#### Abstract

As the key part of internal combustion engines, crankshaft with high efficiency and accuracy processing has always been the target of the engine manufacturer’s pursuit. Grinding is used to obtain the ultimate dimensional accuracy and surface finish in the crankshaft machining. Grinding of the main journals and the pin journals can be accomplished in a single clamping operation by CNC Tangential Point Tracing grinding technology. However, the chatter in the grinding process is harmful to the precision and surface quality. In this paper, stability lobe diagram is developed to predict the grinding chatter. First the dynamic model of Tangential Point Tracing grinding system is established. Then the limit formula of the critical grinding depth is calculated and the stability lobe diagram of the grinding system is presented. Finally, the validation experiments are carried out on the crankshaft grinding machine and the results are consistent with the calculation.

#### 1. Introduction

Crankshaft is one of the key components of engines in automotive industry and its rotation is the power source of the engine. It consists of two important parts: the crankshaft main journal and crankpin. The crankshaft main journal is mounted on the cylinder, while the crankpin is connected to the big end hole of the connecting rod and its other end hole is connected to the cylinder piston. It is a typical slider crank mechanism and it turns the reciprocating motion of the connecting rod into rotating motion. The quality of crankshaft determines the performance of the engine. The crankshaft of an engine is shown in Figure 1.

The traditional crankshaft grinding process of main journal is similar to the cylindrical grinding. The crankpin is adjusted to the center of the grinding by the eccentric fixture and each crankshaft needs special fixture. It has long auxiliary hours and low processing precision when the clamp is adjusting by the operator [1].

Nowadays, the crankshaft is mechanized by the method of CNC Tangential Point Tracing grinding. Grinding of the main journals and the pin journals can be accomplished in a single clamping operation. The grinding method can avoid the positioning error caused by multiple loadings and save the adjustment time. High machining flexibility, accuracy, and efficiency are also improved. CNC Tangential Point Tracing Grinding Crankshaft is high technology processing and it was called Oscillate grinding [2] or Chasing the pin [3] by some scholars. Tangential Point Tracing grinding mode is composed of linkage between the workpiece rotation ( axis) and reciprocating motion of the grinding wheel ( axis) to achieve the eccentric circle machining.

The chatter in the grinding process can result in increased tolerance of dimension and position, the surface roughness and waviness, which seriously affect dimensional accuracy and surface finish of the crankshaft. The chatter in the machining is usually accompanied by considerable noise [4].

There are lots of measures that can effectively control chatter such as using the drive to improve the dynamic stiffness and damping of the grinding machine system to reduce the regenerative phase [5], but these methods need to change the structure of machine tools and they are* not* suitable for the users.

The stability of the stable region and the unstable region can be visually described by the stability lobe diagram [6, 7]. In order to avoid the grinding chatter, the Tangential Point Tracing grinding was set as the research object. The stability lobe chart was used to visually describe the grinding stability region and the unstable region. To predict the grinding stability, the machining quality and the machining efficiency of the grinding process are ensured [8].

#### 2. The Tangential Point Tracing Grinding

With increased demands of industry, Tangential Point Tracing grinding process has been developed to machine nonround shaped parts such as crankshaft and camshaft allowing reduction of nonproductive time and reclamping inaccuracies [9].

The grinding point moves along the surface of the crankpin, while the grinding wheel is always tangent to the crankpin in the grinding process. Figure 2 illustrates the concept of Tangential Point Tracing grinding [3]. The crankpin rotates around the axis followed by horizontal movement of the wheel head along the axis. All of the main journals and crankpins of a concentrically clamped crankshaft can be machined in one fixture that improves the efficiency and accuracy of the products.

Figure 3 is the schematic of Tangential Point Tracing grinding motion model. The crankpin is rotated around the point, and the wheel is followed by the movement. The trajectory equation of the tangent points coordinates can be expressed as follows:

In Figure 3, is the crankshaft turning center. denotes the center of the crankpin. is the center of the grinding wheel.

The eccentric distance of the crankpin is . and correspond to the radius of the crankpin and the grinding wheel, respectively.

Here, and denote the rotation angle of the crankshaft and the angle between and .

and is describe the position of the tangent point in rectangular coordinate system, where is origin of coordinate system.

The movement equation of wheel center is as follows:

#### 3. The Dynamic Model of Tangential Point Tracing Grinding System

##### 3.1. Modeling and Formulation

The crankshaft is supported by the centers of the headstock and tailstock when it is machining. In order to reduce the deformation, main journals are supported by center rests. The headstock provides the low speed rotational drive to the crankshaft.

The crankpin moves around point in the plane while the grinding wheel is doing a reciprocating movement in direction as shown in Figures 3 and 4(a). In this study, a simplified dynamic model of Tangential Point Tracing grinding machine and analysis are presented in Figure 4(c).

