Addition of translational and rotational movements. screw movement
Consider the complex motion of a rigid body, which is composed of translational and rotational movements. The corresponding example is shown in Fig. 78. Here the relative motion of the body 1 is the rotation with angular velocity around the axis Ah fixed on the platform 2, and portable - the translational movement of the platform with a speed. At the same time, the wheel participates in two such movements. 3, for which the relative motion is the rotation around its axis, and the portable one is the motion of the same platform. Depending on the value of the angle α between the vectors and (for the wheel this angle is 90°), three cases are possible here.
1. The speed of translational movement is perpendicular to the axis of rotation ( ). Let the complex motion of the body be composed of rotational motion around the axis Ah with an angular velocity ω and translational motion with a speed perpendicular (Fig. 79). It is obvious that this movement represents (with respect to the plane P, perpendicular to the axis Ah) plane-parallel motion.
If you count the point BUT pole, then the motion under consideration, like any plane-parallel motion, will indeed be composed of translational with speed, i.e. with the speed of the pole, and from rotational around the axis Ah passing through the pole.
The vector , according to Section 6.2, can be replaced by a pair of angular velocities and , assuming , and . At the same time, the distance AR is determined from the equality , whence .
The vectors and give zero when added and, therefore, the motion of the body in this case can be considered as an instantaneous rotation around the axis RR with angular speed. Thus, the rotation of the body around the axes Ah and RR occurs with the same angular velocity, i.e., the rotational part of the motion does not depend on the choice of the pole.
2. Screw movement ( ). If the complex motion of the body is composed of a rotation around the axis Ah with angular velocity and translational with velocity directed parallel to the axis Ah(Fig. 80), then such a movement of the body is called screw. Axis Ah called screw axis. When the vectors and are directed in the same direction, then with the image rule we have adopted, the screw will be right; if in different directions - left. The distance traveled during one revolution by any point of the body lying on the axis of the screw is called step h screw. If the values and are constant, then the pitch of the screw will also be constant. Denoting the time of one revolution through T, we obtain in this case and , whence .
With a constant step, any point M body, not lying on the axis of the screw, describes helical line. Point speed M, located from the axis of the screw at a distance r, is composed of the translational speed and the speed perpendicular to it, obtained in rotational motion, which is numerically equal to ω r. Consequently .
The speed is directed tangentially to the helix. If the cylindrical surface along which the point moves M, cut along the generatrix and unfold, then the helical lines will turn into straight lines inclined to the base of the cylinder at an angle , where .
3. The speed of translational movement forms an arbitrary angle with the axis of rotation. The complex movement performed by the body in this case (Fig. 81, a) can be considered as a general case of the movement of a free rigid body.
We decompose the vector (Fig. 81, b) into components: directed along (), and perpendicular () . The speed can be replaced by a pair of angular velocities and , after which the vectors and can be discarded. Distance AU find by formula.
Then the body is left with rotation with angular velocity and translational motion with velocity . Consequently, the distribution of the velocities of the points of the body at a given moment of time will be the same as with a helical motion around the axis ss with angular velocity and translational velocity.
After performing the transformations (Fig. 81, b), we moved from the pole BUT to the pole FROM. The result confirms that in the general case of motion of a rigid body, the angular velocity does not change when the pole changes (), but only the translational velocity () changes.
Since during the motion of a free rigid body the quantities , α will change all the time, the position of the axis will also change continuously ss, which is therefore called instant helical axis. In this way, the motion of a free rigid body can also be considered as a sum of a series of instantaneous helical motions around continuously changing helical axes.
Conclusion
The role and place of theoretical mechanics in engineering education is determined by the fact that it is the scientific basis for many areas of modern technology. The assimilation of theoretical mechanics is complicated by the fact that modeling and mathematical representation of the studied natural phenomena play an essential role in this science. Therefore, when solving engineering problems, students often experience significant difficulties. The problem of forming a research approach to the tasks set by students (from the section "Kinematics" of the course of theoretical mechanics) can be solved by the proposed textbook. The manual covers the main topics of the "Kinematics" section in an accessible way with all the necessary evidence. Are given guidelines to solving problems and examples of their solution are given. Tasks for mastering and consolidating the material presented will help independent work given at the end of the chapters in the manual.
If the body simultaneously participates in translational translational motion with speed and relative rotational motion with angular velocity , then depending on their relative position It is useful to consider three separate cases.
1. The speed of translational movement is perpendicular to the axis of relative rotation. In this case, the vectors and are perpendicular (Fig. 53). On the line OS, perpendicular to the plane in which and are located, there is a point FROM, whose speed is zero. Determine its distance from the point O.
