In everyday language, the word “backlash” sounds as undesirable as its meaning: a strong adverse reaction or a violent backward movement. In engineering, the situation is no different. Backlash, the excessive play between machine parts, as often occurs in gears and flexible couplings, is undesirable and usually exists with compliance. It gives rise to inaccuracies in the position and velocity of a machine and to delays and oscillations.

Figure 1 shows the physical system under investigation with the accompanying assumptions. In addition, assume that collisions due to backlash are sufficiently plastic to avoid bouncing. It is critical that the model capture the fact that the output from the backlash element causes a torque on—not a displacement of—the load inertia. This model also captures situations in which the assumed massless compliant element has damping (Reference 1). The following demonstration shows the importance of this model: Imagine that you are compressing with your hand a massless spring that possesses no internal damping. If you were to suddenly move your hand away, the spring would stay in contact because its response is instantaneous; being pure, it has no mass or damping. If the spring has damping and you repeat the experiment, however, the spring’s response would not be instantaneous, and it would start to lose contact.