This work centered on the double-toggle clamping mechanism with diagonal-five points for the high-speed precise plastic injection machine. Based on Lagrange equations, the differential equations of motion for the beam elements are established, in a rotating coordinate system and an absolute coordinate system, respectively. 43 generalized coordinates and a model matrix for the mechanism are created and some coordinate matrices are derived. By coupling the coordinate transformation and matrix manipulation, a high nonlinear and strong time-variant elastic dynamic model is obtained. Based on the dynamic model, a Kineto-Elasto Dynamics (KED) analysis and a Kineto-Elasto Static (KES) analysis are carried out, respectively. By comparing and analyzing the simulation results of KED and KES, the regularity of elastic vibration of the clamping mechanism in high-speed clamping process has been revealed.

The high-speed precision injection molding machine is an advanced molding equipment, which represents the top technical level in the field of plastic molding. The clamping mechanism is the key unit of the machine, which is used for opening, clamping, and locking the plastic mould, and accordingly affects the product quality directly. Due to the good dynamic characteristics, high system stiffness, and large stroke of movable platens, the double-toggle clamping mechanisms are used widely in high-speed precision injection molding machines. And the researches on double-toggle clamping mechanisms are getting more and more attention. Aiming at the complicated dynamic characteristics of the double-toggle clamping mechanism, Ren et al. [

With the increasing of the moving velocity and the inertial force during clamping process, the clamping mechanism in the precise plastic injection machine could cause enormous impact, bringing about elastic deformation and excessive vibration. Obviously, the traditional research method for rigid body is unsuited for the new working conditions. Therefore, this work centered on the double-toggle clamping mechanism for high-speed precision injection molding machine began to study the angle of nonlinear vibration, elastic dynamics, and damping theory and built the mathematical model of the clamping mechanism, to provide a theoretical basis for the development of high-speed precision injection molding machines.

In order to build a dynamic model, which can reflect the dynamic characteristics of the clamping mechanism and can be calculated conveniently, some simplification and hypothesis should be made as follows.

All the materials of the parts in the clamping mechanism are uniform and isotropic.

Compared with the nominal displacement of the rigid mechanism obtained by the rigid body kinematics analysis method, the elastic displacement of the elastic mechanism caused by elastic deformation is very small, having little influence on the nominal motion of the mechanism.

The movable platen, stationary platen, and tail platen are considered as rigid bodies and their deformations can be ignored.

All the toggle rods are considered as elastic bodies and are defined as beam elements.

Figure

The schematic plot of double-toggle clamping unit.

The finite element model is used in the elastic dynamics analysis of the clamping mechanism. First, the clamping mechanism was divided into a number of discrete finite elements. And then a finite element method for mechanism analysis [

According to the above hypotheses, the actual displacement can be seen as the superposition of the nominal displacement and the elastic displacement. The nominal displacement can be obtained by the rigid body kinematics analysis method, while the elastic displacement can be calculated by the elastic dynamic analysis method [

The vibration performances of the clamping system depend on the structure, inertia, elasticity, and dampness of the system itself. In order to build an elastic dynamic model and to reflect the dynamic characteristics of the clamping mechanism, the dynamic substructure analysis method was adopted during the modeling process. First, the clamping system was divided into substructures and elements. Then, the equations of motion of the substructures and elements were built. Finally, these equations of motion were combined into equations of motion for the whole system.

During the dynamic analysis and optimal design process by using the dynamic substructure analysis method, the dynamic models for each substructure must be established firstly. And then according to the mutual connection conditions between the substructures, these dynamic models of substructures could be integrated into the dynamic model for the whole clamping system.

The detail modeling steps for the double-toggle clamping mechanism with diagonal-five points, shown in Figure

(1) According to the shape of the parts and the computational accuracy required, determine the type and number of the elements. For the rod parts of the clamping mechanism, the beam element can be chosen as the element type. By setting nodes at specified points, the double-toggle clamping mechanism can be divided into 13 elements.

