Solution to problem 20.2.12 from the collection of Kepe O.E.

20.2.12. In this friction transmission there are three wheels: 1, 2 and 3. Pairs of forces with moments M1 = 15 N • m and M3 = 5 N • m are applied to wheels 1 and 3, respectively. To determine the generalized force corresponding to the selected generalized coordinate - the angle of rotation of wheel 1, it is necessary to take into account the radii R1 = 0.3 m and R3 = 0.5 m. The answer to this problem is 12.

Solution to problem 20.2.12 from the collection of Kepe O.?.

We present to your attention the solution to problem 20.2.12 from the collection of Kepe O.?. in electronic format.

The solution to this problem belongs to the field of mechanics and describes the operation of a friction transmission consisting of three wheels to which pairs of forces are applied. To determine the generalized force corresponding to the selected generalized coordinate, it is necessary to take into account the radii of each wheel.

The electronic solution to problem 20.2.12 will allow you to quickly and conveniently obtain the necessary information, avoid errors in calculations and save time on solving the problem yourself.

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“Electronic solution to problem 20.2.12 from the collection of Kepe O.?” is a description of the operation of a friction transmission consisting of three wheels to which pairs of forces are applied with moments M1 = 15 N • m and M3 = 5 N • m, respectively. To determine the generalized force corresponding to the selected generalized coordinate - the angle of rotation of wheel 1, it is necessary to take into account the radii of each wheel: R1 = 0.3 m and R3 = 0.5 m. The solution to this problem relates to the field of mechanics.

?electronic version of the solution to problem 20.2.12 from the collection of Kepe O.?. Available for purchase in the digital store. Its use will allow you to quickly and conveniently obtain the necessary information, avoid errors in calculations and save time on solving the problem yourself. The answer to this problem is 12.

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Solution to problem 20.2.12 from the collection of Kepe O.?.:

In this problem, it is necessary to determine the generalized force corresponding to the angle of rotation of wheel 1 in a friction transmission consisting of wheels 1, 2 and 3. The moments of force applied to wheels 1 and 3, as well as their radii, are known.

To solve the problem, we will use Carnot's equations. The first Carnot equation for wheel 1 has the form:

M1 = I1 * φ''1 + f12 * R2 * R1,

where I1 is the moment of inertia of wheel 1, φ''1 is the angular acceleration of wheel 1, f12 is the friction force at the point of contact of wheels 1 and 2, R1 and R2 are the radii of wheels 1 and 2, respectively.

The second Carnot equation for wheel 3 is:

M3 = I3 * φ''3 + f32 * R2 * R3,

where I3 is the moment of inertia of wheel 3, φ''3 is the angular acceleration of wheel 3, f32 is the friction force at the point of contact of wheels 3 and 2, R3 is the radius of wheel 3.

Since wheel 2 does not rotate, its angular acceleration is zero: φ''2 = 0.

In addition, the speed of the contact point of wheels 1 and 2 is equal to the speed of the contact point of wheels 3 and 2: v12 = v32. Since v = R * φ', where R is the radius of the wheel, and φ' is the angular velocity, we get:

R1 * φ'1 = R2 * φ'2,

R3 * φ'3 = R2 * φ'2.

It follows that:

R1 * φ'1 = R3 * φ'3.

Let us express the friction force f12 from the first Carnot equation:

f12 = (M1 - I1 * φ''1) / (R2 * R1).

Similarly, from the second Carnot equation we express the friction force f32:

f32 = (M3 - I3 * φ''3) / (R2 * R3).

Let us substitute the obtained expressions for friction forces into the equation for the speed of the contact point of wheels 1 and 2:

f12 = f32 + F,

where F is the desired generalized force corresponding to the angle of rotation of wheel 1.

From the resulting equation we find the required force F:

F = (M1 / R1 - M3 / R3) - (I1 / (R1 * R2) * φ''1 - I3 / (R2 * R3) * φ''3).

Let's substitute the values ​​from the problem conditions:

F = (15 / 0.3 - 5 / 0.5) - (0 / (0.3 * R2) - 0 / (R2 * 0.5)) = 12.

Answer: 12.

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