Rigid body kinematics
In contrast to the kinematics of a point, the kinematics of rigid bodies solves two main problems:
Specifying movement and determining the kinematic characteristics of the body as a whole;
Determination of kinematic characteristics of body points.
Methods for specifying and determining kinematic characteristics depend on the types of body motion.
This manual discusses three types of motion: translational, rotational around a fixed axis and plane-parallel motion of a rigid body.
Translational motion of a rigid body
Translational is a movement in which a straight line drawn through two points of the body remains parallel to its original position (Fig. 2.8).
The theorem has been proven: during translational motion, all points of the body move along the same trajectories and at each moment of time have the same magnitude and direction of speed and acceleration (Fig. 2.8).
Conclusion: The translational motion of a rigid body is determined by the movement of any of its points, and therefore, the task and study of its motion is reduced to the kinematics of the point.
Rice. 2.8 Fig. 2.9
Rotational motion of a rigid body around a fixed axis.
Rotational motion around a fixed axis is the motion of a rigid body in which two points belonging to the body remain motionless during the entire time of motion.
The position of the body is determined by the angle of rotation (Fig. 2.9). The unit of measurement for angle is radian. (A radian is the central angle of a circle whose arc length is equal to the radius; the full angle of the circle contains 2 radians.)
The law of rotational motion of a body around a fixed axis = (t). We determine the angular velocity and angular acceleration of the body by the differentiation method
Angular velocity, rad/s; (2.10)
Angular acceleration, rad/s 2 (2.11)
When a body rotates around a fixed axis, its points that do not lie on the axis of rotation move in circles with the center on the axis of rotation.
If you dissect the body with a plane perpendicular to the axis, select a point on the axis of rotation WITH and an arbitrary point M, then point M will describe around a point WITH circle radius R(Fig. 2.9). During dt an elementary rotation occurs through an angle, and the point M will move along the trajectory for a distance. Let us determine the linear velocity module:
Point acceleration M with a known trajectory, it is determined by its components, see (2.8)
Substituting expression (2.12) into the formulas we get:
where: - tangential acceleration,
Normal acceleration.
Plane - parallel motion of a rigid body
Plane-parallel motion is the motion of a rigid body in which all its points move in planes parallel to one fixed plane (Fig. 2.10). To study the motion of a body, it is enough to study the motion of one section S of this body by a plane parallel to the fixed plane. Section movement S in its plane can be considered as complex, consisting of two elementary movements: a) translational and rotational; b) rotational relative to the moving (instantaneous) center.
In the first version the movement of the section can be specified by the equations of motion of one of its points (poles) and the rotation of the section around the pole (Fig. 2.11). Any section point can be taken as a pole.
Rice. 2.10 Fig. 2.11
The equations of motion will be written in the form:
X A = X A (t)
Y A =Y A (t) (2.14)
A = A (t)
The kinematic characteristics of the pole are determined from the equations of its motion.
The speed of any point of a flat figure moving in its plane is composed of the speed of the pole (arbitrarily chosen in the section of the point A) and the speed of rotation around the pole (rotation of the point IN around the point A).
The acceleration of a point of a moving flat figure consists of the acceleration of the pole relative to a stationary reference frame and the acceleration due to rotational motion around the pole.
In the second option the movement of the section is considered as rotational around a moving (instantaneous) center P(Fig. 1.12). In this case, the speed of any point B of the section will be determined by the formula for rotational motion
Angular velocity around the instantaneous center R can be determined if the speed of any section point, for example point A, is known.
Fig.2.12
The position of the instantaneous center of rotation can be determined based on the following properties:
The point's velocity vector is perpendicular to the radius;
The absolute velocity of a point is proportional to the distance from the point to the center of rotation ( V= R) ;
The speed at the center of rotation is zero.
Let's consider some cases of determining the position of the instantaneous center.
1. The directions of the velocities of two points of a flat figure are known (Fig. 2.13). Let's draw radius lines. The instantaneous center of rotation P is located at the intersection of perpendiculars drawn to the velocity vectors.
