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Appendix III
List of Symbols

A	Transverse inertia moment of the projectile, cross‐section area, linearization coefficient
	Matrix of system, rotation matrix
a	Acceleration of the mass center of the projectile
a_p	Axial acceleration of original point o₀ of rocket launcher coordinate system o_ox_oy_oz_o relative to the launching guider
B_z,	Linearization coefficients
C	Polar inertia moment of the projectile, mass center of the rigid body, linearization coefficient
C _j	Damping coefficient of damper j
	Damping coefficient of torsional damper j
	Damping matrix
D	Action point of exterior forces, interphase drag
D¹, D³	Operator
D_x, D_y, D_xy	Bending stiffness of the plate
d_p	Equivalent diameter of the propellant grain
D_p	Diameter of the propellant grain
D_z,	Linearization coefficients
d¹, d³	Operator
d₀	Rotating band diameter of the projectile
E	Young’s modulus
	Transform matrix
EA	Tensile stiffness
EI	Bending stiffness of the beam
E_X, E_Y, E_Z	Longitudinal, vertical, lateral mean square
e₁	Half the web size of the propellant
e_g	Internal energy of gas
	External forces matrix acting on the system
	External force (including external torque) matrix acting on element i
f_p	Impetus of the propellant
F_i,	External forces acting on element i
	Contact force between the front bourrelet and the launching tube
	Contact force between the rear bourrelet and the launching tube
F_p	Engine thrust
G	Shear elastic modulus
GJ_p	Torsional rigidity of axis
GA_s	Shearing rigidity
g	Gravity acceleration
h	Thickness of plate
	Coordinate transformation matrix
H_c	Burned released enthalpy of principal charge per unit mass
h_p	Convective heat transfer coefficient
h_re	Radiative heat interchange coefficient
i, i	Number of element, imaginary unit
I	Input point of element
	Unit matrix of order n
j	Number of element
J_p	Polar product of inertia of axial section
	Inertia matrix
k	Modal order
	Augmented operator
K_j	Stiffness coefficient of the spring j
	Stiffness coefficient of the torsional spring j
K ^*	Winckler (elastic) basic modulus
K′^*	Rotating basic modulus
k_p	Coefficient of thermal conductance of the propellant
	Eccentricity of thrust
	Eccentricity of mass
L_p	Length of the propellant grain
	Position coordinate matrix
	Skew symmetric matrix of position vector
l₁	Distance between the mass center of the projectile and the front end plane of front bourrelet of the projectile
l_R	Distance between the mass center of the projectile and the rear end plane of rear bourrelet of the projectile
l_α	Distance between the interface point of the combined rifling and rifling initial point
	Bending moment, moment, augmented operator, internal moment matrix under modal coordinate
M_j	Mass parameter matrix of the body j
m	Mass of body element, mass of the projectile
	Internal moment matrix under physics coordinate
	Rate of gas mass production of principal charge per unit volume
	Linear density of the beam
o	Rear bourrelet center of the projectile
O	Output point of element
	m × n zero matrix
O₁	Geometric center of the projectile
o₂	Front bourrelet center of the projectile
O₃	Intersection point of the bore axis and the vertical plane of the bore axis with the mass center
P	Gas pressure
P_i,j	Connection point in the system
p	Distributed load intensity
	Internal force matrix under modal coordinate
	Internal force matrix under physics coordinate
q^k	Generalized coordinate
Q_s	Heat exchange capacity between solid phase and gas phase per unit volume
R	Gas constant
	Linear displacement matrix under modal coordinate
	Linear displacement matrix under physics coordinate
r_b	Radius of neutral circularity of guide pin
Re	Gas motion Reynold’s number
R_p	Stress of solid phase intergranular
T	Temperature
t	Time
ΔT	Time step
t_g	The time when the projectile flies out of the muzzle (or the muzzle of the launching tube)
,	Transfer matrix of the element
	Extended transfer matrix
	Complex transfer matrix
	Complex extended transfer matrix
	Overall transfer matrix of the system
	Augmented eigenvector corresponding to the kth eigenfrequency of the system
u₁	Burning rate coefficient of the propellant
u_p	Solid phase rate
u_g	Gas phase rate
v	Augmented eigenvector matrix under physics coordinate
v	Burning rate exponent of the propellant
v₀	Muzzle velocity of the projectile
v_p	Velocity of the origin o₀ of rocket launcher coordinate system o_ox_oy_oz_o relative to the launching tube
v_g	Axial component of the velocity of the mass center of the projectile relative to the muzzle (muzzle of the launching tube)
w	Deflection of transverse vibration plate
X′	Artillery recoil displacement
x_q	Longitudinal displacement of the projectile relative to the barrel
,	Vertical and lateral component of displacement of the mass center of the projectile relative to the artillery coordinate system
, ,	Longitudinal, vertical and lateral components of displacement of the mass center of the rocket relative to the rocket launcher coordinate system
Z	Relative burned thickness of the propellant
	State vector of connection point P _i,j or (i,j) under modal coordinate
	State vector of connection point P _i,j or (i,j) under physics coordinate
α	Rifling (guide slot of launching tube) twist angle
α₀	Twist angle of the rifling initial point
α_g	Muzzle rifling twist angle
	Dynamic unbalance angle
	Thrust malalignment angle
γ	Spin angle of projectile, gas adiabatic exponent
	Spin angular speed of projectile
γ₀	Initial orientation angles of guide slot
δ	Nutation angle
Δ	Attack angle
	Vertical swing angle of the projectile axis relative to the launching tube
	Lateral swing angle of the projectile axis relative to the launching tube
ε	Strain
	Angular displacement matrix under modal coordinate
	Angular displacement matrix under physics coordinate
θ₁	Departure angle of the rocket launcher
θ_x, θ_y, θ_z	Generalized angle of the rigid body for describing orientation
μ	Friction coefficient between the guide pin and the tube wall of the launching tube, Poisson’s ratio of material
μ_g	Kinematic viscosity coefficient of gas
ρ	Material density, distributed density
ρ_g	Gas phase density
ϕ	Voidage
ϕ₀	Criticality voidage of propellant charge with free loading
	Projectile oscillation angle
φ₁	Vertical swing angle of the projectile
φ₂	Lateral swing angle of the projectile
	Deflection angle of the projectile
ψ₁	Vertical deflection angle of the projectile
ψ₂	Lateral deflection angle of the projectile
	Vertical spin angle of barrel (launching tube)
	Lateral spin angle of barrel (launching tube)
ω	Angle velocity
ω_k	The kth order eigenfrequency of system
Ω	Excitation frequency