Dynamic modelling and simulation of rail car suspension systems using classic controls

Figures & data

Figure 1. Rail car body and its suspension assembly (Zolotas & Goodall, 2007)

Figure 2. The free body diagram and its degree of freedom

Table 1. Input parameter for rail car system modelling

S/N	Parameter	Notation	Value	Unit
1.	Average mass of the rail car	$M_r$	50,500	kg
2.	Average mass of bogie	$M_b$	2,410	kg
3.	Mass of primary suspension system	$M_p$	30,000	kg
4.	Mass of secondary suspension system	$M_s$	30,000	kg
5.	Moments of inertia	$I_i$	56,900	kg m^2
6.	Rail car roll inertia	$I_r$	68,200	kg m^2
7.	Rail car pitch inertia	$I_p$	71,000	kg m^2
8.	Average mass of first wheelset and axle	$m_1$	1,300	kg
9.	Average mass of second wheel	$m_2$	1,300	kg
10.	Distance between the center of gravity and the front position of the rail car	$d_i$	6	m
11.	Distance between the center of gravity and the middle position of the rail car	$d_2$	6	m
12.	Distance between the center of gravity and the rear position of the rail car	$d_3$	6	m
13.	Spring constant of the primary suspension system	$k_1$	$2.4\times$10^6	N/m
14.	Spring constant of the secondary suspension system	$k_2$	5.6$\times$10^5	N/m
15.	Spring constant of the wheel	$k_3$	4.0$\times$10^5	N/m
16.	Damping constant of the primary suspension system	$b_1$	1.2$\times$10^3	Ns/m
17.	Damping constant of the secondary suspension system	$b_2$	2.95$\times$10^4	Ns/m
18.	Damping constant of the wheel	$b_3$	5.0$\times$10^4	Ns/m

Source: (Sharma & Kumar, 2017; Ulum et al., 2017).

Figure 3. Design of the control of the rail car suspension system

Table 2. Zeigler-Nichols tuning rules

S/N	Type of controller	Kp	Ki	Kd
1	P	0.5 Kcr	$\mathrm{\infty }$	0
2	PI	0.45 Kcr	0.83 Pcr	0
3	PID	0.6 Kcr	0.5 Pcr	0.125 Pcr

Figure 4. Step response of the semiactive system for 3 s (primary suspension system)

Figure 5. Adjustment of the control of the rail car suspension system

Figure 6. Adjusted step response for the active system for 3 s (primary suspension system)

Figure 7. Oscillation response of rail car to load disturbances (primary suspension system)

Figure 8. Response to a 0.1 m step under the PID control (primary suspension system)

Figure 9. Response to a 0.1 m step with high gain PID (primary suspension system)

Table 3. Effects of controller parameters on closed-loop system

S/N	Controller response	Rise time	Over-shoot	Setting time
1	Kp	Decrease	Increase	Small change
2	Ki	Decrease	Increase	Increase
3	Kd	Small change	Decrease	Decrease

Figure 10. Bode diagram (primary suspension system)

Figure 11. Oscillation response of rail car to rail disturbances (secondary suspension system)

Figure 12. Response to a 0.1 m step under the PID control (secondary suspension system)

Figure 13. Final oscillation step response to rail disturbance (secondary suspension system)

Figure 14. Bode diagram (secondary suspension system)

Figure 15. Vertical acceleration for the controlled and uncontrolled systems

Figure 16. Pitch acceleration for the controlled and uncontrolled systems

Figure 17. Output disturbance rejection

Figure 18. Input disturbance rejection

Table

Parameters	Notation	Parameter	Notation
Average mass of the rail car	$M_r$	Sprung mass	$m_s$
Average mass of bogie	$M_b$	Unspring mass	$m_u$
Mass of primary suspension system	$M_p$	Linear component of the spring constant	$k_s^l$
Mass of secondary suspension system	$M_s$	Nonlinear component of the spring constant	$k_s^{nl}$
Moments of inertia	$I_i$	Wheel spring constant	$k_t$
Rail car roll inertia	$I_r$	Linear component of the damping coefficient	$b_s^l$
Average mass of first wheelset and axle	$m_1$	Nonlinear component of the damping coefficient	$b_s^{nl}$
Average mass of second wheel	$m_2$	Deflection of primary suspension	$y_1$
Dissipative energy	$R_d$	Deflection of secondary suspension	$y_2$
Distance between the center of gravity and the front position of the rail car	$d_i$	Body deflection (m),	$x_1$
Distance between the center of gravity and the middle position of the rail car	$d_2$	Wheel deflection	$x_2$
Distance between the center of gravity and the rear position of the rail car	$d_3$	Rail disturbance	$w$
Spring constant of the primary suspension system	$k_1$	Actuating force	$F$
Spring constant of the secondary suspension system	$k_2$	Spring force on the body	$F_{k_s}$
Spring constant of the wheel	$k_3$	Damping force	$F_{b_s}$
Damping constant of the primary suspension system	$b_1$	Spring force acting on the wheel	$F_{k_t}$
Damping constant of the secondary suspension system	$b_2$	System’s input	$u\left(t\right)$
Damping constant of the wheel	$b_3$	System’s output	$y\left(t\right)$
Vertical displacement of the suspension system	$a_j$	Current state	$x\left(t\right)$
Stiffness of the spring	$k_j$	Moments of deviation to the axes passing through the center of gravity C.	$d_{xy}$ and $d_{yx}$
Vertical component of the velocity at point $j$	${\dot{a}}_j$	Moments of inertia to the axes through the center of gravity $C_1$ and $C_2.$	$M_{a1}$, $M_{b1}$ and $M_{a2}$, $M_{b2}$
Damping intensity	$F_j$	Moments of deviation to the axes through the center of gravity $C_1$ and $C_2.$	$d_{xy1},$ $d_{yx1,}{d}_{xy2},$ $d_{yx2,}$
Operating frequency of the rail disturbance	$\omega$	Vertical direction	$a_1,a_{2,}{a}_3$
Natural frequency of the rail car system	${\omega }_n$	Longitudinal direction	$l_1,{l_{2,}}_3$
Transfer function	$\frac{x}{y}$	Lateral directions	${}_{1,}{m}_{2,}{m}_3$
Damping coefficient	$\tau$	Roll, pitch and yaw angles	${\phi }_x,{\phi }_y,{\phi }_{x1},{\phi }_{y1,}{\phi }_{x2},{\phi }_{y2}$
Phase angle	$\mathrm{\varnothing }$	Proportional gain	Kp
Output response	$x\left(t\right)$	Integral gain	Ki
		Derivative time	Kd

Zolotas, A. G., & Goodall, R. M. (2007). Modelling and con-trol of railway vehicle suspensions. In M. C. Turner & D. G. Bates (Eds.), Lecture notes in control and information sciences, mathematical methods for robust and nonlinear control (pp. 373–412). New York: Springer.

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Sharma, S. K., & Kumar, A. (2017). Ride performance of a high-speed rail vehicle using controlled semi active suspension system. Smart Materials and Structures, 26(1–19), 055026. doi:10.1088/1361-665X/aa68f7

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Ulum, Z., Affaf, M., Salmah, & Suparwanto, A. (2017) Active suspension systems design of a light rail vehicle using MPC with preview information disturbance”. 5th International Conference on Instrumentation, Control, and Automation (ICA) Yogyakarta, Indonesia, August 9-11, 2017. pp. 18–23.

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