Fib

F in

forces

F 2 ... F n (Fig. 21)

How old is the module

υ = dS, where S is the arc coordinate.

Projection of accelerating on dotichno dovnyu a =

Vrakhovuuchi, what swidk_st

A folding function for an hour, that is. υ = f(S(t)),

a τ = d υ

D υ

= u d u.

The main equalization of the dynamics of the projection on the dotist may look

matτ = ∑ Fi τ

υd υ

= ∑ F i τ.

points M 1

and M 2

mυ dυ = dS∑ Fi τ

m ∫ υ d υ = ∑ ∫ F i τ dS, stars

mυ 2

= ∑ Ai.

mυ 2

Half of the additional weight of the material point per square of the speed

called the kinetic energy of the point.

mυ 2 2

− kinetic energy of the point after displacement,

− kinetic energy of the point before moving,

mυ 2

Theorem about changing the kinetic energy of a mechanical system

Primary nutrition:

1. Robot force.

2. Kinetic energy of a point and a mechanical system.

3.Theorem about changing the kinetic energy of a point.

4. The theorem about changing the kinetic energy of a mechanical system.

5. Potential force field and potential energy.

1. Robot force.

The elementary work of force is an infinitely small scalar quantity that is equal to the scalar addition of the force vector to the vector of an infinitely small displacement of the force reporting point:

.

-increment of the radius-vector points of the report of the force, the hodograph of which is the trajectory of the points. Elementary relocation
points along the trajectory
by virtue of their children. Tom

so yak
- projection of force on a straight line moving point (with a curvilinear trajectory - on a dot to the trajectory, then

,

so that the robot has no more sufficient force, and the robot of normal force is equal to zero.

Yakscho
then

yakscho
then

yakscho
then
.

Imagine a vector і
through their projections on the axes of Cartesian coordinates:

,

robot force on the last move the cost of the integrated sum of elementary labor on whom you move

.

.

As soon as the force has become, and the point її zastosuvannya moves rectilinearly, then

.

Gravity robot

de h- moving the point of stagnation of force vertically down (height).

When the point is moved, the gravity force is uphill
(dot, dot
- at the bottom,
- Vgori). Otzhe,

.

The robot of the force of gravity lies in the form of a trajectory. With Russia a closed trajectory (
Zіvpadє z
) of the robot is equal to zero.

Working force springs.

The spring expands less than the axle X

,

de - The amount of deformation of the spring. When moving the point of stagnation of force
from the lower position in the upper direction, the forces that move in a straight line are shifted, also
.

To that robot force of springiness

.

The work of forces that reach the solid body.

but) Work of internal forces

For two k - x point: , t. to.
i (be brought to the kinematics) (Fig. 80).

The elementary work of all internal forces in a solid body is equal to zero:

.

Otzhe, on the end of the moving body

.

b) The work of the outer forces.

Progressive movement of the body.

Elementary robot k-ї force

For all powers

.

Oskіlki with translational Russian, then

,

de
- projection of the head vector of external forces directly moving.

The work of forces at the end of the movement

.

The body wrap around the indestructible axis .

Elementary robot k - th force

de
,
і
- storage forces behind natural axes

so yak
,
, then the work of these forces to move
the points of the report of the force are equal to zero. Todi

.

Elementary robot k - y zovnіshnoї force to improve the moment of strength
on an elementary turn
the body is about the axis.

Elementary work of all calling forces

,

de
- The main moment of zovnіshnіh forces schodo osі.

The work of forces at the end of the movement

.

Yakscho
, then

de
- Kіntsevy ku turn;
, de P- The number of body wraps is about the axis.

Tension - tse robot, vikonan by force for one hour. Like a robot feels equally, then tightness

,

de BUT– a robot, vikonan by force on the last move, for an hour t.

In a wild mood, the intensity of strength is possible as a setting of elementary robotic strength dA up to an elementary interval dt, for some kind of vikonan tsya robot, scho є pokhіdnoyu vіd robi for an hour. Tom

With the wrapping of the body on a slightly indestructible axis

,

de
- Kutova shvidkіst body wrap.

Alone in the world of work and tightness. The CI system has a single vimir robotic force - Joule (1 J= 1 Nm),

Loneliness vimiru poguzhnosti vіdpovidno - wat (1 Tue = 1 j/s)

75 kGm/s = 1 l. h. (Kіnska strength).

1 kW= 1000 Tue= 1,36 l. h.

Let's take a look at two points of a solid body M 1 and M 2 - a part of the mechanical system. We will conduct a prompt (div. Fig. 14.13).

Internal forces P J 1 , P J 2 , that there are one points on the side on the other side, on the basis of the law of equality of two and opposite equals behind the module and opposite straight P J 1 = - P J 2 .

