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.
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
Crankshaft.
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 [
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 [
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 [
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 [
The stability of the stable region and the unstable region can be visually described by the stability lobe diagram [
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 [
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
Tangential Point Tracing crankshaft grinding machine. 1: headstock, 2: grinding wheel, 3: wheel head, 4: tailstock, 5: worktable, 6: center rest, and 7: crankshaft.
Figure
Crankshaft Tangential Point Tracing grinding movement schematic diagram.
In Figure
The eccentric distance of the crankpin is
Here,
The movement equation of wheel center
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
A simplified dynamic model of Tangential Point Tracing grinding machine.
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
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 (
Schematic diagram of grinding force decomposition.
The relationship between normal grinding force and tangential grinding force can be written as follows:
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 [
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 [
Dynamic grinding force is usually proportional to the removal rate of material, which has the following formula:
After combining (
Laplace transformations to the upper formula can be
Then
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,
According to Euler equation,
After rearranging (
The real part of (
To make (
Bring (
It is shown that the presented approach can be used to predict the crankshaft grinding stability [
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:
The complicated phenomena in engineering cannot be concluded only by theoretical analysis and the accuracy of theoretical analysis results need be verified by experiment.
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 D06A10130 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
The test experiment of crankshaft circumferential stiffness.
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
The deformation and stiffness of crankpin.
The angle of crankpin (°)  The deformation (mm)  Stiffness of crankpin (N/mm) 

0  0.0213  9389.671 
5  0.0214  9375.934 
10  0.0213  9389.671 
15  0.0217  9216.590 
20  0.0218  9174.312 
25  0.0219  9132.420 
30  0.0218  9174.312 
35  0.0221  9049.774 
40  0.0228  8771.930 
45  0.0229  8733.624 
50  0.0232  8620.690 
55  0.0243  8230.453 
60  0.0240  8333.333 
65  0.0252  7936.508 
70  0.0259  7722.008 
75  0.0256  7812.500 
80  0.0260  7692.308 
85  0.0259  7722.008 
90  0.0261  7662.835 
95  0.0253  7905.138 
100  0.0257  7782.101 
105  0.0255  7843.137 
110  0.0247  8097.166 
115  0.0237  8438.819 
120  0.0234  8547.009 
125  0.0237  8438.819 
130  0.0237  8438.819 
135  0.0228  8771.930 
140  0.0228  8771.930 
145  0.0220  9090.909 
150  0.0216  9259.259 
155  0.0213  9389.671 
160  0.0214  9375.934 
165  0.0212  9433.962 
170  0.0215  9302.326 
175  0.0214  9375.934 
180  0.0218  9174.312 
185  0.0225  8888.889 
190  0.0226  8849.558 
195  0.0223  8968.610 
200  0.0225  8888.889 
205  0.0226  8849.558 
210  0.0224  8928.571 
215  0.0230  8695.652 
220  0.0227  8810.573 
225  0.0237  8438.819 
230  0.0249  8032.129 
235  0.0250  8000.000 
240  0.0255  7843.137 
245  0.0257  7782.101 
250  0.0255  7843.137 
255  0.0257  7782.101 
260  0.0252  7936.508 
265  0.0260  7692.308 
270  0.0254  7874.016 
275  0.0268  7462.687 
280  0.0270  7407.407 
285  0.0270  7407.407 
290  0.0268  7462.687 
295  0.0270  7407.407 
300  0.0268  7462.687 
305  0.0263  7604.563 
310  0.0250  8000.000 
315  0.0241  8298.755 
320  0.0241  8298.755 
325  0.0228  8771.930 
330  0.0231  8658.009 
335  0.0218  9174.312 
340  0.0218  9174.312 
345  0.0213  9389.671 
350  0.0215  9302.326 
355  0.0210  9523.810 
360  0.0213  9389.671 
Crankpin Stiffness of the experiment.
According to the experimental results and the structural characteristics of the crankshaft, we use cosine curve fitting its stiffness, as shown in Figure
Crankpin Stiffness of the experiment and fitting curve.
If we plug the parameters of Table
The parameter of the grinding process.






1025  (8.503903 + 
0.053  439.82  30 
Stability lobe diagram.
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 [
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.
In order to verify the correctness of the stability limit diagram, we carried out the relevant grinding experiments [
The specific parameters of the workpiece and grinding wheel.
Name of the workpiece  Material of the workpiece  Type of grinding wheel 

Diesel engine crankshaft  CK45  AWA(19A)white corundum mixed abrasive 
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
Crankshaft Tangential Point Tracing grinding setup is shown in Figure
Experiment setup.
The position of acceleration sensors.
The normal signals.
The chatter signals.
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
Crankpin surface after chatter in grinding.
Vibration Waviness of crankpin
Vibration Waviness of crankpin
The crankpin profile.
The normal
The grinding chatter
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:
The dynamic equation of the grinding system has been constructed by the dynamic analysis of the grinding system.
Expression of the critical crankshaft grinding depth has been developed based upon the work of Altintaş and Budak [
Through the experimental study, the law of crankshaft rigidity is obtained.
The stability of the grinding system can be predicted by using the method of drawing the stability diagram.
Experimental results show that the prediction method is consistent with the experimental data.
Equivalent stiffness of the support system of
Equivalent stiffness of the right support system of
Equivalent stiffness of the left support system of
Equivalent damping of the support system of
Equivalent damping of the right support system of
Equivalent damping of the left support system of
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
The equivalent damping of the grinding wheel system in the
Masses of the crankshaft
Masses of the grinding wheel
Equivalent stiffness of the support system of the grinding wheel of
Equivalent damping of the support system of the grinding wheel of
The force of the grinding system in the direction of
The force of the grinding system in the direction of
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
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.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is supported by National Science and Technology Major Project of China under Grant no. 2009ZX04001111. 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.