Open AccessArticle

New Mixed Skyhook and Displacement–Velocity Control for Improving the Effectiveness of Vibration Isolation in the Lateral Suspension System of a Railway Vehicle

Yaojung Shiao

^1,2,*

and

Tan-Linh Huynh

Department of Vehicle Engineering, National Taipei University of Technology, Taipei 10608, Taiwan

Railway Vehicle Research Center, National Taipei University of Technology, Taipei 10608, Taiwan

Author to whom correspondence should be addressed.

Appl. Sci. 2024, 14(24), 11680; https://doi.org/10.3390/app142411680

Submission received: 29 October 2024 / Revised: 10 December 2024 / Accepted: 12 December 2024 / Published: 14 December 2024

(This article belongs to the Special Issue Novel Advances in Noise and Vibration Control)

Download

Browse Figures

Figure 1
The quarter railway vehicle model. "> Figure 2
(a) The velocity region division and (b) the contribution coefficient for the new SH control under the vibration velocity of the carbody. (a): the regions (1) 0 to 0.0075 m/s, (2) 0.0075 to 0.025 m/s, (3) 0.025 to 0.1 m/s, and (4) ≥0.1 m/s, assumed to correspond the carbody vibration at the high frequency domain, medium frequency domain, around the second-order resonance frequency domain, and around the first-order resonance frequency domain, respectively. A, C, D, E, G, and H are the intersections of region thresholds and line graph of vibration velocity of carbody; B and F are the peak at the first and second-order resonance frequencies, respectively. "> Figure 3
Comparisons of the vibration transmissibility under different controls. "> Figure 4
The RMS of (a) displacement, (b) velocity, and (c) acceleration of the carbody. "> Figure 5
Damping contribution coefficient for the new DV control according to the vibration velocity of the carbody. "> Figure 6
Comparisons of the vibration transmissibility of different controls under sinusoidal signal input with amplitudes of ±5 mm. "> Figure 7
The RMS of (a) displacement, (b) velocity, and (c) acceleration of the carbody under sinusoidal signal input with amplitudes of ±5 mm. "> Figure 8
Comparisons of the vibration transmissibility of the carbody under three control algorithms under sinusoidal signal input with amplitudes of ±5 mm. "> Figure 9
The RMS of (a) displacement, (b) velocity, and (c) acceleration of the carbody under sinusoidal signal input with amplitudes of ±5 mm. ">

Versions Notes

Abstract

Demands for increasing the velocity and load carrying capacity of railway vehicles are a challenge to the passive suspension systems used for isolating the lateral vibrations of the carbody of a railway vehicle, especially under a wide range of vibration frequencies. Semiactive suspension systems, especially systems with a magnetorheological damper (MRD), have been investigated as promising alternatives. Many control algorithms have been developed for fine-tuning the damping force generated by MRDs, but they have been ineffective in isolating carbody vibrations at or around the resonance frequencies of the carbody and bogie. This study aims to develop a mixed control algorithm for a new skyhook (SH) control and a new displacement–velocity (DV) control to improve the effectiveness of vibration isolation in resonance frequency regions while producing high performance across the remaining frequencies. The damping coefficient of the new SH controller depends on the vibration velocity of the components of the suspension system and the skyhook damping variable, whereas that of the new DV controller depends on the velocity and displacement of the components of the suspension system and the stiffness variable. The values of the skyhook damping variable and stiffness variable were identified from the vibration velocity of the carbody using the trial and error method. The results of a numerical simulation problem indicated that the proposed control method worked effectively at low frequencies, similar to the conventional SH–DV controller, whereas it significantly improved ride comfort at high frequencies; at the resonance frequency of the bogie (14.6 Hz), in particular, it reduced the vibration velocity and acceleration of the carbody by 50.85% and 45.39%, respectively, compared with the conventional mixed SH–DV controller. The simplicity and high performance of the new mixed SH–DV control algorithm makes it a promising tool to be applied to the semiactive suspension of railway vehicles in real-world applications.

Keywords:

lateral suspension; magnetorheological damper; mixed SH–DV control; ride comfort; semiactive control

1. Introduction

Railway vehicles are crucial for economic development and social connectivity because they transport goods and people. With increasing usage, their speed and load-carrying capacity need to be upgraded [1], posing challenges to the quality of train transportation, which is evaluated on the basis of ride comfort and safety. To address these challenges, passive suspension systems are commonly used in railway vehicles, and their structures and underlying principles are continually being optimized. For example, passive inertial suspensions have been used to expand the scope of both vibration isolation ability and road friendliness, however, the adaptation range of such passive suspension systems is relatively low compared with the range of all possible vibration excitation frequencies from the track encountered during the operation of a train [2]. Semiactive suspension systems have been explored as alternatives, offering some advantages, such as (1) being easy to embed into the original passive suspension structure, (2) being easy to implement using simple algorithms of conventional controls, (3) being attainable at a reasonable cost, (4) exhibiting low energy consumption, and (5) demonstrating high effectiveness in vibration isolation under a wide range of vibration frequencies [3,4,5,6,7]. Of all the devices used in semiactive suspension systems, magnetorheological dampers (MRDs) have garnered widespread attention because of their quick response, reasonable cost, and compact structure [4,5,6,7,8,9,10]. By controlling external magnetic fields, MRDs generate damping forces that vary over a large range depending on dynamic signals from suspension system components.

