Addition of angular velocities of a rigid body. Addition of rotations of a rigid body. What will we do with the received material?
Let us consider the case when the relative motion of the body is rotation with angular velocity around the axis aa", mounted on the crank bа (Fig. 74, a), and the portable motion is rotation of the crank bа around an axis parallel to the angular velocity. Then the movement of the body will be plane-parallel in relative to the plane perpendicular to the axes. Three special cases are possible here.
1. Rotations are directed in one direction. Let us depict the section S of the body with a plane perpendicular to the axes (Fig. 74, b). The traces of the axes in the section S will be denoted by the letters A and B. Point A, as lying on the axis, receives speed only from rotation around the axis Bb", therefore, . In the same way. In this case, the vectors and are parallel to each other (both perpendicular to AB) and directed in different directions. Then point C is the instantaneous center of velocities (), and therefore the axis Cc, parallel to the axes Aa and Bb, is the instantaneous axis of rotation of the body.
a)b)Fig. 74. Addition of rotations around two parallel axes(rotations are directed in one direction)
To determine the angular velocity ω of the absolute rotation of the body around the axis Сс" and the position of the axis itself, i.e. point C, we use the equality
The last result comes from the properties of proportion. Substituting into these equalities, we finally find:
So, if a body simultaneously participates in two rotations directed in the same direction around parallel axes, then its resulting motion will be an instantaneous rotation with absolute angular velocity around an instantaneous axis parallel to the data; the position of this axis is determined by proportions.
Over time, the instantaneous axis of rotation Сс" changes its position, describing a cylindrical surface.
2. Rotations are directed in different directions. Let us again depict the section S of the body (Fig. 75) and assume, for definiteness, that. Then, reasoning as in the previous case, we find that the velocities of points A and B will be numerically equal, ; at the same time, they are parallel to each other and directed in the same direction. Then the instantaneous axis of rotation passes through point C (Fig. 75), and
The last result also comes from the properties of proportion. Substituting the values and into these equalities, we finally find:
So, in this case, the resulting motion is also an instantaneous rotation with absolute angular velocity around the axis Cc", the position of which is determined by the proportions.
3. A couple of spins. Let's consider special case, when rotations around parallel axes are directed in different directions (Fig. 76), but in modulus. Such a set of rotations is called a pair of rotations, and the vectors and form a pair angular velocities.
Rice. 75. Addition of rotations around two parallel axes (rotations are directed in different directions) Fig. 76. A couple of spins
In this case we get that and, i.e. . Then the instantaneous center of velocities is at infinity and all points of the body are at this moment time have the same speed.
Consequently, the resulting motion of the body will be translational (or instantly translational) motion with a speed numerically equal to and directed perpendicular to the plane passing through the vectors and; the direction of the vector is determined in the same way as in statics the direction of the moment of a pair of forces was determined. In other words, a pair of rotations is equivalent to translational (or instantaneous translational) motion with a speed equal to the moment of a pair of angular velocities of these rotations.
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From the content of the previous paragraphs it is clear that the simplest kinematic elements introduced above - the angular velocities of rotation of the body (or coordinate system) and the velocities of translational movements - obey the same laws as forces and couples in statics. In fact, pairs of rotations or translational movements are analogous to pairs of forces. As in statics, a set of kinematic pairs is equivalent to a pair whose moment (or speed of the resulting translational motion) equal to the sum of the moments of the terms of the pairs.
The angular velocities of rotation around axes intersecting at one point are replaced by one angular velocity in the same way as a converging system of forces in statics is reduced to one force (resultant). The analogy between the angular velocities of the rotation components and forces is not limited to this. We will now establish that the addition of rotations around parallel axes is completely analogous to the addition of parallel forces.
Let us assume that the body rotates with angular velocity ω 2 around the axis O 2 z 2 relative to the coordinate system O 2 x 2 y 2 z 2, and the latter rotates with angular velocity ω 1 around the axis O 1 z 1 relative to the coordinate system O 1 x 1 y 1 z 1 , and the axes O 1 z 1 and O 2 z 2 are parallel (Fig. 14.7).
