- Detail

Force analysis and motion characteristics of V-belt transmission

I. force analysis of belt transmission

when installing belt transmission, the belt must be tensioned, that is, it must be tightly sleeved on two pulleys with a certain initial tension. At this time, the tension in the transmission belt is equal, which is the initial tension F0 (see Figure 7 – 8a)

figure stress of belt transmission

a) when it is not working b) when it is working with a little additive

when the belt transmission is working, due to the friction on the contact surface between the belt and the pulley, the side of the belt winding into the driving wheel is further tensioned, and the tension increases from F0 to F1, which is called the tight side; The other side is relaxed, and the tension is reduced from F0 to F2. This side is called loose side (see Figure 7 – 8b). The difference between the pull forces on both sides is called the effective pull force, which is expressed in F, that is,

F = F1 – F2 (7 – 4)

the effective pull force is the effective circumferential force that the belt drive can transmit. It is not the concentrated force acting on a fixed point, but the sum of the friction generated on the contact surface between the belt and the pulley. When the belt transmission works, the circumferential resistance f generated by the working resistance moment T2 on the driven wheel is

F = 2 t'2/d2

when it works normally, the effective tension f is equal to the circumferential resistance F. under certain conditions, the friction force that can be generated on the contact surface between the belt and the pulley has a limit value, that is, the maximum friction force (the maximum effective circumferential force) Fmax. When Fmax ≥ F, the belt transmission can operate normally. If the circumferential resistance to be transmitted exceeds this limit, the drive belt will slip on the pulley

at the beginning of slipping, there is the following relationship between tight edge tension F1 and loose edge tension F2, that is,

F1 = F2 EF a (7 – 5)

where e – – the base of natural logarithm (E ≈ 2.718)

f – – friction coefficient between belt and rim

a – – the wrap angle (RAD) of the drive belt on the pulley

the above formula is flexible body rub 3 Use the fastening screw with stirring plate to fasten the specimen on the spindle and rub the Euler formula

Derivation of():

next, take the flat belt as an example to study the relationship between tight edge tension and loose edge tension when the belt is about to slip on the driving wheel

it is assumed that the belt has no elastic elongation in operation, and the effects of bending, centrifugal force and the quality of the belt are ignored

as shown in Figure 7 – 9, take a micro segment transmission belt DL, and use DN to represent the positive pressure of the micro segment transmission belt of the pulley pair. The tension at one end of the micro segment transmission belt is f, the tension at the other end is F + DF, and the friction force is f · DN. F is the friction coefficient between the transmission belt and the pulley (for V-belt, use the equivalent friction coefficient FV, and F is the groove angle of the pulley). Then

because Da is very small, so sin (da/2) da/2, and omit the second-order trace DF sin (da/2), get

dn=f da

and

take cos (da/2) 1, get f dn=df or dn=df/f, so you can get

f da=df/f or df/f=f da

integral on both sides

f1 = F2 EF a

if it is approximately believed that the total length of the transmission belt during operation remains unchanged, Then the increase of the tight edge tension should be equal to the decrease of the loose edge tension, that is,

f1-f0 = f0-f2

or

F1 + F2 = 2f0()

substituting formula (7 – 4) into formula (7 – 6) to get (7 – 7)

substituting formula (7 – 7) into formula (7 – 5) after finishing, the maximum effective circumferential force (7 – 8) that can be transmitted by the belt transmission can be obtained

from formula (7 – 8), the belt transmission is the most effective.Circumferential force and F0 A and belt and pulley materials. The greater F0, a, F, etc., the greater the maximum effective circumferential force. F0 has the greatest impact, which directly affects the working capacity of the belt drive, but if F0 is too large, the service life of the belt will be shortened. Therefore, F0 value must be reasonably determined in the design of belt transmission

II. Elastic sliding and slipping of belt transmission

belt is an elastomer, which will produce elastic elongation under the action of tension, and the amount of elastic elongation will increase or decrease with the increase or decrease of tension. In the working process of belt drive, the tension of tight side and loose side is different. When the belt wraps around the driving wheel at point a, the speed V of the belt is equal to the circumferential speed V1 of the driving wheel. However, in the process of the belt from point a to point B, the tensile force gradually decreases from F1 to F2, and the elastic elongation also decreases correspondingly. In this way, the belt moves forward with the pulley and shrinks backward on the driving wheel. Therefore, the speed of the belt is lower than the peripheral speed of the driving wheel, causing relative sliding between the two. On the driven wheel, the situation is just the opposite, that is, the speed V of the belt is greater than the peripheral speed V2 of the driven wheel, and relative sliding also occurs between the two. This kind of sliding between the belt and the pulley caused by the elastic deformation of the belt is called elastic sliding