According to the speed addition theorem for a point FROM we have
since when rotating around the axis
Considering that the velocities and are opposite in direction, we get
Since , then and, therefore, the points FROM and O are at a distance
Other points with velocities equal to zero are located on the line passing through the point FROM, parallel to the axis of rotation of the body with an angular velocity . Thus, there is an instantaneous axis of rotation parallel to the axis of relative rotation and passing through the point FROM.
When adding the translational translational and rotational relative motions of a rigid body, in which the translational velocity is perpendicular to the axis of relative rotation, the equivalent absolute motion is rotation around the instantaneous axis parallel to the axis of relative rotation with an angular velocity coinciding with the angular velocity of relative rotation.
2. Screw movement. The movement in which the speed of the portable translational motion of the body is parallel to the axis of relative rotation is called screw motion of a solid body (Fig. 54). The axis of rotation of the body in this case is called the in and o o o y axis. In helical motion, the body moves translationally parallel to the axis of helical motion and rotates around this axis. The helical motion is not reduced to any other single simple equivalent motion.
With a helical motion, the vectors and can have both the same and opposite directions. The helical movement of the body is characterized by the parameter of the helical movement, which is considered the value . If and change over time, then the parameters of the helical motion are also variable. In the general case , and , i.e. p is the displacement of the body along the axis of helical motion when the body is rotated by one radian.
For point M we have
But where r is the distance of the point to the screw axis. The speeds and are perpendicular. Consequently,
Considering that , we get
If the body rotates at a constant angular velocity and has a constant translational speed, then such a body motion is called a constant screw motion. In this case, the point of the body during movement is always on the surface of a circular cylinder with a radius r. The trajectory of the point is a helix. In addition to the parameter in the case under consideration, enter screw pitch, i.e., the distance that any point of the body will move during one revolution of the body around the axis of helical motion. The angle of rotation of the body at is calculated by the formula . For one body revolution. The time required for this.
During T the point will move in the direction parallel to the helical axis by a helical pitch.
Hence the dependence of the screw pitch on the screw motion parameter is obtained.
Point motion equations M bodies along a helix (Fig. 102) in Cartesian coordinates are expressed in the following form:
In these equations, the quantities and are constant.
3. General case. Let the speed of translational translational motion and the angular speed of relative rotation form an angle . The case when , and , have already been considered. have all points of the body. Thus, a helical motion was obtained with a helical axis spaced from the original axis of rotation by .
Parameter of the resulting helical motion .
The general case of translational translational and relative rotational motion of a rigid body turned out to be equivalent to instantaneous screw motion.
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SCREW MOVEMENT. If the motion of an unchanging system (for example, a rigid body) is composed of rotation about an axis and translational motion along this axis, then such motion of the body is called helical motion; the indicated axis is called the helical axis, or the axis of rotation - sliding. If two arbitrary positions of a body moving in space are given, then the transition from position I to II can be performed with one helical movement around a definitely located helical axis (Chal's theorem); while the rotational and translational movements can be performed either simultaneously or sequentially in any order. Considering everything given movement bodies in space as consisting of infinitesimal elementary displacements and applying to each of them Shall's theorem, we obtain the following position: any movement of a body in space is a series of infinitely small helical displacements around instantaneous helical axes, changing their position and direction in space at every moment .
Helical elementary displacements of the body around each instantaneous axis are movements equivalent to infinitesimal real displacements of the body, and represent them up to infinitesimal values of higher orders. The laws of screw movement, equivalent to any movement of a rigid body, were established by Mozzi (Giulio Mozzi, 1768). The addition of two helical movements also results in a helical movement.
forward movement,
- rotation around a fixed axis,
- flat movement,
- spherical movement,
- free movement.
Translational motion of a rigid body - this is a movement in which any straight line associated with the body, during its movement, remains parallel to its initial position.
Examples of translational motion: the movement of the pedals of a bicycle relative to its frame, the movement of pistons in the engine cylinders internal combustion relative to the cylinders, the movement of the Ferris wheel cabins relative to the Earth, etc.
The problem of the kinematics of the translational motion of a rigid body is reduced to the problem of the kinematics of a material point.
Theorem . In translational motion, all points of the body describe the same (coincident when superimposed) trajectories and have at each moment of time the same magnitude and direction of speed and acceleration.
Proof.
If we select two points of a rigid body BUT and AT, then the radius vectors of these points are related by the relation
Point trajectory BUT is a curve that is given by the function , and the trajectory of the point B is the curve given by the function . The trajectory of point B is obtained by translating the trajectory of point A in space along the vector AB, which does not change its magnitude and direction in time (AB = const). Therefore, the trajectories of all points of the rigid body are the same.
