Department of Theoretical Mechanics Engineering and Robotic Systems

Department of Theoretical Mechanics, Engineering and Robotic Systems (202) Methodic room 227 m. b. Teachers room 236 m. b. Khai 202. ho. ua Assistant Ternovskaya Svetlana Sergeevna

Theoretical mechanics. Lecture 1. Introduction Theoretical mechanics is the branch of the physical sciences that deals with the mechanical motion of bodies, i. e. changing of relative position of bodies in space in the course of time.

Statics is concerned with bodies that are at rest or have uniform motion. Such bodies are said to be in equilibrium. Statics provides methods for determination of support reactions and relationships between internal force distribution and external loads for structures.

1. Basic conceptions: rigid body, particle, force system, rigid body equilibrium • • A rigid body (абсолютно твердое тело) A body is called rigid if the distance between any two points of the body does not change during its interaction with other solids. Consequently, the angle between any two straight lines in the body remains constant A rigid body system (система твердых тел) A set of rigid bodies connected among themselves is called rigid body system

Example of rigid body system: the mechanism used to raise the bucket of a bulldozer. In engineering practice particle systems or rigid body systems are more often called structures.

• A particle (материальная точка) If the dimensions of a body are irrelevant to the description of its position the body may by treated as a particle. • A particle system (система материальных точек, материальная система) A particle system is a set of particles whose motions are interconnected. So, a position, velocity and acceleration of particle in the system are connected with same parameters of other particles. The sample of particle system is system of planets that rotates about the Sun.

• Force (сила) Force is one of the main fundamental quantities of mechanics. A force is measure of mechanical interaction between two bodies. So when we say “a force acts on the body” we know that there is another body acting as a source of the force. As a result of this interaction bodies can be accelerated or deformed.

Force classification The physical nature of forces is not studied in mechanics. We distinguish forces solely by the mode of their interaction : • a force may be exerted at a distance as in the case of gravitational or magnetic attraction (long-range interaction); • a force may act through actual contact as the lift force acting on the airplane wing in incident flow (close-range interaction)

A force is vector value. It has three main characteristics: a magnitude, a direction and a point of application.

• System of forces (система сил) A set of forces applied to a material object (particle, rigid body, system of rigid bodies) and treated as group is called system of forces. Example. Forces that act on an airplane: tractive force, lift force, weight, drag force.

• Equivalent force systems (эквивалентные системы сил) Two force systems are said to be equivalent if they produce the same external effect when applied in turn to a given body. • Resultant of force system (равнодействующая системы сил) If a force system is equivalent to only single force, this force is called a resultant.

A resultant of a force system is the simplest equivalent system to which the original system can be reduced. The process of reducing a force system to a simpler equivalent system is called composition (приведение). The process of expanding a force or a force system into a less simple equivalent system is called resolution (разложение на составные части). A component (компонента) of a force is one of the two or more forces into which the given force may be resolved.

• Balanced force system (equilibrated force system) (уравновешенная система сил, система сил эквивалентная нулю) A system of forces is called balanced if the force system resultant is zero. A rigid body under the action of balanced force system is in the state of equilibrium, i. e. body is at the rest or moves translational with uniform velocity along straight line.

2. Axioms of static • Axiom 1. A rigid body which is acted upon by two forces will be in equilibrium if and only if the two forces have the same magnitude and the same line of action but opposite sense.

• Axiom 2. The action of a given force system on a rigid body remains unchanged if another balanced force system is added to, or subtracted from, the original system. In a special case, in accordance with Axiom 1, this balanced force system can consist of two equal and opposite forces acting along the common line. • It follows from Axiom 2 (corollary) that a force may be applied at any point on its given line of action without altering the resultant effects of the force, external to the rigid body on which it acts. The corollary is named the principle of transmissibility.

Principle of transmissibility: A force point of application can be shifted along force’s line of action without change to the state of the body. Force is sliding vector. Без изменения состояния тела точка приложения силы может быть перемещена вдоль линии действия силы. Сила – скользящий вектор

• Axioms 3 (parallelogram law). Two forces applied at one point of a body have as their resultant a force applied at the same point and represented by the diagonal of a parallelogram constructed with the two given forces as its sides.

• Axiom 4 (principle action reaction). of and The forces of action and reaction existing between contacting bodies are equal in magnitude and act along the same line in opposite directions

• Axiom 5 (principle of (принцип отвердевания) solidification) If a deformable body is in the state of static equilibrium, it would also be in static equilibrium if the body were rigid.

Three nonparallel forces theorem If under the action of three coplanar forces a body is in the state of equilibrium therefore lines of action of these forces are intersected at a common point, i. e. the force system is concurrent.