Many control algorithms have been developed for fine-tuning the damping force generated by MRDs. These control algorithms can be divided into two main groups: traditional control (e.g., skyhook (SH) [11], groundhook (GH) [12], acceleration-driven damping (ADD) [13], displacement–velocity (DV) [8], mixed SH–GH [14], mixed SH–ADD [15], mixed SH–DV [8], and advanced hybrid SH–GH control using semiactive inertial suspension to address the control-phase deviation [16]); and modern control (e.g., fuzzy [17] and neural network [18,19] control). The algorithms in the modern control group are generally more effective in isolating undesirable vibrations of sprung mass parts; however, they require actual data on desirable damping forces generated by MRDs under real-world operating conditions, and obtaining such data is both time-consuming and laborious. By contrast, the algorithms in the traditional control group are easier to implement because they only require the dynamic signals obtained by accelerometers to control the external magnetic field. However, traditional control algorithms have their own limitations. Singular control-type algorithms can effectively attenuate vibrations with frequencies only within a certain range (i.e., low or high-frequency regions). Mixed control type algorithms inherit the effective working region of their singular control components and also expand this region; however, these algorithms are unable to reach the asymptotic effectiveness of their individual singular control components. Moreover, both the traditional singular and mixed control-type algorithms perform ineffectively around the resonance frequency of the sprung and unsprung mass parts [20].

In this study, we propose a new mixed SH–DV control method to isolate undesirable lateral vibrations of the carbody of a railway vehicle, that offers the advantages of a traditional control algorithm while addressing the aforementioned limitations. The proposed method comprises two main stages. First, the conventional singular controllers (SH and DV) are upgraded to improve their vibration isolation performance. The damping coefficient of the new SH controller depends on the vibration velocity of the secondary lateral suspension system components and the skyhook damping variable, whereas the damping coefficient of the new DV controller depends on the velocity and displacement of the secondary lateral suspension system components and the stiffness variable. Second, the two new singular controllers are combined into a new mixed SH–DV controller that selects between the two control states (new SH control or new DV control) depending on the conversion constant and the vibration acceleration and velocity of the carbody of the railway vehicle. In summary, the new mixed SH–DV controller is expected to enhance the effectiveness of conventional singular and mixed controllers in vibration isolation for the carbody of a railway vehicle.

2. Definition of the Simulation Problem

2.1. Quarter Railway Vehicle Model

To demonstrate the effectiveness of the new SH–DV controller in suppressing undesirable vibrations, a numerical simulation problem was designed and implemented, with the lateral oscillation of the railway vehicle’s carbody being the study object. In the field of vehicle transportation, railway vehicles have been represented by several numerical models, including the quarter, half, and whole railway vehicle models. Of these, the quarter railway vehicle model is the simplest in terms of implementation, while providing results comparable to those obtained from the whole railway vehicle model. Yang et al. [8] simplified the quarter railway vehicle model by ignoring node stiffness in the dynamic formula. Given that the damping coefficient was the control object in this study and that the characteristics of ride comfort were assumed to be retained under different suitable stiffnesses, the current study adopted the simplified quarter railway vehicle model for its analyses. The configuration and dynamic equation of the quarter railway vehicle model with the traditional secondary lateral suspension system are shown in Figure 1 and Equation (1), respectively [8].

\{\begin{matrix} m_{c} {\ddot{y}}_{1} + k_{s} (y_{1} - y_{2}) + c_{s} ({\dot{y}}_{1} - {\dot{y}}_{2}) & = 0 \\ m_{b} {\ddot{y}}_{2} - k_{s} (y_{1} - y_{2}) - c_{s} ({\dot{y}}_{1} - {\dot{y}}_{2}) + k_{p} (y_{2} - y_{w}) & = 0 \end{matrix}

(1)

where

y_{1}, y_{2},

and

y_{w}

are the absolute displacement of the carbody, bogie, and wheel, respectively;

{\dot{y}}_{1}

and

{\dot{y}}_{2}

are the absolute velocity of the carbody and bogie, respectively; and

{\ddot{y}}_{1}

and

{\ddot{y}}_{2}

are the absolute acceleration of the carbody and bogie, respectively. Other structural parameters of the traditional passive lateral suspension system are listed in Table 1.

2.2. Simulation Conditions

After the quarter railway vehicle model was selected for analysis, the vibration excitation source was analyzed. Vibrations are generated through contact between the rough surfaces of the rail and the wheels when the wheels roll on the surface of the rail. Sources of vibrations can be represented using several numerical data inputs, such as the sinusoidal function, German low-interference track irregularity, and Chinese high-speed railway track irregularity. Data on the latter two functions are collected and converted from real-world vibration data, making these functions more applicable to the whole or half railway vehicle models. Conversely, the sinusoidal function is usually applied to the quarter railway vehicle model [8]. Given that this study aimed to evaluate the effectiveness of vibration isolation across a wide range of vibration frequencies, which contains both the natural resonance frequencies of the carbody and the bogie, a sinusoidal function with a vibration frequency range of 0.1–20 Hz was selected. The amplitude of the sinusoidal vibration was selected to be

\pm

5 mm, considering that the amplitude of actual wheel vibrations can vary within a range of few millimeters. In addition, the numerical simulation was set up with a fixed step size of 0.001 s and a total simulation time of 100 s [21].

2.3. Evaluation of Effectiveness of Vibration Isolation

The effectiveness of the proposed controller in isolating vibrations was directly evaluated through the absolute displacement, velocity, and acceleration of the train body. Furthermore, the acceleration frequency response characteristics of the carbody were considered to more comprehensively evaluate ride comfort. Because of its simplicity, the vibration transmissibility characteristics

H (f)

from the disturbance input excitation of the rail and the wheel to the vibration acceleration of the carbody was used to represent acceleration frequency response characteristics, as shown in Equation (2) [8].

H (f) = \sqrt{\frac{\int_{0}^{t} {\ddot{y}}_{1}^{2} d t}{\int_{0}^{t} y_{w}^{2} d t}}

(2)

In addition, the natural resonance frequencies of the carbody and bogie components were theoretically calculated using Equation (3), referring to Table 1 for the values of the parameters of the railway vehicle’s lateral suspension system. The natural resonance frequencies of the carbody and bogie were calculated to be 1.57 and 14.2 Hz, respectively.

