PART I. KINEMATICS.--PRELIMINARY ELEMENTARY MOVEMENTS OF INVARIABLE
POINTS AND SYSTEMS.
LESSONS 1-2.
Object of kinematics, under the geometrical and experimental point of view. Its principal divisions.
Re-statement of the notions relative to the motion of a point, its geometrical representation, and more especially the determination of its velocity.
_Simultaneous Velocities of a Point and the Increments of its Velocities._
Ratio of the elementary displacement and the velocity of a point to the displacement, and velocity of its projection upon a straight line or plane. Use of infinitesimals to determine these ratios. Example:--Oscillatory motion of the projection upon a fixed axis of a point moving uniformly upon the circumference of a circle.
Analogous considerations for polar co-ordinates. Relations of the velocity of a point, of its velocity of revolution and its angular velocity about a fixed pole; of its velocity in the direction of the radius vector; of the velocity of increase of the area which this radius describes.
_Simple Motions of Solids, or Rigid Systems._
1. Motion of rectilinear or curvilinear translation; simultaneous displacements, and velocities of its different points.
2. Motion of rotation about a fixed axis; relation of the velocities of different points to the angular velocity.
Geometrical notions and theorems relative to the _instantaneous center_ of rotation of a body of invariable figure and movable in one plane, or to the _instantaneous axis_ of rotation of a rigid system situated in space, and movable parallel to a fixed plane. Relation of the velocities of different points to their common angular velocity. Use of the instantaneous center of rotation for tracing tangents; examples--and amongst others--that of the plane curve described by a point in a straight line of given length, whose extremities slide upon two fixed lines. Rolling of a curve upon another fixed curve in a plane. Descartes’ theorems upon the intersection of the normals at the successive points of contact: cycloids, epicycloids, involutes, and evolutes. Extension of the preceding motions to the instantaneous axis of rotation of a rigid system movable about a fixed point.
COMPOSITION OF MOTIONS.
LESSONS 3-6. _Composition of the Velocities of a Point._
Polygon of velocities. Example of movements observed relatively to the earth. Particular cases; composition of velocities taken along three axes; composition of the velocity of a point round a fixed pole, and its velocity along the radius vector. Method of Roberval for tracing tangents.
_Composition of the Simple Motions of a Solid System._
Composition of any number of translatory displacements of a solid. Composition of two rotations about two intersecting axes. Composition of any number of rotations about axes cutting one another at the same point; parallelopiped and polygon of rotations. Composition of two simultaneous rotations about parallel axes; case where the rotations are equal and of opposite kinds. Decomposition of a rotation about an axis into an equal rotation about any axis whatever parallel to the first, and a translation perpendicular to the direction of this axis. Direct and geometrical decomposition of the most general motions of a body into a rotation about, and a translation along, an axis called the _instantaneous axis_. Composition of any two motions whatever. Every movement of an invariable system is at each instant of time decomposable into three movements of rotation, and three movements of translation with respect to three axes, which are neither parallel nor lying in the same plane, but otherwise arbitrarily chosen.
_Relative or Apparent Motions._
Relative motion of two points whose absolute motions are given graphically _à priori_. Trajectory of the relative motions, relative velocities, and displacements upon curves or upon the direction of the mutual distance of the two points; use of the parallelogram to determine its amount. Relative motion of a point in motion in respect of a body turning about a fixed axis; relative motion of two bodies which turn about parallel or converging axes, and in general of two rigid bodies or systems impelled by any motions whatever. How this problem is immediately reduced to that of the composition of given motions.
The most general continued motion of an invariable figure in a plane is an _epicycloidal_ motion, in which the instantaneous center describes a curve fixed in relation to absolute space, and traces relatively to the proposed figure a movable curve, which is rigidly connected with that figure and draws it along with it in its motion of rolling upon the other fixed curve. Case of space or spherical figures.
ON THE ACCELERATED MOTION OF A POINT.
LESSONS 7-9. _Accelerated Rectilinear Motion._
Re-statement of the motions acquired relatively to the acceleration in the variable rectilinear motion of a point. Brief indication of the solution of six problems arising out of the investigation of the laws of the motion in terms of the space, time, velocity, and accelerating force. For the most part these solutions may be brought to depend on exact or approximate quadratures. Numerical exercises.
_Accelerated Curvilinear Motions._
Re-statement of the notions acquired relative to the composition of accelerating forces; the resulting acceleration, the normal and tangential acceleration animating a point in motion on a curve. The total acceleration of a point upon an axis or plane is the projection upon this axis or plane of the acceleration of the moving body in space. In uniform curvilinear motion the total or resultant acceleration becomes normal to the curve. Particular case of the circle; value of the normal acceleration in terms of the velocity of revolution or the angular velocity of the radius vector. Case of any curve whatever; geometrical expression of the total or resultant acceleration.
_Accelerated Compound and Relative Motions._
Geometrical investigation of the simple and compound accelerations arising out of the hypothesis in which the motion of any system of points whatever is referred to another system of invariable form, but also in motion. Geometrical and elementary explanations of the results obtained by means of the transformation of co-ordinates.
_Examples or Exercises chosen from among the following Questions:_--
Projection of circular and uniform motion upon a fixed straight line or plane; motion of a circle which rolls uniformly on a straight line; comparison of the motions of the planets relatively to each other, treating them as circular and uniform: comparison of the accelerating force on the moon with that of bodies which fall to the earth.
GEOMETRICAL THEORY AND APPLICATION OF MECHANISMS OR CONTRIVANCES FOR THE TRANSFORMATION OF MOTION.
LESSONS 10-19.
Succinct notions on the classification of elementary motions and organs for transmission of motion in machines after Monge and Hachette, Lanz and Bétancourt.
The most essential details upon this subject are set forth in the following order, and made clear by outline drawings previously distributed among the pupils.
