Medical, Pharma, Engineering, Science, Technology and Business

Ho Chi Minh City University of Technology – Vietnam National University of Ho Chi Minh City, Vietnam

- *Corresponding Author:
- Pham HH

Ho Chi Minh City University of Technology

Vietnam National University of Ho Chi Minh City

Vietnam

**Tel:**+84 8 3864 7256;

**E-mail:**[email protected]

**Received date: ** May 14, 2017; **Accepted date: ** June 30, 2017; **Published date: ** July 04, 2017

**Citation: ** Pham HH, Nguyen PV (2017) Dynamic Analysis of Cam Manufacturing.
J Appl Mech Eng 6: 274. doi: 10.4172/2168-9873.1000274

**Copyright:** © 2017 Pham HH, et al. This is an open-access article distributed under
the terms of the Creative Commons Attribution License, which permits unrestricted
use, distribution, and reproduction in any medium, provided the original author and
source are credited.

**Visit for more related articles at** Journal of Applied Mechanical Engineering

In cam milling process, cutting force is a variant factor during every time period and cam has a quite complex profile that leads to alternate force direction. These consequently, create machine vibration. The dynamic behaviour of machine can be predicted approximately if it is represented by a mathematical model. This paper shows result of cam cutting machine’s dynamic, which used Lagrange’s equation to solve. In this case, the machine vibration is surveyed only dimensions such as X and Y through using cutting condition with alloy cutting tool to mill a 10 mm thickness steel cam. The machine is modelled into the two degree of freedom vibrating system follow X and Y direction. Each of X and Y table equal to the compound: stiffness, damper and mass, which applied as constant coefficients in Lagrange’s equation. On the other hand, analysing cam characteristic and milling process in detail provides the resultant cutting follow X and Y in order to become external force of previous equation. After giving data in sufficient that necessary for problem, Matlab Simulink displays the vibration of X, Y for two states tangent force factor Kt=299.3 and Kt=598.6. At the end, it gives a comparison between these states.

Cam mechanism; Milling; Dynamics

A cam – a part of the cam – follower mechanical system, which complex peripheral profile depends on follower movement rule is an important mechanism and used popular in automatic mechanical. Designers always expect the accuracy position of follower motion during all operating periods. It means that cam machining has to achieve a tolerance deviation. Vibration of cutting machine has a significant influence on affecting the success of machining. As an illustration, a high value cutting force ease to create a large amplitude of fluctuation that of course decrease the surface quality and dimensional tolerance [1-8]. Moreover, vibration causes the claim and machine duration to be harmful. As a result, the machine dynamic should be considered studying like a main manufacturing factor to handle issues above [9- 17]. There are several documents refer to cam machining. For instance, Rothbart [6] illustrated cam manufacturing method, tolerance and errors. In addition, Altintas [17] represent the mechanics of metal cutting. Stephen and Radze [14] mentioned kinematic geometry of surface machining in 2014. They play a role just like the basic theory for this search. The paper also supplies some new acknowledge such as improving the cutting force formulations and analysing the cam machine dynamic, which others do not yet. Furthermore, it provides the essential data to evaluate and choose suitable cutting condition in order to increase the accuracy of cam.

**Physical model the dynamic of cam machining**

A cam machine model is explored, include table X carries milling
cutting tool and table Y bears table X, both of them can only move one
direction which same their name (**Figure 1**). Tool has clockwise spins
around its axis and cam peripheral in the chosen cutting condition. In
contrast, cam is fixed stationary. In mathematical model, tables X–Y
become mass (m_{1}, m_{2}), damping (c_{1}, c_{2}) spring (k_{1}, k_{2}) and displacements
(x, y). Likewise cutter gives the information of cutter’s diameter (d)
feed rate (f) (or cutting centre velocity VB1), angular velocity ωc the
resultant cutting force Fc. Last of all, follower-cam displays S: follower
displacement (S= (φ)) follower’ angle rotation φ=φ (t), follower offset
e, roller’s diameter (D), cutting centre velocity, tangent velocity and
transitive velocity of follower (vc, vs) (**Figure 2**).