\{\begin{matrix} ω_{n} = \sqrt{\frac{k}{m}} \\ f_{n} = \frac{1}{2 π} \sqrt{\frac{k}{m}} \end{matrix}

(3)

3. Conventional Skyhook, Displacement–Velocity, and Skyhook and Displacement–Velocity Control Methods

The new SH–DV control algorithm was derived by upgrading the conventional SH, DV, and SH–DV control algorithms. First introduced by Karnopp et al. [11], the SH control algorithm was developed on the basis of the operating principle of a suspension system using an ideal sky damper. SH control is defined by the damping coefficient

c_{S H}

, which can be altered continuously. Note that

c_{S H}

is proportional to the skyhook damping constant

c_{s k y}

and the ratio of the absolute and relative vibration velocities of the sprung mass part. The activating condition for SH control is given by the product of the absolute and relative vibration velocities of the sprung mass part. In addition, considering the real-world applications of MRDs, the minimum

c_{m i n}

and maximum

c_{m a x}

values of the damping coefficient were introduced;

c_{m i n}

represents the scenario wherein an external magnetic field is absent (i.e., the MRD functions as a hydraulic damper), whereas

c_{m a x}

represents the scenario wherein the MRD operates at maximum capacity under its allowable assembly space constraints. The damping coefficient for SH control, designed in the context of practical application, has been described in another study [22] and can be mathematically represented as Equation (4):

c_{S H} = \{\begin{matrix} \max (c_{m i n}, m i n (c_{m a x}, |c_{s k y} \frac{{\dot{y}}_{1}}{{\dot{y}}_{1} - {\dot{y}}_{2}}|)) & {\dot{y}}_{1} ({\dot{y}}_{1} - {\dot{y}}_{2}) \geq 0 \\ c_{m i n} & {\dot{y}}_{1} ({\dot{y}}_{1} - {\dot{y}}_{2}) < 0 \end{matrix}

(4)

The DV control algorithm was derived from the mathematical analysis of a minimization problem. Yang et al. [8] constructed the DV control algorithm to minimize the acceleration of the carbody by minimizing the displacement and velocity functions of the carbody and bogie, as shown in Equation (5). The damping coefficient for DV control,

c_{D V}

, is directly proportional to the spring force [i.e.,

F_{s} = k_{s} (y_{1} - y_{2})

] and inversely proportional to the relative velocity of the carbody [i.e.,

({\dot{y}}_{1} - {\dot{y}}_{2})

]. The activating condition for DV control is given by the product of the relative displacement and velocity of the carbody. In particular, when the relative displacement and relative velocity have opposite signs, the DV control algorithm activates a damping coefficient, which is given by

c_{D V} = |F_{s} / ({\dot{y}}_{1} - {\dot{y}}_{2})|

, implying that the damping force of DV control cancels out the spring force, so that the carbody’s acceleration approaches zero. In other scenarios, the DV control algorithm does not activate a damping coefficient. Considering the applications of MRDs, Yang et al. defined the damping coefficient for DV control as follows:

c_{D V} = \{\begin{matrix} \max (c_{m i n}, m i n (c_{m a x}, |\frac{k_{s} (y_{1} - y_{2})}{{\dot{y}}_{1} - {\dot{y}}_{2}}|)) & (y_{1} - y_{2}) ({\dot{y}}_{1} - {\dot{y}}_{2}) < 0 \\ c_{m i n} & (y_{1} - y_{2}) ({\dot{y}}_{1} - {\dot{y}}_{2}) \geq 0 \end{matrix}

(5)

A mixed control algorithm—such as SH–GH, SH–ADD, or SH–DV—is constructed by combining two conventional singular control algorithms. The mixed SH–DV control algorithm was used in this study. Yang et al. [8] constructed the SH–DV control algorithm by using a function of the vibration acceleration and velocity of the carbody and the conversion constant c, which serves as a switching condition for the control state. The conversion constant is given by

c = 2 π f_{c}

, where

f_{c}

, representing the crossover frequency constant, is defined as the frequency intersection point between the vibration transmissibility of SH control and DV control. The mixed control algorithm is presented in Equation (6) [8].

c_{S H - D V} = \{\begin{matrix} c_{S H} {\ddot{y}}_{1}^{2} - c^{2} \cdot {\dot{y}}_{1}^{2} < 0 \\ c_{D V} {\ddot{y}}_{1}^{2} - c^{2} \cdot {\dot{y}}_{1}^{2} \geq 0 \end{matrix}

(6)

where

y_{1}

{\dot{y}}_{1}, {\ddot{y}}_{1}

, and

y_{2}

{\dot{y}}_{2}, {\ddot{y}}_{2}

denote lateral displacements, velocities, and accelerations of the body and the bogie, respectively. The following values were selected for the maximum and minimum values of the damping coefficient and skyhook damping constant for the MRD: c_max = 50,000 Ns·m⁻¹, c_min = 3333 Ns·m⁻¹, and c_sky = 50,000 Ns·m⁻¹, respectively.

4. New Skyhook and Displacement–Velocity Control Method

4.1. Design and Analysis of the New Skyhook Control Algorithm

A new SH control algorithm is proposed to address the large vibration transmissibility at and around the natural resonance frequencies of the suspension system. The concept of this algorithm was first considered in our previous study [23]. Similar to the conventional SH control, the damping coefficient for the new SH control,

c_{n_S H}

, depends on the absolute and relative vibration velocities of the carbody. In addition, it also depends on the skyhook damping variable

V_{s k y}

, as shown in Equation (7). The skyhook damping variable is proportional to the skyhook damping constant

c_{s k y}

and the contribution coefficient

α

, as given by the equation

V_{s k y} = α c_{s k y}

. Note that

α \geq 1

, and

α

increases as the absolute vibration velocity of the carbody increases, and vice versa. When the vibration velocity of the carbody is low,

V_{s k y}

is low, resulting in a low

c_{n_S H}

. By contrast, when the vibration velocity of the carbody is high,

V_{s k y}

is high, and

c_{n_S H}

is higher than

c_{S H}

by a factor of

α

under the same vibration velocity of the carbody. When

α = 1

, the new SH control and conventional SH control algorithms are identical.

