Design of Scotch Yoke Mechanisms with Improved Driving Dynamics

Original Article

Proc IMechE Part K: J Multi-body Dynamics 2016, Vol. 230(4) 379–386 Design of Scotch yoke mechanisms with ! IMechE 2015 Reprints and permissions: improved driving dynamics sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/1464419315614431 pik.sagepub.com

Vigen Arakelian1,2, Jean-Paul Le Baron2 and Manuk Mkrtchyan1

Abstract Input torque balancing through addition of an auxiliary mechanism is a well-known way to improve the dynamic behavior of mechanisms. One of the more efficient methods used to solve this problem is creating a cam-spring mechanism. However, the use of a cam mechanism is not always possible or desirable because of the wear effect due to the contact stresses and high friction between the roller and the cam. The Scotch yoke mechanism is most commonly used in control valve actuators in high-pressure oil and gas pipelines, as well as in various internal combustion engines, such as the Bourke engine, SyTech engine and many hot air engines and steam engines. This mechanism does not create lateral forces on the piston. Therefore, the main advantages of applications include reducing friction, vibration and piston wear, as well as smaller engine dimensions. However, the input torque of the Scotch yoke mechanism is variable and can be balanced. This paper proposes to balance the input torque of Scotch yoke mechanisms without any auxiliary linkage just by adding linear springs to the output slider. It is shown that after cancellation of inertial effects the input torque due to friction in joints becomes constant, which facilitates the control of the mechanism. An optimal control is considered to improve the operation of balanced Scotch yoke mechanisms. The efficiency of the suggested technique is illustrated via simulations carried out by using ADAMS software.

Keywords Scotch yoke mechanism, input torque, balancing, optimal control, dynamics

Date received: 16 May 2015; accepted: 6 October 2015

Introduction applications include reducing friction, vibration and It is obvious that whatever the power of control, even piston wear, as well as smaller engine dimensions. today, one cannot correctly operate a machine with The analysis of a Scotch yoke mechanism shows poor mechanics. If the input torque, that is, the torque that its input torque is highly variable. The input ensuring the constant speed, is highly variable, the torque may be reduced by optimal redistribution of resulting drive speed fluctuation will be substantial. moving masses.7–11 or by using non-circular gears.12 Therefore, highly variable input torques might excite One of the more efficient methods used to solve the torsional vibration, while input torques with frequent problem of input torque balancing is creating a cam- sign changes present a very unfavorable loading case spring mechanism, in which the spring is used to for the gears that are possibly present between the absorb the energy from the system when the torque mechanism and its driving actuator. is low, and release energy to the system when the This paper provides a simple and efficient input required torque is high. It allows reducing the fluctu- torque balancing method, which can be applied to ation of the periodic torque in the high-speed mech- Scotch yoke mechanisms. The Scotch yoke mechan- anical systems.13–21 ism is subject to a wide range of applications and The input torque balancing technique proposed in various publications have been devoted to its this paper is achieved by adding linear springs. study.1–5 This mechanism is most commonly used in control valve actuators in high-pressure oil and gas pipelines, as well as in various internal combustion engines, such as the Bourke engine, SyTech engine 1IRCCyN, Nantes Cedex, France and many hot air engines and steam engines. It is 2I.N.S.A. Rennes, Rennes, France also used in testing machines to simulate vibrations 6 Corresponding author: having simple harmonic motion. The Scotch yoke Vigen Arakelian, I.N.S.A., Rennes 20, avenue des Buttes de Coesmes CS mechanism does not create lateral forces on the 70839, Rennes, F-35708, France. piston. Therefore, the main advantages of Email: [email protected]

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Figure 2. Input torque of a Scotch yoke mechanism.

