= ⋅π⋅ω⋅τ ⋅ − ⋅π⋅ω⋅τ ⋅ 2 3 Total contact shoulder 2 9 3 + ⋅ω⋅τ ⋅ ⋅ + ⋅π⋅ω⋅τ ⋅ contact contact probe contact 2 9 3 2 3 1 2 = ⋅π⋅ω⋅τ ⋅ + ⋅ω⋅τ ⋅ ⋅ 1 2 Total contact contact probe shoulder ⋅ ⋅ ⎛⎝ ⎜ 2 3 = ⋅π⋅ω⋅τ ⋅ + 3 4 Total contact shoulder probe π ⋅ ⋅ ⎛⎝ ⎜ 2 3 9 4 The energy per unit length of the weld can be calculated as dividing Equation 6 by weld speed () as ⋅ ⋅ ⎛⎝ ⎜ ⋅ ⋅ ⎛⎝ ⎜ Where contact = friction = p = , pressure equals the force divided by the projected area. Analytical Heat Generation Equation for Square Pin Profile The heat generation from the shoulder surface is calculated by subtracting the heat generated due to probe tip (Q3) from the heat generated due to the shoulder (QShoulder) where a = Rprobe ·2 is the side of the SQ. Top Left Probe Side (Q2), Bottom Left Square Probe (Q3), and Right Shoulder Surface Similarly, the heat generation from the tip surface (Q3) is given by the general expression (1) from Fig. 3 (bottom left), an infinitesimal segment having the width dx and height dy by considering only 1/8th part, which is integrated and converted from rectangular to polar coordinates with the limits given in the below expression as follows: R probe 2 θ=π 4 = = ∫ ∫ ω⋅τ ⋅ ⋅ ⋅ θ⋅ 3 (8) The multiplication term 8 denotes the eight different regions that form a SQ surface. Therefore, Q1 is calculated as Qshoulder – Q3, i.e., Equations 2–8 = ⋅π⋅ω⋅τ ⋅ −π⋅ω⋅τ ⋅ 2 3 contact 3 3 2 3 1 (9) Q R a contact shoulder With the heat generation from the probe surface, considering rectangular coordinates, we get = = H H R a probe probe = ∫ ∫ ω⋅τ ⋅ ⋅ ⋅ Q R dx dy = = 0 0 a H R probe 2 2 =ω⋅τ ⋅ ⋅ H contact probe contact probe 4 (10) The multiplication term 4 denotes the four sides of the SQ probe. From Equations 8–10, QTotal can be calculated as QTotal = Q1 + Q2 + Q3 2 3 3 2 Total contact shoulder contact contact 3 2 2 3 2 3 Total contact shoulder contact probe ⋅ ⋅ ⎛⎝ ⎜ 3 2 Total contact shoulder probe ⋅ ⋅ ⎛⎝ ⎜ contact probe 2 3 3 The energy per unit length of the weld can be calculated by dividing Equation 11 by the weld speed () as ⋅ ⋅ ⎛⎝ ⎜ ⋅ ⋅ ⎛⎝ ⎜ Analytical Heat Generation Equation for Pentagonal Pin Profile The heat generation from the shoulder surface is calculated by subtracting the heat generated due to the probe tip (Q3) from the heat generated due to shoulder (QShoulder). a = Rprobe/0.8506 is the side of the PEN pin profile. The simplified tool design for the PEN pin is presented in Fig. 4. Therefore, Q1 is calculated as Qshoulder – Q3, i.e., Equations 2–15 Q R R d dr R a contact probe probe probe 0 0 = θ= = ⋅ ω⋅μ⋅ υ⋅ ⋅ + π ⎞⎠ ⎟ = ⋅ ω⋅μ⋅ υ⋅ ⋅ + π ⎞⎠ ⎟ Q F R R a H or Q F R R R H Energy/length shoulder shoulder probe Energy/length shoulder shoulder probe probe 2 3 3 4 2 3 9 4 (7) 2 3 2 2 3 2 ⎞⎠ ⎟ = ⋅π⋅ω⋅τ ⋅ + π ⎞⎠ ⎟ Q R a a H a Q R a H Q R a H or Q R R H Total contact shoulder probe probe (6) 3 3 2 3 3 2 3 2 3 2 = ⋅π⋅ω⋅τ ⋅ −π⋅ω⋅τ ⋅ +ω⋅τ ⋅ ⋅ + π⋅ω⋅τ ⋅ = ⋅π⋅ω⋅τ ⋅ +ω⋅τ ⋅ ⋅ = ⋅π⋅ω⋅τ ⋅ + π ⎞⎠ ⎟ = ⋅π⋅ω⋅τ ⋅ +π ⎞⎠ ⎟ Q R a a H a Q R a H Q R a H or Q R R H Total contact shoulder probe probe (11) 3 3 2 3 3 2 3 2 3 2 = ⋅ ω⋅μ⋅ υ⋅ ⋅ + π ⎞⎠ ⎟ = ⋅ ω⋅μ⋅ υ⋅ ⋅ + π ⎞⎠ ⎟ Q F R R a H or Q F R R R H Energy/length shoulder shoulder probe Energy/length shoulder shoulder probe probe 2 3 3 2 2 3 3 (12) 2 3 2 2 3 2 WELDING RESEARCH 120-s WELDING JOURNAL / APRIL 2015, VOL. 94
Welding Journal | April 2015
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