EntranceDifferential equations of motion of a material point. Two point dynamics problems
NON-VISCOUS LIQUID
In this section we will establish the general laws of motion of an inviscid fluid. To do this, in the flow of an inviscid fluid, we select an elementary volume in the form of a parallelepiped with edges dx, dy, dz parallel to the coordinate axes (Fig. 4.4).
Rice. 4.4. Scheme for deriving differential equations
motion of inviscid fluid
The mass of liquid in the volume of the parallelepiped is equally affected by mass forces, proportional to the mass, and surface pressure forces of the surrounding liquid, distributed along the faces of the parallelepiped, perpendicular to them and proportional to the areas of the corresponding faces.
Let us denote by the distribution density of the resultant mass forces and by , its projections onto the corresponding coordinate axes. Then the projection onto the direction OX of the mass forces acting on the isolated mass of liquid is equal to 
.
Let us denote by p the pressure at an arbitrary point with coordinates x, y, z, which is one of the vertices of the parallelepiped. Let this be point A in Fig. 4.4.
Due to the continuity of the liquid and the continuity of the pressure function p = f (x, y, z, t) at point B with coordinates (x + dx, y, z), the pressure will be equal to within infinitesimals of the second order.
The pressure difference is and will be the same for any pair of points selected on the faces with the same y and z coordinates.
The projection onto the OX axis of the resulting pressure force is equal to . Let us write the equation of motion in the direction of the OX axis
or after dividing by mass we get
. (4.15)
Similarly, we obtain the equations of motion in the direction of the OY and OZ axes. Then the system of differential equations of motion of an inviscid fluid has the form
(4.16)
These differential equations were first obtained by L. Euler in 1755.
The terms of these equations represent the corresponding accelerations, and the meaning of each of the equations is as follows: the total acceleration of a particle along the coordinate axis is the sum of the acceleration from mass forces and the acceleration from pressure forces.
Euler's equations in this form are valid for both incompressible and compressible fluids, as well as for the case when, along with gravity, other mass forces act during the relative motion of the fluid. In this case, the values of R x , R y and R z must include the acceleration components of the portable (or rotary) movement. Since the derivation of equations (4.6) did not impose stationary motion conditions, they are also valid for unsteady motion.
Considering that for unsteady motion the components (projections) of the velocity V are functions of time, we can write the acceleration of the selected fluid mass in expanded form:
Since Euler’s equations (4.16) can be rewritten in the form
. (4.18)
For the case of a fluid at rest 
equations (4.16) coincide with the differential equations of fluid equilibrium (2.5).
In fluid dynamics problems, body forces are usually considered given (known). The unknowns are the pressure functions
p = f (x,y,z,t), velocity projections V x = f (x, y, z, t), Y y = f (x, y, z, t),
V z = f (x, y, z, t) and density r = f (x, y, z, t), i.e. only five unknown functions.
To determine unknown variables, a system of Euler equations is used. Since the number of unknowns exceeds the number of equations, the continuity equation and the equation of state of the medium are added to the Euler system.
For an incompressible fluid, the equation of state p = const and the continuity equation
. (4.19)
In 1881, Professor of Kazan University I.S. Gromeka transformed Euler’s equations and wrote them in a different form. Let's consider equations (4.18).
In the first of them, instead of and we substitute their expressions from (3.13):
And 
. (4.20)
Having adopted the designation 
, we can write
Having similarly transformed the other two equations of system (4.7), we obtain a system of equations in the form given by Gromeka
(4.23)
If the mass forces acting on the fluid have potential, then the projections of the mass force distribution density R x , R y , R z are represented as partial derivatives of the potential function P:
DP = R x dx + R y dy + R z dz .(4.25)
Substituting the values of R x , R y , R z into system (4.8), we obtain a system of differential equations of motion of an incompressible fluid under the action of forces having a potential:
(4.26)
In steady motion, the partial derivatives of the velocity components with respect to time are equal to zero:
. (4.27)
Then the equations of system (4.10) take the form
(4.28)
Multiplying each of the equations of system (4.11) by the corresponding projections of elementary displacement equal to dx = V x dt; dy = V y dt;
dz = V z dt, and add up the equations. Will have
The right side of the resulting expression can be rewritten as a determinant, i.e.
(4.29)
If the determinant is equal to zero, i.e.
(4.30)
. (4.31)
This is Bernoulli's equation for an elementary stream with steady motion of an inviscid fluid.