(2) Choose nodal displacement mode, create generalized coordinates, and derive the element’s differential equations of motion according to Lagrange equations. The elastic displacements and angular displacements at the nodes can be chosen as the generalized coordinates. All the generalized coordinates can be combined as an array, which is the unknown quantity needed to be solved. The number of the generalized coordinates can be described as the DOF of the dynamic model, which determines the scale of the problem solving.

(3) By coordinate transformation, the substructure equations of motion in the physical coordinate can be transformed into ones in the modal coordinate and in uncoupled form. By use of the coordinate conditions of joint points in the joining interface between the substructures, all the mode coordinates of the substructure can be transformed into the coupling mode coordinate of the whole structure. And then the decoupling mathematical model of the whole structure can be obtained by further coordinate transformation. The equation of motion for the clamping system is similar to the one of the vibration system with multiple DOF, as

The toggle rods are defined as beam elements and there are two nodes in each beam element. It is assumed that the mass of each element is focused on the axle of the beam and the kinetic energy of the revolving section is ignored. The bending deformation of the beam at the bending moment and the tensile or compressive deformation at the axial load are considered, while the other deformations, such as shearing deformation, are ignored. Therefore, the beam elements are only subjected to the loads from the axial and lateral direction, and there are overall 6 nodal displacements at each endpoint of the beam elements, as shown in Figure

The beam element and the nodal displacements.

In the figure,

In order to simplify the solving process, three calculating steps were taken as follows.

First, assume that there are only axial loads applied to the beam element and an axial deformation occurred accordingly. The mass matrix and rigid matrix of this loading condition can be calculated.

Then, assume that there is only lateral loads applied to the beam element and there are only lateral displacements and angular displacements in each node. The mass matrix and rigid matrix can be calculated too.

Finally, superimpose the mass matrices and rigid matrices of the two loading conditions together, respectively, and then the mass matrix and rigid matrix of the beam element subjected to the loads from the axial and lateral direction can be obtained [

The nodes of the 13 beam elements of the clamping mechanism can be represented by capital letters

For the clamping mechanism discussed in this work, the geometry was meshed with beam elements, as shown in Figure

Division of clamping mechanism of beam elements.

The instantaneous kinematic analysis method was adopted, which defines the direction and magnitude (translational and rotational) of impending motion of the clamping mechanism at a special point in time. At any special instantaneous time, the mechanism has a special discrete orientation, just like being frozen.

Figure

The kinematic relations of a beam element before and after deformation.

The element generalized coordinates are the measures in the element coordinate system. In order to facilitate the following derivation, the absolute velocity and absolute acceleration in the absolute coordinate system are transformed to the ones in the rotary coordinate system. The transformation relations of velocities are described by

The vectors in (

Similarly, the transformation relations of accelerations can be described by

The vectors in (

In the above equations, the vectors

Equations (

The following equation is Lagrange’s equation:

The following is the deriving process of dynamic equations for the clamping mechanism by means of Lagrange’s equations.

(1) Differential equations of motion of beam elements in the rotary coordinate system

Based on Lagrange’s equation, the differential equations of motion of beam elements in rotary coordinate system

(2) Differential equations of motion of beam elements in the absolute coordinate system

In order to assemble the equations of motion of element into the equations of motion of system, it is needed to choose the nodal linear displacements, parallel to the coordinate axes of the absolute coordinate system, as generalized coordinates. Accordingly, a new element generalized coordinate array is introduced as the following array and is shown in Figure

Coordinate transformation.

Here, it is necessary to illustrate the significance of generalized coordinate array

Obviously, the two groups of coordinates in the rotary coordinate system and in the absolute coordinate system, respectively, have the following relationships as in the formula

In formula (

Figure

The schematic diagram of the clamping mechanism.

