Abstract

The PID classic control systems are often employed for rail car systems to reduce the vibrations and disturbance rate during movement. In this study, the dynamic modeling and simulation of PID controls for rail car systems were carried out. Using 9 degrees of freedom, the modeling process comprises the representation of the rail car system and the rail track followed by the generation of equations of motion as well as differential equations for the rail car body, wheel sets and bogie. The represented systems are simulated in the MATLAB Simulink 2018 environment based on the equations of motion generated, and subsequently vibration analysis was carried out. The PID control system tuned according to the Nichols-Ziegler rule was introduced to minimize the vibrations and disturbance rate. The performance of the control and the rail car system in terms of the input step response, bandwidth, frequency, phase margin, frequency and input and output rejections was evaluated. The control system demonstrated significant robustness in providing the required active control for the system, while there was improved stability and reduction in noise and vibration under control action of the PID, thus improving ride comfort.

Keywords: advanced controls, modeling, rail car, simulation, vibration

## 1. Introduction

The rail car system uses steel wheels moving on steel rails, and its suspension system comprises the bogie, primary and secondary suspension elements as well as springs and dampers. The rail car has the power bogies in the front and rear positions each having four wheels arranged in pairs and rigidly connected through an axle. The sets of wheel, which rotates at the same speed, are connected to the bogie through the primary suspension system, which is harder and stiffer in order to minimize load disturbances and uneven weight distributions. This is to maintain a good balance of the rail car while moving along its track. The stiff primary suspension system connects the wheel set to the bogie, while the soft secondary suspension system connects the bogie to the rail car body in order to isolate it from disturbances that stem from rail irregularities, uneven track profiles and its associated vibrations. While the primary suspension system is designed to provide guidance and rail car stability amidst load and weight variations, the secondary suspension system is to enhance comfortable ride by isolating rail car body from rail track irregularities. These design requirements of both suspension systems are to dampen the effect of vibration and increase the overall system's performance [1, 2]. For a rail car, when vibrations are not kept within the permissible limits, the comfort, safety and health of passengers are at risk, and the time interval for maintenance activities for the components will likely increase. Hence, undesirable vibration in the rail car system is the cause of noise, considerable energy loss, reduction in the system's performance, fatigue or fracture of some component, instability of a moving rail car and displacement of the rail track, amongst others. Over the years, there are three main types of suspension systems often employed to check vibration in the rail car system, namely, the passive, semi-active and fully active suspension systems [3, 4]. The design requirements of the suspension system of the rail car are to provide support against the dynamic loads and weight of the railcar; prevent load and rail disturbances; isolate the rail car body disturbances that could offset its balance; prevent irregular motions such as bouncing, yawing, etc.; provide guidance along the rail track; and optimize the curving, braking and other car maneuvering performances. The passive suspension system, which consists of springs, absorbs and stores the energy absorbed, while the dampers mounted on each wheel act as the shock absorber that dissipates the energy stored in the spring and reduces the vibrations from the rail transmitted to the vehicle ([5–7]. The passive suspension system is cost-effective and driven by simple technology, but its demerit lies in the fact that it cannot measure some critical parameters such as the velocity, displacement, acceleration, etc. of the rail car system in real time in order to effect the needed adjustments to stabilize the rail car system; hence, it is rigid and cannot reach the compromise between alternate hard primary and soft secondary suspension systems amidst load and rail irregularities. On the other hand, the active suspension system uses sensors, comparators and actuator for measuring and monitoring some critical parameters that influence rail car stability in real time [8, 9]. The measured system's parameters are compared with the threshold already pre-set on the controller, which can guarantee comfortable ride. The steady-state errors generated are eliminated through adjustment and compensation for such errors, and as such, there is high-level damping without compromising the system's performance. The third category of suspension system is the semi-active suspension system, which employs a spring and controllable damper. The spring element stores the energy, while the controllable damper dissipates the energy stored. The semiactive suspension systems combine the features of the passive and active suspension systems such as the use of passive damper and an actively controlled spring. The merit of the semi-active suspension is that it is cost and energy effective [9–11].

