherwise stated
A single-threaded screw is made by cutting a single helical groove on the cylinder For
a single thread, the lead is the same as the pitch Should a second thread be cut in the space between the grooves of the first (imagine two strings wound side by side around a pencil),
a double-threaded screw would be formed For a multiple (two or more)-threaded screw,
Unthreaded shank
Washer face 1 Thread length ¬ Nat
Head — Shank or bolt length |
Figure 15.2 Hexagonal bolt and nut illustrate the terminology of threaded fasteners
Note: p = pitch, 4 = helix or lead angle, a = thread angle, d = major diameter, dp = pitch
CHAPTER 15 © PowER SCREWS, FASTENERS, AND CONNECTIONS
in which L = lead, n = number of threads, and p = pitch We observe from this relation- ship that a multiple-threaded screw advances a nut more rapidly than a single-threaded screw of the same pitch Most bolts and screws have a single thread, but worms and power screws sometimes have multiple threads Some- automotive power-steering screws occasionally use quintuple threads
UNIFIED AND ISO THREAD ForM
For fasteners, the standard geometry of screw thread shown in Figure 15.3 is used This is essentially the same for both the Unified National Standard (UNS), or so-called unified, and International Standards Organization (ISO) threads The UNS (inch series) and ISO (metric series) threads are not interchangeable In both systems, the thread angle is 60° and the crests and roots of the thread may be either flat (as depicted in the figure) or rounded
The major diameter d and root (minor) diameter d, refer to the largest and smallest diame- ters, respectively The diameter of an imaginary cylinder, coaxial with the screw, intersect- ing the thread at the height that makes width of thread equal to the width of space is called
the pitch diameter dp
Tables 15.1 and 15.2 furnish a summary of the various sizes and pitches for the UNS and ISO systems We see from these listings that the thread size is specified by giving the number of threads per inch N for the unified sizes and giving the pitch p for the metric sizes
The tensile stress area tabulated is on the basis of the average of the pitch and root
diameters This is the area used for calculation of axial stress (P/A) Extensive information for various inch-series threads may be found in ANSI Standards [1, 2]
Coarse thread (designated as UNC) is most common and recommended for ordinary applications, where the screw is threaded into a softer material It is used for general as- sembly work Fine thread (denoted by UNF) is more resistant to loosening, because of its smaller helix angle Fine threads are widely employed in automotive, aircraft, and other ap- plications where vibrations are likely to occur In identifying threads, the letter A is used for external threads, and B is used for internal threads The UNS defines the threads according
to fit Class 1 fits have the widest tolerances and so are the loosest fits Class 2 fits are most commonly used Class 3 fit is the one having the least tolerance and is utilized for highest
Figure 15.3 Unified and ISO thread forms The portion of basic profile of the external thread is shown: h = depth of thread, b = thread thickness at the root
601
Trang 12
Table 15.4 Dimensions of unified screw threads Table 15.2 Basic dirnensions of ISO (metric) screw threads
diameter, perinch, diameter stress area, perinch, diameter, stress area, diameter, d (mm) p Gam) | am?) ng Tensile stress
a - Sa - SOURCE: [1]
precision applications Clearly, cost increases with higher class of fit An example of Notes: Metric threads are specified by nominal diameter and pitch in millimeters; for example, M10 x 1.5 The
approved identification symbols: letter M, which proceeds the diameter, is the clue to the metric designation
Root or minor diameter d, + d ~ 1.227p
1 in.-12 UNF-2A-LH
i This defines !-in diameter x 12 threads per inch, unified fine thread series, class 2 fit,
ị external, left-handed thread Metric thread specification is given in Table 15.2
Power SCREW THREAD FoRMS
Figure 15.4 depicts some thread forms used for power screws The Acme screw is in wide- spread usage They are sometimes modified to a stub form by making the thread shorter
