lexible Mechanical Elements
INTRODUCTION Belts, ropes, chains, and other similar elastic or flexible machine elements are used in conveying systems and in the transmission of power over comparatively long distances. In often happens that these elements can be used as a replacement for gears, shafts, bearings, and other relatively rigid power transmission devices.
lexible mechanical elements Chapter outline 1. Belts.lat and round belt drives 3.lat metal belts 4.V belts 5.Timing belts
Belt
Belt Belts are the cheapest utility for power transmission between shafts that may not be parallel. They run smoothly and with little noise, and cushion motor and bearings against load changes, albeit with less strength than gears or chains. However, improvements in belt engineering allow use of belts in systems that only formerly allowed chains or gears.
Belt Belt drive, moreover, is simple, inexpensive, and does not require parallel shafts. It helps protect the car from overload, and damping it from noise and vibration. Load fluctuations are shock-absorbed (cushioned). They need no lubrication and minimal maintenance. They have high efficiency (90-98%, usually 95%), high tolerance for misalignment, and are inexpensive if the shafts are far apart. Clutch action is activated by releasing belt tension.
Advantages Cheap Allows misalignment (parallel shafts) Protects from overload Absorbs noise and vibrations Cushion load fluctuations Needs little maintenance High efficiency (90-98%, usually 95%), Disadvantages Speed ratio is not constant (slip & stretch) Heat accumulation Speed limited 000 m/min, Power limited - 700 kw Endless belts needs special attention to install
Belt
Belt types
lat-belt geometry Open belt sin sin 1 1 d D C d D C d D π θ π θ + Crossed belt [ ] ) ( 1 ) ( 4 1/ d D d D d D C L C θ θ + + [ ] θ π θ ) ( 1 ) ( 4 sin 1/ 1 d D d D C L C d D + + + +
0 ) ( 0 ) (, ) ( ω θ ω θ θ θ θ θ θ θ θ ω ω θ fmr f d d fmr fd fds fd fdn d d fdn ds d dn ds dn d d d mass the belt m speed belt V d d mv d mr r mrd ds t r c + + > + + + ω θ fmr f d
V m dn V f f mr f mr mr mr f mr mr f mr loose side starts at t cons A mr A mr f A c c c c c 1 1 1 1 1 sec / ) exp( 1 ) exp( ) ( ) exp( ) )exp( ( ) )exp( (, tan ) exp( ω π φ φ ω φ ω ω ω φ ω ω θ ω θ ω ω θ φ θ > > + + > > + the belting equation g c
i initial tension c hoop tension due to centrifugal force Δ tension due to the transmitted torque T D diameter of the pulley H ( ) V 1 H the transmitted power
The difference between 1 and is related to the pulley torque. Subtracting adding Dividing
If i equals zero, then T equals zero: no initial tension, no torque transmitted. or satisfactory flat-belt drive, the initial tension must be: (1) provided, () Sustained (3) in the proper amount (4) Maintained by routine inspection. rom the above equation:
Similarly,
Plot of initial tension i against belt tension 1 or, showing the intercept c, the equations of the curves, and where T/D is to be found.
Manufacturers provide specifications for their belts that include allowable tension a (or stress σall), (N/unit width) Belt life is usually several years. The severity of flexing at the pulley and its effect on life is reflected in a pulley correction factor C p. Speed in excess of 600 ft/min and its effect on life is reflected in a velocity correction factor C v. or polyamide and urethane belts use C v 1. A service factor Ks is used for excursions of load from nominal, applied to the nominal power as H d H nom K s n d, where n d is the design factor for exigencies. ( 1 ) a ba C p Cv b belt width, mm a manufacturer s allowed tension, N/mm C p pulley correction factor C v velocity correction factor ( 1 ) a allowable largest tension, N
( The steps in analyzing a flat-belt drive 1. ind exp(fø) from belt-drive geometry and friction.. rom belt geometry and speed find c. 3. rom find necessary torque. 4. rom torque T find the necessary 5. ind from 6. rom Equation find necessary initial tension i. 7. Check the friction development, f <f. Used Equation solved for f : 8. ind the factor of safety from
Example 17-1 A polyamide A-3 flat belt 150 mm wide is used to transmit 11 kw under light shock conditions where Ks 1.5, and a factor of safety equal to or greater than 1.1 is appropriate. The pulley rotational axes are parallel and in the horizontal plane. The shafts are.4 m apart. The 150 mm driving pulley rotates at 1750 rev/min in such a way that the loose is on top. The driven pulley is 450 mm in diameter. The factor of safety is unquantifiable exigencies. a) Estimate the centrifugal tension c and the torque T. b) Estimate the allowable 1,, i and allowable power Ha. c) Estimate the factor of safety. Is it satisfactory?
