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Effect of Joint Stiffness on the Dynamic Performance of Machine Tool

This paper presents the effect of the joint stiffness on the dynamic performance of the machine tool by using the frequency response function (FRF) of the tool-spindle system, structure, and measurement of vibration (power spectrum) during heavy milling, and analysis using the stability lobe diagrams (SLD). These methods of measurement show how non-uniformed scraped surfaces and improper threaded holes are the cause for weak joint stiffness and poor machine tool performance.


Figure 1
FIGURE 1. Tool marks generated on the workpiece are from the tool itself
Figure 2
FIGURE 2. Bridge type machine tool. (Courtesy of Vision Wide Tech Co., Ltd.)

Machine tool performance has many influencing factors from feed rates, spindle rotation, various systems that turn on and off during the machining operation, thermal coefficients, and the stiffness of the machine tool and more. Joint stiffness is one factor that is typically overlooked in evaluating the performance of machine tools.

Machine Tool Chatter Problem

Machine tool manufacturers typically will perform an in-house test that increases the amount of material removal beyond normal specifications to investigate the shape and surface accuracy on the workpiece to determine if the machine tool is working properly prior to shipping. The bridge type machine tool in our case study is designed with a high material removal rate (MRR) but exhibits chatter in the workpiece. The milling chatter occurs when the spindle head is at the position (Z= -800mm) and only in one feed direction (-X). The specifications of the machine tool and the milling conditions are listed below.

Machine tool main specifications:

  • X / Y/ Z travel: 2200/ 2300/ 920 mm
  • Spindle: Gear type/ BT50 (6,000 RPM)/ 22KW

Milling Conditions

  • Spindle speed: S= 500 RPM= 8.33Hz
  • Feed rate: F= 1500mm/min, Ft= 0.375 mm/flute
  • Tool: Sandvik R200-140Q40-20M 160mm, RCKT2006M0-PH4240 with 8 flutes
  • Workpiece material: S45C
  • Axial depth of cut (DOC): 5 mm
  • Radial width of cut (WOC): 100mm
  • Z axis position: -800mm

FRF Test on Spindle and Structure

Equipment used: 1-axis accelerometer (Endevco 27AM1-10), 3-axis accelerometer (Kistler 8762A50), small modal impact hammer (Kistler 9726A5000), big modal impact hammer (Kistler 9728A20000) and 4 channel real time FFT analyzer (Benstone impaq Elite).

Figure 3
FIGURE 3. FRF of the tool-spindle system.
Figure 4
FIGURE 4. FRF of the structure at Z=-800mm.
Figure 5
FIGURE 5. X-axis FRF of machine tool structure at Z= -650mm, Z= -700mm, Z= -800mm.
Figure 6
FIGURE 6. SLD of the tool-spindle system FRF only -- showing expected DOC of 6.227 mm and milling force of 5 mm DOC milling (no chatter).[6]
Figure 7
FIGURE 7. SLD of 3.138 mm of the structure and spindle. Simulated chatter observed at 5 mm DOC cutting force. [6]

The FRF of the tool-spindle system is measured by the 1-axis accelerometer which is placed directly on the tool itself and the small modal hammer strikes the tool tip on the opposite side. The measured results of the FRF as displayed in the graph (fig. 3) shows that the tool-spindle system has high natural frequency at 609Hz (X), 591Hz (Y), and 308Hz (Z).

The FRF of the structure is then measured by using the 3-axis accelerometer, a big impact hammer and FFT analyzer. The 3-axis accelerometer is placed directly on the spindle housing and the modal hammer strikes the spindle housing on the opposite side. The measured results of the FRF as displayed in the graph (fig. 4) shows that the structural stiffness of X-direction is lower than the Y-direction or Z-direction.

The spindle position is then moved from Z= -800 mm to Z= -700 mm, and Z= -650 mm, and the test repeated. The X-axis first natural frequency of FRF changes from 53.8Hz to 54.1Hz and 56.3Hz respectively. The dynamic stiffness (2k (1- 2) ) and the damping ratio changes dramatically, indicating that the dynamic stiffness of the structure is sensitive to the Z axis position (fig. 5). The X-axis dynamic stiffness of the first natural frequency is 12.8 N/ m at Z= -800mm (fig. 4).

Stability Lobe Diagram (SLB)

The stability expression was first obtained by Tlusty [1]. Tobias [2] presented a similar solution. Altintas [3] did much more detailed research in the field. Stability lobe diagrams show the maximum depth of cut that you can expect from the machine tool. The SLD and cutting force graphs are obtained by using CUTPRO software [6] (Manufacture Automation Laboratory Inc.) by inputting the measured FRF readings. For high speed machining or heavy machining, if the dynamic stiffness of the structure is higher than the tool-spindle, then the cutting chatter is dominated by the tool-spindle dynamic stiffness, called regenerative chatter. So, we consider firstly the tool-spindle system FRF for evaluation using the SLD.

A machine tool very often has structural stiffness greater than 50N/ m. However, our results show a weak structural dynamic stiffness. When we add the structural dynamic stiffness together with the tool-spindle FRF to create a new SLD, it shows the expected maximum depth of cut to be reduced to 3.138 mm (fig. 7).

This tool-spindle system is estimated to be capable of milling a 6.227 mm DOC without chatter, but we cannot achieve this depth of cut. The estimated maximum DOC is only 3.138 mm for the structure (fig. 7). Our conclusion is based on the SLD and FRF analysis that the chatter problem is a result from the structural dynamic weakness, not from the spindle.