This results in a larger minor diameter and a slightly stronger screw A square thread pro- vides somewhat greater strength and efficiency but is rarely used, due to difficulties in manufacturing the 0° thread angle The 5° thread angle of the modified square thread par- tially overcomes this and some other objections Standard sizes for three power screws (@) ®) ()
thread forms are listed in Table 15.3 The reader is referred to ANSI Standards for further Figure 15.4 Typical power screw thread forms All threads shown are external
Trang 13
Threads per inch
Major diameter, Acme, Square and
d (in) Acme stub modified square
45.3 MECHANICS OF POWER SCREWS
As noted previously, a power screw, sometimes called the linear actuator or translation screw, is in widespread usage in machinery to change angular motion into linear motion, to _exert force, and to transmit power Applications include the screws for vises, C-clamps, presses, micrometers, jacks (Figure 15.5), valve stems, and the lead screws for lathes and other equipment In the usual configuration, the nut rotates in place, and the screw moves axially In some designs, the screw rotates in place, and the nut moves axially Forces may
be large, but motion is usually slow and power is small In all the foregoing cases, power screws operate on the same principle
A simplified drawing of a screw jack having the Acme thread is shown in Figure 15.6
The load W can be lifted or lowered by the rotation of the nut that is supported by a washer, called a thrust collar (or a thrust bearing) Itis, of course, assumed that the load and screw are prevented from turning when the nut rotates Hence, there needs to be some friction at the load surface to prevent the screw from turning with the nut Alternatively, the power serew could be turned against a nut that is prevented from turning to lift or lower the load
In either case, there is significant friction between the screw and nut as well as between the nut and the collar Ordinarily, the screw is a hard steel, while the nut is made of a softer ma- terial (e.g., an alloy of aluminum, nickel, and bronze) to allow the parts to move smoothly
representation of power screw used as
a screw jack Note: Only the nut rotates in this model: d,, = mean thread diameter, d.= mean collar diameter
Figure 15.5 Worm-gear screw jack
(Courtesy of Joyce/Dayton Corp.)
; In this section, we develop expressions for ascertaining the value: r
to lift and lower the load using a jack We see from Figure 15.6 that vomning the oat fovces each portion of the nut thread to climb an inclined plane This plane is depicted by un- wrapping or developing one revolution of the helix in Figure 15.7a, which includes a small block representing the nut being slid up the inclined plane of an Acme thread The forces acting on the nut as a free-body diagram are also noted in the figure Clearly, one edge of the thread forms the hypotenuse of the right triangle, having a base as the circumference of the mean-thread-diameter circle and as the lead Therefore,
Trang 14The preceding notation is the same as for worms (see Section 12.9) except that unnecessary subscripts are omitted
Torque To LIFT THE LOAD
The sum of all loads and normal forces acting on the entire thread surface in contact are de- noted by W and N, respectively To lift or rise the load, a tangential force Q acts to the right and the friction force ƒN acts to oppose the motion (Figure 15.7) The quantity f represents
the coefficient of sliding friction between the nut and screw, or the coefficient of thread
friction The thread angle increases the frictional force by the wedging action of the threads The conditions of equilibrium of the horizontal and vertical forces give
F,=0: Q-N(f cosa + cosa, sind) = 0
(a)
» F,=0: W + N(f sind — cosa, cosa) = 0
where «, is the normal thread angle and the other variables are defined in the figure Inas- much ag we are not interested in the normal force N, we eliminate it from the foregoing equations and solve the result for Q In so doing, we have
9= cosa, Cosi, ~ ƒ sind
The screw torque required to move the load up the inclined plane, after dividing the numerator and denominator by cosA, is then
Te i Ody = Wd„ ƒ + cosơa TRÀ
2 2 cose, — /†anA (15.4)
