KINEMATICAL MODELS OF THE LOCK MOTIONS

In this study, mathematical modelling of lock and key mechanisms is focused and the kinds of motions of the structures are studied. The basic process principle of the lock and key mechanisms mathematically modelled in mechanic and kinematic are brought up. In this modelling, it is shown mutually as the movement the moving part produces or the movement the part setting into motion (K;A). 1. INTRODUCTION 1.1. Rotation. If X = (x; y) are the coordinates of a point P 2 R in the moving body measured in the coordinate frame M , then the coordinates of P measured in coordinate frame F can be given by D : F !M: This transformation is given by ~ X = [A]~x+ d; where ~x is the coordinate vector of a point in M and ~ X is the coordinate vector of the same point but measured in F . If the dimension of the moving body is n ( usually n = 2 or 3 ), then [A] is an n n matrix and d is n-dimensional vector. Let n = 2 , so [A] = cos sin sin cos ; d = ><>: d1 d 2 >=>; : The pair (A; d) denes this transformation, and is called a planar displacement [2]. The matrix [A] has [A] as its inverse, therefore it is an orthogonal matrix; and, because its determinant is 1, it denes a rotation. It is interesting to examine the 2 2 orthogonal matrices that are not rotations [2]. Received by the editors Oct. 14, 2009, Accepted: May. 25, 2010. 2000 Mathematics Subject Classication. 70B15. Key words and phrases. Lock, key, motion, kinematic, rotation, transition. c 2010 Ankara University 25 26 SEMRA SARAÇO 1⁄4 GLU AND BÜLENT KARAKAŞ Let R (0) be the degree rotation leaving the O point xed. Then the rotation equations for n=2 can be given by x0 = x cos + y sin y0 = x sin + y cos x0 y0 = cos sin sin cos x y In brief, it can be written as ~ X = R ~x where R matrix is a matrix having the R = R 1 property and R is an orthogonal matrix. Figure 1. The plane evident with X2; X3. The dx line evident with ~ X vector is called the rotation axis of the rotation R. Denition 1.1. Let f : I R ! R show a given function. The matrix A determined by [A] = cos(( 1)[jf(x)j]: (x)) sin (( 1)[jf(x)j]: (x)) sin (( 1)[jf(x)j]: (x)) cos(( 1)[jf(x)j]: (x)) is called code matrix. KINEMATICAL MODELS OF THE LOCK MOTIONS 27 Example 1.2. Any rotation series for ve handled can be shown by identied code matrix as: If f : R! R, let f(x) = 8><>>>>: 1; x = 0 2; x = 1 2 3; x = 1 4; x = 2 1; x = 14 and then, [A] = cos(( 1)([jf(x)j]:2 x) sin (( 1)[jf(x)j]:2 x) sin (( 1)[jf(x)j]:2 x) cos(( 1)[jf(x)j]:2 x) [A1] = cos 0 sin 0 sin 0 cos 0 = 1 0 0 1 [A2] = cos sin sin cos = 1 0 0 1 [A3] = cos ( 2 ) sin ( 2 ) sin( 2 ) cos ( 2 ) = 1 0 0 1 [A4] = cos (4 ) sin (4 ) sin(4 ) cos (4 ) = 1 0 0 1 [A5] = cos ( =2) sin ( =2) sin( =2) cos ( =2) = 0 1 1 0 is obtained. 2. THE MATHEMATICAL MODELLING OF THE LOCK MOTIONS In this chapter, rst, the lock mechanisms will be categorized by being studied in terms of motion kinds. There are basically two motions fulllling the locking process. These are the motions that the moving part does and that put into motion does.Accordingly, the lock and key duet makes up a mechaism by moving bound to each aother. The mutual motion of this duet is kinematically based on the rotational and transitional motions. As the combination of rotations and translations, it is seen that di¤erent kinds of motions appear. Accordingly, if the key and lock duet (K;A) is expressed, it is understood that (K;A) : (H1;H2) H1 : The motion that the tumbler in the lock does H2: The motion that the mechanism ensuring the tumbler to move 28 SEMRA SARAÇO 1⁄4 GLU AND BÜLENT KARAKAŞ Here H1 and H2 are combination of one or more of the motions of T : transition, R: rotation, RT : rotational transition, DH: degenerate motion, V : screw motion. According to all those, the lock mechanisms can be grouped like the following in point of kinematic view with 6 di¤erent motion kinds: 1. (K;A) : (T; T ) Lock motion, 2. (K;A) : (R;R) Lock motion, 3. (K;A) : (T;RT ) Lock motion, 4. (K;A) : (R; V ) Lock motion, 5. (K;A) = (DH;T ) Lock motion, 6. ((K1;K2;K3;K4); A) : ((T1; T2; T3; T4); R) Lock motion. 7. Multi and total motion of the lock mechan 3. THE STUDY OF THE KINEMATICAL MODELS OF THE LOCK MOTIONS 3.1. (K;A) : (T; T ) Lock motion. The lock is the one having the simplest structure in the lock mechanism motion kinds. In this kind of motion, both the part taking over the locking function and the motion applied on this part are directional motion. When whole of the mechanism is regarded, it is observed that the motion appeared is a transitional motion. The model of the concerning mechanism is,


