|Fig. 1: Relationship between mechanical input energy and electrical output energy of PZT. (Data from )|
Though often considered separate subjects, the disciplines of mechanics and electromagnetism find themselves unified in the study of piezoelectricity. The piezoelectric effect describes the generation of electric current in certain materials when they are exposed to mechanical stress. It was first proposed by Charles-Augustin de Coulomb in the late eighteenth century, but not formally discovered until 1880, when Pierre and Jacques Curie demonstrated the phenomena in various crystalline substances.  The array of materials that display some degree of piezoelectricity is vast - crystals, ceramics, even bone - and such materials can be used in an equally varied number of applications, from commercial stereo tweeters to radiation detectors.
The piezoelectric effect is a reversible process, meaning piezoelectric materials can either generate electricity when compressed or stretch when a voltage is applied across them. This enables them to be used as either small-scale generators or motors, but for the purposes of energy generation, the so-called direct piezoelectric effect (electricity created from compression) is far more useful. The idea of owning, say, a pair of running shoes that can charge a cell phone with electricity generated from a morning jog is enticing, and has been the subject of much research. This paper will examine the physics behind and feasibility of such ideas, using a nightclub in Rotterdam as a case study in the application of piezoelectricity to power generation.
The piezoelectric effect appears in crystalline materials that have asymmetric unit cells. Because of this asymmetry, when a mechanical force deforms a piezoelectric material, the (polarized) unit cells shift into a different pattern - one that is generally more aligned and regular. As a result, dipole effects build up, and a potential difference is generated across the crystal. 
The effect can be seen as a combination of Hooke's law, which governs the relationship between stress and strain, and the electric displacement field, which relates dielectric permittivity and electric field. Hooke's law states:
Where sigma is stress, a measure of the force applied to the crystal divided by the area it is applied over; S is the compliance of the material, related to its modulus of elasticity; and lambda is strain, a measure of the elongation or compression of the crystal divided by its height at zero load. For an anisotropic material, this can be written as:
The axial scheme here must take into account six dimensions: x, y, and z, along with shear around each axis.
Analogously to Hooke's law, the equation for electric displacement states:
Where D is the electric displacement field, epsilon is the dielectric constant, and E is the electric field in the material. Again, this can be written for an anisotropic material:
These equations can be combined into a generator equation and motor equation: 
The bold type face indicates that these equations are matrix equations. d is a matrix of piezoelectric coefficients, inherent to a material and its lattice structure. The subscripts on epsilon and S indicates that stress and electric field is held constant, respectively. Since this paper is concerned, after all, with solving the world's energy shortage, it focuses on generation.
While the Curies' original piezoelectricity experiments were done with quartz, more recent research has led to the discovery that ferroelectric materials can have a much larger piezoelectric effect associated with them. One of the most prominent piezoelectric materials used today is lead zirconate titanate (PZT), a ferroelectric ceramic developed in the mid-twentieth century. Experiments by Xu et al. have tested the energy generation of small pieces of PZT impacted by a steel ball, and found that the electrical energy created is several orders of magnitude less than the mechanical energy of the steel ball.  For a circular sample of PZT 24 mm in diameter and 3 mm thick, the impact energy data can be linearly fitted around the line (see Figure 1):
Where Ee is the output electrical energy and Em is the input mechanical impact energy.
For the case of slowly applied stress, where a steel plate is pushed onto the sample of PZT at a rate of 1 mm/min, much more energy was generated (on the order of hundreds of microjoules).
One of the most intriguing applications of piezoelectric technology is championed by the company Sustainable Dance Club, whose Rotterdam nightclub Watt generates part of its power from the dance moves of its patrons. The dance floor is suspended on a series of springs and piezoelectric crystals, and as dancers move, the crystals compress, generating electricity.
According to the manufacturers of Watt's dance floor, each module generates 20 W of power while an adult dances on it.  A maximum of 160 of these modules can be wired together, giving an entire dance floor 3200 W of energy-generating capacity as people dance. For a typical night (say, four hours of dancing), this means the floor will generate approximately 4.6 × 107 J of energy.
To test this claim, the dance floor will be modeled with 700 dancers (the club has a capacity of 1400, so the assumption here is that half of the people at any given time are enjoying the bar, lounge, or any other stationary facility). Assuming the dancers are moving vigorously, each will be jumping approximately 10 cm in the air with a frequency of 1 Hz (a standard techno song will have around 120 beats per minute; here the dancers move every other beat). With an average weight of 70 kg, this gives each dancer a kinetic energy upon impact of approximately 70 J. If each person compresses ten spring/crystal systems - not unreasonable given the small size of the crystals examined - then using the formula presented in (7), this translates to a generated electrical energy per person per jump of 781.5 microjoules.
Assuming all of this energy can be stored, a typical night in the club (four hours of dancing) would generate a total of only 7800 J of energy - much less than the manufacturers claims. However, it is worth noting that these numbers come from a 3 mm thick sample of PZT, and the nightclub would presumably use larger crystals to increase power generation. In addition, if modeled using Xu et al.'s data for slowly applied stress, the total energy generated jumps up five orders of magnitude, so that the total electrical energy made is around 4 × 107 J - just as claimed. 
While 4 × 107 J is a much more respectable number, it is still less than the energy stored in only a third of a gallon of crude oil. 
The phenomena of piezoelectricity is perfect for applications in sensors and the like, yet it has a very small ratio of mechanical input to electrical output. Therefore, piezoelectric materials may be more useful for small scale generation (cf. ) than powering a night club, and will likely remain so in the foreseeable future. One of the largest problems with the piezoelectric effect is that only crystalline materials exhibit it, and any crystal will not deform much under stress, so the generation of electricity is fundamentally limited. As an idea for promoting alternative energy technologies and the quest to find a method to generate electricity in new and innovative ways, however, Watt's idea of a piezoelectrically-powered night club is just the right kind of thinking.
© Seth Winger. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.
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