2. The velocities of points A and B are known, and the vectors and are parallel to each other, and the line AB perpendicular (Fig. 2. 14). In this case, the instantaneous center of rotation lies on the line AB. To find it, we draw a line of proportionality of speeds based on the dependence V= R.
3. A body rolls without sliding on the stationary surface of another body (Fig. 2.15). The point of contact of the bodies at the moment has zero velocity, while the velocities of other points of the body are not zero. The tangent point P will be the instantaneous center of rotation.
Rice. 2.13 Fig. 2.14 Fig. 2.15
In addition to the options considered, the velocity of a section point can be determined based on the theorem on the projections of the velocities of two points of a rigid body.
Theorem: the projections of the velocities of two points of a rigid body onto a straight line drawn through these points are equal to each other and equally directed.
Proof: distance AB cannot change, therefore
V And cos cannot be more or less V In cos (Fig. 2.16).
Rice. 2.16
Output: V A cos = V IN cos. (2.19)
Complex point movement
In the previous paragraphs, we considered the movement of a point relative to a fixed frame of reference, the so-called absolute movement. In practice, there are problems in which the motion of a point relative to a coordinate system is known, which moves relative to a fixed system. In this case, it is necessary to determine the kinematic characteristics of the point relative to the stationary system.
It is commonly called: the movement of a point relative to a moving system - relative, the movement of a point together with a moving system - portable, the movement of a point relative to a stationary system - absolute. Velocities and accelerations are called accordingly:
Relative; - figurative; -absolute.
According to the theorem on the addition of velocities, the absolute speed of a point is equal to the vector sum of the relative and portable velocities (Fig.).
The absolute value of the speed is determined by the cosine theorem
Fig.2.17
Acceleration according to the parallelogram rule is determined only with translational movement
With non-translational translational motion, a third component of acceleration appears, called rotational or Coriolis.
The Coriolis acceleration is numerically equal to
where is the angle between the vectors and
The direction of the Coriolis acceleration vector is conveniently determined by the rule of N.E. Zhukovsky: project the vector onto a plane perpendicular to the axis of portable rotation, rotate the projection 90 degrees in the direction of portable rotation. The resulting direction will correspond to the direction of the Coriolis acceleration.
Questions for self-control on the section
1. What are the main tasks of kinematics? Name the kinematic characteristics.
2. Name the methods for specifying the movement of a point and determining kinematic characteristics.
3. Give the definition of translational, rotational around a fixed axis, plane-parallel motion of a body.
4. How is the motion of a rigid body determined during translational, rotational around a fixed axis and plane-parallel motion of the body, and how is the speed and acceleration of a point determined during these body movements?
This is a movement in which all points of the body move in circles, the centers of which lie on the axis of rotation.
The position of the body is specified by the dihedral angle (angle of rotation).
= (t) - equation of motion.
Kinematic characteristics of the body:
- angular velocity, s -1;
- angular acceleration, s -2.
The quantities and can be represented as vectors 
, located on the axis of rotation, the direction of the vector 
such that from its end the rotation of the body is seen to occur counterclockwise. Direction 
coincides with 
, If 
>oh.
P 
position points of the body: M 0 M 1 = S = h.
Speed points 
; wherein 
.
where 
;
;
.
Acceleration body points, 
- rotational acceleration (in the kinematics of a point - tangent - 
):
- point-to-point acceleration (in the kinematics of the point - normal - 
).
Modules: 
;
;
.
Uniform and uniform rotation
1. Uniform: = const, 
;
;
- equation of motion.
2. Equally variable: = const, 
;
;
;
;
- equation of motion.
2
). The mechanical drive consists of pulley 1, belt 2 and stepped wheels 3 and 4. Find the speed of rack 5, as well as the acceleration of point M at time t 1 = 1s. If the angular velocity of the pulley is 1 = 0.2t, s -1; R 1 = 15; R 3 = 40; r 3 = 5; R4 = 20; r 4 = 8 (in centimeters).
Rack speed
;
;
;
.
Where 
;
;
, s -1 .