where IS1 is the axial inertia moment of link 1; mi are Figure 1. Scotch yoke mechanism. the masses of the corresponding links (i ¼ 1, 2, 3); rS1 is the distance between the centre of the joint O and Input torque of a Scotch yoke the centre of mass S1 of link 1. mechanism Substituting equation (5) into equation (1), the input torque of the mechanism is The Scotch yoke mechanism is a reciprocating motion 2 2 mechanism, converting the linear motion of a slider MIN ¼0:5m3lOAðÞ’_ sin 2’ ð6Þ into rotational motion of a crank or vice versa (Figure 1). In the present study, it has been considered The obtained result shows that the input that the gravitational forces are perpendicular to the torque of a Scotch yoke mechanism varies according motion plane. to sin2u (Figure 2). As is mentioned by Berkof,7 the input torque of a It means that the average value of the input torque single degree of freedom mechanism due to inertial is equal to zero, and the correction moment created by effects can be found from equation the spring system should be similar to the input torque of the mechanism. Thus, in balancing the system for 1 dT the periods ’ 2 ½0; =2 and ’ 2 ½; 3=2 the spring M ¼ ð1Þ , IN ’_ dt must to absorb and accumulate the energy from the Scotch yoke mechanism because the input torque is where T is the total kinetic energy of the mechanism low. With regard to the periods ’ 2 ½=2; and and ’_ is the input angular velocity. ’ 2 ½3=2; 2 , the spring should release energy to The relationship between the rotation of link 1 and the Scotch yoke mechanism because the required the translation of link 3 can be written as torque is high. It should be noted once again that in the present s ¼ lOA sin ’ ð2Þ paper, the input torque due to inertial effects is considered. In the case of the presence of combustion where ’ is the rotating angle of link 1; lOA is the length forces, the input torque balancing will be different. of link 1, i.e. the distance between the joints O and A; For the case of engines please see combustion-induced s is the translational displacement of slier 3. torque variation in literature.22 The slider velocity can be found by differentiating equation (2) Input torque balancing s_ ¼ lOA’_ cos ’ ð3Þ The spring system should ensure the following Considering that the input angular velocity is con- condition stant and differentiating equation (3), the slider accel- eration can be written as Fspdx þ MINd’ ¼ 0 ð7Þ 2 s€ ¼lOAðÞ’_ sin ’ ð4Þ

The kinetic energy of the mechanism can be where Fsp ¼ kx is the elastic force of the spring; k is written as the stiﬀness coeﬃcient of the spring, x is the displacement of the spring. 2 2 2 2 2 T ¼ 0:5ðÞ’_ IS1 þ m1rS1 þ m2lOA þ m3lOA cos ’ It should be noted that Fsp has a minus sign during ð5Þ the accumulation of energy and a plus sign during the restitution of energy.

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For the period of the accumulation of potential energy Z ’ Aaccum ¼ MINd’ Z 0 ’ 2 2 ¼ 0:5m3lOAðÞ’_ sin 2’d’ 0 ¼ 0:5m l2 ðÞ’_ 2sin2 ’ ð8Þ 3 OA Figure 3. Displacements of the balancing spring.

The maximal value of the accumulate potential With regard to the spring it can be written energy for both periods mentioned above is Z xmax Z 2 2 =2 Arest ¼ kxdx ¼ 0:5kðxmax x Þð17Þ x Amax ¼ MINd’ Z 0 =2 or 2 2 2 2 ¼ 0:5m3lOAðÞ’_ sin 2’d’ ¼ 0:5m3lOAðÞ’_ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 2Arest ð9Þ x ¼ x2 ð18Þ max k

