I.5 Contour Equations 0Contents5 Contour Equations 15.1 Contour Velocity Equations . . . . . . . . . . . . . . . . . . . 25.2 Contour Acceleration Equations . . . . . . . . . . . . . . . . . 45.3 Independent Contour Equations . . . . . . . . . . . . . . . . . 75.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17I.5 Contour Equations 15 Contour EquationsThis section aims at providing an algebraic method to compute the veloci-ties and accelerations of any closed kinematic chain. The classical methodfor obtaining the velocities and accelerations involves the computation of thederivative with respect to time of the position vectors. The method of con-tour equations avoids this task and utilizes only algebraic equations [4, 55].Using this approach, a numerical implementation is much more efficient. Themethod described here can be applied to planar and spatial mechanisms.Two rigid links (j) and (k) are connected by a joint (kinematic pair) at A,Fig. 5.1. The point Ajof the rigid body (j) is guided along a path prescribedin the bo dy (k). The points Ajbelonging to body (j) and the Akbelongingto body (k) are coincident at the instant of motion under consideration. Thefollowing relation exists between the velocity vAjof the point Ajand thevelocity vAkof the point AkvAj= vAk+ vrAjk, (5.1)where vrAjk= vrAjAkindicates the velocity of Ajas seen by an observer atAkattached to body k or the relative velocity of Ajwith respect to Ak,allowed at the joint A. The direction of vrAjkis obviously tangent to the pathprescribed in body (k).From Eq. (5.1) the accelerations of Ajand Akare expressed asaAj= aAk+ arAjk+ acAjk, (5.2)where acAjk= acAjAkis known as the Coriolis acceleration and is given byacAjk= 2 ωk× vrAjk, (5.3)where ωkis the angular velocity of the body (k).Equations (5.1) and (5.2) are useful even for coincident points belonging totwo links that may not be directly connected. A graphical representation ofEq. (5.1) is shown in Fig. 5.1(b) for a rotating slider joint.Figure 5.2 shows a monocontour closed kinematic chain with n rigid links.The joint Ai, i = 0, 1, 2, ..., n is the connection between the links (i) and(i − 1). The last link n is connected with the first link 0 of the chain. Forthe closed kinematic chain, a path is chosen from link 0 to link n. At thejoint Aithere are two instantaneously coincident points: 1) the point Ai,iI.5 Contour Equations 2belonging to link (i), Ai,i∈ (i), and 2) the point Ai,i−1belonging to body(i − 1), Ai,i−1∈ (i − 1).5.1 Contour Velocity EquationsThe absolute angular velocity, ωi= ωi,0, of the rigid body (i), or the angularvelocity of the rigid body (i) with respect to the ‘fixed” reference frame Oxyzisωi= ωi−1+ ωi,i−1, (5.4)where ωi−1= ωi−1,0is the absolute angular velocity of the rigid body (i − 1)(or the angular velocity of the rigid body (i − 1) with respect to the ‘fixed”reference frame Ox yz) and ωi,i−1is the relative angular velocity of the rigidbody (i) with respect to the rigid body (i − 1).For the n link closed kinematic chain the following expressions are ob-tained for the angular velocitiesω1= ω0+ ω1,0ω2= ω1+ ω2,1.................................ωi= ωi−1+ ωi,i−1.................................ω0= ωn+ ω0,n. (5.5)Summing the expressions given in Eq. (5.5), the following relation is obtainedω1,0+ ω2,1+ ... + ω0,n= 0, (5.6)which may be rewritten asX(i)ωi,i−1= 0. (5.7)Equation (5.7) represents the first vectorial equation for the angular velocitiesof a simple closed kinematic chain.The following relation exists between the velocity vAi,iof the point Ai,iand the velocity vAi,i−1of the point Ai,i−1vAi,i= vAi,i−1+ vrAi,i−1, (5.8)I.5 Contour Equations 3where vrAi,i−1= vrAi,iAi,i−1is the relative velocity of Ai,ion link (i) with respectto Ai,i−1on link (i − 1). Using the velocity relation for two particles on therigid body (i) the following relation existsvAi+1,i= vAi,i+ ωi× rAiAi+1, (5.9)where ωiis the absolute angular velocity of the link (i) in the reference frameOxyz, and rAiAi+1is the distance vector from Aito Ai+1. Using Eqs. (5.8)and (5.9) the velocity of the point Ai+1,i∈ (i + 1) is written asvAi+1,i= vAi,i−1+ ωi× rAiAi+1+ vrAi,i−1. (5.10)For the n link closed kinematic chain the following expressions are obtainedvA3,2= vA2,1+ ω2× rA2A3+ vrA2,1vA4,3= vA3,2+ ω3× rA3A4+ vrA3,2............................................................vAi+1,i= vAi,i−1+ ωi× rAiAi+1+ vrAi,i−1............................................................vA1,0= vA0,n+ ω0× rA0A1+ vrA0,nvA2,1= vA1,0+ ω1× rA1A2+ vrA1,0. (5.11)Summing the relations in Eq. (5.11)hω1× rA1A2+ ω2× rA2A3+ ... + ωi× rAiAi+1+ ... + ω0× rA0A1i+hvrA2,1+ vrA3,2+ ... + vrAi,i−1+ ... + vrA0,n+ vrA1,0i= 0. (5.12)Because the reference system Oxyz is considered “fixed”, the vector rAi−1Aiis written in terms of the position vectors of the points Ai−1and AirAi−1Ai= rAi− rAi−1, (5.13)where rAi= rOAiand rAi−1= rOAi−1. Equation (5.12) becomes[rA1× (ω1− ω0) + rA2× (ω2− ω1) + ... + rA0× (ω0− ωn)] +hvrA1,0+ vrA2,1+ ... + vrAi,i−1+ ... + vrA0,ni= 0. (5.14)Using Eq. (5.5), Eq. (5.14) becomes[rA1× ω1,0+ rA2× ω2,1+ ... + rA0× ω0,n] +hvrA1,0+ vrA2,1+ ... + vrA0,ni= 0. (5.15)I.5 Contour Equations 4The previous equation is written asX(i)rAi× ωi,i−1+X(i)vrAi,i−1= 0. (5.16)Equation (5.16) represents the second vectorial equation for the angular ve-locities of a simple closed kinematic chain.EquationsX(i)ωi,i−1= 0 andX(i)rAi× ωi,i−1+X(i)vrAi,i−1= 0, (5.17)represent the velocity equations for a simple closed kinematic chain.5.2 Contour Acceleration E quationsThe absolute angular acceleration, αi= αi,0, of the rigid body (i) (or theangular acceleration of the rigid body (i) with respect to the ‘fixed” referenceframe Oxyz) isαi= αi−1+ αi,i−1+ ωi× ωi,i−1, (5.18)where αi−1= αi−1,0is the absolute angular acceleration of the rigid body(i−1) (or the angular acceleration of the rigid body (i−1) with respect to the‘fixed” reference frame Oxyz) and αi,i−1is the relative angular accelerationof the rigid body (i) with respect to the rigid body (i − 1).For the n link closed kinematic chain the following expressions are ob-tained for the angular accelerationsα2= α1+ α2,1+ ω2× ω2,1α3= α2+ α3,2+ ω3×