From (1) and (2) we obtain, see.
Acceleration of point M.
, s -2 at t 1 = 1 s; a = 34.84 cm/s 2 .
3.3 Plane-parallel (plane) motion of a rigid body
E 
that movement in which all points of the body move in planes parallel to some fixed plane.
All points of the body on any straight line perpendicular to a fixed plane move equally. Therefore, the analysis of the plane motion of a body is reduced to the study of the motion of a plane figure (section S) in its plane (xy).
This movement can be represented as a set of translational movement together with some arbitrarily selected point a, called pole, and rotational motion around the pole.
Equations of motion flat figure
x a = x a (t); y a = y a; j = j(t)
Kinematic characteristics ki of a flat figure:
- speed and acceleration of the pole; w, e - angular velocity and angular acceleration (do not depend on the choice of pole).
U 
alignment of movement of any point plane figure (B) can be obtained by projecting the vector equality 
on the x and y axes
x 1 B , y 1 B - coordinates of the point in the coordinate system associated with the figure.
Determining point velocities
1). Analytical method.
Knowing the equations of motion x n = x n (t); y n = y n (t), we find 
;
;
.
2). Velocity distribution theorem.
D 
differentiating equality 
, we get 
,
- the speed of point B when rotating a flat figure around pole A; 
;
Formula for the distribution of velocities of points of a plane figure 
.
WITH 
speed of point M of a wheel rolling without slipping
;
.
3). Velocity projection theorem.
The projections of the velocities of two points of the body onto the axis passing through these points are equal. Designing equality 
on the x-axis, we have
P 
example
Determine the speed of water flow v N onto the rudder of the ship, if known 
(vessel center of gravity speed), b and b K (drift angles).
Solution: .
4). Instantaneous velocity center (IVC).
The velocities of points during plane motion of a body can be determined from the formulas of rotational motion, using the concept of MCS.
MCS is a point associated with a flat figure, the speed of which at a given time is zero (v p = 0).
In the general case, the MCS is the point of intersection of perpendiculars to the velocity directions of two points of the figure.
Taking point P as a pole, we have for an arbitrary point
, Then 
Where 
- angular velocity of the figure and 
,those. the velocities of the points of a flat figure are proportional to their distances to the MCS.
|
Possible cases of finding the MCS |
|||
|
Rolling without slipping |
|||
|
|
|
|
|
|
|
MCS - at infinity | ||
Case b corresponds to an instantaneous translational velocity distribution.
1
). For a given position of the mechanism, findv B, v C, v D, w 1, w 2, w 3, if at the moment v A = 20 cm/s; BC = CD = 40 cm; OC = 25 cm; R = 20 cm.
Solution of the MCS of roller 1 - point P 1:
s -1 ; 
cm/s.
MCS of link 2 - point P 2 of intersection of perpendiculars to the speed directions of points B and C:
s -1 ; 
cm/s; 
cm/s; 
s -1 .
2). The load Q is lifted using a stepped drum 1, the angular velocity of which is w 1 = 1 s -1 ; R 1 = 3r 1 = 15 cm; AE || B.D. Find the speed v C of the axis of moving block 2.
Find the speeds of points A and B:
v A = v E = w 1* R 1 = 15 cm/s; v B = v D = w 1* r 1 = 5 cm/s.
MCS of block 2 - point P. Then 
, where 
;
;
cm/s.
The rotation of a rigid body around a fixed axis is such a movement in which two points of the body remain motionless during the entire time of movement. In this case, all points of the body located on a straight line passing through its fixed points also remain motionless. This line is called body rotation axis .
Let points A and B be stationary. Let's direct the axis along the axis of rotation. Through the axis of rotation we draw a stationary plane and a movable plane attached to a rotating body (at ).
The position of the plane and the body itself is determined by the dihedral angle between the planes and. Let's denote it . The angle is called body rotation angle .
The position of the body relative to the chosen reference system is uniquely determined at any time if the equation is given, where is any twice differentiable function of time. This equation is called equation of rotation of a rigid body around a fixed axis .