As mention above, machine’s kinematic depends on cam profile,
so firstly it needs to determine some factors relative (**Figure 2**). Cam
profile coordinates B at cam angle φ:

x coordinate of cam surface profile: xB=x=ecosφ-Ssinφ

y coordinate of cam surface profile: yB=y=esinφ+Scosφ

Radius of the roller center’s curvature s at B:

(1)

Cam profile radius at B:

(2)

Profile radius of milling cutting centre B1:

(3)

The pressure angle will be: α=tan-1[(S′-e)]/S (4)

The velocity of milling cutting centre: (5)

Therefore, follower’s angular velocity at t is:

(6)

Denominate determinated by: β=2π-π/2-α-(π- φ)=π/2-α+

Cam centre coordinate at A:

x coordinate of cam surface profile: xA=ecosφ-Ssinφ-ρsin (α-φ)

y coordinate of cam surface profile: yA=esinφ+Scosφ-ρcos (α-φ)

**Dynamic of milling processing**

Milling cutters can be considered to have two orthogonal degrees
of freedom as shown in **Figure 3**. The cutter is assumed to have number
of teeth with a zero helix angle. The cutting forces excite the structure
in the radial force F_{r} and tangential force Ft, causing displacement X
and Y. The dynamic displacements are carried to rotating tooth in the
radial or chip thickness direction with the coordinate transformation
of fdx.sin (φ_{1}+τ)+dy.cos (φ_{1}+τ)where 1 is the instantaneous angular immersion of tooth measured
anticlockwise from the cutting edge starts to contact work piece
andτ=sin^{-1} (0.5s/r). The resulting chip thickness consists of a static
part h, attributed to rigid body motion of the cutter, and a dynamic
component caused by the vibrations of the tool at the present and
previous tooth periods. Because the chip thickness is measured in the
radial direction, the total chip load [17] can be expressed by

h (t)={h+dxsin (-τ)+dycos (φ_{j}-τ)}g (j) (7)

The function g(φ_{j}) is a unit step function that determines whether
the tooth is in or out of cut, that is

The static chip load h (7) divided into 2 period:

If ,then

(8)

If, then

(9)

Where is the
angle between perpendicular direction work plane with h_{max} and out
of cutting area. In addition, start angle:

φ_{st}=-π/2+τ, cutting angle: φ_{st}=-π/2+τ-φ_{1}, φ_{ex}=-π/2-η, exit angle: and
φ_{p}=2π/Z, cutting pitch angle:

When mill on plane, the cutting force includes tangent and radial
cutting force F_{t}, F_{r}

F_{t}=K_{t}th=Ktt {h+dxsin (φ_{1}-τ)+dycos(φ_{1}-τ)}g(1) (10)

Fr=Krth=Krt {h+dxsin (φ_{1}-τ)+dycos(φ_{1}-τ)}g(φ_{1}) (11)

Project Ft (10), F (11) into X and Y direction get Fx, Fy

Fx=Ftcos [τ-φ_{j}]-Frsin [τ-φ_{j}] (12)

Fy=Ftcos[τ-j]-Frsin [τ-φ_{j}] (13)

Apply (12), (13) into cam machining in **Figure 4**, X and Y direction
force become

Fx=Ftcos [φ-α+τ-φ_{j}]-Frsin [φ-α+τ-φ_{j}] (14)

Fy=Ftcos [φ-α+τ-φ_{j}]-Frsin [φ-α+τ-φ_{j}] (15)

**Dynamics of cam cutting**

To survey the oscillation of this machine, Lagrange’s equation [2] is compatible to use

(16)

F is the total external force, T [2] and V [2] are the system kinetic and potential energy, respectively. They are defined in (17) and (18)

(17)

(18)

Combine equations (10) and (11) with equations (16-18) to derive the dynamic equation of cam machining (19) and (20):

(19)

(20)

**Example Model**

The input value includes machine parameters in **Table 1** and
follower displacement rule [4] in **Figures 5 and 6**.