INTRODUCTION
1.1.Rotation.If X = (x; y) are the coordinates of a point P 2 R 2 in the moving body measured in the coordinate frame M , then the coordinates of P measured in coordinate frame F can be given by D : F ! M: This transformation is given by where x is the coordinate vector of a point in M and X is the coordinate vector of the same point but measured in F .If the dimension of the moving body is n ( usually n = 2 or 3 ), then [A] is an n n matrix and d is n-dimensional vector.Let n = 2 , so The pair (A; d) de…nes this transformation, and is called a planar displacement [2].
The matrix [A] has [A] T as its inverse, therefore it is an orthogonal matrix; and, because its determinant is 1, it de…nes a rotation.It is interesting to examine the 2 2 orthogonal matrices that are not rotations [2].
Let R (0) be the degree rotation leaving the O point …xed.Then the rotation equations for n=2 can be given by x 0 = x cos + y sin In brief, it can be written as where R matrix is a matrix having the R T = R 1 property and R is an orthogonal matrix.
Figure 1.The plane evident with X 2 ; X 3 .
The d x line evident with X vector is called the rotation axis of the rotation R. is called code matrix.

THE MATHEMATICAL MODELLING OF THE LOCK MOTIONS
In this chapter, …rst, the lock mechanisms will be categorized by being studied in terms of motion kinds.There are basically two motions full…lling the locking process.These are the motions that the moving part does and that put into motion does.Accordingly, the lock and key duet makes up a mechaism by moving bound to each aother.The mutual motion of this duet is kinematically based on the rotational and transitional motions.As the combination of rotations and translations, it is seen that di¤erent kinds of motions appear.Accordingly, if the key and lock duet (K; A) is expressed, it is understood that The motion that the tumbler in the lock does H 2 : The motion that the mechanism ensuring the tumbler to move Here H 1 and H 2 are combination of one or more of the motions of T : transition, R: rotation, RT : rotational transition, DH: degenerate motion, V : screw motion.According to all those, the lock mechanisms can be grouped like the following in point of kinematic view with 6 di¤erent motion kinds:  Accordingly, the motion that the tumbler brings up is a rotation.The model of this system can be given with The matrix form of the motion with center 0 = (0; 0) is given by The motion belonging to this system is as in Figure 3  Let the rotation axis of the key be as z axis, the rotation point be as the origin point and the rotation plane as x y plane.The closing point of the lock goes from the location A to location B by the e¤ect of the lock.Since the joint will move in a certain slide on the lock, it creates a transition motion.The transition vector is (L; 0).Each point on the key joint has the rotation of (x; y) !(x L; y): Accordingly, the matrix of this motion is as Since the degree is evident, (d 1 ; d 2 ) transition vector must be calculated, x cos + y 0 sin + d 1 = x L sin x sin + y 0 cos + d 2 = y 0 And then, d 1 and d 2 transitions are obtained as, Accordingly, the model is evident as (K; A) : (T; RT ): The key entering the system with a helix motion results in releasing squeezing the string mechanism.Such a motion allows the system to be opened.Therefore, (K; A) : (R; V ) equality can be written for this motion.The piece having the role of a key in this mechanism leaves the pieces hindering the mechanism to be opened inactive by forcing on the free positioned the string.Therefore, the pieces in the mechanism can be separated from each other and the system gets opened.A di¤erent kind of motion which we can also call obstacle canceller, kind of lock shows itself in this lock system.
3.6.((K 1 ; K 2 ; K 3 ; K 4 ); A) : ((T 1 ; T 2 ; T 3 ; T 4 ); R) Lock Motion.It is possible to name this kind of motion as multi motioned-single motioned mechanism.A multi joined lock mechanism with a motion from a single center is simultaneously a mechanism in which a four-directed lock motion is done.Here four systems are put into motion in a single stage with a single lock motion shown in type 3.This system has not only sprang mechanisms but also pinned and obstacled key kinds developed today.The spring in the system allows the mechanism to be opened even without using a lock pick.However, pinned and obstacled keys cancel this problem Rotational transition motion in four di¤erent direction is taken as K 1 ; K 2 ; K 3 ; K 4 .In this case we have ÖZET: Bu çal¬ şmada kilit ve anahtarlar¬n matematik ve kinematik modellemeleri çal¬ ş¬ld¬.Matematik ve kinematik aç¬dan modellemenin temel işlemi öncelikle mekanizmalar¬s¬n¬ ‡and¬rmakt¬r.Modellemede kilit ve anahtarlar (K; A) gösterimi kullan¬larak s¬n¬ ‡and¬r¬ld¬.

1 .
(K; A) : (T; T ) Lock motion, 2. (K; A) : (R; R) Lock motion, 3. (K; A) : (T; RT ) Lock motion, 4. (K; A) : (R; V ) Lock motion, 5. (K; A) = (DH; T ) Lock motion, 6. ((K 1 ; K 2 ; K 3 ; K 4 ); A) : ((T 1 ; T 2 ; T 3 ; T 4 ); R) Lock motion.7. Multi and total motion of the lock mechan 3. THE STUDY OF THE KINEMATICAL MODELS OF THE LOCK MOTIONS 3.1.(K; A) : (T; T ) Lock motion.The lock is the one having the simplest structure in the lock mechanism motion kinds.In this kind of motion, both the part taking over the locking function and the motion applied on this part are directional motion.When whole of the mechanism is regarded, it is observed that the motion appeared is a transitional motion.The model of the concerning mechanism is, (K; A) : (T; T ):

Figure 4
Figure 4 Rotational transition motion system

5 x[y 0 ;Figure 8 .
Figure 7. Rotational transition motion in four di¤erent direction from a single center