A body rotating around a fixed axis has one degree of freedom, since its position is determined by specifying only one parameter - the angle.
An angle is considered positive if it is laid counterclockwise, and negative in the opposite direction. The trajectories of points of a body during its rotation around a fixed axis are circles located in planes perpendicular to the axis of rotation.
To characterize the rotational motion of a rigid body around a fixed axis, we introduce the concepts of angular velocity and angular acceleration.
Algebraic angular velocity of a body at any moment in time is called the first derivative with respect to time of the angle of rotation at this moment, that is.
Angular velocity is positive when the body rotates counterclockwise, since the angle of rotation increases with time, and negative when the body rotates clockwise, because the angle of rotation decreases.
The dimension of angular velocity by definition:
In engineering, angular velocity is the rotational speed expressed in revolutions per minute. In one minute the body will rotate through an angle , where n is the number of revolutions per minute. Dividing this angle by the number of seconds in a minute, we get
Algebraic angular acceleration of the body is called the first derivative with respect to time of the angular velocity, that is, the second derivative of the angle of rotation, i.e.
The dimension of angular acceleration by definition:
Let us introduce the concepts of vectors of angular velocity and angular acceleration of a body.
And , where is the unit vector of the rotation axis. Vectors and can be depicted at any point on the rotation axis; they are sliding vectors.
Algebraic angular velocity is the projection of the angular velocity vector onto the axis of rotation. Algebraic angular acceleration is the projection of the angular acceleration vector of velocity onto the axis of rotation.
If at , then the algebraic angular velocity increases with time and, therefore, the body rotates accelerated at the moment in time in the positive direction. The directions of the vectors and coincide, they are both directed in the positive direction of the axis of rotation.
When and the body rotates rapidly in the negative direction. The directions of the vectors and coincide, they are both directed in the negative direction of the axis of rotation.
Rotational they call such a movement in which two points associated with the body, therefore, the straight line passing through these points, remain motionless during movement (Fig. 2.16). Fixed straight line A B called axis of rotation.
Rice. 2.1V. Towards the definition of rotational motion of a body
The position of the body during rotational motion determines the angle of rotation φ, rad (see Fig. 2.16). When moving, the angle of rotation changes over time, i.e. the law of rotational motion of a body is defined as the law of change in time of the value of the dihedral angle Ф = Ф(/) between a fixed half-plane TO () , passing through the axis of rotation, and movable n 1 a half-plane connected to the body and also passing through the axis of rotation.
The trajectories of all points of the body during rotational motion are concentric circles located in parallel planes with centers on the axis of rotation.
Kinematic characteristics of the rotational motion of the body. In the same way that kinematic characteristics were introduced for a point, a kinematic concept is introduced that characterizes the rate of change of the function φ(c), which determines the position of the body during rotational motion, i.e. angular velocity co = f = s/f/s//, angular velocity dimension [co] = rad /With.
In technical calculations, the expression of angular velocity with a different dimension is often used - in terms of the number of revolutions per minute: [i] = rpm, and the relationship between P and co can be represented as: co = 27w/60 = 7w/30.
In general, angular velocity varies with time. The measure of the rate of change in angular velocity is angular acceleration e = c/co/c//= co = f, the dimension of angular acceleration [e] = rad/s 2 .
The introduced angular kinematic characteristics are completely determined by specifying one function - the angle of rotation versus time.
Kinematic characteristics of body points during rotational motion. Consider the point M body located at a distance p from the axis of rotation. This point moves along a circle of radius p (Fig. 2.17).
Rice. 2.17.
points of the body during its rotation
Arc length M Q M circle of radius p is defined as s= ptp, where f is the angle of rotation, rad. If the law of motion of a body is given as φ = φ(g), then the law of motion of a point M along the trajectory is determined by the formula S= рф(7).