_Organs fitted to regulate the direction of the circular or rectilinear motion of certain pieces_.
Axle; trunnions, gudgeons; pivots and bearings; couplings of axes; adjustment of wheels and of their arms. Joints with hinges, &c.; sheaves and pulleys; chains, ropes, and straps; means of securing them to the necks. Grooves and tongue-pieces. Eyelet-holes sliding along rectilinear or curvilinear rods. Advantages and disadvantages of these different systems of guides under the point of view of accuracy.
Rapid indication of some of their applications to drawbridges and to the movable frames or wagons of saw-works and railways.
_Transmission at a Distance of Rectilinear Motion in a determinate Direction and Ratio._
Inclined plane or wedge guiding a vertical rod. Wedge applied to presses. Rods, winch-handles, &c. Disposition of drums or pulleys in the same plane or in different planes; geometrical problem on this subject. Fixed and movable pulleys. Blocks to pulleys. Simple and differential wheel and axle moved by cords. Transmission through a liquid. Ratios of velocities in these different organs.
_Direct Transformation of circular progressive motion into progressive and intermittent rectilinear motion._
Rod conducted between guides: 1º, by the simple contact of a wheel; 2º, by cross-straps or chains; 3º, by a projecting cam; 4º, by means of a helicoidal groove set upon the cylindrical axis of the wheel. To-and-fro movement, and heart-shaped or continuous cam, waves, and eccentrics. Simple screw and nut. Left and right handed screws; differential screw of Prony, called the micrometric screw. Ratio of the velocities in these different organs.
The example of the cam and pile-driver will be particularly insisted upon; 1º, in the case where this cam and the extremity of the rod have any continuous form given by a simple geometrical drawing; 2º, in the case where this form is defined geometrically by the condition, that the velocity is to be transmitted in an invariable ratio, as takes place for cams in the form of epicycloids or involutes of circles.
_Transformation of a circular progressive motion into another similar to the first._
1º, by contact of cylinders or cones, the two axes being situated in the same plane; 2º, by straps, cords, or endless chains, the axes being in the same situation; 3º, by cams, teeth, and grooves, at very slight intervals; 4º, by the Dutch or universal joint. Case, where the axes are not situated in the same plane; use of an intermediate axis with beveled wheels or a train of pulleys; idea of White or Hooke’s joint in its improved form. Endless screw specially employed in the case of two axes at right angles to one another. Combinations or groupings of wheels. Idea of differential wheels. Relations of velocities in the most important of these systems of transmission.
_Transformation of circular progressive Motion into rectilinear or alternating circular motion._
Ordinary circular eccentric. Eccentrics with closed waves or cams. Examples and graphical exercises in the class-rooms relative to the alternate action of the traveling frames of saw-mills, of the slides or entrance valves of steam-engines. Cams for working hammers and bellows.
_Transformation of alternating circular motion into alternating rectilinear motion, or into intermittent and progressive circular motion._
Pump rods with or without circular sectors, &c. Examples taken from large exhausting pumps, fire-engines, and common pumps. Suggestions as to the best arrangement of the parts. Lagarousse’s lever, &c. Application of the principle relative to the instantaneous center of rotation to give the relations of the velocities in certain simple cases.
_Transformation of alternating circular or rectilinear motions into progressive circular motion._
The knife-grinder’s treadle. System of great machines worked with connecting rods, fly-wheel, &c. Watt’s parallelogram, and the simplest modifications of it for steamboats, for instance. The most favorable proportions for avoiding the deviation of piston-rods. Simplification of parts in the modern steam-engines of Maudsley, Cavé, &c. Variable ratios of the velocities.
_Of organs for effecting a sudden change of motion._
Suspendors or moderators, &c. Dead wheels and pulleys, &c. Mechanisms for stretching cords or straps, and make them change pulleys during the motion. Brakes to windmills, carriages, &c., &c. Case where the axes are rendered movable. Means for changing the directions and velocity of the motions. Coupled and alternate pulleys; alternate cones; castors moving by friction and rotation upon a plate or turning-cone; eccentric and orrery wheels. Means of changing the motion suddenly and by intervals; wheels with a detent pile-drivers; Dobo’s escapement for diminishing the shock, &c.
_Geometrical Drawing of Wheel-work._
General condition which the teeth of toothed wheels must satisfy. Consequence resulting from this for the determination of the form of the teeth of one of two wheels, when the form of the teeth of the other wheel is given.
_Cylindrical action of toothed wheels_ or toothed wheels with parallel axes. External engagement of the teeth; internal engagement. Particular systems of toothed wheels; lantern wheels, flange wheels, involutes of circles. Reciprocity of action; case where the action can not be rendered reciprocal. Pothook action. Details as to the form and dimensions given in practice to the teeth and the spaces which separate them.
_Conical action of toothed wheels_, or toothed wheels with converging axes. Practical approximate method of reducing the construction of a conical to that of a cylindrical engagement of toothed wheels.
_Means of observation and apparatus proper for discovering experimentally the law of any given movement._
Simple methods practiced by Galileo and Coulomb in their experiments relative to the inclined plane and the motion of bodies sliding down it. Various means of observing and discovering the law of the translatory and rotatory motion of a body according as the motion is slow or rapid. Determination of the angular velocity, &c. The counter in machines. Apparatus of Mattei and Grobert for assigning the initial velocity of projectiles (musket balls.) Colonel Beaufoy’s pendulum apparatus. Chronometrical apparatus for continuous indications by means of a pencil. Eytelwein’s apparatus with bands, and its simplest modifications. Apparatus with cylinders or revolving disks. Use of the tuning-fork for measuring with precision very small fractions of time.
(The principal sorts of the apparatus above described are made to act under the eyes of the pupils.)