Nonlinear dynamic responses of high-speed railway vehicles under combined self-excitation and forced excitation considering the influence of unsteady aerodynamic loads

Abstract

The dynamic characteristics of a railway vehicle system under unsteady aerodynamic loads are examined in this study. A dynamic analysis model of the railway vehicle considering the influences of aerodynamic loads was established. The model not only considers the forced excitation effect of unsteady aerodynamic loads but also accounts for the effect of unsteady aerodynamic loads on the change of the wheel–rail contact normal forces as well as changes of the wheelset creep coefficients and creep forces/moments. Therefore, this model also considers the influences of unsteady aerodynamic loads on the self-excited vibration characteristics of the vehicle system. The time-history curves, phase trajectory diagrams, Poincaré sections, and Lyapunov exponents of the vehicle system running on a smooth straight track under unsteady aerodynamic loads were determined. The results show that when the critical speed is exceeded, the vehicle system usually performs quasi-periodic motion under unsteady aerodynamic loads, which is significantly different from the periodic motion under steady aerodynamic loads. In different cases, the amplitude and phase of motion are significantly different. The amplitude of the motions can be increased by more than 159%, and the difference of phase can be up to 173°. (The phase is almost reversed.) The dynamic responses of the vehicle system under unsteady aerodynamic loads contain abundant frequency components, including the frequency of the self-excited vibration, the frequency of the forced excitation, and combinations of their integer multiples. The vibration forms corresponding to the main harmonic components under unsteady and steady aerodynamic loads were compared, and the self-excited vibration component of the vehicle system under unsteady aerodynamic loads was identified. The variations in the critical speed with various parameter combinations were computed. The variation range of the critical velocity can reach 73%.

Appendices

Appendix A: Details of equations of motion

The force diagrams and dynamics equations of each component of the vehicle system are listed in detail below [35].

For the i-th wheelset (i = 1–4) (Fig. 

Fig. 26

Schematic diagram of the force of the i-th wheelset

26).

For the front frame (n = 1) (Fig. 

Fig. 27

Schematic diagram of the force of the front frame

27).

The rear frame (n = 2) is similar to the front frame.

For the car body (Fig. 

Fig. 28

Schematic diagram of force of the car body

28).

The primary suspension longitudinal force is as follows (i = 1–4):

$$ F_{{x{\text{f}}\left( {{\text{L}},{\text{R}}} \right)i}} = K_{{{\text{p}}x}} \left( { \pm d_{{\text{w}}} \psi_{{{\text{f}}n}} + H_{{{\text{fw}}}} \beta_{{{\text{f}}n}} \mp d_{{\text{w}}} \psi_{{{\text{w}}i}} } \right) + C_{{{\text{p}}x}} \left( { \pm d_{{\text{w}}} \dot{\psi }_{{{\text{f}}n}} + H_{{{\text{fw}}}} \dot{\beta }_{{{\text{f}}n}} \mp d_{{\text{w}}} \dot{\psi }_{{{\text{w}}i}} } \right) $$

(A.1)

n = 1 when i = 1, 2 and n = 2 when i = 3, 4. The subscript i = 1–4 represents the i-th wheelset, and n = 1–2 represents the n-th frame, where the upper and lower sign (‘ + ’ or (‘ − ’) of ± and ∓ apply to the left and right wheel, respectively. This convention is also followed in similar cases later.

