Mechanics is a branch of physics that studies the motion of bodies, as well as the interactions between these material bodies. This section of physics includes dynamics - one of the subsections of mechanics that is devoted to the study of the causes of mechanical motion. One of the basic principles of dynamics is called the d'Alembert principle. It makes it possible to formulate dynamic problems through static problems, which greatly simplifies the calculations.
Kinetostatic method
Dynamic problems are often solved through Newton's laws. However, this is not the only way. The principles of mechanics for solving such problems are developed - these are some starting points that underlie the ways of solving dynamic problems. One of these principles is the d'Alembert principle, which is interconnected with the kinetostatic method. This method is one of the ways to solve dynamic problems, which is based on writing dynamic equations in the form of equilibrium equations. The method of kinetostatics is used in the theory of mechanisms and machines, the resistance of materials (sopromat), in other areas of theoretical mechanics. It is used to simplify the solution of a number of general technical problems. It is most convenient for solving the first problem of dynamics (determining the effective force or one of several forces on a material point, provided that its mass and motion are specified).
Statement of principle for a material point
The d'Alembert principle, or also called the kinetostatic principle, can be applied both to a material point and to a mechanical system. This principle allows the use of static solution methods to solve dynamics problems. A material point is considered to be a body whose dimensions are taken equal to zero, but at the same time its mass is preserved. D'Alembert made a proposal, which implied the conditional application of an inertia force to a body that moves with acceleration, that is, actively accelerates. In this case, the system of forces that act on the point becomes balanced, which allows us to solve the problems of dynamics using the equations of statics. The d'Alembert principle for a material point is formulated as follows:
If we apply its inertia force to a non-free material point moving under the action of applied active forces and bond reaction forces, then at any time the resulting system of forces will be balanced, i.e. the geometric sum of the indicated forces will be zero.
In other words, if the force of inertia is conditionally added to the forces acting on a material point, then the result will be a balanced system.
The procedure for using the kinetostatic principle
There is a certain procedure for solving problems using the principle of kinetostatics - the d'Alembert principle. The following sequence of actions is carried out:
- The calculation scheme is drawn up.
- The coordinate system is selected.
- The direction of acceleration and its magnitude are clarified.
- The force of inertia is applied (conditionally).
- A system of equilibrium equations with unknowns is compiled.
- Unknown quantities are determined by solving a compiled system of equations.
The mechanical system, the principle of kinetostatics for it
A mechanical system is the community of material points, provided that their movements are interconnected. A more detailed definition says that a mechanical system is a set, a community of material points that move according to the laws of classical mechanics, while they interact not only with each other, but also with bodies that are not part of this set of points. The d'Alembert principle for a mechanical system is as follows:
For a moving mechanical system at any time, the geometric sum of the main vectors of external forces, bond reactions, inertia forces is zero and the geometric sum of the main moments from external forces, bond reactions, inertia forces is zero.
For a mechanical system (as well as for a material point), the equations of motion can be written as equations of equilibrium, from which later unknown quantities (forces) can be determined, which include bond reactions. The derived formulas for solving problems by the D'Alembert principle are second-order differential equations due to the fact that in each of them there is acceleration, which is the second derivative of the law of motion of a point, a body.
The combination of the principle of analytical statics and the principle of kinetostatics
The principle of analytic statics is the principle of possible displacements - the Lagrange principle. This principle, or rather its formulation, states that for the balance of the system it is necessary and sufficient that the sum of the forces applied to the system is zero for any possible movement of the system, accompanied by its exit from the state of equilibrium.
The D'Alembert principle and the Lagrange principle can easily be combined into one, which allows us to express the general equation of dynamics. The result is an equation for a system with perfect connections. The d'Alembert-Lagrange principle is formulated as follows:
When a mechanical system with perfect connections moves at each moment of time, the sum of the elementary work of all the applied active forces and inertia forces at any possible movement of the system will be zero.
From the general equation of dynamics it is possible to derive all the theorems of dynamics presented in theoretical mechanics. This equation puts in importance the work of inertia forces and the work of active forces on the same level, that is, these works are considered on a par with each other.