Over the years, researchers have employed several approaches ranging from classic to advance systems to control the suspension system in order to minimize the effect of vibrations. Such approaches include the use of proportional-integralderivative (PID) control, Fuzzy PID, Linear Quadratic Regulator vibration controller, Adaptive Neuro-Fuzzy Inference System control and magnetorheological dampers, amongst others [12–16].

and yaw rotational motions. Figures 1 and 2 illustrate the free body diagram for the

In order to give a complete vibration analysis and control, the model design and simulation of the suspension systems are considered as both linear and non-linear systems. There are three forces acting on the rail car suspension system, namely, the spring force, rolling resistance and forces due to the wheel-track interactions. The spring force and the rolling resistance act on the rail car body with mass m<sup>1</sup> in the horizontal directions, while the spring force, rolling resistance and the forces due to the wheel-track interactions act on the bogies with mass m<sup>2</sup> due to load in the horizontal directions. The masses of the rail car and its bogies are represented by m<sup>1</sup> and m2, respectively, while the bodies are connected to the rail car through the secondary suspension with couplings having stiffness K: If force Fd is the force generated as a result of wheel-track interaction, Fu is the control force and μ is the coefficient of rolling friction, then Newton's second law of motion holds; thus,

F ¼ ma (1)

∑F<sup>1</sup> ¼ K xð Þ� <sup>1</sup> � x<sup>2</sup> μm<sup>1</sup> gx\_ <sup>1</sup> ¼ m1x€<sup>1</sup> (2) ∑F<sup>2</sup> ¼ Fd � K xð Þ� <sup>1</sup> � x<sup>2</sup> μm<sup>2</sup> gx\_ <sup>2</sup> ¼ m2x€<sup>2</sup> (3)

∑ð Þ¼ F<sup>1</sup> þ F<sup>2</sup> m1x€<sup>1</sup> þ m2x€<sup>2</sup> (4)

rail car and track system in their degrees of freedom.

The free body diagram for the rail car model [18].

Figure 1.

Figure 2.

121

The rail car and its suspension system [17].

DOI: http://dx.doi.org/10.5772/intechopen.85654

Vibration Analysis and Control in the Rail Car System Using PID Controls

Eq. (3) expresses the summation of the forces

## 2. Materials and method

The analysis of vibration and control in the rail car system starts with the schematic representation of the rail car and track system including their degrees of freedoms and subsequent generation of equations of motion. The modeling is done with the masses of the system (rail car, body and wheel sets) having 6 degrees of freedom in the longitudinal, lateral and vertical directions as well as in the roll, pitch Vibration Analysis and Control in the Rail Car System Using PID Controls DOI: http://dx.doi.org/10.5772/intechopen.85654

#### Figure 1.

track irregularities. These design requirements of both suspension systems are to dampen the effect of vibration and increase the overall system's performance [1, 2]. For a rail car, when vibrations are not kept within the permissible limits, the comfort, safety and health of passengers are at risk, and the time interval for maintenance activities for the components will likely increase. Hence, undesirable vibration in the rail car system is the cause of noise, considerable energy loss, reduction in the system's performance, fatigue or fracture of some component, instability of a moving rail car and displacement of the rail track, amongst others. Over the years, there are three main types of suspension systems often employed to check vibration in the rail car system, namely, the passive, semi-active and fully active suspension systems [3, 4]. The design requirements of the suspension system of the rail car are to provide support against the dynamic loads and weight of the railcar; prevent load and rail disturbances; isolate the rail car body disturbances that could offset its balance; prevent irregular motions such as bouncing, yawing, etc.; provide guidance along the rail track; and optimize the curving, braking and other car maneuvering performances. The passive suspension system, which consists of springs, absorbs and stores the energy absorbed, while the dampers mounted on each wheel act as the shock absorber that dissipates the energy stored in the spring and reduces the vibrations from the rail transmitted to the vehicle ([5–7]. The passive suspension system is cost-effective and driven by simple technology, but its demerit lies in the fact that it cannot measure some critical parameters such as the velocity, displacement, acceleration, etc. of the rail car system in real time in order to effect the needed adjustments to stabilize the rail car system; hence, it is rigid and cannot reach the compromise between alternate hard primary and soft secondary suspension systems amidst load and rail irregularities. On the other hand, the active suspension system uses sensors, comparators and actuator for measuring and monitoring some critical parameters that influence rail car stability in real time [8, 9]. The measured system's parameters are compared with the threshold already pre-set on the controller, which can guarantee comfortable ride. The steady-state errors generated are eliminated through adjustment and compensation for such errors, and as such, there is high-level damping without compromising the system's performance. The third category of suspension system is the semi-active suspension system, which employs a spring and controllable damper. The spring element stores the energy, while the controllable damper dissipates the energy stored. The semiactive suspension systems combine the features of the passive and active suspension systems such as the use of passive damper and an actively controlled spring. The merit of the semi-active suspension is that it is cost and energy effective [9–11]. Over the years, researchers have employed several approaches ranging from classic to advance systems to control the suspension system in order to minimize the effect of vibrations. Such approaches include the use of proportional-integralderivative (PID) control, Fuzzy PID, Linear Quadratic Regulator vibration controller, Adaptive Neuro-Fuzzy Inference System control and magnetorheological

Noise and Vibration Control - From Theory to Practice

dampers, amongst others [12–16].