To bring equation (4.14) to the form of the Bernoulli equation obtained in (4.1), we determine the form of the potential function P for the case when only one mass force acts - gravity. In this case, R x = R y = 0 and R z = - g (OZ axis is directed upward). From (4.9) we have
or . (4.32)
Substituting this expression P into (4.14), we obtain
or 
.
The last expression fully corresponds to the Bernoulli equation (4.4).
Let us find out in what cases of steady motion of an inviscid incompressible fluid the Bernoulli equation is valid or, in other words, in what cases the determinant on the right side of equation (4.13) vanishes.
It is known that a determinant is equal to zero if two rows (or two columns) are equal or proportional to each other or if one of its rows or one of its columns is equal to zero. Let's consider these cases sequentially.
A. The terms of the first and third lines are proportional, i.e. Bernoulli's equation is valid if
.
This condition is satisfied on streamlines (3.2).
B. The terms of the first and second rows are proportional, i.e. Bernoulli's equation is valid if
.
This condition is satisfied on vortex lines (3.16).
B. The terms of the second and third lines are proportional:
. (4.16)
Then ω x = a V x ; ωy = a Vy ; ω z = a Vz.
Galileo–Newton's laws of mechanics
Dynamics is based on laws (axioms), which are a generalization of practical human activity. Various principles of mechanics are logically derived from these laws. These laws were generalized by Galileo and Newton and formulated in relation to a material point.
Newton's first law(law of inertia). A material point that is not acted upon by forces or is acted upon by an equilibrium system of forces has the ability to maintain its state of rest or uniform and linear motion.
In both the first and second cases, the acceleration of the point is zero. This kinematic state of the point is called inertial.
All reference systems in relation to which the law of inertia holds are called inertial.
Newton's second law(basic law of dynamics). The acceleration of a material point relative to the inertial frame of reference is proportional to the force applied to the point and is directed along this force (Fig. 1).
This law can be expressed in the form
(1)
Where m a positive coefficient characterizing the inertial properties of a material point is called the mass of the point. Mass in classical mechanics is considered a constant quantity. The SI unit of mass is the kilogram (kg); – point acceleration; – force applied to a point.
| Rice. 1 | Rice. 2 |
Mass is usually determined by the force of gravity and the acceleration due to gravity at the Earth's surface. According to (1), we have
Newton's third law(law on equality of forces of action and reaction). The forces of interaction between two material points are equal in magnitude and opposite in direction (Fig. 2), i.e. 
Fourth Law(the law of independence of the action of forces). With the simultaneous action of several forces, a material point acquires an acceleration equal to the geometric sum of those accelerations that it would acquire under the action of each of these forces separately. Thus, the forces applied to a material point act on it independently of each other.
Let a system of forces be applied to a material point 
then, according to Newton’s second law, the acceleration from the action of each force is determined by expression (1):
Acceleration with simultaneous action of all forces
(3)
Summing (2) and using (3), we obtain the basic equation for the dynamics of a point:
But the point acquires the same acceleration under the influence of one force
Since the system of forces 
and the force impart the same acceleration to the point, then this system of forces and the force are equivalent.
Differential equations of motion of a material point
3.1.2.1. Differential equations of motion of a free point
| Rice. 3 |
Let a free material point be acted upon by a system of forces that has a resultant, see Fig. 3. Then, according to the basic law of dynamics,
(4)
The acceleration of a point can be represented as 
, therefore equality (4) takes the form:
. (5)
Equation (5) is a vector differential equation of motion of a material point. If we project it on the axes of a Cartesian coordinate system, we will obtain differential equations of motion of a material point in projections onto these axes:
When a point moves in a plane Oxy system of equations (6) takes the form:
When a point moves in a straight line along an axis Ox we obtain one differential equation of motion:
Having projected equality (5) onto the natural coordinate axes, we obtain differential equations of motion of a point in projections onto the natural coordinate axes:
1.2.2. Differential equations of motion of a non-free point
Based on the principle of liberation from connections, a non-free point can be turned into a free point by replacing the action of connections with their reactions. Let be the resultant of the bond reactions, then the basic equation of the dynamics of the point will take the form:
(7)
Having projected (7) on the axes of the Cartesian coordinate system, we obtain differential equations of motion of a non-free point in projections onto these axes:
To solve problems, it is necessary to add constraint equations to these equations.