At frequencies equal to or near the resonance frequency of the system, the suspension system oscillates with great intensity, and

c_{n_S H}

is higher than

c_{S H}

by a factor of

α

, which contributes to the suppression of vibrations in a timely manner, making the new SH control algorithm more effective than the conventional SH control algorithm. Note that

c_{n_S H}

is constrained between the upper and lower bounds of the damping coefficient, which is represented by the max–min function in Equation (7).

c_{n_S H} = \{\begin{matrix} \max (c_{m i n}, m i n (c_{m a x}, |V_{s k y} \frac{{\dot{y}}_{1}}{{\dot{y}}_{1} - {\dot{y}}_{2}}|)), & {\dot{y}}_{1} ({\dot{y}}_{1} - {\dot{y}}_{2}) \geq 0 \\ c_{m i n}, & {\dot{y}}_{1} ({\dot{y}}_{1} - {\dot{y}}_{2}) < 0 \end{matrix}

(7)

The construction process for the new SH control algorithm can be summarized by three main steps: (1) determining the vibration velocity domain of the carbody through the numerical simulation problem for the railway vehicle model under a passive suspension system; (2) defining the thresholds and regions of the velocity from the velocity domain of the carbody; and (3) determining the contribution coefficients

α

for the new control algorithm by using the trial and error method. As shown in Equation (8), this study identified three velocity thresholds,

T_{1}

T_{3}

, corresponding to the four velocity regions—

(0 - T_{1}], (T_{1} - T_{2}], (T_{2} - T_{3}],

and

(T_{3} - \infty)

—and thus, to four values of the contribution coefficient,

α_{1}

α_{4}

c_{n_S H} = \{\begin{matrix} \max (c_{m i n}, m i n (c_{m a x}, |{α_{1} c}_{s k y} \frac{{\dot{y}}_{1}}{{\dot{y}}_{1} - {\dot{y}}_{2}}|)), & |{\dot{y}}_{1}| \leq T_{1} \cap {\dot{y}}_{1} ({\dot{y}}_{1} - {\dot{y}}_{2}) \geq 0 \\ \max (c_{m i n}, m i n (c_{m a x}, |{α_{2} c}_{s k y} \frac{{\dot{y}}_{1}}{{\dot{y}}_{1} - {\dot{y}}_{2}}|)), & T_{1} < |{\dot{y}}_{1}| \leq T_{2} \cap {\dot{y}}_{1} ({\dot{y}}_{1} - {\dot{y}}_{2}) \geq 0 \\ \max (c_{m i n}, m i n (c_{m a x}, |{α_{3} c}_{s k y} \frac{{\dot{y}}_{1}}{{\dot{y}}_{1} - {\dot{y}}_{2}}|)), & T_{2} < |{\dot{y}}_{1}| \leq T_{3} \cap {\dot{y}}_{1} ({\dot{y}}_{1} - {\dot{y}}_{2}) \geq 0 \\ \max (c_{m i n}, m i n (c_{m a x}, |{α_{4} c}_{s k y} \frac{{\dot{y}}_{1}}{{\dot{y}}_{1} - {\dot{y}}_{2}}|)), & T_{3} < |{\dot{y}}_{1}| \cap {\dot{y}}_{1} ({\dot{y}}_{1} - {\dot{y}}_{2}) \geq 0 \\ c_{m i n}, & {\dot{y}}_{1} ({\dot{y}}_{1} - {\dot{y}}_{2}) < 0 \end{matrix}

(8)

Using trial and error, we determined three velocity threshold values of the new SH control, i.e., 0.0075, 0.025, and 0.1 m/s (Figure 2a). These thresholds define four vibration velocity regions of the carbody during the operating process: (1) 0 to 0.0075 m/s, assumed to be corresponding to carbody vibration at a high frequency domain (after the second-order resonance frequency, i.e., from H backwards); (2) 0.0075 to 0.025 m/s, assumed for the carbody vibration at a medium frequency domain (after the first-order resonance and before the second-order resonance frequency, i.e., D to E and G to H); (3) 0.025 to 0.1 m/s, assumed for the carbody vibration at around the second-order resonance frequency, i.e., from A forwards, C to D, and E to G; and (4) ≥0.1 m/s, assumed for the carbody vibration at around the first-order resonance frequency, i.e., A to C. Contribution coefficients associated with these four vibration velocity regions are presented in Figure 2b, ranging between 1 and 8. As a result, when the vehicle body oscillates around the second resonance frequency region, the contribution coefficient increases by a factor of six, leading to a sixfold increase in the instantaneous damping force. Similarly, around the first resonance frequency region, the contribution coefficient increases by a factor of eight, causing the instantaneous damping force to rise by eight times, compared with the conventional SH control algorithm. Both the first and second resonance frequency regions require hard damping (i.e., high damping force) as clearly stated by [24] regarding the ride comfort problem. Therefore, based on the aforementioned rationale, the proposed control method is expected to work effectively in suppressing the undesirable oscillation of the train body.

Figure 3 presents a comparison of the transmissibility of vibrations from the rail to the train body under various control methods. Conventional SH control performed better at isolating vibrations than did passive control, but it was ineffective at approximately 14.6 Hz, which is the resonance frequency of the bogie. At low frequencies, which include the resonance frequency of the carbody (1.5 Hz), the absolute vibration velocity of the carbody was often greater than that of the bogie, resulting in the product of the relative and absolute oscillation velocities of the carbody being positive. By contrast, at around the resonance frequency of the bogie, the absolute oscillation velocity of the bogie was greater than the absolute oscillation velocity of the carbody, resulting in the product of the relative and absolute vibration velocities of the carbody usually being negative. This implies that the number of instances when the MRD generated damping forces was lower at frequencies near the resonance frequency of the bogie than at other frequencies, leading to reduced vibration isolation performance. The same trend was observed with the new SH control algorithm at frequencies near the resonance frequency of the bogie. Nevertheless, the resulting damping force in this case was larger than that generated through conventional SH control by a factor of

α

, which helped the MRD generate sufficient energy in a timely manner to suppress the bogie’s resonance. Therefore, the new SH control algorithm is more effective over the entire frequency domain, including the resonance frequencies of the bogie and carbody.