The analysis of vibration and control in the rail car system starts with the schematic representation of the rail car and track system including their degrees of freedoms and subsequent generation of equations of motion. The modeling is done with the masses of the system (rail car, body and wheel sets) having 6 degrees of freedom in the longitudinal, lateral and vertical directions as well as in the roll, pitch

2. Materials and method

120

The rail car and its suspension system [17].

Figure 2. The free body diagram for the rail car model [18].

and yaw rotational motions. Figures 1 and 2 illustrate the free body diagram for the rail car and track system in their degrees of freedom.

In order to give a complete vibration analysis and control, the model design and simulation of the suspension systems are considered as both linear and non-linear systems. There are three forces acting on the rail car suspension system, namely, the spring force, rolling resistance and forces due to the wheel-track interactions. The spring force and the rolling resistance act on the rail car body with mass m<sup>1</sup> in the horizontal directions, while the spring force, rolling resistance and the forces due to the wheel-track interactions act on the bogies with mass m<sup>2</sup> due to load in the horizontal directions. The masses of the rail car and its bogies are represented by m<sup>1</sup> and m2, respectively, while the bodies are connected to the rail car through the secondary suspension with couplings having stiffness K: If force Fd is the force generated as a result of wheel-track interaction, Fu is the control force and μ is the coefficient of rolling friction, then Newton's second law of motion holds; thus,

$$F = ma \tag{1}$$

$$
\sum F\_1 = K(\varkappa\_1 - \varkappa\_2) - \mu m\_1 g \dot{\varkappa}\_1 = m\_1 \ddot{\varkappa}\_1 \tag{2}
$$

$$
\Sigma F\_2 = F\_d - K(\varkappa\_1 - \varkappa\_2) - \mu m\_2 \mathbf{g} \\
\dot{\varkappa}\_2 = m\_2 \ddot{\varkappa}\_2 \tag{3}
$$

Eq. (3) expresses the summation of the forces

$$
\sum (F\_1 + F\_2) = m\_1 \ddot{\mathbf{x}}\_1 + m\_2 \ddot{\mathbf{x}}\_2 \tag{4}
$$

Noise and Vibration Control - From Theory to Practice

$$
\sum (F\_1 + F\_2) = M\ddot{\mathbf{x}}\tag{5}
$$

$$
\sum (F\_1 + F\_2) = \sum (F\_d + F\_u) \tag{6}
$$

The spring deflection is expressed as Eq. (7):

$$\delta = K(\mathfrak{x}\_1 - \mathfrak{x}\_2) \tag{7}$$

$$
\sum (F\_d + F\_u) = M\ddot{\mathbf{x}} + b\dot{\mathbf{x}} + K\mathbf{x} \tag{8}
$$

$$[F\_d] + [F\_u] = [M]\ddot{\varkappa} + [b]\dot{\varkappa} + [K]\varkappa \tag{9}$$

$$\text{Hence}, \mathfrak{x} = \left[\mathfrak{x} \dots \mathfrak{x}^{\mathbf{1}}\right]^T \tag{10}$$

$$\dot{\mathbf{x}} = \frac{d\mathbf{x}}{dt} = f(\mathbf{x}(t), \boldsymbol{\mu}(t), t) \tag{11}$$

where.

Modeling of the rail car and its suspension system.

Figure 3.

Lr1,Lr2,Lr3,Lr4,Lr5,Lr<sup>6</sup> are the lateral rail disturbance functions; Vr1,Vr3,Vr5,Vr7,Vr9,Vr<sup>11</sup> are the vertical right rail disturbance functions;

Vibration Analysis and Control in the Rail Car System Using PID Controls

DOI: http://dx.doi.org/10.5772/intechopen.85654

distance between the contact points of the wheel-rail.

y t \_ðÞþ ξωny t \_ðÞþ <sup>ω</sup><sup>2</sup>

ms<sup>2</sup> <sup>þ</sup> bs <sup>þ</sup> <sup>k</sup> <sup>¼</sup> kdω<sup>2</sup>

the second-order systems:

G sðÞ¼ X sð Þ

123

F sð Þ <sup>¼</sup> <sup>1</sup>

Vr2,Vr4,Vr6,Vr8,Vr10,Vr<sup>12</sup> are the vertical left rail disturbance functions; Xr,Lr,Vr are the longitudinal, lateral and vertical displacements of the rail car body; Xbi,Lbi,Vbi are the longitudinal, lateral and vertical displacements of the rail car bogie; Xwj,Lwj,Vwj are the longitudinal, lateral and vertical displacements of the rail car wheel sets; θr,∅r,ψr, are the roll, pitch and yaw displacements of the rail car body; θbi,∅bi,ψbi, are the roll, pitch and yaw displacements of the rail car bogie; θwj, ∅wj, ψwj, are the roll, pitch and yaw displacements of the rail car wheel set (i ¼ 1…::3; and j ¼ 1…::6Þ; p<sup>1</sup> is the lateral distance between the vertical primary suspensions; p<sup>2</sup> is the lateral distance between the vertical secondary suspensions; d<sup>1</sup> is the vertical distance between the wheel set and the bogie mass centre; d<sup>2</sup> is the vertical distance between the bogie mass centre and lateral secondary suspension; d<sup>3</sup> is the vertical distance between the lateral secondary suspension and railcar body mass centre; Lo is the longitudinal distance between the bogies; and a is the lateral