Differential equations of motion of a point in projections onto natural coordinate axes:
1.2.3. Differential equations for relative motion of a point
Basic equation of point dynamics 
valid for an inertial reference frame where the acceleration is absolute. According to the Coriolis theorem, the absolute acceleration
where is the acceleration of portable motion; – relative acceleration of the point relative to the moving coordinate system; – Coriolis acceleration.
Substituting the expression for absolute acceleration into the basic equation of the dynamics of a point, we obtain
Let us introduce the following notation: 
– portable inertia force; 
– Coriolis inertial force.
Then equation (9) takes the form
(10)
The resulting equality expresses the dynamic Coriolis theorem.
Coriolis theorem. The relative motion of a material point can be considered as absolute if the transfer and Coriolis inertia forces are added to the forces acting on the point.
Let us consider the case of relative equilibrium of the point 
Then the Coriolis acceleration 
Substituting these values into equation (10), we obtain the condition for the relative equilibrium of a point:
In order for the basic law of dynamics for the relative motion of a point to coincide with the basic law of its absolute motion, the following conditions must be met:
This condition is satisfied if the moving coordinate system moves translationally 
straight and even 
In relation to these reference systems, as well as in relation to stationary ones, when 
the law of inertia will be fulfilled. Therefore, all reference systems moving translationally, rectilinearly and uniformly, as well as those at rest, are inertial.
Since the laws of dynamics are the same in all inertial reference systems, then in all these systems mechanical phenomena proceed in exactly the same way if the same event is taken as the reference point. This follows the principle of relativity of classical mechanics.
The principle of relativity of classical mechanics. No mechanical experiments can detect the inertial motion of the reference system, participating with it in this motion.
SECTION 3. DYNAMICS.
Dynamics Material body- a body that has mass.
Material point
Material
A - bV -
Inertia
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Body mass
Force -
,
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. A - b- - traction force of the electric locomotive; V- -
System Inertial
Movement Space Time
System
TOPIC 1
First Law(law of inertia).
Isolated
For example: -
body weight, -
- starting speed).
Second Law(basic law of dynamics).
Mathematically, this law is expressed by the vector equality
When accelerating, the movement of the point is uniformly variable (Fig. 5: A - movement - slow; b - movement - accelerated, . - point mass, - acceleration vector, - force vector, - velocity vector).
When - the point moves uniformly and rectilinearly or when - it is at rest (law of inertia). The second law allows us to establish a connection between body weight, located near the earth's surface, and its weight , , where is the acceleration of free fall.
Third Law(law of equality of action and reaction).
Two material the points act on each other with forces equal in magnitude and directed along the straight line connecting these points in opposite directions.
Since forces are applied to different points, the system of forces is not balanced (Fig. 6). In its turn 
- the ratio of the masses of interacting points is inversely proportional to their accelerations.
Fourth Law(the law of independence of the action of forces).
Acceleration, received by a point when several forces act on it simultaneously, is equal to the geometric sum of those accelerations that the point would receive when each force was applied to it separately.
Explanation (Fig. 7). The resultant force is defined as . Since 
, That .
Second (inverse) problem.
Knowing the current on the point of force, its mass and initial conditions of motion, determine the law of motion of the point or any of its other kinematic characteristics.
Initial the conditions for the motion of a point in the Cartesian axes are the coordinates of the point, , and the projection of the initial velocity onto these axes, and at the moment of time corresponding to the beginning of the point’s movement and taken equal to zero.
Solving problems of this type comes down to compiling differential equations (or one equation) of the motion of a material point and their subsequent solution by direct integration or using the theory of differential equations.
TOPIC 2. INTRODUCTION TO MECHANICAL SYSTEM DYNAMICS
2.1. Basic concepts and definitions
Mechanical a system or system of material points is a collection of material points interacting with each other.
Examples of mechanical systems:
1. a material body, including an absolutely solid one, as a collection of interacting material particles; a set of interconnected solids; a set of planets in the solar system, etc.
2. A flock of flying birds is not a mechanical system, since there is no force interaction between the birds.
Free a mechanical system is a system in which no connections are imposed on the movement of points. For example: movement of the planets of the solar system.
Unfree mechanical system - a system in which connections are imposed on the movement of points. For example: movement of parts in any mechanism, machine, etc.
Classification of forces
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The classification of forces acting on a non-free mechanical system can be presented in the form of the following diagram:
External forces - forces acting on points of a given mechanical system from other systems.