For a more in-depth assessment of the new SH control algorithm, the absolute displacement, velocity, and acceleration of the train body were compared under various control methods (Figure 4). The displacement of the carbody was much lower under conventional and new SH control than under passive control, especially at low frequencies [Figure 4a]. The velocity of the carbody under conventional SH control was overall lower than that under passive control, except at frequencies near the resonance frequency of the bogie (14.6 Hz). Unlike conventional SH control, the new SH control algorithm performed highly across the entire domain of vibration frequencies, including the resonance frequencies of the bogie and carbody [Figure 4b]. The trends in the acceleration of the carbody were similar to those in the velocity of the carbody under the three control methods [Figure 4c]. Compared with conventional SH control, the new SH control algorithm effectively attenuated vibrations in the lateral suspension system, reducing the vibration velocity and acceleration of the carbody by 51.86% and 51.21%, respectively, at a frequency of 14.6 Hz.

4.2. Design and Analysis of New Displacement–Velocity Controller

Similar to the new SH controller, the new DV controller was designed with the objective of mitigating the transmissibility of vibrations with frequencies near the resonance frequency of the bogie and with low frequencies. Unlike the damping coefficient of the conventional DV control algorithm, which depends only on the ratio of relative displacement and the relative velocity of the carbody, the damping coefficient of the new DV controller,

c_{n_D V}

, also depends on the stiffness variable

V_{s}

, as shown in Equation (9).

V_{s}

is proportional to the stiffness constant

k_{s}

and the contribution coefficient

β

and can be expressed using the equation

V_{s}

β k_{s}

. Note that

β \geq 1

, and

β

increases as the absolute oscillation velocity of the carbody increases, and vice versa. When the vibration velocity of the carbody is low,

c_{n_D V}

is also low. By contrast, at higher carbody’s vibration velocities,

c_{n_D V}

is higher than

c_{D V}

, the damping coefficient of the conventional DV controller, by a factor of

β

. When

β = 1

, the new DV control algorithm resembles the conventional DV control algorithm. In other cases, similar to the new SH control algorithm, the new DV control algorithm generates higher values of

c_{n_D V}

by a factor of

β

compared with the damping coefficient of the conventional DV controller.

c_{n_D V} = \{\begin{matrix} \max (c_{m i n}, m i n (c_{m a x}, |V_{s} \frac{(y_{1} - y_{2})}{{\dot{y}}_{1} - {\dot{y}}_{2}}|)), & (y_{1} - y_{2}) ({\dot{y}}_{1} - {\dot{y}}_{2}) < 0 \\ c_{m i n}, & (y_{1} - y_{2}) ({\dot{y}}_{1} - {\dot{y}}_{2}) \geq 0 \end{matrix}

(9)

Similar to the process adopted by the new SH control algorithm for determining

α

, the new DV control algorithm uses a three-step trial and error method to obtain five values of the contribution coefficient,

β_{1}

β_{5}

, corresponding to the five velocity regions

(0 - L_{1}], (L_{1} - L_{2}], (L_{2} - L_{3}], (L_{3} - L_{4}]

, and

(L_{4} - \infty)

, where

L_{1}

L_{4}

represent four velocity thresholds in Equation (10).

c_{n_D V} = \{\begin{matrix} \max (c_{m i n}, m i n (c_{m a x}, |{β_{1} k}_{s} \frac{(y_{1} - y_{2})}{{\dot{y}}_{1} - {\dot{y}}_{2}}|)), & |{\dot{y}}_{1}| \leq L_{1} \cap (y_{1} - y_{2}) ({\dot{y}}_{1} - {\dot{y}}_{2}) < 0 \\ \max (c_{m i n}, m i n (c_{m a x}, |{β_{2} k}_{s} \frac{(y_{1} - y_{2})}{{\dot{y}}_{1} - {\dot{y}}_{2}}|)), & L_{1} < |{\dot{y}}_{1}| \leq L_{2} \cap (y_{1} - y_{2}) ({\dot{y}}_{1} - {\dot{y}}_{2}) < 0 \\ \max (c_{m i n}, m i n (c_{m a x}, |{β_{3} k}_{s} \frac{(y_{1} - y_{2})}{{\dot{y}}_{1} - {\dot{y}}_{2}}|)), & L_{2} < |{\dot{y}}_{1}| \leq L_{3} \cap (y_{1} - y_{2}) ({\dot{y}}_{1} - {\dot{y}}_{2}) < 0 \\ \max (c_{m i n}, m i n (c_{m a x}, |{β_{4} k}_{s} \frac{(y_{1} - y_{2})}{{\dot{y}}_{1} - {\dot{y}}_{2}}|)), & L_{3} < |{\dot{y}}_{1}| \leq L_{4} \cap (y_{1} - y_{2}) ({\dot{y}}_{1} - {\dot{y}}_{2}) < 0 \\ \max (c_{m i n}, m i n (c_{m a x}, |{β_{5} k}_{s} \frac{(y_{1} - y_{2})}{{\dot{y}}_{1} - {\dot{y}}_{2}}|)), & L_{4} < |{\dot{y}}_{1}| \cap (y_{1} - y_{2}) ({\dot{y}}_{1} - {\dot{y}}_{2}) < 0 \\ c_{m i n} & (y_{1} - y_{2}) ({\dot{y}}_{1} - {\dot{y}}_{2}) \geq 0 \end{matrix}

(10)

