Design & Development of Function Generator Linkage for Reclining Chair

Transcription

1 Design & Development of Function Generator Linkage for Reclining Chair Alkesh A. Chaure 1, Dr. Vinod N. Bhaiswra 2, Siddharth K. Undirwade 3, M. K. Sompimple 4 1 Student of M. Tech, Prityadarshini College of Engineering, Nagpur 2,3,4 Assistant Professor, Priyadarshini College of Engineering, Nagpur Abstract This paper considers design of four-barr linkages for reclining chair. The objective of the present invention is to provide a foldable chair that can attract the attention of consumers as well as convenient of use. Concept of combination of two four bar linkages in series is used for functioning back rest and foot rest of chair which generates straight and parallel motion of back and foot rest with the operation of one lever arm only. This mechanism can be utilized for various commercial and official purposes. The advantage of this mechanism is that the leg remains straight and back remain inclined and because of its parallel motion it help in keeping body s stability statically and dynamically. Function Generation method of synthesis is used to find optimal angles and lengths of linkages. In this correlation of input arm (link) with the output link i.e. foot rest and back rest is been done. This work is related to the conventional chair that having linkage mechanism for tilting back rest & foot rest Keywords: Reclining chair, function generation, four- & foot rest bar linkage, linkages synthesis, back rest dimension & inclination, Introduction A recliner is called to be an armchair that having functioned of inclined back rest and re-inclinerest to the original position. Function Generator is back the term applied to the means by which a designer accomplishes the coordination of the motion of an output lever in a machine with the motion of an input lever. An armchair whose back can be lowered and foot can be raised to allow the sitter to recline in it.the footrest that may be extended by means of a lever provided on the side of the chair. A re-inclination of back rest and footrest is also known as a Reclining chair The purpose of this type of chair is to just provide the comfort to the human being in rest. The desired relationship between the link motion is Function which is to be Generated by whatever mechanical system the designer uses to connect the link. The linkages of chair pivoted with the back rest thereby causing the chair from an upright state to collapsible state. The object of the present invention is to provide a foldable hair chair that can attract the attention of consumers as well as convenient of use. The invention comprises of chair frame, seat unit mounted on chair frame & coupling link unit. 1. Concept of function generator in standard chair The lever arm will generatee the function of motion through the linkages that are hinged from the back of chair which gives backward motion to backrest and upward motion to foot rest Fig: 1. Concept of function generation in chair Page 75

2 Fig: 4. Proposed arrangement of linkages in chair 3. Forming Freudenstein' 's equation for three position function generation Considering R Ø & R ψ as the unknown input and output link angle of rotation This equation can be derived from the loop closer (closed loop) equation and in this project all the two four bar link mechanism is in the form of closed loop. Fig: 2. Concept of reclining back rest & foot rest 2. Schematic & position of linkages of Reclining Chair Mechanism Fig: 5. Freudenstein's equation for three position function generation Equn (1) Where Z1, Z2, Z3, Z4 are the length of linkages Ѳ is the input & output link angle Fig: 3. Line diagram of linkages in chair A 0, A, B, B 0 Back Rest Linkage. A 0, A, B B 0, - Foot Rest Linkage This complex equation is then converted into the real components, two algebraic equation are produced, they are Z 1 CosѲ 1 + Z 2 CosѲ 2 + Z 3 CosѲѲ 3 - Z 4 CosѲ 4 = Equn (2) Z 1 SinѲ 1 + Z 2 SinѲ 2 + Z 3 SinѲ3-3 Z 4 SinѲ 4 = Equn (3) Assuming that the ground link is along the X axis, as COS 180 Deg = -1 and SIN 180 deg = 0, Then the function equation will be -Z 1 + Z 2 CosѲ 2 + Z 3 CosѲ 3 - Z 4 CosѲ 4 = Equn (4) Page 76