Differentiate with respect to time the expression
We get
Let us differentiate the velocity with respect to time and obtain the expression a B = a A . Consequently, the speeds and accelerations of all points of a rigid body are the same.
To set the translational motion of a rigid body, it is enough to set the motion of one of its points
rotational movement- a type of mechanical movement. During the rotational motion of a material point, it describes a circle. During the rotational motion of an absolutely rigid body, all its points describe circles located in parallel planes. The centers of all circles lie in this case on one straight line, perpendicular to the planes of the circles and called the axis of rotation. The axis of rotation can be located inside the body and outside it. The axis of rotation in a given frame of reference can be either movable or fixed. For example, in the reference frame associated with the Earth, the axis of rotation of the generator rotor at the power plant is fixed.
When choosing some axes of rotation, you can get a complex rotational movement - a spherical movement, when the points of the body move along the spheres. When rotating around a fixed axis that does not pass through the center of the body or a rotating material point, the rotational motion is called circular.
Rotation is characterized by angle, measured in degrees or radians, angular velocity (measured in rad/s), and angular acceleration(unit - rad/s²).
6. Relationship between angular and linear parameter
To change the radius vector drawn to point A from an arbitrary point O of the axis of rotation of the body, we have . Let us divide both parts of this expression by taking into account that and , - Euler formula.
Speed module. Let's find the total acceleration of the point A from the Euler formula, using the rule of differentiation of the product of two functions or .
Let us determine which term is normal and which is tangential acceleration:
- the second term, - the first term;
or, arguing differently: since the axis of rotation is fixed, then - this is; - .
These projections are equal; ,
a full acceleration module - .
The total acceleration vectors of the points of a rigid body lying on the same radius drawn perpendicular to the axis of rotation are parallel to each other, and their modulus grows in proportion to the distance from the axis. The angle characterizes the direction relative to the radius and is equal to
, it does not depend on .
So, linear and angular parameters are related in the following way :
You can do the following analogy between translational and rotational types of motion: so, at : , ; at : , .
7. Dynamics. Mass and momentum of a body. Basic laws of dynamics.
Dynamics – This is a branch of mechanics that studies the movement of bodies under the action of forces applied to them.. When studying quantities that are characterized not only by magnitude, but also by direction (for example, speed, acceleration, force, etc.), their vector image is used.
Weight
Weight- physical quantity, which is a measure of the inertia of bodies ( inertial mass) and their gravitational properties ( gravitational mass)
inertia - compliance of the body to a change in its speed (modulo or direction).
Units masses in SI:
mass properties:
- additivity: - the mass of the system is equal to the sum of the masses of its individual elements;
- independence from the speed of movement;
- mass constancy for an isolated system of bodies and independence from the processes occurring in them: - law of conservation of mass.
body momentum
- amount of movement(according to Newton) ; pulse(modern name).
At the heart of classical dynamics in mechanics (the main section of mechanics) are Newton's three laws.
Newton's first law: any material point (body) maintains a state of rest or uniform rectilinear motion until impact from other bodies will not force her to change this state.
The desire of a body to maintain a state of rest or uniform rectilinear motion is called inertia. Therefore, Newton's first law is also called the law of inertia.
Mechanical motion is relative, and its nature depends on the frame of reference. Newton's first law is not valid in any frame of reference, and those systems in relation to which it is performed are called inertial reference systems.
An inertial frame of reference is such a frame of reference, relative to which a material point, free from external influences, either at rest or moving uniformly and in a straight line. Newton's first law states the existence of inertial frames of reference.
From experience it is known that under the same influences, different bodies change their speed of motion unequally, that is, in other words, they acquire different accelerations. Acceleration depends not only on the magnitude of the impact, but also on the properties of the body itself (on its mass).
To describe the effects mentioned in Newton's first law, the concept of force is introduced. Under the influence of forces
bodies either change their speed of movement, i.e., acquire accelerations (dynamic manifestation of forces), or deform, i.e., change their shape and dimensions (static manifestation of forces).
At each moment in time, the force is characterized by a numerical value, a direction in space, and a point
applications. So, strength - this is a vector quantity, which is a measure of the mechanical impact on the body from other bodies or fields, as a result of which the body acquires acceleration or changes its shape and size.
Newton's second law- the basic law of the dynamics of translational motion - answers the question of how the mechanical motion of a material point (body) changes under the action of forces applied to it.
If we consider the action of different forces on the same body, it turns out that the acceleration acquired by the body is always proportional to the resultant of the applied forces: .