The clamping mechanism is divided into 13 beam elements, and accordingly there are 13 coordinate transformation matrices with different horizontal angles

For one mechanism, when different sets of generalized coordinate are chosen, different equations of motion will be obtained and different coupling modes will be adopted. The coupling of equations of motion is not the form of the motion system but the result of the selection of the generalized coordinates. Natural coordinate is a set of generalized coordinate, which will obtain equation of motion in uncoupled form. Any set of generalized coordinate can be transformed into natural coordinate by linear transformation using a modal transformation matrix, which will make the uncoupling of the equations become more convenient.

Define the following array:

By (

In order to keep the symmetry of the equation, we can multiply “on the left” on both sides of the equation by

Equation (

Equation (

The above matrix transformations can transform the coordinates between different coordinate systems but cannot change the nature of the motion system. The transformations of mass matrix and rigid matrix, however, could change the coupling situation of the equation of motion. Equation (

Based on the generated mesh of beam element, the finite element model for the double-toggle clamping mechanism was established, which includes 43 generalized coordinates, as shown in Figure

The finite element model of the clamping mechanism.

The 43 generalized coordinates can be denoted as

As for the double-toggle clamping mechanism discussed in this work, the model matrix is a

The subscript numbers in the above differential equations of motion of beam elements are local numbers. Such equations of motion can be written for each element. Therefore, there are 13 such element equations for the 13 beam elements. By gathering the 13 element equations, the differential equations of motion of the clamping system can be obtained. For the

The generalized coordinate array of the clamping system can be defined as

The rigid velocity array of the clamping unit can be defined as

The arrays

If

The rigid acceleration array of the clamping system can be defined as

Similarly, the arrays

If

As for the double-toggle clamping mechanism discussed in this work, the rigid acceleration array can be written as formula

For the 13 beam elements of the double-toggle clamping mechanism, 13 coordinate matrices can be derived as formula

Similarly, there is the same relation between

By substituting formula (

Formula (

The key of elastic dynamic analysis for the clamping mechanism is to solve the differential equation of motion of the system. Formula (

In this paper, we mainly focus on the establishment of the dynamic mode of the clamping mechanism but not on the solution of the dynamic mode, which will be explored in another paper.

Here, the complicated solving process is neglected, and the results by the method of Kineto-Elasto Dynamics (KED) and Kineto-Elasto Static (KES) are given directly. The comparison and analysis between the KED and KES results were done.

KED analysis is used to study the vibration of the clamping mechanism, which is considered as an elastic motion system and bears inertial forces and external loads, and then to calculate the displacement, velocity, acceleration, strain, and stress of each part in the mechanism. The fundamental difference between KES and KED is that, in KES analysis, the inertial forces and external loads are supposed as static loads.

In this study, a simulation case at different clamping speed was given, and the results from KED analysis and KES analysis are compared. Dimensions of the 13 beam elements for the computation of the clamping unit are shown in Table

Dimensions of beam elements for the computation case.

Elements number | Length | Sectional dimension |
---|---|---|

Element (1) | 168 mm | 50 × 20 mm rectangle |

Element (2) | 50 mm | 45 × 20 mm rectangle |

Element (3) | 227.5 mm | 50 × 20 mm rectangle |

Element (4) | 101.5 mm | 45 × 20 mm rectangle |

Element (5) | 115 mm | 50 × 20 mm rectangle |

Element (6) | 115 mm | 50 × 20 mm rectangle |

Element (7) | 101.5 mm | 45 × 20 mm rectangle |

Element (8) | 168 mm | 50 × 20 mm rectangle |

Element (9) | 50 mm | 45 × 20 mm rectangle |

Element (10) | 227.5 mm | 50 × 20 mm rectangle |

Element (11) | 98 mm | 14 mm radius circle |

Element (12) | 120 mm | 32 mm radius circle |

Element (13) | 360 mm | 32 mm radius circle |

The following figures (from Figures

Vibrating displacement of

Clamping speed = 100 mm/s

Clamping speed = 300 mm/s

Clamping speed = 500 mm/s

Vibrating displacement of

Clamping speed = 100 mm/s

Clamping speed = 300 mm/s

Clamping speed = 500 mm/s

Vibrating displacement of

Clamping speed = 100 mm/s

Clamping speed = 300 mm/s

Clamping speed = 500 mm/s

Vibrating displacement of

Clamping speed = 100 mm/s

Clamping speed = 300 mm/s

Clamping speed = 500 mm/s

Figures

Figures

The following can be seen from these figures.

