I recently attended a joint workshop on the analysis, design, and testing of wave-energy converters (WECs) by the Pacific Marine Energy Institute (PMEC), Sandia National Labs, and OSU. A quote during one of opening presentations caught my attention: “It is estimated that if 0.2% of the ocean’s untapped energy could be harnessed, it could provide power sufficient for the entire world.” (Brekken, 2020)

WECs convert wave power into electricity as a source of renewable energy. At first, WEC design may seem far removed from earthquake-resistant design. However, the following equations for a WEC modeled as a single-degree-of-freedom oscillator may look suspiciously familiar to structural engineers:

$latex F_{e}+F_{r}+F_{h}+F_{m}-F_{pto} = m\ddot{u}$

where $latex u=$ displacement response of the body and $latex F_e$, $latex F_r$, $latex F_h$, $latex F_m$, $latex F_{pto}$, and , $latex m\ddot{u}$ are defined as the excitation, radiation, hydrostatic (buoyancy), mooring, power-take off, and inertial forces acting on the body, respectively.

The waves affect the motion of the WEC (defined by $latex F_e$). The WEC, in turn, affects the wave excitation [defined by $latex F_r=-(A\ddot{u}+B\dot{u})$ as a function of the body acceleration (added mass effect, $latex A$) and velocity (radiation damping)].

The goal is then resonance, something that we are quite familiar with in seismic design. Dynamic magnification (output response being magnified relative to the input motion) is then leveraged maximize the power delivered from the ocean to the device (defined by the power, $latex P_{pto} = F_{pto}\dot{u}$).

In its simplest form, the power take-off force can be written as a function of displacement and velocity, $latex F_{pto}(\dot{u},u)=B_{pto}\dot{u}+K_{pto}u$. Putting everything together results in the following equation of motion:

$latex F_{e}-F_{pto} = (m+A)\ddot{u}+(B+B_{pto})\dot{u}+(K+K_{pto})u$

where $latex m=$ mass; $latex A=$ added mass; $latex B=$ damping coeffient; $latex K=$ “stiffness”-like matrix coming from the hydrostatic (buoyancy) force.

If $latex B_{pto}=B$ and $latex K_{pto}= -K+(m+A)\omega_n^2$, the combined contributions from $latex (m+A)\ddot{u}$ and $latex (K+K_{pto})u$ cancel, resulting in resonance.

Resonance (depending on the mass, geometry, and damping of the WEC) is then leveraged to maximize the power delivered from the ocean to the device. Sounds a bit familiar, doesn’t it?

Brekken, T. (2020) “Wave Energy Resource Overview”, *Integrated WEC Design: Theory and Practice Workshop*, Sandia National Laboratories (SNL), the Pacific Marine Energy Center (PMEC), and Oregon State University (OSU)

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