# Mechanical Tuning

Most vibration energy-harvesting devices are based on a cantilever spring struc­ture, which can be used to highlight the possibilities for mechanical tuning. The principles described are generally applicable to all types of mechanical resonator structures. Mechanical tuning can be achieved by:

• Altering the dimensions of the beam;

• Moving the center of gravity of the proof mass;

• Varying the spring stiffness;

• Straining the structure.

The resonant frequency of a cantilever with an inertial mass, m, at the free end is given by (4.36), where E is Young’s modulus of the cantilever material, w, h, and l are the width, thickness, and length of the cantilever, respectively, and mc is the mass of the cantilever [30]:

1 / Ewh3

2n 4l3 (m + 0.24mc)

The resonant frequency can be adjusted by altering the dimensions but, in practice, it is difficult to change the width and thickness of the beam. Changing the length is feasible and is suitable for intermittent tuning. This also allows a significant change in the resonant frequency, since this is inversely proportional to l3/2. Modifying the length will require the cantilever base clamp to be released and reclamped at a new location along the length of the beam. This can, however, result in inconsistent damping losses through the clamp. Once adjusted, no additional power is required to maintain the new resonant frequency.

Equation (4.36) also shows that by varying the inertial mass, the resonant fre­quency will change. Once a device has been fabricated, however, it is often difficult to subsequently change m. The resonant frequency of a cantilever structure can be adjusted by moving the center of gravity of the inertial mass (i. e., by moving the mass along cantilever beam length).

Another approach is to vary the spring stiffness by placing an adjustable spring (ka) in parallel with the mechanical spring (km). The effective spring constant of such device, kf, is given by km + ka. The adjustable spring can be achieved by elec­trostatic, piezoelectric, magnetic, or thermal mechanisms. The majority of variable spring stiffness devices are continuously operated. Many examples of electrostatic tuning have been demonstrated in tuneable micromechanical resonators and have not necessarily been applied to vibration energy generators [31, 32]. Electrostatic generators can be tuned by adjusting the voltage on the plates, as discussed earlier. The presence of the inertial mass in an energy harvester reduces the tuning effec­tiveness and increases the power required for tuning. Piezoelectric tuning has been demonstrated by using two piezoelectric elements on the energy harvester spring element. One of these elements is used to harvest the energy, while the other ele­ment has a tuning bias applied to it [33]. Thermal techniques utilize the variation in Young’s modulus of the spring material with the temperature or the thermal expansion of the material. This approach, however, requires relatively high powers and is thus avoided in practical energy harvesting applications.

Magnetics have been used to alter the spring stiffness, by applying external forces to the device. For example, the resonant frequency of a cantilever structure can be tuned by applying an axial load. The resonant frequency of a uniform can­tilever with an associated buckling force, Fy, operating in the fundamental flexural mode with an axial load, F (positive for a tensile load and negative in the compres­sive case), is given by [30]:

_ n2 • E • w • h3

F =__________

b 48 • l2

The change in resonant frequency of a cantilever with an applied axial load is shown in Figure 4.9. The compressive load produces a larger frequency shift than the tensile load. If a large tensile force is applied (i. e., much greater than the buck­ling load), the resonant frequency will approach that of a straight tensioned cable as the force associated with the tension in the cantilever dominates the beam stiff­ness. A magnetically applied axial load has been used for energy harvesting, which will be discussed in Section 4.6.7.