## Thursday, 21 January 2016

### Negative spring is a mysterious thing

Negative spring may well be the answer to making wave energy economic, so it would be good if it were less obscure and mysterious. This post describes what negative spring is, and how it can be used to control wave energy converters.

### What exactly is negative spring?

Positive spring results in a stabilising force; so moving away from the equilibrium position causes a force towards the equilibrium. Negative spring does the opposite: moving away from the centre causes a destabilising force away from the centre.

Imagine a marble and a bowl. If you place the marble into the bowl and give it a nudge, it will always return to the centre of the bowl. Gravity is providing positive spring. A good flick of the bowl will send the marble rolling up and down the sides of the bowl in an oscillatory fashion. If instead you balance the marble on top of the upturned bowl, gravity will provide negative spring. The system will be stable as long it is not moved. Any movement will cause the marble to roll off the bowl to its new equilibrium position. There will be no oscillation, which shows why negative spring is less well known to engineers than positive spring. It is not only engineers who need to know about different types of spring; this is also a useful concept in describing human relationships

### How does negative spring solve the challenge of making wave power economic?

To become more cost effective, wave energy needs to increase the ratio of revenue to costs. There are two ways that negative spring can contribute to this challenge.

1) Raise the natural period: The response curve, which we can plot against wave frequency or period, determines the proportion of the incoming wave power that can be converted to revenue. The design challenge is to improve the response curve with respect to the costs. The highest point in the response curve, the natural period, depends on the mass and spring of the system: $$T_n = \frac{1}{f_n} = 2\pi\sqrt{\frac{m}{k}}$$. For a system with a lower natural period than the wave energy period, increasing the natural period results in better performance (more power extracted). To increase the natural period one must either increase the mass ($$m$$), or reduce the spring ($$k$$). Increasing the mass has an associated cost; not just the cost of the increased material, but also the costs of the increased strength to resist the increased forces, and the bigger vessels needed to install and maintain the larger structure. Optimisation of mass is complex, and it may well be that the critical design drivers are factors other than the response curve.

The other option is to design for low spring, but again there are design constraints. Buoyancy spring depends on the mode in which the device moves. Heave is high spring, pitch is low spring, and surge has no spring. Reducing the water plane area of a surface-piercing device would lower spring. There are geometric limits to how low the spring can go without a complete redesign of the concept. This is where negative spring could help. If negative spring is introduced to a given degree of freedom, it subtracts from the natural system spring, giving a lower overall spring.

2) Reduce the ratio of operational to storm loads: The revenues (power generated) have an associated cost: the operational loads (forces and torques). However, the main thing that drives the project costs is the extreme loads. The extreme loads we anticipate drive the capital costs, and the extreme loads we don’t anticipate drive the operational costs, availability and device life. With some experience we will identify seas in which it is not possible to operate without attracting loads that are not economic to design for. These may include sea states that are not considered ‘storms’ from a mariner’s perspective. From the perspective of a wave energy converter, ‘storm loads’ are those experienced in sea states where the economics require a load-avoiding state.

The ideal situation is a wave energy converter that appears big in operational seas but small in storm mode. For sure, you’d want the fail-safe position to be the storm-mode. Negative spring gives us a way of achieving this. One way to appear small in failsafe mode is to be small to start with, but operate in a way that increases wave interaction. If negative spring turns on during power extraction, and turns off during failsafe and storm-modes, then the ratio of mean to extreme loads can be improved.

### What are the physical implementations?

Negative spring can be applied with the following mechanisms:

1) A force produced by a driven actuator. To avoid duplication of physical components this is usually the same mechanism that is supplying the power take off (PTO) damping, driven in reverse. However, it could be a separate mechanism; the important point is that energy is put into the system to create the negative spring. This is only worthwhile when the increased power capture more than covers the power consumed to create negative spring. In the literature this is referred to as ‘reactive’ or ‘damper-spring’ control. This type of control has been investigated for wave power since the 1970s. The challenges are the high ratio between mean and extreme loads in the PTO. This leads to poor use of equipment and high losses due to part load operation.

2) A force produced by an unstable mechanism. It is possible to arrange a network of springs to act as a physical spring mechanism that does not require a power input. For example, two precompressed compression springs in series and would supply negative spring at right angles to the line of springs. In terms of implementation, the springs could be a single beam under compression that bends elastically. Alternatively, if pneumatic springs rather than metal coils were used, it would be possible to turn off the springy behaviour during storm-mode. At least one wave energy company is considering this option.

3) A force produced by a buoyancy instability. The device itself could provide negative spring. A restoring moment (buoyancy - positive spring) in pitch is caused when the centre of mass and the centre of volume move out of vertical alignment. Movement in the opposite direction will cause instability in pitch. Water sloshing inside a pitching tank is one example: the centre of mass moves with pitch. Another way is move the centre of volume, by designing so that pitch causes a change in shape. For example, the Coventry Clam was a ring of air-filled bags. Pitch causes air to shift between bags. The change in shape shifts the centre of volume in the opposite direction of that required for a restoring moment.

### Controlling a wave energy converter with negative spring

There are several implementation considerations:

1) negative spring can be less than the device spring, so that the total spring is small and positive, giving an oscillating system with an increased natural period.

2) it can be used as an amplitude-limiting spring: the negative spring is only active over a chosen range of displacements; outside this range the total system spring is much higher, so extreme excursions are discouraged. Another way of looking at this, is that when a particular large wave hits the device, it acts as though it has a lower natural frequency.

3) If the total system spring is negative over a limited range of displacements, this could result in a very energetic oscillator. However, there is also the risk of getting stuck at a new, less energetic equilibrium. The usefulness of this depends on the loads experienced in the new equilibrium position.

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