3 Z 2 SinѲ 2 + Z 3 SinѲ 3 - Z 4 SinѲ 4 = Equn (5) The four bar mechanism after ground link is along the X axis Next, Equn.(6) and (7) are squared and added together to eliminate Ѳ3. The resulting equation is like ((a +b -c) 2 ) Z 3 2 = Z 1 2 +Z 2 2 +Z 4 2-2Z 1 Z 2 cosѳ 2 +2 Z 1 Z 4 cosѳ 4-2 Z 2 Z 4 (cosѳ 2 cosѳ 4 + sinѳ 2 sinѳ 4 ) Equn (8) Since, (cosѳ 2 cosѳ 4 + sinѳ 2 sinѳ 4 ) = cos (Ѳ 2 - Ѳ 4 ) So, Eq. (8) can be rearranged as In a more compact form, Freudenstein s equation will be Fig: 6. Notation for four bar mechanism for forming Freudenstein's equation 3.1 Three position linkage synthesis for function generation = First position = Second position = Third position Where, Z 1 = length of ground link Z 2 = input link Z 3 = Coupler link / Intermediate link Z 4 = output link Ψ 1 = output link angle when input link will rotate with respect to the Ѳ 1 in horizontal plane Ψ 2 = output link angle when input link will rotate with respect to the Ѳ 2 in horizontal plane Ψ 3 = output link angle when input link will rotate with respect to the Ѳ 3 in horizontal plane Since we wish to synthesize a function generator, Ѳ 3 is not of interest and will be eliminated by transferring the Z 3 terms to the right-hand side, and then the function equation will be: -Z 1 + Z 2 cosѳ 2 - Z 4 cos Ѳ 4 = -Z 3 cosѳ Equn (6) Z 2 SinѲ 2 - Z 4 sin Ѳ 4 = - Z 3 sinѳ Equn (7) K 1 cosѳ 2 + K 2 cosѳ 4 + K 3 =- cos(ѳ 2 - Ѳ 4 ) ----Equn 10) Where, ---Equn (11) Notice that the Ks are three independent algebraic expressions containing the three unknown length of the links. Freudenstein s is a displacement equation for the fourbar linkage which holds true for each position of the linkage. Thus, for three prescribed positions, the equation can be written for each position. The notation will be changed at this point to avoid double subscripts: The three angles for the prescribed position of Z 2 with respect to the fixed X axis will be Ѳ 1, Ѳ 2, and Ѳ 3, while those of Z 4 will be ψ 1, ψ 2, and ψ 3 as in above fig. Thus Freudenstein s equation for three prescribed position is K 1 cosѳ 1 + K 2 cos ψ 1 + K 3 = - cos(ѳ 1 - ψ 1 ) K 1 cosѳ 2 + K 2 cos ψ 2 + K 3 = - cos(ѳ 2 - ψ 2 ) Page 77

4 International al Journal of Mechanical Engineering and Computer Applications, Vol 1, Issue 4, K 1 cosѳ 3 + K 2 cos ψ 3 + K 3 = - cos(ѳ 3 - ψ 3 ) ---Equn (12) Cramer rule may be used to solve Eq.(12). To find the Zs, one length say, Z 1 is arbitrarily picked to scale the function generator. Dealing with third order determination may be avoided by first subtracting the second and third equation from the first eliminating K3. K 1 (cosѳ 1 cosѳ 2 ) + K 2 (cos ψ 1 - cos ψ 2 ) = - cos(ѳ 1 - ψ 1 ) + cos(ѳ 2 - ψ 2 ) Equn (13) K 1 (cosѳ 1 cosѳ 3 ) + K 2 (cos ψ 1 - cos ψ 3 ) = - cos(ѳ 1 - ψ 1 ) + cos(ѳ 3 - ψ 3 ) Equn (14) And solving the resulting system of two equation for K 1 and K Equn (15) In which, ω 1 = cos Ѳ 1 cos Ѳ 2, ω 2 = cos ψ 1 - cos ψ 2 ω 3 = - cos (Ѳ 1 - ψ 1 ) + cos(ѳ 2 - ψ 2 ) ω 4 = cos Ѳ 1 cos Ѳ 3 ω 5 = cos ψ 1 - cos ψ 3 ω 6 = - cos (Ѳ 1 - ψ 1 ) + cos(ѳ 3 - ψ 3 ) Now substituting values of K 1 and K 2 into any part of Equn. (12), then K 3 = - cos(ѳ 1 - ψ 1 ) K 1 cosѳ 1 - K 2 cos ψ i i= 1, 2, or 3 The link length may be expressed in term of the known Ks by using Eq.(11) (By assuming the length of any one of Z1): 4. Determination of the dimension of link for Back Rest Mechanism By using above formed Freudenstein s equation to achieve the required function between input & output link in the Interval of 0 X 20. Function equation y = Sin x considering Where x is the input independent variable. y is the back rest output dependent variable. Ѳs is starting angle of the input link (arm). Øs is the starting angle of the output link. Ѳf is final angle of the input link (arm). Øf is the final angle of the output link. For degree of comfertness and standard inclined angle of back rest with respect to the input lever arm inclination was Ѳs = 0 Deg. And Ѳf = 20 Deg. Xs = 0 deg and Xf = 20 deg. X = 20-0 = 20. Where ( X = Xf Xs) Ys = sin Xs, sin0 = 0. Yf = sin Xf, sin 20 = Y =1-0=1. Where ( Y = Yf Ys) According to the chebychev s equation. Xj = (Xs + Xf) / 2 ( X)/2 Cos π ((2j 1) / 2n) For three precision points, n = 3 for j = 1 X1 = ((Xs + Xf) / 2) ( X)/2 Cos π ((2j 1) / 2n) = ((0 +20) / 2) -20/2 Cos π ( (2x 1 1 ) / 2 x 3) = X1 =1.4 Then for Page 78

