1. Marshall V. Hall Kingsgrove, New South Wales 2208, Australia
Analytical model for the effect of a nonlinear piledriver cushion on underwater sound pressure waveforms radiated from offshore piledriving
Marshall V. Hall
Kingsgrove, New South Wales 2208, Australia

2. Introduction and aim
2009: a piledriver operated in Puget Sound (near Seattle); radiated sound signals were recorded
2011 & 2013: Dahl & Reinhall of University of Washington published data for underwater SPL
RD2011 stated “No cushion between the weight and pile was used”
2015: I published a paper which applied version 1 of my model CAMPRADOP to the Puget Sound acoustic data, assuming no cushion.
2017: Dahl advised that cushions had been used
AIM: To apply version 2 of CAMPRADOP (which can take account of a cushion) to the Puget Sound data

4. Examples of pile-driver cushions
Solid disks
Annular disks

5. Mass (kg) Radius (m) Length (m)
(length truncated)
Radius (m)
Length (m)
Ram
𝑣=7.6 𝑚/𝑠
6200
0.27
3.8
1285
Anvil
0.35
0.43 17
0.16
0.14
Cushion
The ram, anvil and helmet are assumed to be incompressible lumped masses. After impact, initial common velocity of ram and anvil is v = 7.6 * 6200 /( ) = 6.3 m/s
1265
Helmet
0.4
0.32
Pile
𝑐 𝑏𝑎𝑟 =5000
Wall thickness 2.5 cm
(length truncated)
Density
𝜌 𝑠 =7800
0.38 32

6. Forces on the cushion
Ram, anvil & cushion top 𝑧= 𝑧 𝑡 move together ( 𝑧 𝑡 is acceleration):
𝑀 𝑟 + 𝑀 𝑎 𝑧 𝑡 =− 𝑧 𝑡 − 𝑧 𝑏 𝐴 𝑐 𝐸 𝑐 𝐿 𝑐
Cushion bottom 𝑧= 𝑧 𝑏 , helmet & pile-head move together ( 𝑧 𝑏 is velocity):
Force 𝑝𝑖𝑙𝑒 = 𝐴 𝑝 𝜎 𝑝 ≅ 𝐴 𝑝 𝐸 𝑝 𝜖 𝑝 = 𝐴 𝑝 𝐸 𝑝 𝑧 𝑏 𝑐 𝑏𝑎𝑟 = 𝐴 𝑝 𝜌 𝑠 𝑐 𝑏𝑎𝑟 𝑧 𝑏
∴ 𝑀 ℎ 𝑧 𝑏 ≅− 𝐴 𝑝 𝜌 𝑠 𝑐 𝑏𝑎𝑟 𝑧 𝑏 − 𝑧 𝑏 − 𝑧 𝑡 𝐴 𝑐 𝐸 𝑐 𝐿 𝑐
𝐸 𝑐 is the “secant Young modulus” of the cushion: 𝐸 𝑐 = 𝜎 𝑐 𝜖 𝑐
𝜖 𝑐 = 𝑧 𝑏 − 𝑧 𝑡 𝐿 𝑐
𝜎 𝑐 is not proportional to 𝜖 𝑐 (hysteresis), so 𝐸 𝑐 depends on 𝜖 𝑐

7. Solution of nonlinear ODEs
The two simultaneous (coupled) 2nd order ODEs may be transformed into four simultaneous 1st order ODE’s.
In order to apply the Runge-Kutta method (4th “order”, 4 dependent variables)
to the resulting system of ODE’s, it is essential that:
(a) four dependent variables be identified (two must be derivatives),
(b) the first-order derivative of each variable be defined, and
(c) no derivative occurs in any definition (RHS).
Identify: 𝑦 1 = 𝑧 𝑡 , 𝑦 2 = 𝑧 𝑏 , 𝑦 3 = 𝑧 𝑡 , 𝑦 4 = 𝑧 𝑏 :
𝑦 1 = 𝑦 3 (1)
𝑀 𝑟 + 𝑀 𝑎 𝑦 3 =− 𝑦 1 − 𝑦 2 𝐴 𝑐 𝐸 𝑐 𝐿 𝑐 (2)
𝑦 2 = 𝑦 4 (3)
𝑀 ℎ 𝑦 4 =− 𝐴 𝑝 𝜌 𝑠 𝑐 𝑏𝑎𝑟 𝑦 4 − 𝑦 2 − 𝑦 𝐴 𝑐 𝐸 𝑐 𝐿 𝑐 (4)

8. Fishbein: solid rod of synthetic resin with linen base
Lowery: Micarta annular disk; internal radius = 0.35 external
(note concave “toe” region near origin)

12. CAMPRADOP: “Close (range) Analytical Model for the Pressure Radiated from a Driven Offshore Pile” Method for obtaining expression for radiated sound pressure based on the “Transform formulation of the pressure field of cylindrical radiators” (Junger & Feit, Sound, Structures, and Their Interaction, 1993)
Pressure waveform at horizontal range 𝑟, receiver depth 𝜁 is denoted by 𝑝(𝑟,𝜁,𝑡) . Its frequency Fourier Transform is denoted by 𝑃(𝑟,𝜁,𝜔)
Junger & Feit analysis uses the vertical wavenumber () Fourier Transform of the depth-dependence of radial vibration: 𝐹(𝛾, 𝜔) = 𝐹𝑇{𝑊 𝑧,𝜔 } ; 𝑊 𝑧,𝜔 is radial displacement spectrum at depth z below the pile head
An expression for 𝑊 𝑧,𝜔 in terms of mechanical properties of the pile driver and pile has been obtained by Hall (2015)
Junger & Feit result for 𝑃(𝑟,𝜁,𝜔) includes an inverse wavenumber- Fourier Transform of a function that contains 𝐹(𝛾, 𝜔) as a factor

13. Measured sound pressure waveforms for underwater radiation
Pressure scale for each waveform: -100 to kPa
Measured at horizontal range 12m from the driven pile described previously.
VLA mounted on seafloor, depth 13 m.
Hydrophone depths (uncorrected) are given at the start of the time series. Tide-corrected depths are 0.6 m greater.
Source: Dahl & Reinhall (2013), JASA Express Letters.
Phase velocity in steel pile ~ 5 m/ms
Pile length 32 m --> Ring-around time 12 ms

15. Errors in model results for PPL
Cushion neglected
Cushion: Fishbein solid resin/linen
Cushion: Lowery annular micarta
Positive peaks
2.0 ± 2.6 dB
-0.7 ± 1.3 dB
-8.6 ± 1.1 dB
Negative peaks
4.1 ± 4.5 dB
2.2 ± 2.5 dB
-7.0 ± 1.8 dB

16. Conclusions
If the cushion is neglected, the disparity between model and measurement is generally between 0 dB and +9 dB.
If the Lowery Stress-Strain Curves for an annular micarta cushion are used, the disparities are generally between -10 dB and -5 dB.
If the Fishbein Stress-Strain Curves for a solid resin cushion are used, the disparities are generally between -2 dB and +5 dB.