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By Alan Emmerson APPENDIX C GEOMETRY OF THE VERGE AND CROWN WHEEL ESCAPEMENT Xd Datum d r h R r Xr Direction of Motion xd xr Figure C1 Figure C2 Escapement at the Instant of Release. The diagrams show, in elevation and plan, an advancing tooth on one side of the crown wheel (nearer to the reader) about to lift the verge pallet clear of the tooth tip and the corresponding tooth on the other side of the wheel about to drop onto the other pallet. The plane containing the axis of the crown wheel and the axis of the verge is taken as a datum plane. The angle between the pallets is The angular position of the pallet at release is The travel or displacement of the axis of symmetry of the pallets from the datum is . The clearance angle corresponding to the drop is The lateral displacement of the tip of the tooth at release is xr The lateral displacement of the tip of the tooth at drop is xd The crown wheel has N teeth We wish to know the angle of swing at release in terms of the design parameters , h, r and h x From Figure C1 : , cos 1 , and sin 1 r 2 r r h cos 1 C1 2 r 2 There is however a constraint on the free choice of , h, and r . There is a requirement that the angle of drop must be positive as shown. From Figure C1 : , (that is ,) 2 2 xd and tan 1 h xd Whence tan 1 and xd h tan C2 2 h 2 In the symmetrical layout usually adopted, the crown wheel must have an odd number of teeth. Thus the angular distance between the tip of a tooth and the tip of the tooth closest to diametrically opposite is ½tooth pitch. Consequently, in Figure C2, 1 r pitch d 2 ie r d N xr r sin 1 R x Also from Figure C2, d sin 1 d R x x sin 1 r sin 1 d R R N Thus for N15 the angles are small so that : xr xd R R N xd ie: x r R( ) N R h Substituting from C2 gives xr R tan N R 2 We may reasonably anticipate that is small so that: 2 h xr R N R 2 xr R h But sin and thus sin r r N R 2 R h or sin r N r2 h R h h that is sin r r N r2 r R r Whence sin C3 hN 2 h h r h R But cos 1 so that sin cos 1 C4 r 2 h r hN r h R Thus the requirement is: sin cos 1 and 0 C5 2 h r hN The requirement 0 can be satisfied by pre setting a value for drop. The practicalities of construction suggest that drop should be expressed as a fraction of the crown wheel tooth pitch. If D is the angular drop RD of the crown wheel, r RD , for small , so that r 2 So that Dk N RD R 2 And thus k r r N r h R R 2 k Substituting in C5 gives sin cos 1 C6 2 h r hN r N Transforming and collecting terms in C6 gives r h R R sincos1 2 k C7 2 h r Nh r h But cos 1 C1 r 2 There are two approaches to the design of the escapement. h r h R R Adding C1 and C7 yields 2 cos1 sincos1 2 k r h r Nh r h and this prescribes cos 1 2 r Alternatively: h r h R R Subtracting C1 from C7 yields 0 cos1 sincos1 2 k r h r Nh r h r h R R ie cos1 sincos1 2k r h r Nh r h resulting in cos 1 r 2 These are the characteristic equations for the escapement. They are accurate only when N 15 and h is small . The design criteria for the escapement include 1 or the crown wheel will run free. 2 r Table C1 shows the values of the angle of swing to release for various pallet angles using practicable values for the other escapement parameters. Pallets Included Angle of Swing at Release Angle degrees degrees N=15 N=29 r r 70 3.0 40 1.7 29 80 3.1 36 18 26 90 3.3 32 1.9 22 100 3.4 27 2.0 18 110 3.5 23 2.2 15 For: R=15mm, h=0.75mm and k=0.1 Table C1 Verge Escapement , Angle of Swing at Release