a) Estimate the centrifugal tension c and the torque T. θ π sin d 3.0165rad 1 (450 150) (400)
b) Estimate the allowable 1,, i and allowable power Ha. ( 1 ) b a a C p C v C v 1.0 (Polyamide &urethane belts) ( 1 ) a the allowable largest belt tension ( 1 ) a 1890N b C a p C (0.15)(18000)(0.70)(1.0) v
c) Estimate the factor of safety. Is it satisfactory? θ d π sin 3.0165rad 1 (450 150) (400) f0.8 f <f>0.314<0.8 no danger of slipping
Belt tension scheme where; d dip, m L center-to-center distance, m ω weight per foot of the belt, N/m i initial tension, N.
A decision set for a flat belt unction: power, speed, durability, reduction, service factor, C Design factor: nd Initial tension maintenance. Belt material Drive geometry, d, D. Belt thickness : t Belt width : b
lat metal belts Thin flat metal belts fabricated by laser welding and thin rolling technology made possible belts as thin as 0.00 in and as narrow as 0.06 in. Thin metal belts exhibit: High strength-to-weight ratio Dimensional stability Accurate timing Usefulness to temperatures up to 700 Good electrical and thermal conduction properties
lat metal belts A thin flat metal belt with the tight tension 1 and the slack side tension revealed. The relationship between 1 and and the driving torque T is the same as in Equation
lat metal belts The tensile stresses (σ) 1 and (σ) imposed by the belt tension 1 and are Where E Young s modulus T belt thickness v Poisson s ratio D pulley diameter σ b bending stress The largest tensile stress is The smallest tensile stress is
Table: Belt Life for Stainless Steel riction Drives
Steps for the selection of a metal flat belt 1. ind exp(fø) from geometry and friction. ind endurance strength; for 301, 30 stainless steel. N p is the number of belt passes. 5. 1a Δ ab Δ 6. i 7. bmin 8. Choose 3. Allowable tension 4. Δ 9. Check frictional development f :
Example 17-3 A friction-drive stainless steel metal belt runs over two 100-mm metal pulleys (f 0.35). The belt thickness is to 0.08 mm. or a life exceeding 1000000 belt passes with smooth torque (Ks 1), a) Select the belt if the torque is to be 3.4 N.m b) ind the initial tension i. Solution a) Select the belt if the torque is to be 3.4 N.m rom step 1, Ø θd π therefore exp(0.35π)1 rom step, rom step 3, 4, 5, and 6
Example 17-3 ) )(0.08)(10 193(10 ) (1 68 0.1 (3.4) 353 ) 9770(10 ) ( 3 ) exp(0.35 ) exp( 3 9 1 0.407 6 10 6 ab tb D v Et S Nm D T MPa S f f a f π φ 47 13 81 13 68 81 81 14796(0.019) 14796 (0.08)(10 )(0.1) 0.85 (1 ) )(0.08)(10 193(10 ) 353(10 1 1 1 3 3 9 6 1 N N N b a a a a + + [ ] ) (19 1.7 6.9 0.0069 1 3 3 14796 68 1 ) exp( ) exp( 14796 14796 168049849 ) 353(10 ) (0.08)(10 )(0.1) 0.85 (1 ) )(0.08)(10 193(10 ) 353(10 min min 6 1 3 3 9 6 1 mm b mm b mm m b f f a b b a bn b a a φ φ ) 0.35( 0.088 ' 0.088 13 81 ln 1 ln 1 ' 47 13 81 1 1 ok f f f N a i < > < + π φ
V Belts
Note that: 1. To specify a V belt, give the belt-section letter, followed by the inside circumference in inches. or example, C60 is a C- section belt having an inside circumference of 60 in.. Calculations involving the belt length are usually based on the pitch length. or any given belt section, the pitch length is obtained by adding a quantity to the inside circumference. or example, a C60 belt has a pitch length of 6.9 in. 3. The groove angle of a sheave is made somewhat smaller than the belt-section angle. This causes the belt to wedge itself into the groove, thus increasing friction. 4. The optimum working speed of the V-belt should be between 5000 ft/min and 1000 ft/min.