Similar to the new SH control, the velocity thresholds, velocity regions, and contribution coefficients were defined for the new DV control. Using the trial and error method, we determined four velocity thresholds of the new DV control, including 0.0075, 0.025, 0.05, and 0.1 m/s, corresponding to five vibration velocity regions of the carbody during the operating process. Contribution coefficients (beta) associated with these five vibration velocity regions are shown in Figure 5, where beta ranges between 1 and 2. When the vehicle body oscillates around the second resonance frequency region, the contribution coefficient increases by a factor of 1.75, leading to a 1.75-fold increase in the instantaneous damping force. Similarly, around the first resonance frequency region, the contribution coefficient increases by a factor of two, causing the instantaneous damping force to rise by two times, compared with the conventional DV control algorithm. Given that the DV control works more effectively in the region of the second-order resonance frequencies, particularly the frequency region associated with the velocity region (0.025–0.1 m/s, Figure 2a), we divided this region into two regions (0.025–0.05 m/s) and (0.05–0.1 m/s) for the new DV control.

Figure 6 presents a comparison of the transmissibility of vibrations from the rail to the train body under the three control methods. Overall, the conventional DV controller exhibited superior vibration isolation performance at high frequencies compared with the passive controller. However, its effectiveness was lower in three frequency regions: (1) low frequencies (less than 1.5 Hz); (2) near the unstable frequency point (6.6 Hz); and (3) near the resonance frequency of the bogie (14.6 Hz). Compared with the conventional DV controller, the new DV controller exhibited higher vibration isolation performance at low frequencies, eliminated the unstable phenomenon observed at approximately 6.6 Hz, and substantially mitigated vibrations at frequencies near the resonance frequency of the bogie.

Figure 7 presents the displacement, velocity, and acceleration of the train body under the three types of control. Overall, the carbody displacement under the new DV control was lower than that under conventional DV control across the entire frequency domain [Figure 7a]. The new DV controller notably reduced the vibration velocity of the carbody at both low and high frequencies but did not result in the instability at approximately 6.6 Hz that was observed with the conventional DV controller [Figure 7b]. The trends in carbody acceleration were similar to those in carbody velocity under the three control algorithms [Figure 7c]. Compared with the conventional DV controller, the new DV controller significantly reduced the vibration velocity and acceleration of the carbody by 35.11% and 28.34%, respectively, at the frequency of 14.6 Hz.

4.3. Analysis of the New Mixed Skyhook and Distance–Velocity Control

The proposed SH and DV controls were combined to yield a new mixed SH–DV control algorithm in Equation (11). The new mixed control has a switch condition that is a function of the vibration acceleration

{\ddot{y}}_{1}

and velocity

{\dot{y}}_{1}

of the carbody and the conversion constant

c_{n}

, where

c_{n} = 2 π f_{c n}

, and

f_{c n}

, referred to as the crossover frequency, is the frequency at which the vibration transmissibility curves of the two singular controls intersect.

c_{n_S H D V} = \{\begin{matrix} c_{n_S H} {\ddot{y}}_{1}^{2} - {c_{n}}^{2} \cdot {\dot{y}}_{1}^{2} < 0 \\ c_{n_D V} {\ddot{y}}_{1}^{2} - {c_{n}}^{2} \cdot {\dot{y}}_{1}^{2} \geq 0 \end{matrix}

(11)

The term

{\ddot{y}}_{1}^{2} - {c_{n}}^{2} \cdot {\dot{y}}_{1}^{2}

is referred to as the frequency selector (FS). When FS

\geq

0, the angular frequency of the carbody response is greater than or equal to the conversion constant (i.e., the carbody is oscillating in a high frequency range). In this case, the new mixed SH–DV control activates the new DV control, due to the fact that the DV control is more effective at high frequencies. In contrast, when FS < 0, the new mixed SH–DV control activates the new SH control, owing to the SH control’s effectiveness at low frequencies.

Figure 8 illustrates the transmissibility of vibrations from the rail to the train body under passive control, conventional mixed SH–DV control, and new mixed SH–DV control. The new mixed SH–DV controller exhibited higher vibration isolation performance than did the conventional SH–DV controller at low frequencies. Moreover, the new mixed SH–DV controller significantly improved the vibration transmissibility performance by 7.37% at frequencies near the resonance frequency of the bogie (14.6 Hz), compared with the conventional mixed SH–DV controller.

Figure 9 provides a more detailed evaluation of the displacement [Figure 9a], velocity [Figure 9b], and acceleration [Figure 9c] of the train body under the different types of control. Overall, the new mixed SH–DV controller had similar performance to the conventional controller at low frequencies, whereas it was more effective at or around the resonance frequency of the bogie (14.6 Hz), reducing the vibration velocity and acceleration of the carbody by 50.85% and 45.39%, respectively, compared with the conventional mixed SH–DV controller.

5. Conclusions

In this paper, we proposed a new mixed control by combining a new SH control with a new DV control. The new mixed SH–DV control algorithm was applied to the MRD of a secondary lateral suspension system of a railway vehicle, where it was found to effectively suppress vibrations of the train body across a wide range of vibration frequencies, as determined through a numerical simulation problem. The proposed method was found to work effectively at low frequencies, similar to a conventional mixed SH–DV control, whereas it significantly improved ride comfort at high frequencies, particularly near the resonance frequency of the bogie, reducing the vibration velocity and acceleration of the carbody by 50.85% and 45.39%, respectively, at 14.6 Hz, compared with a conventional mixed SH–DV control. With its simplicity, robustness, and ease of implementation, the application of the proposed algorithm is not restricted to railway vehicles but can also be expanded to other vehicles that use suspension systems. Future studies are needed to validate the proposed algorithm in more complex models, such as a whole railway vehicle model of a numerical simulation problem or an actual model of an experimental problem.