Name | Value | Unit |
---|---|---|

Cam characteristic | ||

Follower offset e | 40 | mm |

Roller diameter D | 40 | mm |

Cam thickness (steel) t | 10 | mm |

Machine parameter | ||

Spring k1, k2 | 537728, 296881 | (N/mm2) |

Damping c1, c2 | 5 | Ns/mm |

Mass m1, m2 | 70, 100 | kg |

Cutting condition (up-milling) | ||

Cutter diameter (alloy) d | 20 | mm |

Main spindle speed n | 660 | rev/m |

Feed rate s | 0.18 | mm/tooth |

Feed rate (F= nSZ) |
3.96 | mm/m |

Thickness | 10 | mm |

Number of teeth Z | 2 | tooth |

Resultant cutting forceFcmax | 383.1 | N |

**Table 1:** The value input.

Use the data in Table 1, the follower angle rotation to (21), (22),
thus force factor K_{t}, K_{r} and follower angular rotation are:

(21)

(22)

Derive milling tool angle velocity ωc=22π (rad/s) 0≤φ1≤cos-1 0.7 so the so cutting time cutting time:

t_{p}=2π/(Zw_{c})=1/22 (s)

Finally, model dynamic equation will become (23) and (24), then they are solved by Matlab Simulink similarly.

(23)

(24)

**Figures 7-11** show the table X and Y oscillation in two different
stations: kt=299.3 and kt=598.6 during one cutting period (173.85s).

Natural circular frequency [1,2]:

(25)

(26)

Damping ratio,

(27)

Therefore, with the aid of Euler’s formula, the general solution x(t) and y(t) can be written in the form underdamping free vibration [1], derives the freedom vibrating equation of machine.

X=0.0023e^{-0.55t}sin [1624.5t] (mm) (28)

Y=0.0027e^{-1.13t}sin [2059.4t] (mm) (29)

Force F_{x} , F_{y} in Figures 6 and 8 is the result of the discontinuous
cutting of milling process. They also have the variant value during each
tool’s revolution.

The results show the relationship between cutting force and oscillation. Assume that cutting force increase 150%, yet other
conditions still unchanged. As anticipated, this boost as two times as
much as the amplitude of fluctuation and displacement of X, Y. In
other words, the amplitude of X change from 1.5 × 10^{-4} (mm) to 3 × 10^{-4} (mm) and other is 3.5 × 10^{-5} (mm) to 0.7 × 10^{-5} (mm). The maximum
X and Y displacements also have the same trend, those rise from 3.1
× 10^{-4}, 3 × 10^{-4}(mm) to 6.2 × 10^{-4} (mm), 6 × 10^{-4} (mm) sequentially.
Their shapes alternate between Fx and Fy In contrast, the frequency
responses are invariable, that means the input load and the table’s
physical characteristic almost affect to the table’s movement. Therefore,
the vibration is reduced quickly by increasing the stiffness of cutting
system: tool, clamp, machine, machine’s operating tightness. The rising
mass and damper of tables benefit for machining in the same way.

In this situation, cam cutting machine’s factors and the hardness of cam work piece are considered being unchanged. To keep machining process smoothly, it needs to decrease cutting force. Feed rate, cutting speed, thickness and width have the influence on cutting force. Tool material and abrasive cutter edge are in the same way. Those have to be controlled in strictness. The results are nearly similar as real milling models such as shoulder milling. It is useful for estimating the machining deviation.

The cam machining is modelled on elements: mass, springs, dampers of Lagrange’s equation. Also, milling force, external force of previous equation, is also analysed thoroughly. Then the machine’s vibration is completely achieved by using Matlab Simulink to solve Lagrange’s equation. The results illustrate explicitly the displacements and frequencies of machine’ table, those are correspondent with a rigid cutting system. Finally, this paper is a beneficial result to study the decline of cam machining deviation. In future, this model should be developed in real testing model to have more completely the evaluation of the dynamic of cam cutting machine.

The paper content is the result of project named “Development a NC milling machine for planar cams” – project code: C2015-20-03. The project is financially sponsored by the Vietnam National University of Ho Chi Minh City.

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