Integrating equation (7) for the period of energy To determine the displacement x of the spring for accumulation, the following relationship can be the period of energy restitution, let us introduce equa- obtained tion (14) and equation (16) into equation (18) Z Z ’ x ¼ ð Þ Aaccum ¼ MINd’ ¼ kx dx ð10Þ x xmax sin ’ 19 0 0 and So for two periods, accumulation and restitution, displacements of the spring are same, see Figure 3. 2 Aaccum ¼ 0:5kx ð11Þ The proposed traditional solution for input torque balancing in Scotch yoke mechanism is to add a From which cam to input crank in order to execute harmonic dis- rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi placements of the spring. However, taking into 2 account particularities of the Scotch yoke mechanism, x ¼ A ð12Þ k accum it will be shown that a simple balancing technique of the input torque can be found. when x ¼ xmax, the accumulation of energy becomes Let us now turn our attention to the displacements maximum of slider 3. The displacements of the slider vary with the sinusoidal law. Therefore, it is possible to balance 2 Amax ¼ 0:5kxmax ð13Þ the input torque of a Scotch yoke mechanism by adding linear springs between the frame and output Now, from equation (9) and equation (13) the stiff- slider 3. The added springs should ensure the condi- ness coefficient of the spring can be determined tion: xmax ¼ lOA. Thus, by adding simple linear springs the input m l2 ðÞ’_ 2 torque due to the inertial forces will be fully cancelled. k ¼ 3 OA ð14Þ 2 Although the described solution is very simple, this is xmax the first time it is proposed. To determine the displacement x of the spring let us introduce equation (8) and equation (14) into equa- The input torque due to friction in joints tion (12) Let us now consider a Scotch yoke mechanism taking x ¼ xmax sin ’ ð15Þ into account the friction in the mechanism’s joints. Let us now consider the period of energy restitution. Several friction models have been proposed having In this case, the following expression concerning different levels of accuracy, and wide variety of con- input torque can be written as trol solutions have been developed for its 23–30 Z compensation. ’ 32 2 2sin 2’ 2 2 2 In Sawyer et al., a nearly ideal two-dimensional Arest ¼ m3lOAðÞ’_ d’ ¼ 0:5lOAðÞ’_ cos ’ =2 2 Scotch yoke mechanism was constructed to test a ð16Þ model of wear depth as a function cycle number. The model originally developed by Blanchet33 was

Downloaded from pik.sagepub.com by guest on December 5, 2016 382 Proc IMechE Part K: J Multi-body Dynamics 230(4) non dimensionalized and simplified under conditions The input torque due to friction in joints of large numbers of cycles. Experiments, given in literature,32 showed a linear progression of wear over The differential equation describing the motion of the two distinct regions, suggesting a sudden transition Scotch yoke mechanism without linear springs to the in wear modes just after 1.5 million cycles. output slider is given by The review showed that friction must be considered ÂÃ 2 2 2 2 in dynamic models in order to optimally control ðtÞ¼ IS1 þ m1rS1 þ m2lOA þ m3lOA cos ’ðtÞ ’€ðtÞ mechanisms. 1 2 2 In the present paper, the friction model developed m3l ’_ ðtÞ sin 2’ðtÞð22Þ 2 OA by Wilson and Sadler34 has been used. The choice of this model is due to the fact that it provides analytical The joint variable is ’ðtÞ and the control torque results. It allows authors to keep the principal struc- is ðtÞ ture of the paper with only analytically tractable The parameters of the Scotch yoke mechanism are solutions. After torque balancing described above, the reac- m l2 l I ¼ 1 OA ; r ¼ OA ; l ¼ 0:1m; tion forces in prismatic joints are cancelled, i.e. S1 12 S1 2 OA ð23Þ ¼ ¼ F23 F03 0, where F23 is the reaction force between m1 ¼ 3kg; m2 ¼ 0:5kg; m3 ¼ 5kg links 2 and 3; F03 is the reaction force between link 3 and the frame (denoted as ‘‘0’’). In order to simplify the expression of ðtÞ, please With regard to the reactions in revolute joints, they note that int are constant due to the condition: F21 þ F01 þ F1 ¼ 0, 8 ÂÃ 2 2 2 2 where F21 is the reaction force between links 2 and 1; <½¼’ðtÞ IS1 þ m1rS1 þ m2lOA þ m3lOA cos ’ðtÞ 40 F01 is the reaction force between the frame and link 1; : 1 2 2 int B½¼’ðtÞ,’_ðtÞ m3l ’_ ðtÞsin2’ðtÞ F1 is the resultant inertia force of link 1. 2 OA Thus, for determination of the input torque of the ð24Þ balanced mechanism, only the bearing friction in revolute joints O and A should be taken into consid- The dynamic model is shown below eration. It is known that the effect of the frictional contact at the bearing surfaces is always a torque B½’ðtÞ, ’_ðtÞ ðtÞ ’€ðtÞ¼ þ ð25Þ which acts in a direction to oppose the relative rota- ½’ðtÞ ½’ðtÞ tion of the two links34 The steady-state for ð frÞ Mji ¼ ejijiFji cos jisgnð’_j ’_iÞð20Þ ðtÞ¼B½’ðtÞ, ’_ðtÞ is : ’€ðtÞ¼0 ð26Þ where eji is the nominal radius of the bearing (in practical mechanisms the difference between the By adding simple linear springs, the input torque radii of the bearing and the shaft or pin is less due to the inertial forces is fully cancelled (null value than 0.2% and thus eji may be taken as the nominal for ðtÞ in steady-state) but ’€ðtÞ¼0: size of the bearing); ji is the coefficient of friction; Now, for the steady-state, let us consider the Fji is the bearing reaction force of link j on link i; Scotch yoke mechanism with friction in joints. It 1 ji is the friction angle (ji ¼ tan ji); ’_j is the should be assumed that the input torque due to fric- angular velocity of link j; ’_i is the angular velocity tion in joints can be represented through an additional of link i. constant disturbance d ðtÞ in the state equation as Therefore, the input torque of the balanced mech- follows anism can be written as ’_ðtÞ 01 ’ðtÞ 0 0 ¼ þ uðtÞþ d ðtÞ ¼ ð frÞ þ ð frÞ ¼ ð _ _ Þ MIN M01 M21 e0101F01 cos 01sgn ’0 ’1 |fflfflffl{zfflfflffl}’€ðtÞ |fflfflfflffl{zfflfflfflffl}00|fflfflffl{zfflfflffl}’_ðtÞ |ffl{zffl}1 |ffl{zffl}1 þ e2121F21 cos 21sgnð’_2 ’_1Þð21Þ x_ðtÞ A xðtÞ B E ð27Þ Thus, after balancing of the inertia forces, the input torque of the Scotch yoke mechanism becomes con- The observed variable is given by stant and can be determined by the some of friction ÂÃ torques in joints O and A. ð Þ¼ ð ÞðÞ y t |fflfflfflffl{zfflfflfflffl}10x t 28 Let us now consider the optimal control of the C Scotch yoke mechanism to ensure the constant input angular velocity and the given input torque due to This double integrator is unstable but completely friction in joints. controllable and observable. It is easily from equation