Domestic- interaction forces between points of one mechanical system.
An arbitrary point of the system (Fig. 1) is affected by: - the resultant of external forces (index - the first letter of the French word exterieur - (external)); - resultant of internal forces (index - from the word interieur - (internal)). The same strength of the connection reaction, depending on the conditions of the task, can be both external and internal.
Property of internal forces
and - interacting points of the mechanical system (Fig. 2). Based on the 3rd law of dynamics
On the other side: 
. Therefore, the main vector and the main moment of the internal forces of the mechanical system are equal to zero:
SECTION 3. DYNAMICS.
BASIC CONCEPTS OF CLASSICAL MECHANICS
Dynamics- a section of theoretical mechanics that studies the movement of material bodies (points) under the action of applied forces. Material body- a body that has mass.
Material point- a material body, the difference in the movement of points of which is insignificant. This can be either a body whose dimensions during its movement can be neglected, or a body of finite dimensions if it moves translationally.
Material points are also called particles into which a solid body is mentally broken down when determining some of its dynamic characteristics.
Examples of material points (Fig. 1): A - movement of the Earth around the Sun. Earth is a material point; b- translational motion of a rigid body. A solid body is a material point, because ; V - rotation of a body around an axis. A particle of a body is a material point.
Inertia- the property of material bodies to change the speed of their movement faster or slower under the influence of applied forces.
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Body mass is a scalar positive quantity that depends on the amount of substance contained in a given body and determines its measure of inertia during translational motion. In classical mechanics, mass is a constant quantity.
Force- a quantitative measure of mechanical interaction between bodies or between a body (point) and a field (electric, magnetic, etc.). Force is a vector quantity characterized by magnitude, point of application and direction (line of action) (Fig. 2: - the point of application is the line of action of the force).
In dynamics, along with constant forces, there are also variable forces, which can depend on time, speed , distance or from the totality of these quantities, i.e.
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Examples of such forces are shown in Fig. 3 . A -- body weight, - air resistance force; b- - traction force of the electric locomotive; V- - the force of repulsion from or attraction to the center.
System reference - a coordinate system associated with a body in relation to which the movement of another body is studied. Inertial system - a system in which the first and second laws of dynamics are satisfied. This is a fixed coordinate system or a system moving uniformly and linearly translationally.
Movement in mechanics, it is a change in the position of a body in space and time. Space in classical mechanics, three-dimensional, subject to Euclidean geometry. Time- a scalar quantity that occurs equally in any reference system.
System units are a set of units of measurement of physical quantities. To measure all mechanical quantities: three basic units are sufficient: units of length, time, mass or force. All other units of measurement of mechanical quantities are derived from these. Two types of systems of units are used: the international system of units SI (or smaller - GHS) and the technical system of units - ICG.
TOPIC 1. INTRODUCTION TO THE DYNAMICS OF A MATERIAL POINT.
1.1. Laws of dynamics of a material point (Galileo-Newton laws)
First Law(law of inertia).
Isolated from external influences, a material point maintains its state of rest or moves uniformly and rectilinearly until applied forces force it to change this state.
The movement performed by a point in the absence of forces or under the action of a balanced system of forces is called movement by inertia.
For example: movement of a body along a smooth (friction force is zero) horizontal surface (Fig. 4: -
body weight, -
normal plane reaction). Since, then.
When the body moves at the same speed; when the body is at rest ( - starting speed).
The basic law of mechanics, as indicated, establishes for a material point a connection between kinematic (w - acceleration) and kinetic ( - mass, F - force) elements in the form:
It is valid for inertial systems that are chosen as the main systems, therefore the acceleration appearing in it can reasonably be called the absolute acceleration of a point.
As indicated, the force acting on a point generally depends on the time of the point’s position, which can be determined by the radius vector and the speed of the point. Replacing the acceleration of the point with its expression through the radius vector, we write the basic law of dynamics in the form:
In the last entry, the fundamental law of mechanics is a second-order differential equation that serves to determine the equation of motion of a point in finite form. The equation given above is called the equation of motion of a point in differential form and vector form.