The primary suspension lateral force is as follows (i = 1–4):

$$ F_{{y{\text{f}}\left( {{\text{L}},{\text{R}}} \right)i}} = K_{{{\text{p}}y}} \left[ {y_{{{\text{w}}i}} - y_{{{\text{f}}n}} + H_{{{\text{fw}}}} \phi_{{{\text{f}}n}} + \left( { - 1} \right)^{i} l_{{\text{f}}} \psi_{{{\text{f}}n}} } \right] + C_{{{\text{p}}y}} \left[ {\dot{y}_{{{\text{w}}i}} - \dot{y}_{{{\text{f}}n}} + H_{{{\text{fw}}}} \dot{\phi }_{{{\text{f}}n}} + \left( { - 1} \right)^{i} l_{{\text{f}}} \dot{\psi }_{{{\text{f}}n}} } \right]. $$

(A.2)

The primary suspension vertical force is as follows (i = 1–4):

$$ \begin{aligned} F_{{z{\text{f}}\left( {{\text{L}},{\text{R}}} \right)i}} & = K_{{{\text{p}}z}} \left[ {z_{{{\text{f}}n}} - z_{{{\text{w}}i}} + \left( { - 1} \right)^{i} l_{{\text{f}}} \beta_{{{\text{f}}n}} \pm d_{{\text{w}}} \phi_{{{\text{w}}i}} \mp d_{{\text{w}}} \phi_{{{\text{f}}n}} } \right] \\ & \quad + \;C_{{{\text{p}}z}} \left[ {\dot{z}_{{{\text{f}}n}} - \dot{z}_{{{\text{w}}i}} + \left( { - 1} \right)^{i} l_{{\text{f}}} \dot{\beta }_{{{\text{f}}n}} \pm d_{{\text{w}}} \dot{\phi }_{{{\text{w}}i}} \mp d_{{\text{w}}} \dot{\phi }_{{{\text{f}}n}} } \right] + \frac{{\left( {2M_{{\text{f}}} + M_{{\text{c}}} } \right)g}}{8}. \\ \end{aligned} $$

(A.3)

The secondary suspension longitudinal force is as follows (i = 1, 2):

$$ F_{{x{\text{t}}\left( {{\text{L}},{\text{R}}} \right)i}} = K_{{{\text{s}}x}} \left( {H_{{{\text{cb}}}} \beta_{{\text{c}}} + H_{{{\text{bf}}}} \beta_{{{\text{f}}i}} \pm d_{{\text{s}}} \psi_{{\text{c}}} \mp d_{{\text{s}}} \psi_{{{\text{f}}i}} } \right) + C_{{{\text{s}}x}} \left( {H_{{{\text{cb}}}} \dot{\beta }_{{\text{c}}} + H_{{{\text{bf}}}} \dot{\beta }_{{{\text{f}}i}} \pm d_{{\text{s}}} \dot{\psi }_{{\text{c}}} \mp d_{{\text{s}}} \dot{\psi }_{{{\text{f}}i}} } \right). $$

(A.4)

The secondary suspension lateral force is as follows (i = 1, 2):

$$ \begin{aligned} F_{{y{\text{t}}\left( {{\text{L}},{\text{R}}} \right)i}} & = K_{{{\text{s}}y}} \left[ {y_{{{\text{f}}i}} - y_{{\text{c}}} + H_{{{\text{bf}}}} \phi_{{{\text{f}}i}} + H_{{{\text{cb}}}} \phi_{{\text{c}}} + \left( { - 1} \right)^{i} l_{{\text{c}}} \psi_{{\text{c}}} } \right] \\ & \quad + \;C_{{{\text{s}}y}} \left[ {\dot{y}_{{{\text{f}}i}} - \dot{y}_{{\text{c}}} + H_{{{\text{bf}}}} \dot{\phi }_{{{\text{f}}i}} + H_{{{\text{cb}}}} \dot{\phi }_{{\text{c}}} + \left( { - 1} \right)^{i} l_{{\text{c}}} \dot{\psi }_{{\text{c}}} } \right]. \\ \end{aligned} $$

(A.5)