Let's have a quick check of speed, the point is equal to u 1 and u 2 and for an hour, the growth of vzdovzh vector_v to establish ds 1 = u 1 dt, ds 2 = u 2 dt.

Since, on the basis of one consequence of the theorem about the slipness of the points of the plane figure, the projections of the vectors of the slips on the straight line M 1 M 2 are equal, then the projections of the elementary displacements of these points will be equal.

To that, calculating the sum of the elementary labors of 2 internal forces on the movement, which is seen, and protecting their equalness and opposition is taken away

P J 1 ds 1 cos(P J1,u 1) + P J 2 ds 1 cos(P J2,u 2) = P J 1 * M 1 M' 1 - P J 1 * M 2 M' 2 = 0.

The shards of the skin's internal forces are stronger, even behind the module and are directed, then the sum of the elementary robots of the necessary internal forces is equal to zero.

Kіntseve remіshchennya є sukupnіstyu elementary remіshchenya, and to that

And j = 0,

tobto. the sum of the works of the internal forces of a solid body on whether it is moved to zero.

Progressive movement of the solid body.

In the translational direction of a solid body, the trajectories of all points are the same and parallel. Therefore, the vectors of elementary revolutions are geometrically equal.

Elementary robot force P E i

d A E i =P E i d r.

For all forces

d A = Sd A E i = SP E i d r= d r SP E = d r R E .

Otzhe,

d A = d r R E . (14-46)

Elementary work of forces applied to a solid body, which is collapsing progressively, more elementary work of the head vector of forces.

A = . (14-47)

An elementary work of forces applied to a solid body, which wraps around an indestructible axis, improves the head moment of the external forces, which wraps around the increase in the turn.

Work at the end of the journey

SA i = , (14-48)

de - the main moment of the ovnіshnіh forces schodo osі wrapping.

As the main moment is postiyny, then

SA i = Ez = E z (j 2 – j 1).(14-49)

In this turn, the sum works on the final displacement of the road to increase the head moment of the outward forces on the change of the final kut to the turn of the body.

Same tightness

N= = ME z dj/dt = ME z w.(14-50)

In a wild mood, the elementary work of the outer forces, applied to a free solid body, is strong

dA = SdA i =R E d r O + M E W da,(14-51)

de M E W- the head moment of ovnіshnіh forces shоdo mittєvoї osі; da- An elementary cut to turn around the mittev axis.

14.10. Opir pid hour of frostbite.

On a cylindrical skating rink, which is located on a horizontal plane in a calm camp (Fig. 14.14, a), there are two forces that are mutually equal: the skating rink G that is the normal reaction of the area N = -G .

Yakshcho under the influence of horizontal forces R, applied in the center of the skating rink C, roll on the flat without forging, then G, N utvoryuyut couple of forces, scho shkodzhaє stiffness (Fig. 14.14 b).

Viniknennya tsієї parity of forces is bound by the deformation of the contacting surfaces of the rink and the area. Line of reaction N vyyavlyatsya zsunutoy on deaku vіdstan vіd linії dії sily G.

Moment of betting forces G, N is called the moment of support of the bone. Yogo value is determined by the creation

M ref = Nd. (14-52)

The stiffness coefficient is observed in linear units, that is. [d]=div. For example, steel bandage from steel lath d= 0.005 div; wood on steel d= 0.03-0.04 cm.

Significantly decrease the horizontal force R , which reaches the center of the kovzanka.

Sob kovzanka began to fold, the moment of the parry of forces, folding by the force of P and the force of the fastening of F ss, there is more support for the moment, tobto.

PR>Nd.

Stars Nd/R.

Because here N = G, then

The work of internal forces on the end displacement is zero.

The work of the force, which is moving on the body, which is progressively collapsing, building up the supply of strength to increase the linear movement.

The work of the force, which is on the body, which is wrapped around, is more expensive to the moment of the force, to the axis of the wrapping to the increment of the turn: ; . Tension:
.

Kinetic energy of a mechanical system for various types of movement.

Kinetic energy of a mechanical system- scalar, which is the sum of the kinetic energy points of the system: .

With progressive Russian:

With overt Russian:

With plane-parallel rusі: de d - go to the center of the mass to the MCS

27. Theorem about changing the kinetic energy of a material point.

Kinetic energy of a material point- a scalar, which is more than half of the additional mass of points per square of її svidkostі.

Basic dynamic dynamics: , multiply by the elementary displacement: ; ; . Integrating negative viraz:

Theorem: Changing the kinetic energy of a material point on a certain moving robotic force that moves to a point, on the same moving.

Theorem about changing the kinetic energy of a mechanical system.

The scales of the robot's internal forces are equal to zero, then:
.