center-to-center distance(c): should not be greater than 3 times the sum of the sheave diameters(d +d) and no less than the diameter of the larger sheave(d). The design power is given by: The number of belts, N b, is usually the next higher integer to H d /H a. The centrifugal tension c is given by:
when the belt is used under other conditions, the tabulated value H tab is adjusted as follows:
V belt tension
Bending induces flexural stresses in the belt; the corresponding belt tension that induces the same maximum tensile stress is b1 at the driving sheave and b at the driven pulley. These equivalent tensions are added to 1 as: the Gates Rubber Company used the tension versus pass trade-off in the form: The Miner rule is used to sum damage incurred by the two tension peaks: The lifetime t in hours is given by:
If N P > 10 9, report that N P 10 9 and t > N P L p /(70V) The analysis of a V-belt drive can consist of the following steps:
PROPERTIES V-BELTS are oil and heat resistant and manufactured from material to prevent formation of static electricity. 1) OIL RESISTANT V-BELTS are resistant to damage from influence of mineral oil, splash and fatty components. This property provides longer belt life. ) HEAT RESISTANT To prevent aging and decomposition of the belts in high temperature, V-BELTS have obtained heat resistant properties. 3) ANTI STATIC V-BELTS are of electrical conductivity which prevents the builtup o static electricity. This property is of greatest importance when working with inflammable materials.
θ D π + sin D d C 1
Timing belts
Timing belts Sophisticated characteristics at low speeds and high torques open up new possibilities of use in sectors where only chains had been a possible solution so far. eatures and Benefits of the Industrial Belt : Exceptional power transmission capability. Versatile. Clean Quite running. No Maintenance.
#1. Design a friction metal flat-belt drive to connect a 1-hp, four-pole squirrelcage motor turning at 1750 rev/min to a shaft 15 in away, running at half speed. The circumstances are such that a service factor of 1. and a design factor of 1.05 are appropriate. The life goal is 10 6 belt passes, f 0.35, and the environmental considerations require a stainless steel belt. Given data unction: H nom 1 hp, n 1750 rev/min, V R, C 15 in, Ks 1., N p 10 6 belt passes. Design factor: n d 1.05 Belt material and properties: 301/30 stainless steel Table 17-8: S y 175 000 psi, E 8 Mpsi, ν 0.85 Drive geometry: d in, D 4 in Belt thickness: t 0.003 in Design variables: Solutions: The transmitting torque T and the design power : Belt width b Belt loop periphery
#1. Design a friction metal flat-belt drive to connect a 1-hp, four-pole squirrelcage motor turning at 1750 rev/min to a shaft 15 in away, running at half speed. The circumstances are such that a service factor of 1. and a design factor of 1.05 are appropriate. The life goal is 10 6 belt passes, f 0.35, and the environmental considerations require a stainless steel belt. Given data unction: H nom 1 hp, n 1750 rev/min, V R, C 15 in, Ks 1., N p 10 6 belt passes. Design factor: n d 1.05 Belt material and properties: 301/30 stainless steel Table 17-8: S y 175 000 psi, E 8 Mpsi, ν 0.85 Drive geometry: d in, D 4 in Belt thickness: t 0.003 in Design variables: Solutions: The transmitting torque T and the design power : Belt width b Belt loop periphery
or full friction development,
Decision #1: b 4.5 in Existing friction
#3. A 60-hp four-cylinder internal combustion engine is used to drive a brickmaking machine under a schedule of two shifts per day. The drive consists of two 6-in sheaves spaced about 1 ft apart, with a sheave speed of 400 rev/min. Select a V-belt arrangement. ind the factor of safety, and estimate the life in passes and hours. H nom 60 hp, n 400 rev/min, K s 1.4, d D 6 in on 1 ft Given data: centers. Design task: specify V-belt and number of strands (belts). Tentative decision: Use D360 belts. Pitch length Center to center distance C Inside circumference L p 360 + 3.3 363.3 in Quantity to be added from table
Table 17-13: orθ 180, K 1 1 Table 17-14: or D360, K 1.10 Table 17-1: H tab 16.94 hp by interpolation b/n 000 and 3000 ft/min
Number of belts, N b N b 5 At fully developed friction