Advanced Manufacturing Processes: formulario / formulary

Numerical aperture = arcsin⁡()

Acceptance angle (half angle with respect to the horizontal) 2

( )

2

=

2

Number of modes in a fiber 02 02

2 2 2 2 2

() = + θ = + 4 ∙ Θ

θ = 2 ∙ Θ,

Caustic equation: is the waist diameter, is the distance at which we want to calculate the spot

0

diameter (generally we calculate it subtracting that is the distance between lens and waist: it is the

0

distance between the waist and our spot diameter; in a high number of exercise it is equal to the thickness).

=

Typical relation between the diameter of the output beam and the diameter of the beam at the lens

≈

⁡is

Consequence of the previous formula: the focal length, which is the distance from the lens plane (from

the incident beam axis in case of mirrors) to the waist. It is equal to the real beam focal length generally

0

because the divergence angle is small.

!!! if we have a collimation lens (optic fibre) we need to use:

≈

Where is the collimation lens diameter that is calculated as:

4

2

≈ ∙ ∙ ≈ ∙

0 0

4

2

= ∙ ∙ = 4 ∙

0

Waist diameter of a non-Gaussian beam. It is also valid to calculate the collimation diameter between the

= =

two lenses of an optical fiber ( and ).

<

0

2

= 0

1

2 2

= ∙ ∙ = ∙ ∙

0 0

Rayleigh distance (from caustic equation): the distance from the beam waist to the position where the beam

=

radius , meaning that in this region the beam diameter is almost constant, leading to constant

√2

0

effectiveness (same intensity).

= ∙

0

Waist diameter of an optical fiber = +

= ∙

(1

= − ) =

Relations among absorptivity, reflectivity, transmitted, incident and reflected power.

= =

Interaction time: is the laser beam velocity, is the delivered energy and the power.

∙

"

= = = ∙

2

4

Absorbed heat flow on the surface = ∙ = ∙ ∙

Transmitted energy density

′

= = [ ( − ) + + ]

∙∙

Laser cutting energy per unit volume: is the average kerf width (generally equal to the spot diameter),

′

the thickness of the workpiece, the cutting speed, are latent heat and the fraction of melt that is

= 1 − ].

vaporized, A is the absorption coefficient [



If we need to verify that power needed to cut a given thickness is feasible (lower than allowed

′

= = [ ( − ) + + ] ≤

by the system):

 ′

= 1 Δ = −

In evaporative cutting don’t consider ( and )

 The energy in J can be obtained multiplying by the volume of the hole (typically a cylinder with height

t) :

′ 3

[ ]

= ∙ ⁡ ∙ ∙ ∙ ∙ [ ( − ) + + ] = ∙ [ ] = []

ℎ ℎ 3

ℎ ℎ

= = ∙ ∙ ∙

ℎ ℎ €

⁡⁡⁡ℎ = ∙

ℎ ℎ

⁡ℎ⁡(€)

⁡ =

= ∙ ∙ %

ℎ

€ ⁡⁡⁡ℎ

=

Useful economic relations

Waterjet 1−

2

√

, = [(1 + ) − 1]

ℎ (1 − )

0

= 300⁡ = 0,1368

Theoretical compressible WJ velocity: and at ambient T

1−

ℎ, =√

Ψ= [(1 + ) − 1]

(1

− )

ℎ

Compressibility coefficient 2

√

=

ℎ

0

Theoretical velocity. Remember to put in and in in order to get .

0 3

2

√

= ∙ Ψ ∙ = ∙ Ψ ∙

ℎ

0

=

ℎ,

Velocity coefficient

= =

0

Contraction coefficient 2

2

√

= ∙ =

ℎ 0 ℎ 4

0 3

, , /

Theoretical water flow rate ( in = primary orifice nominal value in in to achieve

0

3

/⁡then / 1000 ∙ 60)

convert in by multiplying it for

= ∙ = ( ∙ )( ∙ )⁡

0 ℎ,

3 3

, , / /⁡then /

Actual flow rate ( in in in to achieve convert in by multiplying it for

0

1000 ∙ 60). ∙

Increase the number of heads: and check if it is feasible (comparison with or

,

generally).

, ̇ = ∙

0

/

Actual flow rate in

= Ψ =

ℎ

Overall discharge coefficient −4

= 0,785 − 1,4 ∙ 10 ∙ − 0,197 ∙

Sapphire orifice with sharp edges empirical formula ∗

= ̇ ∙ |Δ | = ( ∙ + ̇ ) ∙ | − ⁡| = ( ∙ + ̇ ) ∙ ( ⁡ − (− ))

⊥ 0 , ,− 0

Jet reactive force (momentum balance): is the angle of incidence and of reflection of the jet (they are

equal). The reflected velocity is assumed as equal to the incident one or most commonly the 80%.

2 2 2

= ∙ ∙ ∙ Ψ ∙ ∙

2

Another possible formulation of the jet reactive force (only pure waterjet at normal incidence).

1

2 3 3 2 1,5

= ∙ ̇ ∙ = ∙ ∙ ∙ ∙

2 8

√ 0

Hydraulic power of the jet (normal impingement). If there is a threshold value to cut more than one sheet,

compare with that threshold. It never has to be higher than the maximum deliverable power of the

3

/

pumping system. TBN: the second equation allows to find the power directly in if is in , in

0

.

and in 1 2

̇ ∙ ∙ Δ = ̇ ∙

2

2 2

Ψ

Δ =

0

Variation of temperature in the workpiece or in the catchers.

2

0 1,5

√

=∙ ∙ =∙ ∙ = Ψ

0 0 0 0

⁡is

Depressurization. the water stiffness, is a volume

0

̇

+

̇

=

̇

1+ ̇

Relation between water and abrasive (abrasive loading ratio)

̇

= ̇

Abrasive-to-water ratio or abrasive loading ratio. Increase of abrasive flow rate will decrease (becoming

far from (pure water).

( − )

+ = 1⁡⁡

( )

−

,

(2 − )

= ∙

ℎ √ ∙

1 ℎ

2 ∙ ∙ [ +

1 − 2 ∙

ℎ

Mixing chamber equation (relation between air and vacuum) and its solution.

−



=

 , , are the section, the length and the diameter of the hose (feeding line)

ℎ ℎ ℎ

 is the flow friction coefficient

=

2 2

2∙ ∙ ∙ ∙ (1 − )

2 2

2

= ⁡

2

{

= ⁡⁡

= ⁡

Critical pressure: maximum pressure beyond which no more vacuum is created.

−

= −

≤ ≤ <

Pressure ratio 2 2

= 2 ∙ ∙ ∙ ∙ (1 − )

2 2

The max flow rate M occurs for frictionless flows, when N=0; increasing the back pressure will always reduce

≥

the suction flow rate if other parameters are held constant; if backflow occurs (no more abrasive

suction) 1 2

( )