When the mechanism is clamping at the speed of 100 mm/s, the vibrating displacements of both

When the mechanism is clamping at the speed of 300 mm/s, the KED curve and KES curve are similar, despite existing a litter difference, which indicates that it is feasible to use KES instead of KED to analyze the clamping process at low clamping speed.

While at the clamping speed of 500 mm/s, there is an obvious and essential difference between the KES curve and KED curve. Namely, the KES curve is smooth, but the KED curve is fluctuant. At the beginning and end of the clamping process, the KED curve changes suddenly with a larger vibrating displacement, due to the larger elastic deformation of the mechanism.

It is possible for the elastic deformation of the toggle rod to make the movable platen vibrating. The lateral elastic vibration of movable platen may intensify the friction and wear between the tie rods and the axle sleeves, while the longitudinal elastic vibration of movable platen may increase the clamping position error, cause impact on movable platen and stationary platen, and even damage the mould. Therefore, the traditional research method, which considers rods of the clamping mechanism as rigid bodies or quasi rigid bodies, is unsuited for the clamping system with high clamping speed. This will provide theoretical supports for the optimal design of clamping mechanism with high clamping speed, reducing the elastic vibration and improving the accuracy and steadiness of the clamping process.

As can be seen from Figures

The elastic dynamic analysis was done to make all the parts of the clamping mechanism work elastically with uniform deformation amounts. The benefit is obvious. According to the solving results of the theoretical model, the elastic deformations of each part of the clamping mechanism are in an order of magnitude, which is especially important for the mechanism at high clamping speed. It is because that the enormous inertial forces, due to high working speed, may make the parts bear unbalanced forces and cause stress concentration, if the difference of strength, rigidity, and elasticity between parts is too large. The unbalanced forces and stress concentration are the major sources that cause damage to the plastic injection machine.

In this paper, based on appropriate assumptions and Lagrange equation, a high nonlinear and strong time-variant elastic dynamic model of the double-toggle clamping mechanisms was established. A simulation study on the elastic deformation and vibration of the clamping process was carried out by using the KED and KES analysis methods, respectively. The simulation results show that, at low clamping speed, the elastic deformations of the toggle rods are very small, and the solving results of the two analysis methods coincide. While when the clamping speed is up to 500 mm/s, the toggle rods cause vibration and the KED curve changes suddenly with a larger vibrating displacement, at the beginning and at the end of the clamping process, due to the larger elastic deformation of the mechanism. This shows that the traditional research method, which considered rods of the clamping mechanism as rigid bodies, is unsuited for the clamping system with high clamping speed. This work will provide theoretical supports for the optimal design of clamping mechanism with high clamping speed, reducing the elastic vibration and improving the accuracy and steadiness of the clamping process.

Mass matrix of system

Damp matrix of system

Rigid matrix of system

Generalized force matrix

Displacement

Fixed coordinate system

Moving coordinate system

Rotary coordinate system

Generalized coordinate array

Element generalized coordinate array

Absolute velocity array of element

Rigid velocity array of element

Elastic velocity array of element

Relative velocity array of element

Relative normal acceleration array of element

Relative tangential acceleration array of element

Coriolis acceleration array of element

Coordinate transformation matrix

Horizontal angle

Rigid acceleration array of element

Model matrix

Rigid velocity array of system

Rigid acceleration array of system

Coordinate matrix

Mass matrix of element

Damp matrix of element

Rigid matrix of element.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is supported by the Natural Science Foundation of Guangdong Province (S2012010010199) and Scientific and Technological Innovation Project of Foshan (2013AG100063), China.