But, the thrust collar also contributes a friction force That is, the normal reactive force acting on contact surface due to W results in an additional force foW Here, f, is the slid- ing coefficient of the collar friction between the thrust collar and the surface that supports
CHAPTER 15° @ Power SCREWS, FASTENERS, AND CONNECTIONS
the screw It is assumed that this frictional force acts at the mean collar diameter d, (Fig- ure 15.6) The torque needed to overcome collar friction is
| Way f + cose, tan
Ty
TORQUE TO LOWER THE LoaD
The analysis of lowering a load is exactly the same as that just described, with the excep- tion that the directions of Q and f N (Figure 15.7b) are reversed This leads to the equation for the total required torque Ty to lower the load as
(15.7)
VALUES OF FRICTION COEFFICIENTS
When a plain thrust collar is used, as shown in Figure 15.6, values of f and f, vary customarily between 0.08 and 0.20 under conditions of ordinary service, lubrication, and
the common materials of steel and cast iron or bronze The lowest value applies for good
workmanship, the highest value applies for poor workmanship, and some in between value for other work quality The preceding range includes both starting and running friction
Starting friction can be about 4/3 times running friction Should a rolling thrust bearing be used, f, would usually be low enough (about 0.008 to 0.02) that collar friction can be omit- ted For this case, the second term in Eqs (15.6) and (15.7) is eliminated
VALUES OF THREAD ANGLE IN THE NORMAL PLANE
A relationship between normal thread angle a,, thread angle a, and helix angle A can be obtained from a comparison of thread angles measured in axial plane and normal plane
Referring to Figures 15.6 and 15.7c, it can readily be verified that
In most applications, A is relatively small, and hence cosA * 1, So, we can set œ„ ^2 œ and
Trang 15particularly in screw jack applications Self-locking refers to a condition in which the screw cannot be turned by applying an axial force of any magnitude to the nut If collar friction is neglected, Eq (15.7) shows that the condition for self-locking is
For a square thread, the foregoing equation reduces to
fom tank (15.10a)
In other words, self-locking is obtained when the coefficient of thread friction is equal to or
greater than the tangent of the thread helix angle Note that Eq (15.10) presumes a static
situation and most power screws are self-locking
Overhauling or back-driving screw is one that has low enough friction to enable the load to lower itself, by causing the screw to spin In this situation, the inclined plane in Fig- are 15.7b moves to the right and force Q must act to the left to preserve uniform motion It can be shown that the torque T, of overhauling screw is
15
2 cose, +f tanr 2 asa)
A negative external lowering torque must now be maintained to keep the load from lowering
(15.12) + de fe
We observe from this equation that efficiency depends on only the screw geometry and the coefficient of friction If the collar friction is neglected, the efficiency becomes
Cosa: = fitank
For a square thread, a, = 0 and Eq (15.13) simplifies to
neighborhood of either 0° or 90° They generally have an efficiency of 30-90%, depending
on the 4 and jf We mention that values for square threads are higher by less than 1% over those for Acme screws in the figure
609
The Torque and Efficiency of a Power Screw
A screw jack with an’ Acme thread of diameter d, similar to that i i › that illustrated in Figure in Fi 15 i lift a load of W Determine igure 15.6, is used to
(a) The screw lead, mean diameter, arid helix angle
(b) The starting torque for lifting and for lowering the load
(c) The efficiency of the jack when lifting the load, if collar friction is neglected
(4): The length of a crank required, if F = 150 N is exerted by an operator
Design Assumptions: The screw and nut a : ‡ re lubricated with oil Coeffic i ith oi i : fricti
estimated as f= 0.12 and f, = 0.09 oelTicrents of faction are
Given? = 30 mm and W = 6 KN The screw is quadruple threaded having a pitch of p = 4 mm
The mean diameter of the collar is đ¿ = 40 mm , Solution:
(a): From Figure 15.3, dy == d — p/2 = 30 ~ 2'== 28 mm Through the use of Eqs (15.1) and
(15.2), we have Lie np = 4(4) = 16 mm
A= tan”! = 10.31°
(28)
EXAMPLE 15.1