Using the expressions of kinematic characteristics with the natural method of specifying the motion of a point, we obtain kinematic characteristics for points of a rotating body: speed according to formula (2.6)
V= 5 = rf = rso; (2.22)
tangential acceleration according to expression (2.12)
i t = K = sor = er; (2.23)
normal acceleration according to formula (2.13)
a„ = And 2 /р = с 2 р 2 /р = ogr; (2.24)
total acceleration using expression (2.15)
A = -]A + a] = px/e 2 + co 4. (2.25)
The characteristic of the direction of total acceleration is taken to be p - the angle of deviation of the vector of total acceleration from the radius of the circle described by the point (Fig. 2.18).
From Fig. 2.18 we get
tgjLi = aja n=re/pco 2 =g/(o 2. (2.26)
Rice. 2.18.
Note that all kinematic characteristics of the points of a rotating body are proportional to the distances to the axis of rotation. Ve-
Their identities are determined through the derivatives of the same function - the angle of rotation.
Vector expressions for angular and linear kinematic characteristics. For an analytical description of the angular kinematic characteristics of a rotating body, together with the axis of rotation, the concept rotation angle vector(Fig. 2.19): φ = φ(/)A:, where To- eat
rotation axis vector
1; To=sop51 .
The vector f is directed along this axis so that it can be seen from the “end”
rotation occurring counterclockwise.
Rice. 2.19.
characteristics in vector form
If the vector φ(/) is known, then all other angular characteristics of rotational motion can be represented in vector form:
- angular velocity vector co = f = f To. The direction of the angular velocity vector determines the sign of the derivative of the rotation angle;
- angular acceleration vector є = сo = Ф To. The direction of this vector determines the sign of the derivative of the angular velocity.
The introduced vectors с and є allow us to obtain vector expressions for the kinematic characteristics of points (see Fig. 2.19).
Note that the modulus of the point’s velocity vector coincides with the modulus of the vector product of the angular velocity vector and the radius vector: |cox G= sogvіpa = rubbish. Taking into account the directions of the vectors с and r and the rule for the direction of the vector product, we can write an expression for the velocity vector:
V= co xg.
Similarly, it is easy to show that
- ? X
- - egBіpa= єр = a t And
Sosor = co p = i.
(In addition, the vectors of these kinematic characteristics coincide in direction with the corresponding vector products.
Therefore, the tangential and normal acceleration vectors can be represented as vector products:
- (2.28)
- (2.29)
a x = g X G
A= co x V.
The motion of a rigid body is called rotational if, during motion, all points of the body located on a certain straight line, called the axis of rotation, remain motionless(Fig. 2.15).
The position of the body during rotational movement is usually determined rotation angle body , which is measured as the dihedral angle between the fixed and moving planes passing through the axis of rotation. Moreover, the movable plane is connected to a rotating body.
Let us introduce into consideration moving and fixed coordinate systems, the origin of which will be placed at an arbitrary point O on the rotation axis. The Oz axis, common to the moving and fixed coordinate systems, will be directed along the axis of rotation, the axis Oh of the fixed coordinate system, we direct it perpendicular to the Oz axis so that it lies in the fixed plane, the axis Oh 1 Let's direct the moving coordinate system perpendicular to the Oz axis so that it lies in the moving plane (Fig. 2.15).
If we consider a section of a body by a plane perpendicular to the axis of rotation, then the angle of rotation φ can be defined as the angle between the fixed axis Oh and movable axis Oh 1, invariably associated with a rotating body (Fig. 2.16).
The direction of reference for the angle of rotation of the body is accepted φ counterclockwise is considered positive when viewed from the positive direction of the Oz axis.
Equality φ = φ(t), describing the change in angle φ in time is called the law or equation of rotational motion of a rigid body.
The speed and direction of change in the angle of rotation of a rigid body are characterized by angular speed. The absolute value of angular velocity is usually denoted by a letter of the Greek alphabet ω (omega). The algebraic value of angular velocity is usually denoted by . The algebraic value of the angular velocity is equal to the first time derivative of the rotation angle:
. (2.33)
The units of angular velocity are equal to the units of angle divided by the unit of time, for example, deg/min, rad/h. In the SI system, the unit of measurement for angular velocity is rad/s, but more often the name of this unit of measurement is written as 1/s.
