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Medical Device & Automotive Polymer Joining, 20/35 kHz Ultrasonic Plastic Welding

Ultrasonic Plastic Welding Horn Gain & Heating Formula

Calculate Acoustic Stack Peak-to-Peak Horn Amplitude (um), Peak Sonotrode Tip Velocity (m/s), Energy-Director Cyclic Strain Amplitude, and Viscoelastic Melt Power (W).

Viscoelastic ultrasonic weld heating scales with the SQUARE of booster-horn amplitude—swapping a 1.5x gold booster for a 1.0x green booster cuts melt power 7.5x!

**Ultrasonic Plastic Welding** (`20`, `30`, or `35` kHz Frequency `f` kHz) converts piezoelectric PZT transducer vibrations of Zero-to-Peak Converter Amplitude `A_0` (`um`, typically `10` um = `20` um peak-to-peak at `20` kHz) through a half-wave Titanium **Booster Gain `G_{boost}`** (`0.6x–2.5x`) and Stepped/Catenoidal **Sonotrode Horn Gain `G_{horn}`** (`1.2x–3.0x`) into **Horn Face Zero-to-Peak Amplitude `A_{0-p} = A_0 * G_{boost} * G_{horn}`** (`Peak-to-Peak = 2 * A_{0-p}` um) and **Peak Vibrational Velocity `v_{max} = 2 * PI() * (f * 1,000) * (A_{0-p} * 10^{-6})`** (`m/s`)! Concentrating `A_{0-p}` into a molded triangular **Energy-Director Rib** of Effective Height `h_{rib}` (`mm`) and Active Melt-Zone Volume `V_{ed}` (`mm³`) imposes Cyclic Strain `epsilon_0 = (A_{0-p} / 1,000) / h_{rib}`, generating **Internal Hysteretic Loss Power `P_{visc} = PI() * (f * 1,000) * (E’’ * 10^6) * epsilon_0^2 * (V_{ed} * 10^{-9})`** (`W`, where Polymer Loss Modulus `E’’ = E’ * tan(delta)` MPa)!

Verify on a `20` kHz Semi-Crystalline Polypropylene / PC-ABS Medical Filter Housing Weld—Converter_Zero-to-Peak `A_0` = 10.0 um (B2), Gold_Booster_Gain `G_{boost}` = 1.50 (B3), Stepped_Titanium_Horn_Gain `G_{horn}` = 2.20 (B4 -> `66.0` um Peak-to-Peak in E2), Frequency `f` = 20.0 kHz (B5 -> Tip Velocity = `4.15` m/s in E3), Polymer_Loss_Modulus `E’’` = 285 MPa (B6), Energy-Director_Zone `V_{ed}` = 48.0 mm³ (B7), and Rib_Height `h_{rib}` = 0.85 mm (B8): Cyclic Strain hits `0.0388` (E4) and Viscoelastic Melt Power delivers a rapid `1,295.6` W (`0.25`-s weld cycle! E5)!

20 kHz Ultrasonic Stack KPI (A0=10 um, Epp=285 MPa, Ved=48 mm³, h=0.85 mm)FormulaVerified Output
Horn Peak-to-Peak Amplitude (um)2 * A0_um * Booster_Gain * Horn_Gain66.0 um p-p (24.0 um p-p @ 1.0x/1.2x stack!)
Peak Sonotrode Tip Velocity (m/s)2 * PI() * (f_kHz * 1000) * (A0p_um * 1e-6)4.15 m/s (1.51 m/s @ low-gain stack)
Energy-Director Strain eps0(A0p_um / 1000) / Rib_Height_mm0.0388 (0.0141 @ low-gain stack)
Viscoelastic Heating Power (W)PI() * f_Hz * Epp_Pa * eps0^2 * Ved_m31,295.6 W (171.3 W stalls unmelted!)

Acoustic Stack Gain Cascade & Dynamic Loss-Modulus Hysteretic Power Formula

Validate `Converter A_0` in `[3, 25]` um, `Gains G_{boost}, G_{horn}` in `[0.4, 4.0]`, `Frequency f` in `[15, 70]` kHz, `Loss_Modulus E’’` in `[20, 1,500]` MPa, `Volume V_{ed} > 1` mm³, and `Rib_Height h_{rib}` in `[0.2, 3.0]` mm!

=LET(az_u, B2, gb_x, B3, gh_x, B4, fk_h, B5, ep_m, B6, vd_m, B7, hr_m, B8, ap_u, az_u * gb_x * gh_x, es_z, (ap_u / 1000) / hr_m, IF(OR(az_u < 3, az_u > 25, gb_x < 0.4, gb_x > 4, gh_x < 0.4, gh_x > 4, fk_h < 15, fk_h > 70, ep_m < 20, ep_m > 1500, vd_m <= 1, hr_m < 0.2, hr_m > 3), "Invalid Ultrasonic Stack", ROUND(PI() * (fk_h * 1000) * (ep_m * 1000000) * (es_z ^ 2) * (vd_m * 0.000000001), 1)))

Under-Driven 1:1 Green Booster + 1.2x Block Horn Cold-Collapse Check (Set B3 = 1.0, B4 = 1.2)

Mount a `1.0x` booster and `1.2x` large-face block horn (`B3 = 1.0`, `B4 = 1.2`): Peak-to-Peak Amplitude E2 drops from `66.0` to `24.0` um (`2.75x` lower), which SQUARES (`2.75^2 = 7.56x`!) to collapse Viscoelastic Heating Power E5 from `1,295.6` down to **`171.3` W**—cold-mashing the energy director flat before reaching polymer melt temperature!

=IF(ISTEXT(E5), E5, LET(ap_u, B2 * B3 * B4, ROUND(2 * PI() * (B5 * 1000) * (ap_u * 0.000001), 2)))

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