Then the absolute speed of any point M body
Speeds v r And v e points M located in a plane perpendicular to the axes O 1 z 1 and O 2 z 2, therefore, the absolute speed v points M lies in a plane perpendicular to these axes. Since the point M is arbitrary, this means that the body is involved in plane motion. Let's find in the plane x 1 O 1 y 1 instantaneous velocity center in the case when ω 1 and ω 2 are directed in the same direction (Fig. 14.7, a).
For a point R, lying on the straight line O 1 O 2, v r And v e collinear, but directed in different directions. In order for their geometric sum to be equal to zero, the equality must be satisfied
(14.11)
Dot R divides a segment O 1 O 2 internally into parts inversely proportional to the modules of the angular velocities of the rotation components.
Let us now move on to the addition of rotations having opposite directions. Let Speed v r And v e this is mine O 1 O 2, located outside the segment O 1 O 2(Fig. 14.7, b). Let's find a point R, in which these speeds are equal:
(14.12)
Dot R divides a segment O 1 O 2 externally into parts inversely proportional to the angular velocity modules. Such a point can always be found if only
In each of the cases considered, the point R has a speed equal to zero, i.e.
Let us now find the speed of an arbitrary point M:
Here r"- radius vector of a point M relative to the instantaneous velocity center R. Opening the brackets on the right side and using equality (14.13), we obtain
Where 
This implies, that the combination of two rotations occurring around parallel axes, but not representing a pair of rotations, is reduced to one rotation, the instantaneous axis of which divides, internally or externally, the distance between the axes of the component rotations into parts inversely proportional to the modules of angular velocities. The angular velocity of the resulting rotation is geometric sum angular velocities of the component movements.
If the angular velocities are directed in one direction, then the instantaneous axis of rotation is located between the axes O 1 z 1 And O 2 z 2 and the module of the resulting angular velocity 
In the case of oppositely directed rotations, the instantaneous axis is located behind the axis around which rotation occurs with a higher angular velocity and 
The resulting angular velocity is directed towards the larger of the angular velocities.
Tasks
Problem 14.3. In the gearbox (Fig. 14.8) there was a carrier OS makes n=720 rpm, and movable gears 2 and 3 rotate around their axis relative to the driver in the same direction with an angular velocity corresponding to n 23 = 240 rpm. Determine radius r 1 fixed wheel 1 and shaft speed II, If OS= 240 mm, r 4 = 40 mm (r 4 is the radius of gear 4).
Moving gears 2 and 3 perform a complex movement. They rotate around an axis MN relative to the driver and together with this axis around the axis of the shaft.
The radius r 1 of the fixed wheel 1 is found from the condition that the instantaneous axis of absolute rotation of gears 2 and 3 is parallel to the axis MN, passes through the point of contact between fixed wheel 1 and movable gear 2. Based on relation (14.11), we can write:
Where ω 23 is the angular velocity of gears 2 and 3 when they rotate around the axis MN, and ω - angular speed of the shaft I.
There is a relationship between the angular velocity and the number of revolutions per minute of the form
hence,
Absolute angular velocity ω a gears 2 and 3 when rotating around the instantaneous axis on the base (14.14) is equal to
ω a = ω+ ω 23
Characterizing the angular velocity by the number of revolutions, we obtain
n a = n + n 23 = 720 + 240 = 960 rpm.
To determine the number of revolutions of gear 4, and therefore the shaft II, Let's take advantage of the fact that the absolute speeds of the points of gears 3 and 4 at the point IN their engagements are equal to each other (there is no relative slip):
Thus,
Problem 14.4. How many revolutions per minute should the drive shaft make? I gearbox (Fig. 14.9) so that the driven shaft II did n 4 =1800 rpm?
The first wheel with internal teeth is stationary. Given: r 1 =150 mm, r 2 = 30 mm, r 4 = 50 mm.