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(27) that the zero-steady-state error optimal control The steady-state optimal observer which allows law is given by estimating xðtÞ and d ðtÞ has the form "#"# uðtÞ¼Gx^ðtÞd^ðtÞð29Þ ’_^ðtÞ 01 ’^ðtÞ 0 0 ¼ þ uðtÞþ d^ðtÞ ’€^ðtÞ 00 ’_^ðtÞ 1 1 The gain matrix G is an appropriate steady-state |fflfflfflffl{zfflfflfflffl} |fflfflfflffl{zfflfflfflffl} |fflfflfflffl{zfflfflfflffl} |ffl{zffl} |ffl{zffl} _ A B E optimal feedback, x^ðtÞ is an estimate of the state- x^ðtÞ 0 x^ðtÞ 1 vector xðtÞ and d^ðtÞ is an estimate of the disturbance B "#C d ðtÞ. ÂÃ B ÂÃ C k1 B ’^ðtÞ C The function of the gain matrix G ¼ g1 g2 is to þ ð Þ By t |fflfflfflffl{zfflfflfflffl}10 _ C stabilize the system by moving the closed-loop poles k2 @ ’^ðtÞ A C |fflfflfflffl{zfflfflfflffl} in the left-half complex plane. x^ðtÞ For d^ ðtÞ¼d ðtÞ¼0 and x^ðtÞ¼xðtÞ, we seek uðtÞ 0 1 that minimizes the cost B "#C B ÂÃ’^ðtÞ C Z _^ B C 1 ÂÃ d ðtÞ¼k3ByðtÞ 10 C ð33Þ 2 2 @ |fflfflfflffl{zfflfflfflffl} ’_^ðtÞ A J ¼ Ly ðtÞþu ðtÞ dt C |fflfflfflffl{zfflfflfflffl} 0 Z ^ð Þ 1 ÂÃ x t T 2 ¼ x ðtÞQCxðtÞþu ðtÞ dt 0 The state-equations of the observer are The matrix L is based on the controllability tran- 2 3 2 3 2 3 2 3 sient gramian defined by _ ’^ðtÞ k1 10 ’^ðtÞ 0 6 7 6 7 6 7 6 7 Z 6 ’€^ðtÞ 7 ¼ 4 5 4 ’_^ðtÞ 5 þ 4 5 ð Þ TP hi 4 5 k2 01 1 u t T G ð0, T Þ¼ eAtBBTeA t dt ð30Þ _ ^ C P d^ðtÞ k3 00 d ðtÞ 0 0 |fflfflfflffl{zfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflffl{zfflfflfflffl} |fflffl{zfflffl} ÂÃ x_ ðtÞ AE xEðtÞ BE T 1 E 2 3 For the matrix L ¼ TPCGCð0, TPÞC , the T k1 matrix QC ¼ C LC is symmetric and semi-definite 6 7 ÂÃ þ 4 k 5 yðtÞ with yðtÞ¼ 100x ðtÞ positive. The parameter Tp assume that poles of 2 |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} E closed-loop system may be placed, in the S plane, at k |fflfflffl{zfflfflffl}3 CE the left or near of the vertical straight with the K abscissa 1=T : P ð34Þ The output equation uðtÞ¼GxðtÞ of the control- ÂÃ T ler is unique, optimal, full state feedback control law The function of the gain matrix K ¼ k1 k2 k3 T with G ¼ B C that minimizes the cost J. is to stabilize asymptotically the observer. The duality The matrix C is the unique, symmetric, positive between the optimal regulator and the optimal obser- definite solution to the algebraic Riccati equation ver (Kalman filter) enables us to transfer from the regu- T T A C þ C A CBB C þ QC ¼ 0: lator to the observer all important results. For the double integrator, the matrix G is The behavior of the Riccati equation can be T T "# rephrased as follows: AEO þ OAE OCECE pffiffiffi pffiffiffiffiffiffiffiffiffipffiffiffi O þ QO ¼ 0 3 2 3 1 G ¼ g1 ¼ g2 ¼ ð31Þ The matrix Q ¼ ½T G ð0, T Þ is based on the T2 T O R O R P P observability transient gramian defined by Z TR hi Then the closed-loop characteristic polynomial is ATt T A t ð Þ¼ E E ð Þ given by GO 0, TR e CECEe dt 35 0 pffiffiffiffiffiffiffiffiffipffiffiffi pffiffiffi 2 3 3 The solution of the observer Riccati equation is P ðsÞ¼s2 þ s þ : C T T2 P P c c c T T 1 2 3 P ðsÞ¼s2 þ ! s þ !2 K ¼ C ¼ k1 ¼ k2 ¼ k3 ¼ If C 2 n n, O E T T2 T3 ppffiffiffiffiffiffiffiffiffiffi pffiffiffi R R R 3 2 pffiffiffi ! ¼ and ¼ 2 n For c1 2c2 ¼ 9 and c3 ¼ 12 5, the numerical TP 2 values are: c1 ¼ 7:198 c2 ¼ 21:408 c3 ¼ 26:83 For obtain the observer, the constant disturbance Then the characteristic polynomial is is the following 3:0735 4:1248 8:7303 P ðsÞ¼ s þ s2 þ s þ _ O 2 d ðtÞ¼0 ð32Þ TR TR TR