**(a)**

**(b)**

**(c)**

According to Newton motion law, four discrete mass dynamic equations are written as follows:

For computer analysis and calculation, the upper type is changed into matrix form; the mass matrix of the system is as follows:

Damping matrix is

Stiffness matrix is

The generalized coordinates and generalized forces are expressed as

And we can express the above kinetic equation as

##### 3.2. The Crankshaft Stiffness Is Different in the Circumferential Direction

The stiffness of the crankshaft varies not only along the axial direction, but also in the radial direction. It leads to the crankpin deformation in the circumferential direction and affects accuracy of the crankpin. In the cylindrical grinding, the direction of the normal grinding force () and the tangential grinding force () is invariable, but they change in the Tangential Point Tracing grinding. The grinding force is shown in Figure 5 where the angle of the crankpin is :

The relationship between normal grinding force and tangential grinding force can be written as follows:

#### 4. Calculation of Critical Grinding Depth

Adjustment of the grinding process and modification of the machine tools structure are two approaches to avoid chatter. In the first approach, the stability lobe diagram that predicts the onset of chatter is used to determine the critical grinding depth and spindle speed to eliminate or minimize chatter behavior in machining [10].

It is important to study chatter mechanisms to predict critical grinding depth. We need to know grinding chatter boundaries and growth rates, which is helpful to design a grinding process without chatter according to the chatter boundaries.

In order to simplify the model of crankshaft cut into grinding system, the kinematic differential equation of the grinding wheel workpiece grinding system dynamics model is expressed as [11]

Dynamic grinding force is usually proportional to the removal rate of material, which has the following formula:Among them are moments of the surface of the workpiece (or wheel) surface ripple amplitude. In this model, only the delay effect from the workpiece is considered. The delay time is assumed to be constant and equals the workpiece’s rotation period () [12]:And then

After combining (11) and (13), we obtain the following expression:

Laplace transformations to the upper formula can be

Thenwhere is input and is the output of the dynamic grinding process transfer function:

Let

To let the divisor be zero, the crankshaft (or wheel) system for the characteristic equation of regenerative chatter is

According to the first discrimination method of stable Lyapunov system, ; the system is in the critical state of stability when is 0:

According to Euler equation, will be substituted into the above equation:

After rearranging (18), we obtain the following expression: will be substituted into (23):

The real part of (21) and (27) is equal to

To make (29) for a partial derivative of , then let the result approach zero:

Bring (30) into (29); then we get the critical grinding depth:

It is shown that the presented approach can be used to predict the crankshaft grinding stability [5, 13].

The grinding wheel is easily worn and the regenerative chatter of both workpiece and grinding wheel should be considered.

Critical grinding depth of the crankshaft:

Critical depth of the grinding wheel wear:

#### 5. Experimental Study

The complicated phenomena in engineering cannot be concluded only by theoretical analysis and the accuracy of theoretical analysis results need be verified by experiment.

##### 5.1. The Test Experiment of Crankshaft Circumferential Stiffness Change

As the crankshaft is an elongate and complex shape shaft, the crankpin deforms during the grinding process. The center frame is generally used in the machining of crankshaft to eliminate the influence of the gravity, so the grinding force is the main cause of crankpin deformation.

In order to further study the influence of grinding force on the deformation of crankpin, we take the crankshaft of D06A-101-30 diesel engine as the experimental object. The number 1 crankpin was applied to the 200 N vertical constant force to simulate the idea that the crankshaft was stressed by the normal grinding force of 200 N. In the experiment, two laser displacement sensors were used to measure the deformation of crankpin in two directions, as shown in Figure 6.

First, the crankshaft is rotated without the weight, and the position of the crankpin is measured at 5 degrees from the angle of crankpin which is 0 degrees. Then, under the weight of 200 N, repeat the above measurement procedure and another group of position dates is obtained. The difference between the two groups is the deformation in the radius direction of crankpin under the grinding force 200 N. Finally we can achieve the deformation and stiffness of crankpin as in Table 1 and Figure 7:where and represent stiffness of crankpin and grinding force, respectively. denotes the deformation in the radius direction of crankpin.

According to the experimental results and the structural characteristics of the crankshaft, we use cosine curve fitting its stiffness, as shown in Figure 8. We can obtain the expression of the stiffness of the crankshaft.

If we plug the parameters of Table 2 into the above formulas, we can obtain the stability lobe diagram shown in Figure 9.

In the diagram, above the lobe line is “The unstable region” of the grinding system and below the lobe line is “The stability region.”

The crankshaft is a complex shape shaft and its circumferential stiffness is different [14, 15]. In the diagram, it is shown that accounting for the uncertainty or variability of the process parameters can influence the stability boundary and the fuzzy stable region has been formed [16].

As the grinding wheel speed increasing, the limit grinding depth also has the tendency of increase. Therefore, by increasing the grinding wheel speed and keeping the grinding depth less than the critical depth, we can make the process in a stable state and the processing efficiency is improved.

##### 5.2. Experimental Results Contrast

In order to verify the correctness of the stability limit diagram, we carried out the relevant grinding experiments [17–19]. The specific parameters of the workpiece and grinding wheel are shown in Table 3.