Movable gears 2 and 3 as one unit perform a complex movement. They rotate around an axis MN relative to the leash and together with it rotate around an axis I.
The instantaneous axis of absolute rotation of these gears passes through the point IN- point of engagement of the moving gear 2 and the fixed gear I. This axis is parallel to the axis MN. Since the instantaneous axis of absolute rotation of gears 2 and 3 lies outside the axes of the component movements, the rotation of these gears around the axis MN occurs in the direction opposite to the direction of shaft rotation I.
Let us consider the case when the relative motion of the body is rotation with angular velocity around an axis fixed on the crank (Fig. 198, a), and the portable motion is rotation of the crank around an axis parallel to angular velocity. Then the motion of the body will be plane-parallel with respect to the plane perpendicular to the axes. Three special cases are possible here.
1. Rotations are directed in one direction. Let us depict the section S of the body with a plane perpendicular to the axes (Fig. 198, b). We denote the traces of the axes in section 5 by the letters A and B. It is easy to see that point A, as lying on the axis, receives speed only from rotation around the Bb axis, therefore, in the same way
In this case, the vectors are parallel to each other (both perpendicular to AB) and directed in different directions. Then point C (see § 56, Fig. 153, b) is the instantaneous center of velocities and, therefore, the axis parallel to the axes and Bb is the instantaneous axis of rotation of the body.
To determine the angular velocity c of the absolute rotation of the body around the axis and the position of the axis itself, i.e. point C, we use the equality [see. § 56, formula (57)]
The last result comes from the properties of proportion. Substituting into these equalities we finally find:
So, if a body simultaneously participates in two rotations directed in the same direction around parallel axes, then its resulting motion will be an instantaneous rotation with absolute angular velocity around an instantaneous axis parallel to the data; the position of this axis is determined by proportions (98).
Over time, the instantaneous axis of rotation changes its position, describing a cylindrical surface.
2. Rotations are directed in different directions. Let us again depict the section S of the body (Fig. 199) and assume, for definiteness, that sсoz. Then, reasoning as in the previous case, we find that the velocities of points A and B will be numerically equal: at the same time, they will be parallel to each other and directed in the same direction.
Then the instantaneous axis of rotation passes through point C (Fig. 199), and
The last result also comes from the properties of proportion. Substituting the values into these equalities we finally find:
So, in this case, the resulting motion is also an instantaneous rotation with absolute angular velocity around an axis whose position is determined by the proportions (100).
3. A couple of spins. Let's consider a special case when rotations around parallel axes are directed in different directions (Fig. 200), but modulo .
Such a set of rotations is called a pair of rotations, and the vectors form a pair of angular velocities. In this case we obtain, Then (see § 56, Fig. 153, a) the instantaneous center of velocities is at infinity and all points of the body at a given moment in time have the same velocities.
Consequently, the resulting motion of the body will be translational (or instantly translational) motion with a speed numerically equal to and directed perpendicular to the plane passing through the vectors; the direction of the vector v is determined in the same way as in statics the direction of the moment of a pair of forces was determined (see § 9). In other words, a pair of rotations is equivalent to translational (or instantaneously translational) motion with a speed v equal to the moment of a pair of angular velocities of these rotations.
There are three cases to consider.
1) Rotations have the same directions. The body participates in two rotations: portable with angular velocity and relative with angular velocity (Fig. 71). Such a body is the disk shown in Fig. 72. Let's intersect the axes of rotation perpendicular to the straight line. We obtain the intersection points and to which the angular velocity vectors and can be transferred. On a segment of the body at the moment under consideration there is a point whose speed is zero. Indeed, by the velocity addition theorem for a point we have
Points of the body for which the transfer and relative velocities are parallel and opposite can only be located on the segment between the points and . The speed of a point is zero if But , . Hence,
A straight line perpendicular to the axes of rotation can be drawn at any distance. Consequently, there is an axis attached to the body and parallel to the axes of rotation, the velocities of the points of which are equal to zero at a given moment. She happens to be instantaneous axis of rotation at the moment in time under consideration.