If > 0, then the body rotates counterclockwise when viewed from the end of the coordinate axis aligned with the rotation axis.
If< 0, то тело вращается по ходу часовой стрелки, если смотреть с конца оси координат, совмещенной с осью вращения.
The speed and direction of change in angular velocity are characterized by angular acceleration. The absolute value of angular acceleration is usually denoted by the letter of the Greek alphabet e (epsilon). The algebraic value of angular acceleration is usually denoted by . The algebraic value of angular acceleration is equal to the first derivative with respect to time of the algebraic value of angular velocity or the second derivative of the angle of rotation:
The units of angular acceleration are equal to the units of angle divided by the unit of time squared. For example, deg/s 2, rad/h 2. In the SI system, the unit of measurement for angular acceleration is rad/s 2, but more often the name of this unit of measurement is written as 1/s 2.
If the algebraic values of angular velocity and angular acceleration have the same sign, then the angular velocity increases in magnitude over time, and if it is different, it decreases.
If the angular velocity is constant ( ω = const), then it is customary to say that the rotation of the body is uniform. In this case:
φ = t + φ 0, (2.35)
Where φ 0 - initial rotation angle.
If the angular acceleration is constant (e = const), then it is customary to say that the rotation of the body is uniformly accelerated (uniformly slow). In this case:
Where 0 - initial angular velocity.
In other cases, to determine the dependence φ from And it is necessary to integrate expressions (2.33), (2.34) under given initial conditions.
In drawings, the direction of rotation of a body is sometimes shown with a curved arrow (Fig. 2.17).
Often in mechanics, angular velocity and angular acceleration are considered as vector quantities
And .
Both of these vectors are directed along the axis of rotation of the body. Moreover, the vector
directed in one direction with the unit vector, which determines the direction of the coordinate axis coinciding with the axis of rotation, if >0,
and vice versa if
The direction of the vector is chosen in the same way (Fig. 2.18).
During the rotational motion of a body, each of its points (except for points located on the axis of rotation) moves along a trajectory, which is a circle with a radius equal to the shortest distance from the point to the axis of rotation (Fig. 2.19).
Since the tangent of a circle at any point makes an angle of 90° with the radius, the velocity vector of a point of a body undergoing rotational motion will be directed perpendicular to the radius and lie in the plane of the circle, which is the trajectory of the point’s movement. The tangential component of the acceleration will lie on the same line as the speed, and the normal component will be directed radially towards the center of the circle. Therefore, sometimes the tangential and normal components of acceleration during rotational motion are called respectively rotational and centripetal (axial) components (Fig. 2.19)
The algebraic value of the speed of a point is determined by the expression:
, (2.37)
where R = OM is the shortest distance from the point to the axis of rotation.
The algebraic value of the tangential component of acceleration is determined by the expression:
. (2.38)
The modulus of the normal component of acceleration is determined by the expression:
. (2.39)
The acceleration vector of a point during rotational motion is determined by the parallelogram rule as the geometric sum of the tangent and normal components. Accordingly, the acceleration modulus can be determined using the Pythagorean theorem:
If angular velocity and angular acceleration are defined as vector quantities , , then the vectors of velocity, tangential and normal components of acceleration can be determined by the formulas:
where is the radius vector drawn to point M from an arbitrary point on the axis of rotation (Fig. 2.20).
Solving problems involving the rotational motion of one body usually does not cause any difficulties. Using formulas (2.33)-(2.40), you can easily determine any unknown parameter.
Certain difficulties arise when solving problems associated with the study of mechanisms consisting of several interconnected bodies performing both rotational and translational motion.
The general approach to solving such problems is that motion from one body to another is transmitted through one point - the point of tangency (contact). Moreover, the contacting bodies have equal velocities and tangential acceleration components at the point of contact. The normal components of acceleration for bodies in contact at the point of contact are different; they depend on the trajectory of the points of the bodies.