The secondary suspension vertical force is as follows (i = 1, 2):

$$ \begin{aligned} F_{{z{\text{t}}\left( {{\text{L}},{\text{R}}} \right)i}} & = K_{{{\text{s}}z}} \left[ {z_{{\text{c}}} - z_{{{\text{f}}i}} \pm d_{{\text{s}}} \phi_{{{\text{f}}i}} \mp d_{{\text{s}}} \phi_{{\text{c}}} + \left( { - 1} \right)^{i} l_{{\text{c}}} \beta_{{\text{c}}} } \right] \\ & \quad + \;C_{{{\text{s}}z}} \left[ {\dot{z}_{{\text{c}}} - \dot{z}_{{{\text{f}}i}} \pm d_{{\text{s}}} \dot{\phi }_{{{\text{f}}i}} \mp d_{{\text{s}}} \dot{\phi }_{{\text{c}}} + \left( { - 1} \right)^{i} l_{{\text{c}}} \dot{\beta }_{{\text{c}}} } \right] + \frac{{M_{{\text{c}}} g}}{4}. \\ \end{aligned} $$

(A.6)

The secondary lateral damper force is as follows (i = 1, 2):

$$ F_{{y{\text{hx}}\left( {\text{L,R}} \right)i}} = K_{{{\text{hx}}}} \left( {y_{{{\text{f}}i}} + H_{{{\text{fhx}}}} \phi_{{{\text{f}}i}} - y_{{{\text{hx}}\left( {\text{L,R}} \right)i}} \pm E\psi_{{{\text{f}}i}} } \right). $$

(A.7)

The secondary yaw damper force is as follows (i = 1, 2):

$$ F_{{{\text{sx}}\left( {\text{L,R}} \right)i}} = K_{{{\text{sx}}1}} \left( { - y_{{{\text{sx}}\left( {\text{L,R}} \right)i}} \pm d_{{{\text{sx}}}} \psi_{{{\text{f}}i}} - H_{{{\text{fsx}}}} \beta_{{{\text{f}}i}} } \right). $$

(A.8)

The bump stop force is as follows (i = 1, 2):

$$ F_{{y{\text{zd}}i}} = K_{{{\text{zd}}}} \left( {y_{{{\text{f}}i}} - y_{{\text{c}}} + H_{{{\text{fzd}}}} \phi_{{{\text{f}}i}} + H_{{{\text{czd}}}} \phi_{{\text{c}}} + \left( { - 1} \right)^{i} l_{{{\text{zd}}}} \psi_{{\text{c}}} } \right). $$

(A.9)

For the wheelset (i = 1–4),

$$ M_{{\text{w}}} \ddot{y}_{{{\text{w}}i}} = - F_{{y{\text{fL}}i}} - F_{{y{\text{fR}}i}} + F_{{{\text{L}}yi}} + F_{{{\text{R}}yi}} + N_{{{\text{L}}yi}} + N_{{{\text{R}}yi}} $$

(A.10)

$$ \begin{gathered} I_{{{\text{w}}z}} \ddot{\psi }_{{{\text{w}}i}} + I_{{{\text{w}}y}} \dot{\phi }_{{{\text{w}}i}} \left( {\dot{\beta }_{{{\text{w}}i}} - \frac{V}{{r_{0} }}} \right) = a_{0} \left( {F_{{{\text{L}}xi}} - F_{{{\text{R}}xi}} } \right) + a_{0} \psi_{{{\text{w}}i}} \left( {F_{{{\text{L}}yi}} - F_{{{\text{R}}yi}} + N_{{{\text{L}}yi}} - N_{{{\text{R}}yi}} } \right) \\ + M_{{{\text{L}}zi}} + M_{{{\text{R}}zi}} + d_{{\text{w}}} \left( {F_{{x{\text{fL}}i}} - F_{{x{\text{fR}}i}} } \right) + a_{0} \left( {N_{{{\text{L}}xi}} - N_{{{\text{R}}xi}} } \right) \\ \end{gathered} $$