Author Contributions

Conceptualization, Y.S.; Methodology, T.-L.H.; Software, T.-L.H.; Validation, T.-L.H.; Formal analysis, T.-L.H.; Investigation, Y.S.; Writing—original draft, T.-L.H.; Writing—review & editing, Y.S.; Supervision, Y.S.; Project administration, Y.S. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by National Science and Technology Council, Taiwan grant number NSTC 112-2221-E-027-078.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Data is contained within the article.

Conflicts of Interest

The authors declare no conflict of interest.

References

Alotaibi, S.; Quddus, M.; Morton, C.; Imprialou, M. Transport investment, railway accessibility and their dynamic impacts on regional economic growth. Res. Transp. Bus. Manag. 2022, 43, 100702. [Google Scholar] [CrossRef]
Lewis, T.D.; Jiang, J.Z.; Neild, S.A.; Gong, C.N.; Iwnicki, S.D. Using an inerter-based suspension to improve both passenger comfort and track wear in railway vehicles. Veh. Syst. Dyn. 2019, 58, 472–493. [Google Scholar] [CrossRef]
Liu, C.; Lai, S.-K.; Ni, Y.-Q.; Chen, L. Dynamic modelling and analysis of a physics-driven strategy for vibration control of railway vehicles. Veh. Syst. Dyn. 2024, 1–31, Advance online publication. [Google Scholar] [CrossRef]
Ripamonti, F.; Chiarabaglio, A. A smart solution for improving ride comfort in high-speed railway vehicles. J. Vib. Control 2019, 25, 1958–1973. [Google Scholar] [CrossRef]
Sharma, S.K.; Kumar, A. Ride performance of a high speed rail vehicle using controlled semi active suspension system. Smart Mater. Struct. 2017, 26, 055026. [Google Scholar] [CrossRef]
Wei, X.; Zhu, M.; Jia, L. A semi-active control suspension system for railway vehicles with magnetorheological fluid dampers. Veh. Syst. Dyn. 2016, 54, 982–1003. [Google Scholar] [CrossRef]
Oh, J.-S.; Shin, Y.-J.; Koo, H.-W.; Kim, H.-C.; Park, J.; Choi, S.-B. Vibration control of a semi-active railway vehicle suspension with magneto-rheological dampers. Adv. Mech. Eng. 2016, 8, 1–13. [Google Scholar] [CrossRef]
Yang, S.; Zhao, Y.; Liu, Y.; Liao, Y.; Wang, P. A new semi-active control strategy on lateral suspension systems of high-speed trains and its application in HIL test rig. Veh. Syst. Dyn. 2022, 61, 1317–1344. [Google Scholar] [CrossRef]
Zhang, C.; Kordestani, H.; Shadabfar, M. A combined review of vibration control strategies for high-speed trains and railway infrastructures: Challenges and solutions. J. Low Freq. Noise Vib. Act. Control 2022, 42, 272–291. [Google Scholar] [CrossRef]
Kim, H.-C.; Shin, Y.-J.; You, W.; Jung, K.C.; Oh, J.-S.; Choi, S.-B. A ride quality evaluation of a semi-active railway vehicle suspension system with MR damper: Railway field tests. Proc. Inst. Mech. Eng. Part F J. Rail Rapid Transit 2016, 231, 306–316. [Google Scholar] [CrossRef]
Karnopp, D.; Crosby, M.J.; Harwood, R.A. Vibration Control Using Semi-Active Force Generators. J. Eng. Ind. 1974, 96, 619–626. [Google Scholar] [CrossRef]
Sulaiman, S.; Samin, P.M.; Jamaluddin, H.; Rahman, R.A.; Burhaumudin, M.S. Groundhook control of semi-active suspension for heavy vehicle. Int. J. Res. Eng. Technol. (IJRET) 2012, 1, 146–152. [Google Scholar]
Savaresi, S.M.; Silani, E.; Bittanti, S. Acceleration-Driven-Damper (ADD): An Optimal Control Algorithm For Comfort-Oriented Semiactive Suspensions. J. Dyn. Syst. Meas. Control 2004, 127, 218–229. [Google Scholar] [CrossRef]
Díaz-Choque, C.S.; Félix-Herrán, L.C.; Ramírez-Mendoza, R.A. Optimal Skyhook and Groundhook Control for Semiactive Suspension: A Comprehensive Methodology. Shock Vib. 2021, 2021, 1–21. [Google Scholar] [CrossRef]
Savaresi, S.M.; Spelta, C. Mixed Sky-Hook and ADD: Approaching the Filtering Limits of a Semi-Active Suspension. J. Dyn. Syst. Meas. Control 2006, 129, 382–392. [Google Scholar] [CrossRef]
Yang, Y.; Liu, C.; Chen, L.; Zhang, X. Phase deviation of semi-active suspension control and its compensation with inertial suspension. Acta Mech. Sin. 2024, 40, 523367. [Google Scholar] [CrossRef]
Bhardawaj, S.; Sharma, R.C.; Sharma, S.K. Development of multibody dynamical using MR damper based semi-active bio-inspired chaotic fruit fly and fuzzy logic hybrid suspension control for rail vehicle system. Proc. Inst. Mech. Eng. Part K J. Multi-Body Dyn. 2020, 234, 723–744. [Google Scholar] [CrossRef]
Ghoniem, M.; Awad, T.; Mokhiamar, O. Control of a new low-cost semi-active vehicle suspension system using artificial neural networks. Alex. Eng. J. 2020, 59, 4013–4025. [Google Scholar] [CrossRef]
Metered, H.; Bonello, P.; Oyadiji, S.O. An investigation into the use of neural networks for the semi-active control of a magnetorheologically damped vehicle suspension. Proc. Inst. Mech. Eng. Part D J. Automob. Eng. 2010, 224, 829–848. [Google Scholar] [CrossRef]
Shiao, Y.; Huynh, T.-L. A new hybrid control strategy for improving ride comfort on lateral suspension system of railway vehicle. J. Low Freq. Noise Vib. Act. Control 2024, 43, 1842–1859. [Google Scholar] [CrossRef]
Zhao, Y.; Yang, S.; Liu, Y.; Liao, Y.; Liu, P. A New Semi-active Control Strategy and Its Application in Railway Vehicles. In Advances in Applied Nonlinear Dynamics, Vibration and Control-2021: The Proceedings of 2021 International Conference on Applied Nonlinear Dynamics, Vibration and Control (ICANDVC2021); Springer: Singapore, 2022; Volume 799, pp. 240–252. [Google Scholar] [CrossRef]
Shiao, Y.; Huynh, T.-L. Suspension Control and Characterization of a Variable Damping Magneto-Rheological Mount for a Micro Autonomous Railway Inspection Car. Appl. Sci. 2022, 12, 7336. [Google Scholar] [CrossRef]
Shiao, Y.; Huynh, T.-L. New Sky-Hook Control Algorithm for Secondary Lateral Suspension System of Railway Vehicle. In Proceedings of the 9th International Conference on Applying New Technology in Green Building, Danang City, Vietnam, 30–31 August 2024; pp. 1–4. [Google Scholar] [CrossRef]
Nie, S.; Zhuang, Y.; Liu, W.; Chen, F. A semi-active suspension control algorithm for vehicle comprehensive vertical dynamics performance. Veh. Syst. Dyn. 2017, 55, 1099–1122. [Google Scholar] [CrossRef]

