Generally, light weight electric motors and servo amplifiers suitable for a portable guitar cannot provide the power necessary to change the tension in a string over a full octave within reasonable response times. Accordingly, the present invention uses a "negative spring" to effectively neutralize the force on the motor from the tensioned spring over a small range of motor travel. As shown below, as the motor increases the tension of the string, the string and a nonlinear spring are stretched providing a compensating force on the motor (plus force). This back force is compensated by the negative spring which provides increasing negative force as the string is tensioned, balancing out the string tension force. Negative force springs, in fact, do not exist but are implemented by a compensator wheel whose design will be described further below.
The compensator wheel provide a lower half having a constant radius about which the tension cord wraps so that the offset spring (which has a very low spring constant and hence provides a substantially constant force) provides a constant torque to the compensator wheel with wheel rotation. The upper half of the compensator wheel is simply a peg which guides the tension cord back to the motor shaft. The constant torque on the compensator wheel translates to a variable tension in the cord between the motor shaft and compensator wheel increasing as the wheel rotates clockwise from the position shown and decreasing as the wheel rotates counterclockwise. The effect is a negative spring that pulls harder as it receives additional tension cord.
The upper diagram shows the tension between the tension cord in the motor as a function of angle. The negative slope of force required for a negative spring is in fact a small section of the cosine function. The height of the slope section is controlled by adjusting the force of the offset spring. A steeper force curve can be obtained by the second diagram for even lower motor load albeit over a smaller angular distance.
A plot of force curves shows that the current compensator wheels can control the total force on the motor to somewhat less than 5 pounds over the full octave range of tension. Change in the tension of the offset spring moves the net force curve of the motor up and down allowing a trade-off between movement range and peak force.
The wheels are constructed of three layers of 1/8 luan (two disks and one 120 degree sector) cut with a fly cutter and glued together and then counterbored for skateboard bearings. A dowel is inserted through the outer two layers to provide the peg providing the cosine lever arm.