(A.11)

$$ \begin{aligned} I_{{{\text{w}}y}} \ddot{\beta }_{{{\text{w}}i}} & = r_{{{\text{L}}i}} F_{{{\text{L}}xi}} + r_{{{\text{L}}i}} \psi_{{{\text{w}}i}} \left( {F_{{{\text{L}}yi}} + N_{{{\text{L}}yi}} } \right) + r_{{{\text{R}}i}} F_{{{\text{R}}xi}} + r_{{{\text{R}}i}} \psi_{{{\text{w}}i}} \left( {F_{{{\text{R}}yi}} + N_{{{\text{R}}yi}} } \right) \\ & \quad + \;M_{{{\text{L}}yi}} + M_{{{\text{R}}yi}} + r_{{{\text{L}}i}} N_{{{\text{L}}xi}} + r_{{{\text{R}}i}} N_{{{\text{R}}xi}} . \\ \end{aligned} $$

(A.12)

For the frame (i = 1, 2),

$$ M_{{\text{f}}} \ddot{y}_{{{\text{f}}i}} = F_{{y{\text{fL}}\left( {2i - 1} \right)}} + F_{{y{\text{fL}}\left( {2i} \right)}} - F_{{y{\text{tL}}i}} + F_{{y{\text{fR}}\left( {2i - 1} \right)}} + F_{{y{\text{fR}}\left( {2i} \right)}} - F_{{y{\text{tR}}i}} - F_{{y{\text{hxL}}i}} - F_{{y{\text{hxR}}i}} $$

(A.13)

$$ M_{{\text{f}}} \ddot{z}_{{{\text{f}}i}} = F_{{z{\text{tL}}i}} - F_{{z{\text{fL}}\left( {2i - 1} \right)}} - F_{{z{\text{fL}}\left( {2i} \right)}} + F_{{z{\text{tR}}i}} - F_{{z{\text{fR}}\left( {2i - 1} \right)}} - F_{{z{\text{fR}}\left( {2i} \right)}} + M_{{\text{f}}} g $$

(A.14)

$$ \begin{aligned} I_{{{\text{f}}x}} \dot{\phi }_{{{\text{f}}i}} & = - \left[ {F_{{y{\text{fL}}\left( {2i - 1} \right)}} + F_{{y{\text{fL}}\left( {2i} \right)}} + F_{{y{\text{fR}}\left( {2i - 1} \right)}} + F_{{y{\text{fR}}\left( {2i} \right)}} } \right]H_{{{\text{fw}}}} \\ & \quad + \;\left[ {F_{{z{\text{fL}}\left( {2i - 1} \right)}} + F_{{z{\text{fL}}\left( {2i} \right)}} - F_{{z{\text{fR}}\left( {2i - 1} \right)}} - F_{{z{\text{fR}}\left( {2i} \right)}} } \right]d_{{\text{w}}} + \left( {F_{{z{\text{tR}}i}} - F_{{z{\text{tL}}i}} } \right)d_{{\text{s}}} \\ & \quad - \;\left( {F_{{y{\text{tL}}i}} + F_{{y{\text{tR}}i}} } \right)H_{{{\text{bf}}}} - \left( {F_{{y{\text{hxL}}i}} + F_{{y{\text{hxR}}i}} } \right)H_{{{\text{fhx}}}} - F_{{y{\text{zd}}i}} H_{{{\text{fzd}}}} + M_{{{\text{r}}i}} \\ \end{aligned} $$

(A.15)

$$ \begin{aligned} I_{{{\text{f}}y}} \ddot{\beta }_{{{\text{f}}i}} & = \left[ {F_{{z{\text{fL}}\left( {2i - 1} \right)}} - F_{{z{\text{fL}}\left( {2i} \right)}} + F_{{z{\text{fR}}\left( {2i - 1} \right)}} - F_{{z{\text{fR}}\left( {2i} \right)}} } \right]l_{{\text{f}}} - \left( {F_{{x{\text{tL}}i}} + F_{{x{\text{tR}}i}} } \right)H_{{{\text{bf}}}} \\ & \quad - \;\left[ {F_{{x{\text{fL}}\left( {2i - 1} \right)}} + F_{{x{\text{fL}}\left( {2i} \right)}} + F_{{x{\text{fR}}\left( {2i - 1} \right)}} + F_{{x{\text{fR}}\left( {2i} \right)}} } \right]H_{{{\text{fw}}}} + \left( {F_{{{\text{sxL}}i}} + F_{{{\text{sxR}}i}} } \right)H_{{{\text{fsx}}}} \\ \end{aligned} $$

