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Spring

plate, elastic, angle, force, power, weight and direction

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SPRING, in Mechanics, is an elastic plate or rod, which is em ployed as a moving-power, or a regulator of the motions of wheel work ; also to ascertain the weights of bodies, or to diminish the effects of concussion.

The elder Bernoulli was the first whose attention was directed to the curvature assumed by elastic bodies, and he succeeded in resolving the problem in the case of a rectilinear plate being fixed at one end and Lent by a weight applied at the other : it being assumed as a principle that, at any point in an elastic body, the force by which the body when bent by any power endeavours to recover its previous position is pro portional to the angle of contact at that point ; that is, to the angle which a tangent to the curve surface of the body makes, at the point, with that surface.

In order to give some idea of the manner in which the effects of elasticity are to be determined mathematically, let A B be a thin elastic plate immovably fixed at A, and bent into the form AB' by a power r applied at a, sod let a, a, a a' be two consecutive elementary portions of the bent plate : let also z represent the unknown force of elasticity acting perpendicularly to ea' by which that element tends to recover the direction a,a, from which it has been made to deviate by the power at B; and for simplicity let it be supposed that this power acts in the direction parallel to A B. Let fall ap perpendicularly on lee; also represent s'p by x and ap by y. Then, by mechanics P. y expresses the momentum of the power at a' to bend the plate at a, and in the case of equilibrium we have r.y =E. But E varies with the angle of contact, or the angle between the element aa' and pro duced, and that angle in any part of A B' varies inversely with the radius of curvature at that part; therefore let r be the known radius of curvature at a point where the force of elasticity is given, and let this force bo represented by e: also let it be the radius of enr vature at a point, as a, where the force is represented by E. Then 1 1 er • e : e : : : E, or = E, and pitting E' for er, we have r.y Substituting in place of it the differential expression for the radius of curvature, the elastic force might be obtained by the processes of integration. The integral, however, can only be obtained approxi

mately.

If the elastic plate were in a vertical position with its lower end A resting on an immovable object, and a weight r, applied at the upper extremity n', were to act in a direction towards A, the equations of the curve, approximately determined, are (the deflection being small) P )4 r ) y= A sin x and L = (I — 4E' where x is any abscissa from n' on the line n'A, y is the corresponding ordinate, is the greatest deflection, or the ordinate at the middle point in B'A, k = n A, and L is the length of the curve line Aalf : also the greatest weight which the plate or spring will bear without bend ing when pressed in that direction is expressed --. (I= 3.1416). Ifs exceed by a small quantity the value of where in is any whole number whatever, the spring will make several bends crossing the straight line A a' in in — I points between the two ex tremities.

When the elastic plate, in a horizontal position, is fixed at one end, as A, and the weight r, applied at the extremity a' acts always vertically, the equation becomes (the deflection being y = 7, I ( 1 PX• t ex where c is the tangent of the angle of contact at n'. If, on account of the smallness of this angle, c be neglected, it will follow, when x is made equal to V°, that the whole deflection A o or B B' will vary nearly with the weight r, and with the cube of the length of the spring.

The use of a spring as a moving-power may be best exemplified in its application to watch-work. The main-spring of a watch is a thin and narrow plate of well-tempered steel, which is coiled in a spiral form : one of its extremities is attached to a pivot or axle, and the ri other to the interior circumference of the cylindrical box in which it is contained. In being wound, the spring closes round the axle, and afterwards, in the effort by its elasticity to recover its former position, it turns the cylinder in a contrary direction : thus the chain which is attached to the exterior circumference of the cylinder and to the fusee causes the latter to revolve.

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