Chaotic operation and chaos control of travelling wave ultrasonic motor

A T-shape linear piezoelectric motor with single foot

“…For the generation of elliptical movement on the driving tip, one classic method is superimposing two standing waves to form a traveling one in a metal ring; and this traveling wave can produce elliptical movements on the surface particles of the ring [4][5][6][7][8][9][10]. Furthermore, as two orthogonal vibrations with the same resonance frequency and a temporal phase shift of 90°can also form elliptical movement, combination of vibrations, such as longitudinal and bending vibrations [11][12][13][14], longitudinal and torsional vibrations [15][16][17], two orthogonal longitudinal vibrations [18][19][20][21][22][23][24], and two orthogonal bending vibrations [25][26][27][28][29], can be utilized for constructing piezoelectric motors.…”

Linear ultrasonic motor for absolute gravimeter

“…While in the lower half of elliptic motion, the slider is not in contact with the driving foot, but continues moving forward due to inertia. By repeating these sequences, the elliptic motion of the stator can be converted into the linear motion of the slider [22]. An opposite motion can be achieved when reversed excitation signals are applied.…”

New Eight Dimensional Hyperchaotic Cryptosystem

“…For any initial condition 0 ∈ and any 0 ∈ and for large enough the system (1) satisfying previous assumptions [10] [11], can be estimated by the following dynamical system:…”

“…The problem of control of a chaotic system is to find a state feedback control law to stabilize a chaotic system around its unstable equilibrium [10].Some popular methods for chaos control are active control [21], adaptive control [17], sliding mode control [15], etc.…”