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2 2 If POðsÞ¼ðÞs þ !1 s þ 2!0s þ !0 taking into account that xmax ¼ lOA ¼ 0:1m. The reac- 2:9547 !0 ¼ and ¼ 0:698 tion forces in joints O and A are jjF01 ¼ 197:39N and TR Figure 4 shows the closed-loop control system jjF21 ¼ 49:35N respectively. which accumulates information about the plant The variations of the reaction forces in revolute during operation and allows a zero steady-state track- joints O and A for balanced and unbalanced mechan- ing error in spite of constant disturbance d ðtÞ defined isms are given in Figure 6. by equation (32). The input torque of the initial mechanism without Let us now consider an illustrative example with balancing springs determined from equation (6) and simulations carried out by using ADAMS software. with them determined from equation (21) is shown in Figure 7. Illustrative example and numerical simulations Let us carry out the torque balancing of a Scotch yoke mechanism with parameters (Figure 4): lOA ¼ 0:1m; 1 m1 ¼ 3kg m2 ¼ 0:5kg m3 ¼ 5kg; ’_ ¼ 10s ; e01 ¼ e21 ¼ 0:01m; 01 ¼ 21 ¼ 0:2. To balance the input torque of the Scotch yoke mechanism shown in Figure 5, two pairs of compression springs were used. The first pair balances the input torque for one haft of the crank rotation and the second pair for other haft of the crank rotation. Figure 5(a) and (b) shows the Scotch yoke mechanism in dead point positions when x ¼ xmax. The stiffness coefficient k of each pair of springs is Figure 6. The reaction forces in the revolute joints of the 4934:8N=m, which are determined from equation (14) Scotch yoke mechanism before and after balancing.