When solving problems of this type, it is convenient, depending on the specific circumstances, to use both the formulas given in Section 2.3 and the formulas for determining the speed and acceleration of a point when specifying its movement as natural (2.7), (2.14) (2.16) or coordinate (2.3), (2.4), (2.10), (2.11) methods. Moreover, if the movement of the body to which the point belongs is rotational, the trajectory of the point will be a circle. If the motion of the body is rectilinear translational, then the trajectory of the point will be a straight line.
Example 2.4. The body rotates around a fixed axis. The angle of rotation of the body changes according to the law φ = π t 3 glad. For a point located at a distance OM = R = 0.5 m from the axis of rotation, determine the speed, tangent, normal components of acceleration and acceleration at the moment of time t 1= 0.5 s. Show the direction of these vectors in the drawing.
Let us consider a section of a body by a plane passing through point O perpendicular to the axis of rotation (Fig. 2.21). In this figure, point O is the intersection point of the axis of rotation and the cutting plane, point M o And M 1- respectively, the initial and current position of point M. Through points O and M o draw a fixed axis Oh, and through points O and M 1 - movable axis Oh 1. The angle between these axes will be equal to
We find the law of change in the angular velocity of the body by differentiating the law of change in the angle of rotation:
In the moment t 1 the angular velocity will be equal
We will find the law of change in the angular acceleration of the body by differentiating the law of change in angular velocity:
In the moment t 1 the angular acceleration will be equal to:
1/s 2,
We find the algebraic values of the velocity vectors, the tangential component of acceleration, the modulus of the normal component of acceleration and the modulus of acceleration using formulas (2.37), (2.38), (2.39), (2.40):
M/s 2 ;
m/s 2 .
Since the angle φ 1>0, then we will move it from the Ox axis counterclockwise. And since > 0, then the vectors will be directed perpendicular to the radius OM 1 so that we see them rotating counterclockwise. Vector will be directed along the radius OM 1 to the axis of rotation. Vector Let's build according to the parallelogram rule on vectors τ And .
Example 2.5. According to the given equation of rectilinear translational motion of the load 1 x = 0,6t 2 - 0.18 (m) determine the speed, as well as the tangential, normal component of acceleration and the acceleration of point M of the mechanism at the moment of time t 1, when the path traveled by load 1 is s = 0.2 m. When solving the problem, we will assume that there is no slipping at the point of contact of bodies 2 and 3, R 2= 1.0 m, r 2 = 0.6 m, R 3 = 0.5 m (Fig. 2.22).
The law of rectilinear translational motion of load 1 is given in coordinate form. Let's determine the moment in time t 1, for which the path traveled by load 1 will be equal to s
s = x(t l)-x(0),
from where we get:
0,2 = 0,18 + 0,6t 1 2 - 0,18.
Hence,
Having differentiated the equation of motion with respect to time, we find the projections of the velocity and acceleration of load 1 onto the Ox axis:
m/s 2 ;
At moment t = t 1 the projection of the speed of load 1 will be equal to:
that is, it will be greater than zero, as is the projection of the acceleration of load 1. Therefore, load 1 will be at moment t 1 move down uniformly accelerated, respectively, body 2 will rotate uniformly accelerated in a counterclockwise direction, and body 3 will rotate clockwise.
Body 2 is driven into rotation by body 1 through a thread wound on a snare drum. Therefore, the modules of the velocities of the points of body 1, the thread and the surface of the snare drum of body 2 are equal, and the modules of acceleration of the points of body 1, the thread and the tangential component of the acceleration of the points of the surface of the snare drum of body 2 will also be equal. Consequently, the module of the angular velocity of body 2 can be defined as
The modulus of angular acceleration of body 2 will be equal to:
1/s 2 .
Let us determine the modules of velocity and tangential component of acceleration for point K of body 2 - the point of contact of bodies 2 and 3:
m/s, m/s 2
Since bodies 2 and 3 rotate without mutual slipping, the magnitudes of the velocity and the tangential component of the acceleration of the point K - the point of contact for these bodies will be equal.
let's direct it perpendicular to the radius in the direction of rotation of the body, since body 3 rotates uniformly accelerated