## 1 Introduction  
A complete mechanical balance or dynamic balance of a mechanism consists of eliminating the dynamic reactions at the base of a mechanism produced by the movement of its structure. These dynamic reactions are the shaking force (ShF) and the shaking moment (ShM). Such balancing is desirable because the ShF and ShM cause vibrations of the supporting frame which latter turn into noise, fatigue, wear, etc. The four-bar mechanism is often taken as example in dynamic balancing because its number of applications [1; 2]Dynamic balancing can be performed in two stages. In the first one, the balancing conditions are commonly obtained using methods that involve Cartesian coordinates and the use of angles; this implies the use of trigonometric functions that derive into complex expressions [3; 4; 5; 6]. In relation to this point, this work exploits the use of fully Cartesian coordinates (also called natural coordinates [7]) to obtain the expressions for optimizing the mechanism balance avoiding any use of angular variables [8; 9] and therefore, simplifying the equations. In the second stage, dynamic balancing can be achieved by different methods. A complete review of methods can be found in [3; 4; 5] and more recently in [6]. All balancing methods imply the addition of masses in some or other form while trying to maintain the total mass as low as possible. Methods based on mass redistribution are very useful from a practical point of view and are very suitable to be implemented. Using optimization techniques, it is possible to minimize the dynamic reactions while maintaining the increment of mass under control. Mass redistribution can be done by optimizing the geometry of the links [10] or by optimizing the size and position of counterweights. Traditionally, when computational resources were not available, dynamic balancing optimization was carried out using analytic or semi-analytic methods [11; 12; 13; 14; 15; 16]. Nowadays, technological advances have made it easy to implement advanced numerical optimization techniques. It is usually considered that this kind of formulation is difficult to solve because of the nonlinear nature. The most common technique used to optimize mechanism balancing is meta heuristics. Some examples can be appreciated in [17; 18; 19]. Herein, techniques like the so-called evolutionary algorithms, Firefly Algorithm, or Differential Evolution, were used, respectively. Some references to methods that transform the objective function to make it convex [20] can also be found. However, they involve a major mathematical analysis. The optimization algorithm proposed in this paper is a simplified version of Projected Gradient Descent [21], a deterministic and iterative algorithm based on the gradient vector direction. The optimization problem is multi-objective with bound constraints (also known as box constraints). The objective function is obtained by the linear combination of the improvement indexes of the ShF and ShM. This methodology is efficient due to the characteristics of the functions obtained by using fully Cartesian coordinates, allowing to reach a convergence at the minimum value with just few iterations. This article presents an extension of our previous work [22] in which, with the use of Fully Cartesian coordinates, dynamic balancing optimization of a four-bar mechanism was achieved through the sole addition of counterweights [23; 24]. As a novelty, this work presents a comprehensive analysis of the gradient on the optimal points obtained after the first optimization, so it is possible to know whether the proposed limits of the optimized variables can be modified (whenever possible) to obtain even better results than those originally obtained. An analysis of the volume and the relation between the area and the thickness of the counterweights completes and confirms the gradient analysis. The proposed methodology can be used in similar problems to analyze the optimization of mechanisms.