> Introduction to Calculation Models
As we delve into the parameters of scissor lifts, we inevitably encounter their associated calculation models. These models not only facilitate an understanding of the lift's operating principles but also provide essential design guidance, ensuring that the lift's performance potential is fully realized.
When calculating the forces acting on the hydraulic cylinder, the scissor lift can be simplified into a rigid-body linkage structure with a single degree of freedom to facilitate analysis. Link AB represents the position of the hydraulic cylinder, which itself can be modeled as a "two-force member"-a structural element subject solely to axial forces. When the cylinder is in a static state, the linkage structure constitutes a statically determinate structure according to the principles of structural mechanics; consequently, the forces acting on the cylinder can be determined by solving the relevant equilibrium equations.
> The Method of Joints and Its Application
The Method of Joints is a fundamental analytical technique in mechanics. In the context of planar structures, three equilibrium equations can be formulated for each joint, corresponding to the force equilibrium in the X and Y directions, as well as the moment equilibrium. However, as the number of joints increases, the complexity of the analysis increases commensurately. However, in this specific case-given the relatively simple structural architecture-we can employ the method of joints to determine the forces acting on the hydraulic cylinder using just a single equation.
Consequently, the horizontal bar is subjected solely to vertical loads and bears no horizontal loads. Assuming the load acts precisely at the midpoint of the horizontal bar, we can leverage structural symmetry to deduce that the vertical reaction forces at both ends of the bar are equal to half of the total load-specifically, F = (1/2) * mg, where *m* represents the mass of the load and *g* denotes the acceleration due to gravity. Based on this simplified model, we can more readily determine the forces exerted on the hydraulic cylinder.
Let *Fx* represent the force exerted by the hydraulic cylinder. According to the principles of force equilibrium, we can establish that the support reaction force is equal to *Fx*-that is, Support Reaction = *F*. Next, we will delve further into the procedure for calculating the cylinder force. Since point O-the central pivot of the scissor lift mechanism-functions as the axis of rotation, no bending moment can be transmitted between the two scissor arms at this specific point. Thus, we obtain the following relationship:
From this, we can derive the formula for calculating the force exerted by the hydraulic cylinder:
Given that F = (1/2) * mg, this formula can also be expressed in the following form:
......(2)
In this expression, |OC| represents the perpendicular distance from point O to the line segment AC. Next, we will examine how to determine the value of |OC|.
By establishing a coordinate system as illustrated in Figure (5)-and setting the Z-coordinate to zero-we can calculate the specific coordinates for points O, A, and B. These coordinates can be represented as column vectors, corresponding to the X, Y, and Z axes, respectively. Drawing upon principles of spatial analytic geometry from advanced mathematics, we can derive the following: utilizing the point coordinates established in Equation (3), we can proceed to derive further relationships. By substituting the coordinates obtained from Equation (3) into Equation (2), we can ultimately derive the functional expression for the force exerted by the hydraulic cylinder. To obtain a specific numerical solution, we must select appropriate parameter values and substitute them into the equation for calculation.
> The Energy Method
The energy method offers an alternative approach for determining the forces acting on the hydraulic cylinder. By integrating principles of spatial analytic geometry from advanced mathematics, we can readily derive the functional expression for the cylinder force. Furthermore, with the aid of mathematical software, we can perform multi-parameter optimization to rapidly identify the optimal mounting position that minimizes the force exerted on the hydraulic cylinder under specific operating conditions. This computational methodology provides significant advantages and efficiencies in the field of engineering design. By applying the method of joints from structural mechanics, we successfully derived a simplified force function for a scissor lift. Notably, the specific positioning of the hydraulic cylinder in this particular case rendered the force calculations relatively straightforward. However, in actual engineering design, the installation of hydraulic cylinders is subject to numerous complex factors, which can make the application of the method of joints-specifically in solving systems of multivariate equations-comparatively challenging.
