Limit Circular Motion of Spinning Projectiles Induced by Backlash of Actuators

Abstract

Divergent coning motions and limit circular motions of spinning projectiles with large amplitude are harmful and have been widely discussed by many researchers. In this paper, a new factor is exposed and its mechanism is elucidated, which is the backlash of actuators that may lead to a limit circular motion of spinning projectiles in free flight phase. The backlash is generated from additional gear drives outside the closed-loop actuators and its nonlinearity is depicted by describing function method. Both the sufficient condition for the existence of a stable limit circular motion and the equation to calculate the amplitude are obtained and then validated by numerical simulations. Meanwhile, the effects of some key parameters on the motion in both free flight and control phase are discussed as well. It is noted that the harmful impact of the backlash in the free flight phase is much greater than that in the control phase. Nomenclatureaxis moment of inertia, kg·m 2 I y = transverse moment of inertia, kg·m 2 K = amplitude of the limit circular motion, deg l = reference length, m m = mass of the airframe, kg p, q, r = roll, pitch, and yaw rates in the non-rolling frame, rad/s Q = 2 1 2 V  , dynamic pressure, N/m 2 S = reference area, m 2 t = time, s u, v, w = velocity components in the non-rolling frame, m/s V = velocity vector of the airframe, m/s  ,  = angles of attack and sideslip in the non-rolling frame, rad   ,   = angles of attack and sideslip in the body-fixed frame, rad  = i    , complex angle of attack in the non-rolling frame, rad   = i      , complex angle of attack in the body-fixed frame, rad  = air density, kg/m 3

Dither in a rolling airframe flight vehicle with a two-position actuator: An amplitude distribution approach

“…The hinge moment effect on the instability of coning motion was investigated by considering the actuators' transfer function by Zhou et al (2013). The backlash of actuators (canards) that may lead to a limited circular motion of the spinning projectiles was analysed in Zhou et al (2014). Nobahari and Mohammadkarimi (2012) injected a sinusoidal dither with a frequency twice the roll rate frequency of the missile to the input of the two-position actuator, and by applying the multiple-input describing functions (DFs) and linear control theory, developed an autopilot for a rolling missile.…”

Coning motion stability and decoupling methods analysis of spinning missiles with angle‐of‐attack feedback autopilot

“…Furthermore, other factors, such as the backlash of actuators, were also studied [19]. Considering the fin actuator dynamics, Zhou et al [19] introduced the fin actuator as a PD controller, and the stable boundary of such a PD controller was proposed through the classic Routh–Hurwitz criterion. Besides, many studies regarded the guidance loop as a proportional component and further analyzed the coning motion stability.…”

“…Sufficient and necessary conditions for the dynamic stability of spinning missiles with rate loops, autopilots, velocity orientation autopilots, proportionalintegral (PI) autopilots, and three-loop autopilots [9,[15][16][17][18] were proposed, and the analytical solutions of the design parameters were investigated. Furthermore, other factors, such as the backlash of actuators, were also studied [19]. Considering the fin actuator dynamics, Zhou et al [19] introduced the fin actuator as a PD controller, and the stable boundary of such a PD controller was proposed through the classic Routh-Hurwitz criterion.…”

A New Closed Form Solution for Dynamic Stability Analysis of Rolling Airframes Having One Pair on-Off Actuator

“…Variables H, I p , M, T p and M c represent effects of aerodynamic damping, gyroscopic motion, aerodynamic stability, Magnus moment and control surface, respectively (Zhou et al, 2014).…”