We apply Bruel & Kjaer 4366 accelerometer to the grinding machine in order to detect chatter vibrations. The sensor was mounted on the tailstock center (Figure 11). In the process of grinding, the acceleration amplitude of regenerative chatter is increased rapidly with the grinding time. Figures 12 and 13 display the normal and the chatter signals.

Crankshaft Tangential Point Tracing grinding setup is shown in Figure 10. Each test was measured more than three times during grinding, so that the signals produced were sufficient to obtain information.

The crankshaft rotation speed is set to 6 r/min and the speed of the grinding wheel is 1250 r/min, 1450 r/min, and 1650 r/min in grinding experiment. In the case of certain rotational speed, the boundary value of the chatter is determined continuously by changing the depth of the grinding. The chatter points are basically over the stability limit of the curve or near and so the experimental results are consistent with the predictions of the stability limit diagram. Therefore, it is proved that the prediction method is effective and reliable.

We can see chatter marks (Vibration Waviness) with naked eye on the crankpin surface as in Figure 14. Roundness measurement results show that the instability of the regenerative chatter caused by the instability of the grinding phenomenon makes the roundness of the workpiece poor, as shown in Figure 15.

**(a) Vibration Waviness of crankpin**

**(b) Vibration Waviness of crankpin**

**(a) The normal**

**(b) The grinding chatter**

#### 6. Conclusions

To study how to avoid chatter in crankshaft Tangential Point Tracing grinding, Stability lobe diagram has been developed based on dynamic model to predict chatter and some conclusions have been drawn as follows:(1)The dynamic equation of the grinding system has been constructed by the dynamic analysis of the grinding system.(2)Expression of the critical crankshaft grinding depth has been developed based upon the work of Altintaş and Budak [5] and Stepan [13].(3)Through the experimental study, the law of crankshaft rigidity is obtained.(4)The stability of the grinding system can be predicted by using the method of drawing the stability diagram.(5)Experimental results show that the prediction method is consistent with the experimental data.

#### Nomenclature

: | Equivalent stiffness of the support system of and direction in generalized coordinates in the time of the grinding of the crankpin, respectively |

: | Equivalent stiffness of the right support system of and direction in generalized coordinates in the time of the grinding of the crankpin, respectively |

Equivalent stiffness of the left support system of and direction in generalized coordinates in the time of the grinding of the crankpin, respectively: ; | |

: | Equivalent damping of the support system of and direction in generalized coordinates in the time of the grinding of the crankpin, respectively |

: | Equivalent damping of the right support system of and direction in generalized coordinates in the time of the grinding of the crankpin, respectively |

: | Equivalent damping of the left support system of and direction in generalized coordinates in the time of the grinding of the crankpin, respectively: ; |

: | Contact stiffness of crankshaft and grinding wheel |

: | Grinding rigidity of crankshaft and grinding wheel |

: | Normal grinding stiffness of crankshaft and grinding wheel |

: | The grinding stiffness and contact stiffness’s equivalent stiffness of the crankshaft and the grinding wheel: |

: | Equivalent support stiffness of the grinding wheel system in the direction of the generalized coordinates |

: | The equivalent damping of the grinding wheel system in the direction of the generalized coordinate system |

: | Masses of the crankshaft |

: | Masses of the grinding wheel |

: | Equivalent stiffness of the support system of the grinding wheel of direction in generalized coordinates |

: | Equivalent damping of the support system of the grinding wheel of direction in generalized coordinates |

: | The force of the grinding system in the direction of of the generalized system |

: | The force of the grinding system in the direction of of the generalized system |

: | The displacement of the crankshaft horizontal and vertical direction shift |

: | The displacement of wheel horizontal direction shift |

: | The rotation angle of crankpin |

: | The angle between and (Figure 3) |

: | The eccentric distance of the crankpin |

: | The radius of the grinding wheel |

: | The radius of the crankpin |

: | The normal grinding force |

: | The tangential grinding force |

: | The additional couple |

: | The friction coefficient between the contact surface of the grinding wheel and the crankpin |

: | The dynamic grinding force |

: | The time |

: | The grinding depth of the workpiece |

: | The grinding depth of the wheel |

The opposite of its direction, assuming the grinding depth of the workpiece is a positive direction | |

: | The grinding force coefficient of workpiece (or grinding wheel) |

: | The grinding depth of workpiece (or grinding wheel) |

: | The grinding contact width |

: | The vibration pattern of the surface of the workpiece (or wheel) for the moment |

: | The rotation period of workpiece (or wheel) |

: | The square of natural frequency for the system |

: | The system equivalent damping |

: | The frequency ratio. |

#### Conflict of Interests

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

#### Acknowledgments

This work is supported by National Science and Technology Major Project of China under Grant no. 2009ZX04001-111. It is also supported by Key Scientific Research Projects of Jiujiang University under Grant no. 2013ZD08. The authors are grateful for the financial support and also would like to thank the anonymous reviewers and the editor for their comments and suggestions.