Figure 4. The closed-loop control system.

Figure 5. The Scotch yoke mechanism in dead point positions.

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Figure 7. The input torque of the Scotch yoke mechanism before and after balancing. Figure 9. Control law uðtÞ¼Gx^ðtÞd^ ðtÞ (solid line) and disturbance d ðtÞ¼0:48 Nm.

Figure 9 shows the control law uðtÞ¼ Gx^ðtÞd^ðtÞ with integral action which allows the disturbance rejection. The control structure that results from combining observer with state feedback law has the property that the constant disturbance is always compensated so that a zero steady-state regulation or zero tracking error results.

Conclusion This paper deals with the input torque balancing of Scotch yoke mechanisms due to inertia eﬀects. The ð Þ ð Þ¼ Figure 8. Angular velocity ’_ t (solid line) and ’_R t 10 input torque balancing in linkages is usually carried (dashed line). out by adding cam-spring mechanisms. In this study it The numerical simulations showed that in compari- is disclosed that Scotch yoke mechanisms can be son with balanced mechanism, 98% reduction in balanced without any auxiliary linkage by adding input torque has been achieved (from 24.7 Nm to linear springs to the output slider. Although the 0.48 Nm). described solution is very simple, this is the ﬁrst Let us now consider the optimal control of the time, it has been proposed. The analysis of the input mechanism to ensure the constant input angular vel- torque showed that the variation of elastic balancing ocity and the given input torque. forces is a function of the slider displacement. The closed-loop control law can be written as Therefore, the balancing of the input torque of a hi Scotch yoke mechanism can be carried out by two _ ^ pairs of springs connected with the output slider. uðtÞ¼’€RðtÞg1½’^ðtÞ’RðtÞ g2 ’^ðtÞ’_RðtÞ dðtÞ The suggested balancing solution has been improved for the Scotch yoke mechanism taking into account ’RðtÞ, ’_RðtÞ and ’€RðtÞ are given by the equations the friction in the mechanism’s joints. Numerical 8 simulations showed that in comparison with balanced <> ’RðtÞ¼10t mechanism, 98% reduction in input torque has been achieved. It has been shown that after balancing the ’_RðtÞ¼10 :> input torque becomes constant, which facilitate the ’€ ðtÞ¼0 R control of the mechanism. It has also been shown that after balancing the reaction forces in two revolute For Tp ¼ TR ¼ 1s, the following results are joints become constant and far less than before bal- obtained: g1 ¼ 1:732, g2 ¼ 1:861, k1 ¼ 7:198, k2 ¼ ancing (for the considered mechanism about 91% in 21:408, k3 ¼ 26:83: joint A and 72% in joint O). The responses, with Matlab software, to disturb- However, the given reduction can be reached if the ance at t ¼ 4s are shown in Figures 8 and 9. input link has a constant angular velocity. To ensure Figure 8 presents the angular velocity ’_ðÞt , which this condition an optimal control has been developed. approaches at t ¼ 12s, the constant reference It was shown that the given optimal control law ensures ’RðÞ¼t 10, independently of the disturbance a constant input angular velocity taking into account d ðtÞ¼0:48Nm: the friction in joints, as well as the given input torque.

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