Theorem: changing the kinetic energy of a mechanical system at the end of the moving road, the sum of the work of outside forces at that very moving one.

The principle of possible movements for a mechanical system.

; , Let the links, overlays on the points of the mechanical system are bilateral, stationary, holonomic and ideal, etc.: .

The principle of possible movement Lagrange principle- For equal mechanical systems with two-way, stationary, holonomic and ideal links, it is necessary and sufficient that the algebraic sum of the working forces, which are set, on a possible displacement is equal to zero.

d'Alembert's principle for a material point.

The geometrical sum of all applications to a dry material point of forces and forces of inertia tsієї point is equal to zero

d'Alembert's principle for an irregular mechanical system.

In an irregular mechanical system that is collapsing, for a skin material point at some point in time, the sum of the forces applied to it, the reaction of the link and the forces of inertia, is equal to zero. Multiplying the offending part of the viraz by r i is taken: ;
.

, the sum of the moment forces, the linkage reaction and the forces of inertia along the coordinate axes are equal to zero.

Bringing the forces of inertia to the point of a solid body to the simplest look.

Up to the system of forces of inertia, the point of a solid body can be fixed by the Punch method, looking at the statics. If so, the system of forces of inertia can be reduced to the head vector of forces of inertia and the head moment of forces of inertia.

With forward speed: Ф=-ma (with forward speed of a solid body, the force of inertia of the first point is directed to the head vector of forces of inertia equal to the module of additional weight of the body, to the acceleration center of the mass applied to the center of the body and directed to the back of the protile acceleration center of the mass).

In case of wrapping rusі: М=-Iε (with wrapping rusі of a solid body, the forces of inertia of the first point are brought up to the head moment of forces of inertia equal to the moment of inertia of the body of the body of the wrapping forces on the apex of the apex.

With flat rusі: Ф=-ma M=-Iε (with flat rusі of a solid body, the forces of inertia and the point are brought to the head vector and the head moment of the forces of inertia).

Zagalne rіvnyannya dynamіki. d'Alembert-Lagrange principle.

d'Alembert's principle: (P i + R i + Ф i) = 0; å(P i + R i + Ф i) Dr i = 0, mind you. that links, overlays on a mechanical system are two-sided, stationary, holonomic and ideal, also: å(R i × Dr i) = 0;

å(P i + Ф i) Dr i = 0 - more dynamic dynamics- for a mechanical system with two-way, stationary, holonomic and ideal links, the sum of the robotic forces and the forces of inertia is the point of the system, which are set, on any possible displacement, to zero.

The work of forces is calculated according to the formulas taken from § 87 and 88.

1. The robot of the forces of gravity, yakі dіyut system. The work of the force of gravity, which will move on a part of the vag, will be more stable de - coordinates, which will signify the beginning and end position of the part (div. § 88). Todі, calling on those that (div. § 32), we know for the sum of the work of the forces of gravity that are on the system, the meaning

Whose result can be seen at a glance

de R - vaga system - vertically moving to the center of the mass (or the center of gravity). Later, the robot of the forces of gravity, which operates on the system, is counted as a robot of the head vector (in times of a solid body is equal) P on the moving center of the mass of the system (or the center of gravity of the body).

2. The work of forces applied to the body, what turns around. Elementary work applied to the body of force F (Fig. 307) is more expensive (div. § 87)

to that, de - elementary cut to the turn of the body.

Ale yak is easy to bachiti,

Let's call the value torque. Todi otrimaєmo

Also, in this moment, the elementary work of the robot will increase the amount of torque for an elementary turn. Formula (46) is valid even if there are a number of forces, so as to improve

When turning to the end of the robot

and at a time of constant moment

If there is a pair of forces on the body that lies near the plane perpendicular to the Oz axis, then in formulas (46)-(47) it will obviously mean the moment of the bet.

Let's just say, how tightness is shown in which depression (div. § 87). Koristuyuchis jealousy (46), we know

Later, when there is a lot of force on the body, which turns around, the tension will increase the cool moment on the top of the body. With the same tightness, the torque will be greater, the less windy.

3. The work of rubbing strength, what to blow on the body, what to wear. On a wheel with a radius R (Fig. 308), which rolls along the active surface (surface) without forging, a force is applied at the point, rubbing, which crosses the forging of the point of the flat surface. Elementary work of strength. Ale dot At this point, zbіgaєtsya z mittєvim center of swidkost (div. § 56) і

So it is for skin elemental movement.

Later, when the robot was stiff without forging, the forces were rubbing, so that the forging was changed, whether the body was moved to zero. Z tієї well cause in tієї vpadku more zero and the robot of the normal reaction N, yakscho vvazhat tіla not deformed due to N, which is added to the point (like in Fig. 308, a).