Active Vibration Control for the Mitigation of Wheel Squeal Noise Based on a Fuzzy Self-Tuning PID Controller

Table 1

Notations and nomenclatures.

Symbol	Unit	Definition
Η	—	Wheel modal damping
	—	State variable vector of the wheel
	M	Modal displacement
	m/s	Modal velocity
	N	Input dynamic force vector on the wheel
	N	Input dynamic force on the wheel in direction i
	m/s	Output dynamic velocity vector of the wheel
	m/s	Output dynamic velocity of the wheel in direction i
[]	—	Wheel system matrix
[]	—	Wheel input matrix
[]	—	Wheel output matrix
	—	State variable vector of the rail
	N	Input dynamic force vector on the rail
	m/s	Output dynamic velocity vector of the rail
[A^r]	—	Track system matrix
[B^r]	—	Track input matrix
[C^r]	—	Track output matrix
F3	N	Vertical contact force
	m	Wail vertical deflection
	m	Rail vertical deflection
	N/m	Hertzian contact spring stiffness
F2	N	Lateral contact force
	—	Lateral friction coefficient
	—	Lateral creepage
N	N	Static vertical contact force
		Static friction coefficient
	N/m^2	Shear strength of the wheel
	N/m^2	Shear strength of the rail
a, b	M	Semi-axis length of the Hertz contact ellipse in the rolling and lateral direction
	—	Rolling friction coefficient
	—	Creepage
V	m/s	Train speed
	—	Normalized creepage in direction i
G	Pa	Material shear modulus
Cii	—	Kalkar constant
	rad/s	Frequency
p	Pa	Acoustic pressure
R	m	Distance of desired point P and acoustic source
c0	m/s	Sound propagation velocity of air
ρ0	kg/m^3	Density of air
	m/s^2	Wheel lateral acceleration
		Wheel input matrix of the actuator
		Voltage vector applied to actuators
		Wheel mode shape gradient
n		Number of nodal diameters
k		Number of nodal circles
Ka		Piezoelectric patch actuator constant
dij		Strain constant of piezoelectric
Fp	N	Force on the rail induced by a piezoelectric actuator
np	—	Number of piezoelectric actuators
Lp	m	Length of piezoelectric stack
tp	m	Thickness of piezoelectric patch
bp	m	Width of piezoelectric patch
Ep	Pa	Young^’s module of piezoelectric
Ap	m^2	Area of piezoelectric stack actuator section
		Piezoelectric compliance matrix element
	m	Lateral contact position
Kp	N/m	Pad stiffness
Cp	N.s/m	Pad viscous damping
Kb	N/m	Ballast stiffness
Cb	N.s/m	Ballast viscous damping
Ms	Kg	Sleeper mass
Kpr	N/m	Pad rotational stiffness
Cpr	N.s/m	Pad rotational viscous damping
E	Pa	Elastic modulus
	Kg/m^3	Density
	—	Poisson’s ratio