To determine the angular velocity of rotation of the body around the instantaneous axis, we calculate the speed of the point, considering its motion complex. We get:
Hence,
For the speed of a point when the body rotates around the instantaneous axis, we have
Equating the point velocities obtained in two ways, we have
According to (138)
Formula (138) can be represented as:
Forming a derivative proportion and using formula (139), we obtain
Thus, when adding two rotations of a body around parallel axes in the same directions, the result is a rotation around a parallel axis in the same direction with an angular velocity equal to the sum of the angular velocities of the component rotations. The instantaneous axis of the resulting rotation divides the segment between the axes of the component rotations into parts inversely proportional to the angular velocities of the rotations, internally. The point with this division is located between the points and.
The opposite is true. Rotation around an axis with angular velocity can be decomposed into two rotations around two parallel axes with angular velocities and .
A body participating in two rotations around parallel axes performs plane motion. The plane motion of a rigid body can be represented as two rotations, portable and relative, around parallel axes. The plane movement of the satellite wheel 2 on the stationary wheel 1 (Fig. 73) is an example of a movement that can be replaced by two rotations around parallel axes in the same direction, for example, counterclockwise. The satellite wheel performs translational rotation together with the crank around an axis passing through a point with angular velocity , and relative rotation around an axis passing through a point with angular velocity . Both rotations have the same directions. Absolute rotation occurs around an axis passing through the point, which is currently the MCS. It is located at the point of contact between the wheels, if the moving wheel rolls without sliding on the stationary one. Angular velocity of absolute rotation
Absolute rotation at this angular velocity occurs in the same direction as the components of motion.
2) Rotations have opposite directions. Let's consider the case when (Fig. 74). We get the following formulas:
To derive these formulas, we decompose the rotation with angular velocity into two rotations in the same direction around two parallel axes with angular velocities and . Let us take the axis of one of the rotations with angular velocity to pass through the point and choose . Another rotation with angular velocity will pass through the point (Fig. 75). Based on (139) and (140) we have
The validity of formulas (141) and (142) has been proven. Thus, when adding two rotations of a rigid body around parallel axes in opposite directions, the result is a rotation around a parallel axis with an angular velocity equal to the difference in the angular velocities of the component rotations in the direction of rotation with a higher angular velocity. The axis of absolute rotation divides the segment between the axes of the component rotations into parts that are inversely proportional to the angular velocities of these rotations internally. The point with this division is located on the segment behind the point through which the axis of rotation passes with a higher angular velocity.
You can also decompose one rotation into two around parallel axes with opposite directions of rotation. An example of the plane motion of a rigid body, which can be represented by two rotations around parallel axes in opposite directions, is the motion of a satellite wheel rolling inside a stationary wheel without sliding (Fig. 76). In this case, what is portable is the rotation of wheel 2 together with the crank with angular velocity around an axis passing through point . Relative rotation of wheel 2 will be around an axis passing through a point with angular velocity, and absolute rotation of this wheel around an axis passing through the MCS, point, with angular velocity. In this case and therefore the angular velocity of absolute rotation is . This rotation in direction coincides with the direction of rotation, which has a high angular velocity. The axis of absolute rotation is located outside the segment behind the axis of rotation with a higher angular velocity.
3) A couple of rotations. A couple of spins is a combination of two rotations of a rigid body, portable and relative, around parallel axes with equal angular velocities in opposite directions (Fig. 77). In this case . Considering the motion of a body as complex, according to the theorem of addition of velocities for a point we have
The components of motion are rotations with angular velocities and . Using Euler’s formula for them we get
After this, for the absolute speed we have
because . Considering that , we get
Because vector product can be called the moment of angular velocity relative to the point, then
It is equal to the vector momentum of a pair of rotations, which can also be expressed as the vector momentum of one of the angular velocities relative to any point located on the axis of rotation of a body with another angular velocity included in the pair of rotations. The speed of translational motion of a body participating in a pair of rotations depends only on the characteristics of the pair of rotations. It is perpendicular to the axes of the pair of rotations. Its numerical value can be expressed as
where is the shortest distance between the axes of the pair or the shoulder of the pair.
