HYBRID SEVEN-BAR PRESS MECHANISM: LINK OPTIMIZATION AND KINETOSTATIC ANALYSIS

An optimization study with kinetostatic analysis is performed on hybrid seven-bar press mechanism. This study is based on previous studies performed on planar hybrid seven-bar linkage. Dimensional synthesis is performed, and optimum link lengths for the mechanism are found. Optimization study is performed by using genetic algorithm (GA). Genetic Algorithm Toolbox is used with Optimization Toolbox in MATLAB®. The design variables and the constraints are used during design optimization. The objective function is determined and eight precision points are used. A seven-bar linkage system with two degrees of freedom is chosen as an example. Metal stamping operation with a dwell is taken as the case study. Having completed optimization, the kinetostatic analysis is performed. All forces on the links and the crank torques are calculated on the hybrid system with the optimized link lengths.


INTRODUCTION
Many industrial applications are potentially used as hybrid mechanisms. Some of them can be used for cutting, printing, and stamping. Hybrid mechanism configurations are seen in literature: five-bar slider crank, and seven-bar, also nine-bar. Two different actuators are combined with a mechanism to give it a programmable motion. A constant velocity (CV) motor and servo motor (SM) are the actuators. However, while introducing different configurations, there is a need for synthesis to get the links lengths as its optimum [1].
Some studies have given kinematics and dynamics of hybrid configurations. Here some of them are taken as the optimization methods for synthesis of a seven-bar press mechanism. Simulated annealing (SA), genetic algorithms (GA), neural networks (NN), particle swarm optimization (PSO), and the others are applied in studies [2]. M. A. Laribi et al. [3] have used a hybrid method GA-FL for solving the problem of path generation in synthesis of a four-bar mechanism. Blackett [4] has performed synthesis of a fivebar mechanism. The Hooke and Rees pattern search technique has been applied. R. McDougall [5] has used Optimization based mechanism synthesis (OBMS) using four-bar and five-bar mechanisms using PSO algorithm. C. Meng et al. [6] have presented a research on a press introducing hybrid input. Inverse kinematics is studied for hybrid driven mechanism. Z. Yuan et al. [7] have studied two different hybrid mechanisms to get an output motion by minimizing torque and power from the servo motor. Their arrangement is called the slider hybrid mechanism. K. Zang [8] has studied hybrid mechanism with a five-bar. A hybrid five-bar configuration is presented with related kinematics and its mathematical model. Li and Zhang [9] have used a seven-bar mechanism press with hybrid driven case for deep drawing. M. E. Kütük [10] has performed a study on inverse kinematics issues on hybrid driven seven-bar mechanism with motion design for press application. The dimensions of a seven-bar hybrid press mechanism are found by using experimental optimization. S. Ebrahimi and P. Payvandy [11] have presented a study on the dimensional synthesis using different heuristic optimization techniques: GA, PSO and DE.
This study is based on an industrial project for a press mechanism designed with a seven-bar mechanism. Motion with a dwell segment is given for a deep drawing study. MATLAB® GA solver is used in Global Optimization Toolbox. Synthesis of a seven-bar mechanism is presented by using eight precision points. Having performed optimization, the mechanism is drawn. Then kinetostatic analysis is achieved by applying a representative force on the ram. The results are presented and discussed.

SEVEN-BAR PRESS MECHANISM
A seven-bar mechanism is used for producing different ram motion scenarios. A hybrid press mechanism is given a detailed motion design procedure in Kütük's study [1]. Forward and inverse kinematics issues are explicitly given. One axis is driven by a CV motor and the other axis is driven by an SM. The seven-bar press mechanism referred is shown below in Fig.  1.
Loops 1 and 2: Here r 1 , r 2 , r 3 , r 4 , r 5 , r 6 , θ and e are described as the dimensional parameters of the seven-bar press mechanism. The slider motion (s) is expressed as s = f(θ 2 , θ 5 ). The angular displacement, angular velocity and angular acceleration of link r 5 are driven by a CV motor. The angular displacement, angular velocity and angular acceleration of link r 2 are driven by an SM.

Synthesis of a Seven-Bar Mechanism
Li et al. [12] have given the classifications of hybrid fivebar linkages by using the unrestraint double-crank type. Kütük et al. have then performed an experimental optimization [10,13]. A traditional trial and error approach is used to design a seven-bar press mechanism. The input link r2 and r 5 are given as unrestraint cranks. r r r r r + + < + r r r r r + + < + The inequality constraints should be satisfied to provide slider mobility condition. X D gives the coordinate of point D in x direction in Eq. (6). The link lengths inequality constraints are given in Eq. (7). S 0 is the point where r 5 , r 6 and r 4 are lined up in a straight line. By using Eq. (8); This study was started by using the values found previously when deciding intervals during optimization. M.E. Kütük found them while performing conceptual design during his M.Sc. thesis [1] as: r 1 = 530 mm, r 2 = 200 mm, r 3 = 650 mm, r 4 = 900 mm, r 5 = 170 mm, r 6 = 800 mm, e = 6.73 mm.

