Design and Analysis of Mechanisms: A Planar Approach

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The mass center coordinates of main linkage and main slider are depicted by G 4 and G 5. The length of moving part and the size of clearance are defined by l 1 and r E , respectively.

The position of mass center for moving part plays an important role, which is associated with velocity and acceleration Erkaya, Therefore, the positions of mass centers are defined as follow:. In the same manner, the positions of mass center for the transmission mechanism are represented as:. The velocities and accelerations of mass centers are given by the time-derivatives of position Olyaei and Gahazavi, We can get the values of angular velocity and angular acceleration for linkages in the following equations:.

The contact force happens in the surface between journal and bearing, which causes the process of contact or impact Flores and Ambrosio, In order to evaluate the dynamic characteristics of mechanism, the equations of motion are defined by the balancing of inertial forces and torques, as shown in Fig. Thus, the equations of motion are expressed by:. M Gi and J Gi are the inertial moment and the mass moment of inertia. The motion equations are achieved in the section 3. The contact process of journal and bearing can cause the contact-impact force, which can be obtained by normal and tangential contact forces.

And relative position between the journal center and the bearing center is calculated by eccentricity vector, which can determine that the state of the bodies is contact or not. Therefore, the model of clearance joint can describe the influence of clearances effectively, which is built based on force interaction Daniel and Cavalca, The components of the eccentricity vector can be given by:.

Design and Analysis of Mechanisms: A Planar Approach - Michael J. Rider - Book |

As proposed in Fig. In order to obtain the relative velocity of the bodies interact, it is necessary to compute the point along the eccentricity line during the process of contact occurs Reis et al. The position of point can be expressed as:. The relative velocity of the point is derived as:. And the relative normal and tangential velocities to plane of collision are defined as following Flores and Ambrosio, :.

The contact force model plays an important role in the dynamic analysis of mechanical system during the contact-impact process. The well-known contact force model of Lankarani and Nikravesh is widely employed by elasticity contact theory, which takes into account the effects of elasticity and damping. Moreover, the influence of damping relates to the dissipated energy during the impact process, along with the dissipative effect correlated with the Coulomb friction on the contact surface Wang et al.

Thus, the contact between journal and bearing can be modeled by the L-N contact force model:. The generalized parameter K depends on the material properties and radius. K can be calculated as Rahmanian and Ghazavi, :. The quantities R i and h i are the radius and the material parameter. Furthermore, the hysteresis damping coefficient is defined as:. Substituting the parameters of K and D from Eqs. Furthermore, the friction force is always a complex phenomenon in the multibody systems, which has the influence on the efficiency and the motion of mechanism.

In recent years, many researchers have established different friction models Wang and Tian, As is well known, the friction force model employed in the numerical simulation is based on Coulomb's friction law, which can effectively describe the highly nonlinear phenomenon Erkaya, It is calculated by:. The dynamic correction coefficient c d is written as:. In this work, it is extremely difficult to measure the performance parameters such as contact force and penetration depth of journal bearing under conditions of different clearance sizes.

Therefore, the experimental tests with the revolute joint clearance of 0. The slider accelerations of the mechanical system with respect to various driving speeds are analyzed by the proposed computation method. If the numerical result is in accordance with that acquired by experimental test under the same operating conditions, the method will be validated.

The experiment consists of testing platform and testing system Flores et al. The composition of testing platform is based on the transmission mechanism of a high speed press system, which includes motor, crank shaft, fly wheel, linkages, main slider and balancing mechanism. The main slider of the high speed press system, as shown in Fig. Due to the acceleration cannot be showed directly, the data processing system is installed into the PC to facilitate the analysis of the testing data. Moreover, the acquisition and analysis systems are established into the BeeTech software, which can transform data at any time.

In order to collect the acceleration data accurately, we calibrate the acceleration sensor before testing. Furthermore, the experiment is conducted under the unloading condition and two periods of acceleration are employed to comparisons between numerical results and experimental data, which is more effective to depict the validity of proposed methodology. The parameters of kinematic and dynamic for the planar multibody system are presented in Table 1.

The purpose of this section is to study the effects of clearance joints on dynamic response of the mechanism in high speed press system.

Mechanical Design (Part 5: Four Bar Linkage)

The planar mechanism is employed as simulation model to analyze the dynamic responce of planar multibody system with joint clearance. Then, the numerical results at different driving speeds are compared with that of the experimental data to demonstrate the computational efficiency of the presented method. In addition, the results of the numerical simulation with different clearance sizes are compared in the section. The results reveal the effects of these parameters on dynamic characteristics of a high speed press mechanical systems. To investigate dynamic behavior of the mechanism with clearance joint at different driving speeds, the model is selected as the case study, as shown in Fig.

The rotational speeds are rpm, rpm, rpm and rpm, respectively. The clearance size is 0. Then, the differences between the experiment and simulation for dynamic behavior plotted in Fig. In addition, the misalignment and lubrication between journal and bearing, which are always employed in actual mechanism system, cannot be considered in the simulation. In the formulation for the mathematic model of clearance joint, the assumption is made that the joint flexibility is neglected.

Furthermore, the values of restitution and friction coefficient play the important roles in the numerical results. However, the values of the restitution and friction coefficient used in the work are based on the published data.

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The same issue occurs and has been observed by many other researchers Bai and Zhao, ; Ma et al. Looking at the curves of slider acceleration, the peak value increases slowly at the lower speed. However, the results of Fig. Moreover, it is clear from the figure that the fluctuation value of acceleration in Fig. This phenomenon appears due to the increase of contact velocity for journal and bearing and the continuous contact motion is the main period in motion period. Meanwhile, the results indicate that the oscillation phenomenon still exists in the operating time of mechanism system.

Design and Analysis of Mechanisms: A Planar Approach

Furthermore, the acceleration of the main slider can be changed by the existence of clearance. The driving speeds are rpm, rpm, rpm and rpm, respectively. When the driving speed is higher than rpm in the calculation, the repeatable precision obviously drops, as shown in curves of the simulations. The value of the repeatable precision reaches the minimum at the angular velocity rpm. It seems that the influence of the driving speed on the repeatable precision in the low-speed conditions become slighter due to the collision velocity of journal and bearing increases weakly.

The value of fluctuation increases from 0.

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The reason for this phenomenon may be that the deformation of bearing and contact-impact force increase with the increase of oscillation frequency. Although there is difference in the results of the repeatable precision, the deviation value of the lowest position for slider is smaller. The dynamic analysis is always conducted based on the ideal joint at the constant running speed.

In fact, the size of clearance joint slowly increases as long-term wear between journal and bearing. It means that the motion of planar multibody system with the various clearance sizes may provide significantly different dynamic responses. To investigate the influence of the clearance size on the dynamic performance of mechanism, the driving speed is rpm.

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The values of revolute joint clearances are chosen to be 0.