(A.16)

$$ \begin{aligned} I_{{{\text{f}}z}} \ddot{\psi }_{{{\text{f}}i}} & = \left[ {F_{{y{\text{fL}}\left( {2i - 1} \right)}} - F_{{y{\text{fL}}\left( {2i} \right)}} + F_{{y{\text{fR}}\left( {2i - 1} \right)}} - F_{{y{\text{fR}}\left( {2i} \right)}} } \right]l_{{\text{f}}} + \left( {F_{{x{\text{tL}}i}} - F_{{x{\text{tR}}i}} } \right)d_{{\text{s}}} - \left( {F_{{{\text{sxL}}i}} - F_{{{\text{sxR}}i}} } \right)d_{{{\text{sx}}}} \\ & \quad + \;\left[ {F_{{x{\text{fR}}\left( {2i - 1} \right)}} + F_{{x{\text{fR}}\left( {2i} \right)}} - F_{{x{\text{fL}}\left( {2i - 1} \right)}} - F_{{x{\text{fL}}\left( {2i} \right)}} } \right]d_{{\text{w}}} + \left( {F_{{y{\text{hxR}}i}} - F_{{y{\text{hxL}}i}} } \right)E \\ \end{aligned} $$

(A.17)

For the car body,

$$ M_{{\text{c}}} \ddot{y}_{{\text{c}}} = F_{{y{\text{tL}}1}} + F_{{y{\text{tL}}2}} + F_{{y{\text{tR}}1}} + F_{{y{\text{tR}}2}} + F_{{y{\text{hxL}}1}} + F_{{y{\text{hxL}}2}} + F_{{y{\text{hxR}}1}} + F_{{y{\text{hxR}}2}} + F_{{y{\text{zd}}1}} + F_{{y{\text{zd}}2}} + F_{2} $$

(A.18)

$$ M_{{\text{c}}} \ddot{z}_{{\text{c}}} = - F_{{z{\text{tL}}1}} - F_{{z{\text{tL}}2}} - F_{{z{\text{tR}}1}} - F_{{z{\text{tR}}2}} + M_{{\text{c}}} g + F_{3} $$

(A.19)

$$ \begin{aligned} I_{{{\text{c}}x}} \ddot{\phi }_{{\text{c}}} & = - \left( {F_{{y{\text{tL}}1}} + F_{{y{\text{tL}}2}} + F_{{y{\text{tR}}1}} + F_{{y{\text{tR}}2}} } \right)H_{{{\text{cb}}}} - \left( {F_{{y{\text{hxL}}1}} + F_{{y{\text{hxL}}2}} + F_{{y{\text{hxR}}1}} + F_{{y{\text{hxR}}2}} } \right)H_{{{\text{chx}}}} \\ & \quad + \;\left( {F_{{z{\text{tL}}1}} + F_{{z{\text{tL}}2}} - F_{{z{\text{tR}}1}} - F_{{z{\text{tR}}2}} } \right)d_{{\text{s}}} - \left( {F_{{y{\text{zd}}1}} { + }F_{{y{\text{zd2}}}} } \right)H_{{{\text{czd}}}} - M_{{{\text{r1}}}} - M_{{{\text{r2}}}} + M_{1} \\ \end{aligned} $$