elastic sliding is a normal physical phenomenon that cannot be avoided in belt transmission. Due to the existence of elastic sliding, friction and wear occur between the belt and the pulley; The peripheral speed V2 of the driven wheel is lower than the peripheral speed V1 of the driving wheel, that is, speed loss occurs. This speed loss also changes with the change of external load, which makes belt transmission unable to ensure accurate transmission ratio

usually based on slip rate ε Indicates the degree of speed loss, i.e. (7 – 9)

general ε＝ 1~2%, under the condition of considering elastic sliding, the transmission ratio of belt transmission is (7 – 10)

where N1, N2 – – the speed of main and driven wheels respectively (r/min)

d1, D2 – – the reference diameter of the main and driven wheels respectively (mm)

generally speaking, elastic sliding does not occur on all contact arcs. The contact arc is divided into two parts: relative sliding (sliding arc) and no relative sliding (static arc). Their corresponding center angles are called sliding angle (a) and static angle (a) respectively. Practice has proved that the static arc always occurs on the side of the belt entering the pulley (see Figure 7 – 10). When the belt transmission does not transmit load, the sliding angle is zero. With the increase of load, the sliding angle gradually increases and the static angle gradually decreases. When the sliding angle is equal to the wrap angle and the static angle is zero, that is, when the elastic sliding expands to the whole contact arc, the effective circumferential force of the belt transmission reaches the maximum value. If the load is further increased, the belt and pulley will slip. When the belt transmission slips, it cannot work normally and the transmission fails. Therefore, the belt drive should avoid slipping in normal operation, that is, the circumferential force to be transmitted should not be greater than the maximum effective circumferential force Fmax

III. stress analysis of the transmission belt

during the working process of the transmission belt, three kinds of stresses will be generated

(I) tight side tensile stress S1 and loose side tensile stress S2

s1 = f1/a (MPA) s2=f2/a (MPA)

where F1, F2 – – tight side and loose side tensile force (n)

a – – sectional area of the belt (mm2)

(II) bending stress sb

when the belt bypasses the pulley, bending stress is generated due to bending. Taking V-belt as an example, it can be seen from material mechanics that the bending stress is

α max = α + 0.003ld

and M = ei/r, w = i/y0, r = D it is said that there are more than 150 million tons of plastic waste/2 in the ocean, so (MPA) (7 – 11)

where e – – modulus of elasticity with material (MPA)

y0 – – distance from the neutral layer to the outermost layer of the transmission belt section (mm)

d – – pulley reference diameter (mm)

r – – radius of curvature of neutral layer (mm)

i – – inertia distance (MM4)

according to formula (7 – 11), the thicker the belt, the smaller the pulley diameter, the greater the bending stress in the belt. Therefore, the bending stress SB1 when the belt is wound on the small pulley is greater than the bending stress SB2 when the belt is wound on the large pulley. In order to avoid excessive bending stress, when designing V-belt transmission, the minimum reference diameter Dmin of V-belt pulley should be limited (Table 7 – 3). Table minimum reference diameter of V-belt pulley and mass of V-belt per meter length

(III) centrifugal tensile stress SC

when the belt bypasses the pulley, it makes a circular motion, thus generating centrifugal force and causing centrifugal tensile stress SC in the belt. As shown in Figure 7-12, let the speed of the belt be V (m/s), take the micro segment belt DL (m), and the centrifugal force on the micro segment belt be C, then in the

formula, Q – ––– the mass per meter of the transmission belt (kg/m). See Table 7 – 3

suppose that the tension caused by centrifugal force in the transmission belt is FC, and take the micro segment belt DL as the separator, then according to the equilibrium conditions,

because Da is very small, sin (da/2) da/2 can be taken, then

fc=qv2 (n) (7 – 12)

the speed of the belt has a great influence on the centrifugal tensile stress. Although the centrifugal force is only produced on the arc segment of the belt in circular motion, the resulting centrifugal tensile stress acts on the whole length of the transmission belt, and the sizes are equal everywhere. The existence of centrifugal force will reduce the positive pressure on the contact surface between the transmission belt and the pulley, and the working capacity of the belt transmission will be reduced

according to the above analysis, when the belt transmission transmits power, tensile stress, bending stress and centrifugal tensile stress are generated in the belt, and the stress distribution is shown in Figure 7-13. It can be seen from the figure that the stress of the belt at the tight edge entering the driving wheel is the largest, and its value is (7 – 14)

as shown in Figure 7 – 13. It can be seen that when the belt is running, the stress acting on a point on the belt changes with its location, so the belt works under variable stress. When the number of stress cycles reaches a certain value, the belt will produce fatigue failure

stress distribution of the strip

(end)

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