Influence of track flexibility and spatial coherence of track irregularity on vehicle-slab track interaction: frequency-domain analysis

Figures & data

Figure 1. Vehicle-track interaction model (a) Side view; (b) End view; (c) Global coordinate system

Figure 2. Wheel-rail interaction model: (a) Side view, (b) Top view

Figure 3. Comparisons between time-domain and frequency-domain solutions: (a) Car body vertical acceleration, (b) Wheel-rail vertical force

Figure 4. Car body response: (a) Rigid track, (b) Track flexibility included

Figure 5. Front bogie response: (a) Rigid track, (b) Track flexibility included

Figure 6. Rear bogie response: (a) Rigid track, (b) Track flexibility included

Figure 7. Carbody vertical motion, normalized with respect to $k_H\cdot A$: (a) Response of car body vertical motion in $\lambda$ and $v$ plane, (b) Frequency response of car boy vertical motion for certain wavelengths of irregularities

Figure 8. Car body pitch motion, normalized with respect to $k_H\cdot A$: (a) Response of car body pitch motion in $\lambda$ and $v$ plane, (b) Frequency response of the car body pitch motion for certain wavelengths of irregularities

Figure 9. First wheelset vertical motion, normalized with respect to $k_H\cdot A$

Figure 10. Second wheelset vertical motion, normalized with respect to $k_H\cdot A$

Figure 11. Third wheelset vertical motion, normalized with respect to $k_H\cdot A$

Figure 12. Fourth wheelset vertical motion, normalized with respect to $k_H\cdot A$

Figure 13. Driving point receptance of rail: (a) Driving point track receptance, without vehicle, the first peak is at 52 Hz, (b) Driving point track receptance, with vehicle, first peak at 33 Hz

Figure 14. Vertical motion of rail under first wheelset, normalized with respect to $k_H\cdot A$

Figure 15. Vertical motion of rail under second wheelset, normalized with respect to $k_H\cdot A$

Figure 16. Vertical motion of rail under third wheelset, normalized with respect to $k_H\cdot A$

Figure 17. Vertical motion of rail under fourth wheelset, normalized with respect to $k_H\cdot A$

Figure 18. Car body response, normalized with respect to $A$: (a) Vertical motion, (b) Pitch motion

Figure 19. Front bogie response, normalized with respect to $A$: (a) Vertical motion, (b) Pitch motion

Figure 20. Rear bogie response, normalized with respect to $A$: (a) Vertical motion, (b) Pitch motion

Figure 21. Driving point displacement FRF, wheel vertical motion, normalized with respect to $A$

Figure 22. Rail vertical receptance, normalized with respect to $A$

Figure 23. Influence of contact spring stiffness on the 1^st resonance frequency $f_r$ of the rail

Table A1. Main parameters of the vehicle

Notation	Parameter (unit)	Values
$M_{\mathrm{c}}$	Car body mass (kg)	49080
$M_{\mathrm{t}}$	Bogie mass (kg)	4420
$M_{\mathrm{w}}$	Wheelset mass (kg)	1680
$I_{\mathrm{c}x}$	Mass moment of inertia of car body about X-axis (kg m^2)	65700
$I_{\mathrm{c}y}$	Mass moment of inertia of car body about Y-axis (kg m^2)	1698400
$I_{\mathrm{c}z}$	Mass moment of inertia of car body about Z-axis (kg m^2)	1447100
$I_{\mathrm{t}x}$	Mass moment of inertia of bogie about X-axis (kg m^2)	1576
$I_{\mathrm{t}y}$	Mass moment of inertia of bogie about Y-axis (kg m^2)	4902
$I_{\mathrm{t}z}$	Mass moment of inertia of bogie about Z-axis (kg m^2)	3551
$I_{\mathrm{w}x}$	Mass moment of inertia of wheelset about X-axis (kg m^2)	773.3
$I_{\mathrm{w}y}$	Mass moment of inertia of wheelset about Y-axis (kg m^2)	130
$I_{\mathrm{w}z}$	Mass moment of inertia of wheelset about Z-axis (kg m^2)	770
$k_{\mathrm{p}x}$	Stiffness coefficient of primary suspension along X-axis (MN/m)	1.24
$k_{\mathrm{p}y}$	Stiffness coefficient of primary suspension along Y-axis (MN/m)	1.24
$k_{\mathrm{p}z}$	Stiffness coefficient of primary suspension along Z-axis (MN/m)	1.27
$k_{\mathrm{s}x}$	Stiffness coefficient of secondary suspension along X-axis (MN/m)	0.11
$k_{\mathrm{s}y}$	Stiffness coefficient of secondary suspension along Y-axis (MN/m)	0.11
$k_{\mathrm{s}z}$	Stiffness coefficient of secondary suspension along Z-axis (MN/m)	0.27
$c_{\mathrm{p}z}$	Damping coefficient of primary suspension along Z-axis (kN s/m)	65
$c_{\mathrm{s}y}$	Damping coefficient of secondary suspension along Y-axis (kN s/m)	5.4
$c_{\mathrm{s}z}$	Damping coefficient of secondary suspension along Z-axis (kN s/m)	16.5
$L_{\mathrm{c}}$	Semi-longitudinal distance between bogies (m)	7.85
$L_{\mathrm{t}}$	Semi-longitudinal distance between wheelsets in bogie (m)	1.25
$R_0$	Wheel radius (m)	0.43

Table A2. Main parameters of the track structures

Notation	Parameter (unit)	Values
$E_r$	Elastic modulus of the rail (N/m^2)	2.059 × 10^11
${\rho }_r$	Mass of the rail per unit length (kg/m)	60.64
$I_{ry}$	Rail second moment of the area about the Y axis (m^4)	3.215 × 10^−5
$I_{rz}$	Rail second moment of the area about the Z axis (m^4)	5.24 × 10^−6
$G_k$	Rail torsional stiffness (N m/rad)	1.9587 × 10^5
$k_{\mathrm{r}\mathrm{z}}$	Fastener stiffness in the vertical direction (N/m)	3 × 10^7
$k_{ry}$	Fastener stiffness in the lateral direction (N/m)	3.5 × 10^7
$c_{\mathrm{r}\mathrm{z}}$	Fastener damping in the vertical direction (N s/m)	2.5 × 10^4
$c_{ry}$	Fastener damping in the lateral direction (N s/m)	2.5 × 10^4
$E_t$	Elastic modulus of the slab (N/m^2)	3.6 × 10^10
${\mu }_t$	Poisson ratio of the slab	0.25
$m_t$	Mass per unit volume of the slab (kg/m^3)	2500
$H_t$	Height of the slab (m)	0.19
$L_t$	Length of the slab (m)	4.95
$W_t$	Width of the slab (m)	2.40
$k_{tz}$	Equivalent vertical stiffness coefficient of the CA layer (N/m)	1.3 × 10^9
$c_{tz}$	Equivalent vertical damping coefficient of the CA layer (N/m)	1.0 × 10^4
$k_{ty}$	Equivalent vertical stiffness coefficient of the CA layer (N/m^2)	6 × 10^7
$c_{ty}$	Equivalent vertical damping coefficient of the CA layer (N/m)	1.0 × 10^4