Differential equation of motion of a point in projections onto Cartesian coordinates
Integrating a differential equation (see above) in the general case is a complex problem and usually to solve it one moves from a vector equation to scalar equations. Since the force acting on a point depends on the time position of the point or its coordinates and the speed of the point or the projection of the speed, then, denoting the projection of the force vector onto a rectangular coordinate system, the differential equations of motion of the point in scalar form will have the form:
Natural form of differential equations of motion of a point
In cases where the trajectory of a point is known in advance, for example, when a connection is imposed on the point that determines its trajectory, it is convenient to use the projection of the vector equation of motion onto the natural axes directed along the tangent, the main normal and the binormal of the trajectory. The projections of the force, which we will call accordingly, will in this case depend on the time t, the position of the point, which is determined by the arc of the trajectory and the speed of the point, or Since acceleration through projections onto natural axes is written in the form:
then the equations of motion in projection onto the natural axes have the form:
The latter equations are called natural equations of motion. From these equations it follows that the projection of the force acting on a point onto the binormal is zero and the projection of the force onto the main normal is determined after integrating the first equation. Indeed, from the first equation it will be determined as a function of time t for a given then, substituting into the second equation we will find since for a given trajectory its radius of curvature is known.
Differential equations of motion of a point in curvilinear coordinates
If the position of a point is specified by its curvilinear coordinates, then by projecting the vector equation of motion of the point onto the directions of the tangents to the coordinate lines, we obtain the equations of motion in the form.
Using differential equations of motion, the second problem of dynamics is solved. The rules for composing such equations depend on how we want to determine the movement of a point.
1) Determination of the movement of a point using the coordinate method.
Let the point M moves under the influence of several forces (Fig. 13.2). Let's compose the basic equation of dynamics and project this vector equality on the axis x, y, z:
But the projections of acceleration on the axis are the second derivatives of the coordinates of the point with respect to time. Therefore we get
a) Assign a coordinate system (number of axes, their direction and origin). Well-chosen axes simplify the solution.b) Show a point in an intermediate position. In this case, it is necessary to ensure that the coordinates of this position are necessarily positive (Fig. 13.3.).
c) Show the forces acting on the point in this intermediate position (do not show inertial forces!).
In example 13.2, this is only the force, the weight of the core. We will not take air resistance into account.
d) Compose differential equations using formulas (13.1): . From here we get two equations: and .
e) Solve differential equations.
The equations obtained here are linear equations of the second order, with constants on the right side. The solution to these equations is elementary.
And 
All that remains is to find the constant integrations. We substitute the initial conditions (at t = 0 x = 0, y = h, 
, 
) into these four equations: u cosa = C 1 , u sina = D 1 , 0 = WITH 2 , h = D 2 .
We substitute the values of the constants into the equations and write down the equations of motion of the point in their final form
Having these equations, as is known from the kinematics section, it is possible to determine the trajectory of the nucleus, the speed, acceleration, and position of the nucleus at any time.
As can be seen from this example, the problem solving scheme is quite simple. Difficulties can only arise when solving differential equations, which can be difficult.
2) Determining the movement of a point in a natural way.
The coordinate method usually determines the movement of a point that is not limited by any conditions or connections. If restrictions are imposed on the movement of a point, on speed or coordinates, then determining such movement using a coordinate method is not at all easy. It is more convenient to use a natural way of specifying movement.
Let us determine, for example, the movement of a point along a given fixed line, along a given trajectory (Fig. 13.4.).
To the point M In addition to the given active forces, the reaction of the line operates. We show the components of the reaction along natural axes
Let's compose the basic equation of dynamics and project it onto natural axes
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Because 
then we obtain differential equations of motion such
(13.2)
Here the force is the friction force. If the line along which the point moves is smooth, then T=0 and then the second equation will contain only one unknown – the coordinate s:
Having solved this equation, we obtain the law of motion of a point s=s(t), and therefore, if necessary, both speed and acceleration. The first and third equations (13.2) will allow you to find the reactions and .
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The scheme for solving the problem is the same as with the coordinate method (example 13.2). The only difference is in the choice of axes. Here are the axes N And T move with the skier. Since the trajectory is a flat line, the axis IN, directed along the binormal, does not need to be shown (projections onto the axis IN The forces acting on the skier will be zero).
Differential equations from (13.2) we get the following
(13.3)
The first equation turned out to be nonlinear: . Because s=r j, then it can be rewritten like this: 
. Such an equation can be integrated once. Let's write it down 
Then in the differential equation the variables will be separated: 
. Integration gives the solution 
Since when t=0 j =
0 and , then WITH 1 =0 and 
A 