Unit 9.

Plane Motion of a Rigid Body.

Equations of Plane Motion. Resolution of Motion Into Translation and Rotation.

Learn the following words and word combinations by heart:

assume arbitrary line connecting rod be sufficient belong to	допускать; полагать; представить; пусть; произвольно выбранная, произвольно  проведённая  линия соединительная тяга; соединительная штанга; шатун быть достаточным принадлежать …
case general ~	обычный случай; пример; факт общий случай
special ~	особый случай
hence	следовательно, в результате; с этих пор
instead of	вместо (чего-либо)
move in the same way	перемещаться точно так же, таким же образом
otherwise	иначе, иным образом, иным способом
plane motion	движение в плоскости, двухмерное движение
pole reciprocating engine resolution of sth into…	полюс поршневой двигатель разложение (вектора); разложение (на составляющие); разложение (сил);
successive	последующий; следующий один за другим; последовательный
whatever	какой бы ни, любой

Plane motion of a rigid body is such motion in which all the points of a body move parallel to a fixed plane P (Fig. 14). Many machine parts have plane motion, for example, a wheel running on a straight track or the connecting rod of a reciprocating engine. Rotation is, in fact, a special case of plane motion.

Let us consider the section S of a body produced by passing any plane Oxy parallel to a fixed plane P (see Fig. 14). All the points of the body belonging to line MM' normal to plane P move in the same way.

Fig. 14 Fig. 15

Therefore, in investigating plane motion it is sufficient to investigate the motion of section S of that body in the plane Oxy. Here we shall always take the plane Oxy parallel to the page and represent a body by its section S.

The position of section S in plane Oxy is completely specified by the position of any line AB in this section (Fig. 15). The position of the line AB may be specified by the coordinates xA and yA of point A and the angle φ between an arbitrary line AB in section S and axis x.

The point A chosen to define the position of section S is called the pole. As the body moves, the quantities x_A, y_A, and φ will change and the motion of the body, i.e., its position in space at any moment of time, will be completely specified if we know

x_A = f1 (t), y_A = f2 (t), f = f3 (t). (48)

Eqs. (48) are the equations of plane motion of a rigid body. Let us show that plane motion is a combination of translation and rotation. Consider the successive positions I and II of the section S of a moving body at instants t₁, and t₂ = t₁ +∆t (Fig. 16). It will be observed that the following method can be employed to move section S, and with it the whole body, from position I to position II.

Fig. 16

Let us first translate the body so that pole A occupies position (line A₁B₁, occupies position A₂B₁') and than turn the section about pole A₂, through angle ∆φ₁. In the same way we can move the body from position II to some new position III, etc. We conclude that the plane motion of a rigid body is a combination of a translation, in which all the points move in the same way as the pole A, and of a rotation about that pole.

The translational component of plane motion can, evidently, be described by the first two of Eqs. (48), and the rotational component by the third.

The principal kinematics characteristics of this type of motion are the velocity and acceleration of translation, each equal to the velocity and acceleration of the pole ( ) and the angular velocity and angular acceleration of the rotation about the pole. The values of these characteristics can be found for any time t from Eqs. (48).

In analyzing plane motion, we are free to choose any point of the body as the pole. Let us consider a point C as a pole instead of A and determine the position of the line CD making an angle φ₁ with axis x (Fig. 17). The characteristics of the translatory component of the motion would have been different, for in the general case and (otherwise the motion would be that of pure translation). The characteristics of the rotational component of the motion remain, however, the same. For, drawing CB, parallel to AB, we find that at any instant of time angle φ₁ = φ –a, where a = const. Hence,

, ,

This result can also be obtained from an examination of Fig. 16: whatever point is taken as the pole, to carry section S from position I to position II line A1B1 must be made parallel to A2B2, i.e., the section must be rotated around any pole through the same angle ∆φ₁, equal to the angle between the two lines. Hence, the rotational component of motion does not depend on the position of the pole.

Fig. 17

Comprehension check.

Ex.1. Answer the following questions .

1. How can we define plane motion?

2. What is the position of section S in plane Oxy specified by?

3. What point is called the pole?

4. How can we prove the plane motion is a combination of translation and rotation?

5. What are the principal kinematics characteristics of plane motion?

Ex.2. Put the questions to the following answers.

1. Any point of the body.

2. They would be different.

3. They remain the same.

4. The rotational component of motion.

Ex.3. Complete the following sentences with the information from the text.

1. Rotation is, in fact, a special case of ……...

2. Therefore, in investigating plane motion it is sufficient to investigate …….

3. The position of section S in plane is completely specified…….

4. The pole is the point…….

5. Plane motion is a combination of……..

6. The principal kinematics characteristics of plane motion……….

Ex.4. Fill in the following verbs in the appropriate form:

Obtain, rotate , consider, make, carry, form.

1. Let us …….. a point c as a pole.

2. The line CD……… an angle with this axis.

3. We can ……… this result from this equation.

4. We need to ……….section S from position I to position II.

5. Line AB must be …….parallel to line A₁B₁.

6. The section must be …….around any pole through the same angle.

Ex.5. Use the correct form of the words in brackets.

Problems of kinematics of rigid bodies are ……..(base) of two types: (1) ………(define) of the……….. (move) and ………(analyze) of the kinematics …….(characterize) of the ……….. (move) of a body as a whole; (2) ………(analyze) of the……….. (move) of every point of the body in particular. We shall begin with the ………(consider) of the ……….. (move) of …….(translate) of a rigid body.

…….(translate) of a rigid body is such a ……….. (move) in which any straight line through the body remains ……..(continue) parallel to itself.

…….(translate) should not be confused with rectilinear ……….. (move) . In …….(translate) the particles of a body may move on any ……(curve) paths.

Ex.6. Match the adjectives in the left column with the nouns in the right one, nouns can be used more than once.

	Adjectives		Nouns
1	arbitrary	a	position
2	different	b	way
3	following	c	motion
4	general	d	body
5	in the same	e	component
6	moving	f	translation
7	new	g	point
8	plane	h	case
9	pure	i	characteristics
10	rigid	j	method
11	rotational
12	special
13	successive
14	translational
15	whole

Ex.7. For the words in Part 1 find the synonyms in Part 2 and antonyms in Part 3. There can be more than one synonym or antonym to the given words.

Part 1. Words. Rigid , rotational, straight, special, fixed, same, arbitrary.

Part 2. Synonyms. Unusual, inflexible, unique, unchanging, similar, random, revolving, stiff, turning around, direct, illogical, particular, rigid, identical.

Part 3.Antonyms. Logical, flexible, average, easily bent, loose, common, curve, ordinary, variable, different, linear, usual.