Figure 1. The quarter railway vehicle model.

Figure 2. (a) The velocity region division and (b) the contribution coefficient for the new SH control under the vibration velocity of the carbody. (a): the regions (1) 0 to 0.0075 m/s, (2) 0.0075 to 0.025 m/s, (3) 0.025 to 0.1 m/s, and (4) ≥0.1 m/s, assumed to correspond the carbody vibration at the high frequency domain, medium frequency domain, around the second-order resonance frequency domain, and around the first-order resonance frequency domain, respectively. A, C, D, E, G, and H are the intersections of region thresholds and line graph of vibration velocity of carbody; B and F are the peak at the first and second-order resonance frequencies, respectively.

Figure 3. Comparisons of the vibration transmissibility under different controls.

Figure 4. The RMS of (a) displacement, (b) velocity, and (c) acceleration of the carbody.

Figure 5. Damping contribution coefficient for the new DV control according to the vibration velocity of the carbody.

Figure 6. Comparisons of the vibration transmissibility of different controls under sinusoidal signal input with amplitudes of ±5 mm.

Figure 7. The RMS of (a) displacement, (b) velocity, and (c) acceleration of the carbody under sinusoidal signal input with amplitudes of ±5 mm.

Figure 8. Comparisons of the vibration transmissibility of the carbody under three control algorithms under sinusoidal signal input with amplitudes of ±5 mm.

Figure 9. The RMS of (a) displacement, (b) velocity, and (c) acceleration of the carbody under sinusoidal signal input with amplitudes of ±5 mm.

Table 1. The parameters of the quarter railway vehicle model.

Parameters	Value	Unit
Quarter mass of the carbody, m_c	9308	kg
Half mass of the bogie, m_b	1981	kg
Stiffness of the secondary lateral suspension, k_s	$9 \cdot 10^{5}$	$N \cdot m^{- 1}$
Damping of the secondary lateral suspension, c_s	$2 \cdot 10^{4}$	$N s \cdot m^{- 1}$
Stiffness of the primary lateral suspension, k_p	$15.69 \cdot 10^{6}$	$N \cdot m^{- 1}$

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

Share and Cite

MDPI and ACS Style

Shiao, Y.; Huynh, T.-L. New Mixed Skyhook and Displacement–Velocity Control for Improving the Effectiveness of Vibration Isolation in the Lateral Suspension System of a Railway Vehicle. Appl. Sci. 2024, 14, 11680. https://doi.org/10.3390/app142411680

AMA Style

Shiao Y, Huynh T-L. New Mixed Skyhook and Displacement–Velocity Control for Improving the Effectiveness of Vibration Isolation in the Lateral Suspension System of a Railway Vehicle. Applied Sciences. 2024; 14(24):11680. https://doi.org/10.3390/app142411680

Chicago/Turabian Style

Shiao, Yaojung, and Tan-Linh Huynh. 2024. "New Mixed Skyhook and Displacement–Velocity Control for Improving the Effectiveness of Vibration Isolation in the Lateral Suspension System of a Railway Vehicle" Applied Sciences 14, no. 24: 11680. https://doi.org/10.3390/app142411680

APA Style

Shiao, Y., & Huynh, T. -L. (2024). New Mixed Skyhook and Displacement–Velocity Control for Improving the Effectiveness of Vibration Isolation in the Lateral Suspension System of a Railway Vehicle. Applied Sciences, 14(24), 11680. https://doi.org/10.3390/app142411680

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Menu

New Mixed Skyhook and Displacement–Velocity Control for Improving the Effectiveness of Vibration Isolation in the Lateral Suspension System of a Railway Vehicle

Abstract

1. Introduction

2. Definition of the Simulation Problem

2.1. Quarter Railway Vehicle Model

2.2. Simulation Conditions

2.3. Evaluation of Effectiveness of Vibration Isolation

3. Conventional Skyhook, Displacement–Velocity, and Skyhook and Displacement–Velocity Control Methods

4. New Skyhook and Displacement–Velocity Control Method

4.1. Design and Analysis of the New Skyhook Control Algorithm

4.2. Design and Analysis of New Displacement–Velocity Controller

4.3. Analysis of the New Mixed Skyhook and Distance–Velocity Control

5. Conclusions

Author Contributions

Funding

Institutional Review Board Statement

Informed Consent Statement

Data Availability Statement

Conflicts of Interest

References

Share and Cite

Article Metrics

Article Access Statistics

Further Information

Guidelines

MDPI Initiatives

Follow MDPI