Milling and Vibration Test

To perform the vibration spectrum measurement, a 3-axis accelerometer is placed on the spindle head with a 5 mm depth of cut and 100 mm width of cut milling, and the results recorded. A tooth passing frequency is observed of (66.7Hz) and a chatter frequency of (51Hz) which is excited when the chatter vibration occurs in the A area (up milling).

Notice that there is no chatter in the C area. The vibration power spectrum shows the tooth passing harmonic frequencies (fig. 8). The milling force is the same in these two areas. But the force direction acting on the spindle is opposite (fig. 9), which means the direction of force changes the structure dynamic stiffness. This is another indication of a weak structure.

Figure 8 Figure 9
FIGURE 8. Vibration spectrum as milling test with 5 mm DOC & 100 mm WOC. FIGURE 9. Milling force and direction acts on the spindle. (A area: milling-force pulls the spindle out from structure, C area: milling-force pushes the spindle into the structure.)

Modal Testing

Modal testing is a proven method to find the location of the mode shapes on the machine tool. The input force and the model are shown in Figure 10. The FRF data is shown in Figure 11.

Figure 10 Figure 11
FIGURE 10. The measuring points and force input of modal testing. [7] FIGURE 11. The FRF and joint mode frequencies as shown on the plot.

The natural frequency, damping ratio and mode shapes are estimated by using ME'scope software [7] based on the modal values obtained from the Benstone Instruments' Impaq elite analyzer. They are listed in Figure 12.

Mode Frequency (Hz) Damping Ratio (%)
 1 19.8  2.1
 2  40.4  1.8
 3  54.3  13.1
 4  62.9  7.7
 5  78.9  1.5
 6  85.6  1.6
 7  92.5  1.4
Figure 13
FIGURE 12. The natural frequencies and damping ratios are shown in the table. FIGURE 13. Mode 3 (54.3Hz, damping ratio 13.1%) is the joint separation motion between the spindle head and the saddle. [7]

The 1st mode (19.8 Hz) is rigid body motion. The 2nd mode (40.4 Hz) is an elastic mode. The double columns twist in this mode. The 3rd mode (54.3Hz) and 4th modes which have high damping ratio (13.1%) are joint separation motion (fig. 13). The damping ratio is normally below 5% for the structure of a machine tool. The 3rd mode (54.3Hz) is the cause of milling chatter in this case study (very close to 53.8 Hz as measured with FRF and vibration power spectrum 51 Hz). Opposite direction of milling force causes the joint to be stiffer or softer. That motion also changes the dynamic performance, the FRF and the SLD.


The spindle head is installed into the hard rail of the saddle. The joint interface includes the bolt, thread, and the scraped surface of the hard rail. If the scraped surface is not uniform or is configured with inadequate contact points, plastic deformation will occur in the contact points when force is applied during machining. Therefore, joint stiffness and heavy milling ability are reduced. The weak joint stiffness is sensitive to the amplitude of milling force. Because the contact area is not uniform a crater lip is formed near the thread upon tightening and milling force (fig. 14).

Figure 14 Figure 15
FIGURE 14. The scraping surface is not uniform and the tightening force form a crater lip. FIGURE 15. Rebuild the scraping surface and the countersink to get a stiffer joint.

To correct this problem, threads are redesigned to start a distance below the surface to avoid forming a crater lip upon bolt tightening and during milling force. Both surfaces of the saddle are finely re-scraped to achieve a more uniform surface and thus proper contact points. The rule of scraping and bolt joint are presented by Moore [5] and Slocum [6].

The FRF of the rebuilt joint has a much higher dynamic stiffness (fig. 16). The dynamic performance is also improved in SLD (fig. 17).

Figure 16 Figure 17
FIGURE 16. The new FRF is stiffer than the previous one. FIGURE 17. The SLD of the rebuilt joint structure shows a higher depth of cut. [6]


This paper describes the approach of using frequency response function (FRF), power spectrum measurements and stability lobe diagrams (SLD) to clearly identify weak joint stiffness that contribute to machine tool performance during machining. Using these techniques also help understand the amount of improvement achieved after correcting the discovered problems.


About our Authors

Faby Fong and Robert J. Benjamin of Benstone Instruments

Benstone Instruments is a maker of signal analyzers for the sound and vibration industry.  They feature portable stand-alone units not requiring computers as well as expandable PC-based systems.

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Impaq Elite

In accomplishing this work, we acknowledge the helpful support of Professor Altintas and the Vision Wide Tech Co., Ltd.


  1. J. Tlusty and M. Polacek. The Stability of Machine Tools Against Self Excited Vibrations in Machining. International Research in Production Engineering, ASME.
  2. S. A. Tobias and W. Fishwick. A Theory of Regenerative Chatter. The Engineer Londonm 1958.
  3. Yusuf Altintas. Manufacturing Automation-Metal Cutting Mechanics, Machine Tool Vibration, and CNC Design.
  4. Wayne R. Moore. Foundations of Mechanical Accuracy. The Moore Tool Company, Inc.
  5. Alexander H. Slocum. Precision Machine Design.
  6. CUTPRO Advanced Machining Process Simulation Software, Manufacturing Automation Laboratories Inc.
  7. ME’scopeVES Modal Analysis Software, Vibrant Technology, Inc,