(A.20)

$$ \begin{aligned} I_{{{\text{c}}y}} \ddot{\beta }_{{\text{c}}} & = \left( {F_{{z{\text{tL}}1}} - F_{{z{\text{tL}}2}} + F_{{z{\text{tR}}1}} - F_{{z{\text{tR}}2}} } \right)l_{{\text{c}}} - \left( {F_{{x{\text{tL}}1}} { + }F_{{x{\text{tL2}}}} { + }F_{{x{\text{tR}}1}} { + }F_{{x{\text{tR}}2}} } \right)H_{{{\text{cb}}}} \\ & \quad \times \;\left( {F_{{x{\text{sL}}1}} + F_{{x{\text{sL}}2}} + F_{{x{\text{sR}}1}} + F_{{x{\text{sR}}2}} } \right)H_{{{\text{csx}}}} + M_{2} \\ \end{aligned} $$

(A.21)

$$ \begin{aligned} I_{{{\text{c}}z}} \ddot{\psi }_{{\text{c}}} & = \left( {F_{{y{\text{tL}}1}} - F_{{y{\text{tL}}2}} + F_{{y{\text{tR}}1}} - F_{{y{\text{tR}}2}} } \right)l_{{\text{c}}} + \left( {F_{{x{\text{tR}}1}} + F_{{x{\text{tR2}}}} - F_{{x{\text{tL}}1}} - F_{{x{\text{tL2}}}} } \right)d_{{\text{s}}} \\ & \quad - \;\left( {F_{{x{\text{sR}}1}} + F_{{x{\text{sR2}}}} - F_{{x{\text{sL}}1}} - F_{{x{\text{sL2}}}} } \right)d_{{{\text{sx}}}} + \left( {F_{{y{\text{hxL}}1}} - F_{{y{\text{hxL}}2}} + F_{{y{\text{hxR}}1}} - F_{{y{\text{hxR}}2}} } \right)l_{{\text{c}}} \\ & \quad + \;\left( {F_{{y{\text{hxL}}1}} + F_{{y{\text{hxL}}2}} - F_{{y{\text{hxR}}1}} - F_{{y{\text{hxR}}2}} } \right)E + \left( {F_{{y{\text{zd}}1}} - F_{{y{\text{zd}}2}} } \right)l_{{\text{c}}} + M_{3} . \\ \end{aligned} $$

(A.22)

For the connecting points of the lateral dampers (i = 1, 2),

$$ K_{{{\text{hx}}}} \left( {y_{{{\text{f}}i}} + H_{{{\text{fhx}}}} \phi_{{{\text{f}}i}} - y_{{{\text{hx}}\left( {\text{L,R}} \right)i}} \pm E\psi_{{{\text{f}}i}} } \right) = C_{{{\text{hx}}}} \left( {\dot{y}_{{{\text{hx}}\left( {\text{L,R}} \right)i}} - y_{{\text{c}}} + H_{{{\text{chx}}}} \dot{\phi }_{{\text{c}}} + \left( { - 1} \right)^{i} \left( {l_{{\text{c}}} \pm E} \right)\dot{\psi }_{{\text{c}}} } \right). $$

(A.23)

For the connecting points of the yaw dampers (i = 1, 2),

$$ K_{{{\text{sx1}}}} \left( { \pm d_{{{\text{sx}}}} \psi_{{{\text{f}}i}} - H_{{{\text{fsx}}}} \beta_{{{\text{f}}i}} - y_{{{\text{sx}}\left( {{\text{L}},{\text{R}}} \right)i}} } \right) = C_{{{\text{sx1}}}} \left( {\dot{y}_{{{\text{sx}}\left( {{\text{L}},{\text{R}}} \right)i}} \mp d_{{{\text{sx}}}} \dot{\psi }_{{\text{c}}} - H_{{{\text{csx}}}} \dot{\beta }_{{\text{c}}} } \right). $$

(A.24)

Appendix B: Nomenclature and vehicle parameters

The nomenclature used in the dynamics equations follows [45] (Tables

Table 5 Nominal values of vehicle parameters

5,

Table 6 Damping characteristics of yaw dampers

6,

Table 7 Damping characteristics of lateral dampers

7,

Table 8 Characteristics of bump stop forces

8).