5 X2 = (Xs + Xf) / 2 ( X)/2 Cos π ( (2j 1 ) / 2n) Here j = 2, & n = 3 X2 = ( (0 + 20) / 2) (20/2) Cos π ( (2x 2 1 ) / 2 x 3) X2 = 10. Then for X3 = (Xs + Xf) / 2 ( X)/2 Cos π ( (2j 1 ) / 2n) Here j = 3, & n = 3 X3 = ( (0 + 20) / 2) (20/2) Cos π ( (2x 3 1 ) / 2 x 3) X3 = X3 = The function Equn is : y1 = Sin x1 y1 = sin 1.4 y1 = 0.98, Similarly y2 = -0.5 & y3 = Now Ѳj = Ѳs + ( Ѳ/ X) (xj xs) Ѳ1 = Ѳs + ( Ѳ/ X) (x1 xs) Ѳ1 = 0 + (20/20) (1.4 0) Ѳ1 = 1.4 deg Similarly Ѳ2 = 0 + (20/20) (10 0) Ѳ2 = 10 deg & Ѳ3 = 0 + (20/20) (18.6 0) Ѳ3 = 18.6 deg Fig: 6. Back rest linkages with Øs as Øj = Øs + ( Ø / Y ) (yj ys) Ø1 = ( 40 / 1 ) ( ) Ø1 = deg Similarly Ø2 = Øs + ( Ø / Y ) (y2 ys) Ø2 = ( 40 / 1 ) ( ) Ø2 = 89 deg & Ø3 = 101. deg Now the Freudenstein s equation for three position synthesis is K 1 + K2 cosѳ 1 + K 3 cos Ø 1 = - cos (Ѳ 1 - Ø 1 ) ---(16) K 1 + K2 cosѳ 2 + K 3 cos Ø 2 = - cos (Ѳ 2 - Ø 2 ) ---(17) K 1 + K2 cosѳ 3 + K 3 cos Ø 3 = - cos (Ѳ 3 - Ø 3 ) ---(18) From Equn (16), (17), (18) By putting the value of Ѳ1, Ѳ2, Ø1 and Ø2 we get K1 = (A) Similarly, K2 = (B) & K3 = (C) Now assuming one of the link of the back rest mechanism, say link 2 i.e. Z2 = 95 mm As we know K3 = (-Z1 / Z2) by putting the values of K3 & Z1 we get Z1 = mm fix point to fix point Similarly, K2 = (Z1 / Z4) by putting the values of K2 & Z1 we get Z4 = mm. K1 Z 2 3 Z 2 1 Z Z 4 = 2 Z 2 Z 4 By putting the values in the above said equation we get Z3 = 550 mm So the overall link dimension get from above equations for the back rest mechanism is, Z1 = mm fit to fit Z2 = 95 mm Page 79

6 Z3 = 550 mm fit to fit coupler link Z4 = mm. Hence with the dimensions of above found linkages, we achieved the following function, Initial position of lever arm (Input Link ) is at the 19 Deg with respect to the vertical axis, in this position the back rest will be at 90 Deg with respect to the horizontal axis. When the lever arm (Input Link) rotates by the 10 Deg anticlockwise the position of the backrest rotates by 20 Deg from vertical axis. Again by the rotation of lever arm (Input Link) rotates by the 10 Deg anticlockwise the position of the backrest rotates by 40 Deg from vertical axis. 5. Determination of the dimension of link for foot Rest Mechanism By using the same procedure as did above for back rest mechanism we get the principal dimension of linkages as Fig: 7. Back rest linkages with Øs as 19 0 Z1 = mm Z2 = 195 mm Z3 = 200 mm Z4 = mm. Hence with the dimensions of above found linkages, we achieved the following function Initial position of lever arm (Input Link) is at the 19 Deg with respect to the vertical axis, in this position the foot rest will be at 90 Deg with respect to the horizontal axis. When the lever arm (Input Link) rotates by the 10 Deg anticlockwise the position of the foot rest rotates by 20 Deg from vertical axis. Again by the rotation of lever arm (Input Link) rotates by the 10 Deg anticlockwise the position of the foot rest rotates by 40 Deg from vertical axis.. 6. Actual position of function generation linkages in reclining chair Conclusion Analytical synthesis of Mechanism is done. Estimated the principle dimensions for back rest and foot rest function generator linkage for reclining chair. Estimated of inclination of back rest & foot rest with respect to the movement of lever arm Fabricated the mechanism for back rest and foot rest with function generator linkage. References 1. Optimization of Watt s Six-bar Linkage to Generate Straight and Parallel Leg Motion Hamid Mehdigholi and Saeed Akbarnejad, Journal of Humanoids, Vol. 1, No. 1, (2008) ISSN , pp Simulation Of Software For Four-Bar Function Generator Mechanism, Suwarna. B. Torgal, K. Tripathi and N. K. Nagar, Paper presented in 11th National Conference on machines & mechanism Page 80

7 3. Teaching Kinematic Synthesis of Linkages Without Complex Mathematics, By Dr. Louis G. Reifschneider, Journal of Industrial Technology, Volume 21, Number 4, October 2005 through December Foldable hair chair using a four bar linkages, Red Lan, Marks & Clark LLP, Published on 21/04/2010 International Journal of Mechanical Engineering and Computer Applications, Vol 1, Issue 4, Page 81