Under the action of the same force on bodies with different masses, their acceleration
are different, namely
Considering that force and acceleration are vector quantities, we can write
The ratio expresses Newton's second law: the acceleration acquired by a material point (body), proportional to the force causing it, coincides with it in direction and is inversely proportional to the mass
material point (body).
In SI, the proportionality factor to - 1. Then or
Considering that the mass of a material point (body) in classical mechanics is a constant value, in the expression it can be introduced under the sign of the derivative:
This expression - a more general formulation of Newton's second law: the rate of change of momentum of a material point is equal to the force acting on it. The expression is also called the equation of motion of a material point.
If several forces act on the body, then in the formulas under F their resulting
(vector sum of forces).
Unit of force in SI - newton (N): 1 N is the force that imparts acceleration 1 to the mass of 1 kg in the direction of the force: 1N = 1kg *. Newton's second law is valid only in inertial frames of reference.
The interaction between material points (bodies) is determined by Newton's third law: any action of material points (bodies) on each other has the character of interaction; the forces with which material points act on each other are always equal in absolute value, oppositely directed and act along the straight line connecting these points: , where - the force acting on the first material point from the second; - the force acting on the second material point from the side of the first. These forces are applied to different material points (bodies), always act in pairs and are the forces one nature.
Newton's third law, as well as the first two, is valid only in inertial frames of reference.
8. Classification of forces. All about strength.
Strength is a vector quantity that characterizes the degree of influence on a material point at any point in time from other material objects.
Dimension strength:
,
The resultant of all forces acting on the point under study, according to superposition principle
Where is the force with which the th body would act on a given point in the absence of other bodies .
line of action force is a straight line along which the force vector is directed.
Two forces equal in magnitude and oppositely directed- if they, attached to the body, do not cause acceleration.
Types of interactions: gravitational, electromagnetic, strong, weak.
Two manifestations of forces:
- static (deformation of bodies),
Dynamic (change in speed).
Force classification
- Fundamental forces:
a) gravity,
b) electrical.
- Approximate forces:
a) gravity;
b) friction force;
c) elastic force (elastic force);
d) resistance force.
a) Gravity in the reference frame associated with the Earth,
Reaction force suspension or support is the force with which other bodies act on the body, limiting its movement.
Body weight- the force with which the body acts on the support or suspension.
If the suspension or support is at rest relative to the Earth (or moves without acceleration):
b) Friction force
1) external (occurs at the points of contact between bodies and prevents their relative movement);
Sliding friction (occurs during the translational movement of one body on the surface of another);
Rolling friction (occurs when one body rolls over the surface of another);
Friction of rest (occurs when trying to cause movement);
2) internal (occurs when moving parts of a liquid or gas)
Empirical law for all types of external friction forces:
Where is the force of normal pressure pressing the surfaces in contact with each other, is the coefficient of sliding (rest, rolling) friction, depending on the nature and condition of the surfaces (roughness, etc.).
in) Elastic force
Where is the radius vector characterizing the displacement of a material point from the equilibrium position, is the coefficient of proportionality. Motion with a variable mass.
t rocket mass t, and her speed v, then after time dt t - dm, and the speed will become equal v+dv. dt
Where and -
The second term on the right side is called reactive force Fp. If a and opposite v in direction, then the rocket accelerates, and if it coincides with v, then it slows down. So we got equation of motion of a body of variable mass , which was first derived by I. B. Meshchersky (1859-1935):
Where - Reactive force, which arises as a result of the action on the body of the attached (separated) mass.
10. Movement of a body with a variable mass. Tsiolkovsky formula.
The movement of some bodies is accompanied by a change in their mass, for example, the mass of a rocket decreases due to the outflow of gases formed during the combustion of fuel, etc. Such a movement is called motion with variable mass.
Let us derive the equation of motion of a body of variable mass on the example of the motion of a rocket. If at the time t rocket mass t, and her speed v, then after time dt its mass will decrease by dm and become equal to t - dm, and the speed will become equal v+dv. Change in the momentum of the system over a period of time dt
Where and - the speed of the outflow of gases relative to the rocket.
If external forces act on the system, then either
Assuming F = 0 and assuming that the velocity of the ejected gases relative to the rocket is constant (the rocket moves in a straight line), we obtain , whence
The value of the constant of integration FROM determine from the initial conditions. If at the initial moment of time the speed of the rocket is zero, and its starting mass , then C= . Consequently,
This ratio is called the Tsiolkovsky formula. It shows that: 1) the greater the final mass of the rocket, the greater should be the launch mass of the rocket; 2) the greater the velocity of the outflow of gases, the greater the final mass can be for a given starting mass of the rocket.
11. Dynamics of rotational motion of a rigid body.
The basic Law.