MOTION EXAMPLE
The displacement of the slider link is given in Fig. 2(a). Stroke of the slider is 687 mm with 10 stroke/minute and the motion has got 3 segments. The motion kinematics is illustrated in Tab. 1. The segment number, time interval and initial and final values of position, velocity and acceleration of each segment are given.
The 5 th link is driven by CV motor. It rotates 2π radians in counterclockwise (CCW) direction. Its angular velocity is a constant value, 1.0472 rad/s in Fig. 2

OPTIMIZATION PROBLEM WITH GENETIC ALGORITHM
The mechanism optimization problem is separated into four parts: the conceptual design; the model development with design variables; building of an objective function; and finding the best values satisfying the design constraints for the mechanism [14]. Optimization is very important in engineering design and can solve many synthesis problems. The link lengths and initial angles describe the design variables of the mechanism. The constrained nonlinear optimization is performed for a seven-bar hybrid mechanism. The optimization objective function and the constraints are determined by considering the press motion characteristics. The following optimization problem represents the constrained nonlinear optimization. Objective function is described first. Forward kinematics equations are used to get the objective function. The objective function is given in Eq. (9) by minimizing fobj as: where i d s represents the design precision points required, and

HYBRID SEVEN-BAR MECHANISM IN MATLAB®
A program based on forward kinematics equations is written in Matlab® in order to see the simultaneous positions of the hybrid seven-bar mechanism. The angular positions of the second and the fifth links must be given as the input values. The hybrid system with eight different values of θ2 and θ 5 are illustrated in Figs. 3-5. The hybrid mechanism tracing precision points are given in Fig. 2(a). Here s = 0 and s = 687 mm are taken by referring to study [1]. Industrial applications will refer to dwell period for the second segment of the motion, which is very important for metal forming operations.

THE DYNAMIC MODEL FROM KINETO-STATIC METHOD
The kinetostatic method is also known as D'Alembert technique. This method rearranges the Newton's second law. A dynamic problem is transformed into a static one by adding fictitious inertial forces and inertial torques onto the system. One fictitious force on each moving body which is equal to the mass of that body times the acceleration of its mass center. The direction is opposite to its acceleration. It is directly applied onto the center of gravity, apart from the already existing real forces. One fictitious torque on each moving body which is equal to the centroidal inertia of that body times its angular acceleration. The direction is opposite to that of acceleration. The already existing torque is also applied. The model is then analyzed with the rules of statics [16]. The free body diagrams of the links are given in Fig. 6. In the dynamic model, the velocity fluctuation of the constant velocity motor, the elastic deformation and friction force of the links are assumed as zero to make the system simple. Kinetostatic diagram of each link in hybrid mechanism is shown in Fig. 6 (a, b, c, d, e, f, g). Inertia forces and inertia torques are acted on every link of the hybrid driven mechanism. They are shown by dashed vectors. Thus, the static equilibrium equations of each link can be constructed [17]. The links used in the mechanism are taken as solid cylinders. The parameters of the links are given in Tab. 2.
where 2 Acg r is the distance between the mass center of the second link and A point.
is the distance between the mass center of the second link and B point.
, , 0 is the distance between the mass center of the fourth link and C point.
, , 0 is the distance between the mass center of the fifth link and F point.

An Example Analysis
The reaction forces in each joint and the cranks torques can be obtained as illustrated below by using a Matlab ® code written for kinetostatic equations. The data including the slider displacement, velocity and acceleration and the angular displacement, velocity and acceleration of the crank driven by the constant velocity motor and forming force of the slider are given as input parameters. Fig. 7 shows the output of the code written in Matlab ® .

CONCLUSION
A hybrid seven-bar press mechanism is explained with link optimization. Genetic algorithm is applied for finding optimum link lengths for the mechanism. The variations of the design variables are checked. They all operate in the operating limits of the press mechanism. One mechanism satisfying eight precision point is taken as an example and presented here. Future work will include more precision points for performing the required motion. An example for kinetostatic analysis showing all forces in each link and torques in the cranks of the hybrid system is presented. Studies on this subject are continuing in Mechatronics laboratory.