A pair of rotations is analogous to a pair of forces acting on a rigid body. The angular velocities of rotation of a body, similar to forces, are sliding vectors. The vector moment of a couple of forces is a free vector. The vector momentum of a pair of rotations has a similar property.
If you fasten a straight segment to gear 2, then it will remain parallel to its original position when the mechanism moves. If this horizontal segment is combined with the bottom of a cup of water, attaching the cup to a movable gear, then the water will not spill out of the cup when the mechanism moves in a vertical plane.
During translational motion, the trajectories of all points of the body are identical. The point describes a circle of radius . The trajectories of all other points of the moving gear will also be circles of the same radius. A body participating in a pair of rotations performs a plane translational motion.
1. Addition of rotations around intersecting axes. Let a rigid body participate simultaneously in two rotations: portable with angular velocity and relative with angular velocity. The axes of rotation intersect at point O (Fig. 49.a)
An example of a body participating in two rotations about intersecting axes is a disk A, loosely mounted on axle OO" and rotating around it with angular velocity. Along with axle OO" the disk is still rotating around another
axes O 1 O 2(Fig.49.b) with angular velocity.
According to the theorem on the addition of velocities for a point M we have
Since it is portable and relative motion are rotations around the axes, then
Where h 1 And h 2 - shortest distances from a point M to the corresponding axes of rotation. The areas of the triangles in the parallelogram are equal, therefore .
By adding two rotations around intersecting axes, one of which is portable and the other relative, the result is a rotation of the body around the instantaneous axis.
To determine the absolute angular velocity of rotation around the instantaneous axis, select a point on the body N and calculate its speed once as the speed of complex motion, and the other as rotation around the instantaneous axis. According to Euler's formula for rotational movements with complex motion we have
For absolute rotation around the instantaneous axis
Equating the speeds, we get
i.e. the angular velocity of absolute rotation is equal to vector sum angular velocities of the rotation components.
2. Addition of rotations around parallel axes. There are three cases to consider.
1) Rotations have the same directions. The body participates in two rotations: portable with angular velocity and relative with angular velocity (Fig. 50). On the segment AB body at the moment under consideration there is a point C, the speed of which is zero. Indeed, by the velocity addition theorem for point C we have
The speed of point C is zero if . But , . Hence,
To determine the angular velocity of rotation of the body around the instantaneous axis, we calculate the speed of the point IN, considering her movement difficult. We get
Hence,
For point speed IN when the body rotates around the instantaneous axis, we have
Equating point velocities IN, obtained in two ways, we have
According to (*),
Formula (*) can be represented as follows:
Forming a derivative proportion and using formula (**), we get
Thus, when adding two rotations of a body around parallel axes in the same directions, a rotation around a parallel axis in the same direction is obtained With angular velocity equal to the sum of the angular velocities of the component rotations. The instantaneous axis of the resulting rotation divides the segment
between the axes of the component rotations into parts inversely proportional to the angular velocities of these rotations, in an internal manner.
2) Rotations have opposite directions. Let's consider the case when . We get the following formulas:
Thus, when adding two rotations of a rigid body around parallel axes in opposite directions, a rotation around a parallel axis is obtained With angular velocity equal to the difference in angular velocities of the components of rotation in the direction of rotation With higher angular velocity. The axis of absolute rotation divides the segment between the axes of the component rotations into parts that are inversely proportional to the angular velocities of these rotations internally.
3. A couple of spins. A pair of rotations is a combination of two rotations of a rigid body, portable and relative, around parallel axes with equal angular velocities in opposite directions (Fig. 52 ).
In this case, Considering the motion of a body as complex, according to the theorem of addition of velocities for a point M we have
Replacing (~) in the formula with , we get accordingly
Combining the results, we have
Thus, if a rigid body participates in a pair of rotations, then the velocities of all points of the body, according to(~~), are identical, i.e. the body performs an instantaneous translational motion.