self-excited vibration ii

7/28/2019 Self-Excited Vibration II

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II. Self-excited vibration in machining operations

Sources of vibration in machining operations:

Forced vibration caused by rotating unbalances in machinery Self-excited vibration caused by positive feedback mechanism leading

to dynamic instability of cutting process

Consequences: Poor surface finish, reduction of tool lifetime, damage toworkpiece, tool or machinery


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Negative velocity dependence of cutting force due to thermal feedback:


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0 1 2 3 4

8

F0

F

C u t

t i n g

f o r c e

Cutting speed v / v1

Example: Exponential velocity dependence of cutting force

]exp[)()( 10 vvF F F vF +=


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II.1 Regenerative vibration in high-speed machining

Observation:

Surface oscillationfrequency at a givencutting speedapproximately matchesnatural frequency offree tool oscillations inperpendicular direction


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Feedback due to dependence of cutting force oncutting depth when cutting into previously machinedsurface

Increased surfaceundulation

Increasing oscillationin cutting force magnitude

Increasing oscillation ofcutting tool

cutting width

Phase lag betweencutting force andtool oscillation


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Mechanical model of regenerative tool vibration:

dh

dF F

t xT t xhF hF kx xc xm

x=

+==++

'

)]()(['' 0&&&

is proportional to cutting width w


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k F

t xT t x x x x

'

)]()([2 20200

=

=++

&&&

After shifting coordinates to eliminate constant force F h0 and transformationto canonical form

Crucial argument: Boundary of stability separates a regime of exponentiallydamped oscillations from a regime of exponentially growing oscillations

At boundary of stability: Oscillations are neither growing nor damped

Consequence: We can assume solution in the form )exp()( t i X t x =


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[ ]

( )[ ] ( )( )[ ]20

20

2220

020

2

12

41

1)exp(21

+=

=

+

M

T ii

Minimal critical value of the feedback parameter: At

Below this parameter, no instability can occur at any frequency. In physicalterms, this means that there is a minimal chip width below which cutting isabsolutely stable irrespective of spindle speed.

( ) 2120 +=

)1(2min +=


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Diagram has the shape of a sequence of lobes. Interpretation of the lobediagram:

Instability occurs first where the spindle speed (revolutions per second)times the number of oscillations on the workpiece surface (the lobenumber)matches the natural frequency of the tool oscillation (in Hertz)

At higher cutting parameters the instability boundaries expand andultimately merge.

For each cutting parameter there is a highest spindle speed wherethe cutting operation becomes absolutely stable.


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Summary: Sources of vibration in machining operations

Forced vibration due to rotating unbalances in machinery -> vibrationamplitude varies weakly with machine speed, vibration amplitude isindependent of cutting parameters, vibration frequency = rotation speed

of machine Self-excited vibration -> vibration amplitude varies strongly with spindlespeed and chip width (vibrations emerge suddenly upon a smallchange of parameters), vibration frequency = natural frequency of tooloscillations

Consequences: Poor surface finish, reduction of tool lifetime, damage toworkpiece and machinery.

Diagnostics: Measure frequency and amplitude of vibration while varying

spindle speed and chip width.

Mitigation: If forced vibration: move rotation speed out of resonance.If self-excited: fine tune rotation speed, reduce chip width.

self-excited vibration ii

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