The mathematical model can be obtained in either the state space or using the transfer function for necessary control actions. The control action is to improve the speed of response, system's balance and stability and to reduce the steady-state error as well as the amplitude of oscillations. Eqs. (15) and (16) express the form of

ny tðÞ¼ kdω<sup>2</sup>

n

<sup>¼</sup> kdω<sup>2</sup>

n s<sup>2</sup> þ 2ξωns þ ω<sup>2</sup> nu tð Þ (15)

d

(16)

n ð Þ <sup>s</sup> <sup>þ</sup> <sup>σ</sup> <sup>2</sup> <sup>þ</sup> <sup>ω</sup><sup>2</sup>

where x tð Þ is the state vector and a set of variables representing the configurations of the system. The modeling of the rail car system and its suspension system as well as the track system is done on the MATLAB Simulink 2018 environment (Figure 3).

The transfer function was used for representing the linear systems, and the inputs are the load changes, applied forces as well as the uneven track profiles, while the output of the system is the acceleration and displacement of the rail car body as well as the deflection of the suspension systems.

According to Sezer and Atalay [19], the vectors for the displacement, rail car disturbance and control forces are expressed as Eqs. (12)–(14), respectively:

x ¼ Xr,Lr,Vr, θr,∅r,ψr,Xb1,Lb1,Vb1, θb1∅b1,ψb1,Xb2,Lb2,Vb2, θb2∅b2,ψb2, Xb3,Lb3,Vb3, θb3∅b3,ψb3,Xw1,Lw1,Vw1, θw1,ψw1,Xw2,Lw2,Vw2, θw2, ψw2,Xw3,Lw3,Vw3, θw3,ψw3,Xw4,Lw4,Vw4, θw4,ψw4,Xw5,Lw5,Vw5, θ<sup>w</sup>5,ψ<sup>w</sup>5,Xw6,Lw6,Vw6, θ<sup>w</sup>6,ψ<sup>w</sup>6�T (12) ½ �¼ Fd ½000000000000000000000000ð Þ ksLYr<sup>1</sup> ð Þ ksVð Þ Vr<sup>1</sup> þ Vr<sup>2</sup> ð Þ ksLR1Lr<sup>1</sup> þ ksVð Þ Vr<sup>2</sup> � Vr<sup>1</sup> 00ð Þ ksLLr<sup>2</sup> ð Þ ksVð Þ Vr<sup>3</sup> þ Vr<sup>4</sup> ð Þ ksLR1Lr<sup>2</sup> þ ksVa Vð Þ <sup>r</sup><sup>4</sup> � Vr<sup>3</sup> 00ð Þ ksLLr<sup>3</sup> ð Þ ksVð Þ Vr<sup>5</sup> þ Vr<sup>6</sup> ð Þ ksLR1Lr<sup>3</sup> þ ksVa Vð Þ <sup>r</sup><sup>6</sup> � Vr<sup>5</sup> 00 ð Þ ksLLr<sup>4</sup> ð Þ ksVð Þ Vr<sup>7</sup> þ Vr<sup>8</sup> ð Þ ksLR1Lr<sup>4</sup> þ ksVa Vð Þ <sup>r</sup><sup>8</sup> � Vr<sup>7</sup> 00 ð Þ ksLLr<sup>5</sup> ð Þ ksVð Þ Vr<sup>9</sup> þ Vr<sup>10</sup> ð Þ ksLR1Lr<sup>5</sup> þ ksVa Vð Þ <sup>r</sup><sup>10</sup> � Vr<sup>9</sup> 00ð Þ ksLLr<sup>6</sup> ð Þ ksVð Þ Vr<sup>11</sup> þ Vr<sup>12</sup> ð Þ ksLR1Lr<sup>6</sup> þ ksVa Vð Þ <sup>r</sup><sup>12</sup> � Vr<sup>11</sup> 0� <sup>T</sup> (13) ½ �¼ Fu ½0 Uð Þ L1 þ UL2 þ UL3 ð Þ UV1 þ UV2 þ UV3 þ UV4 þ UV5 þ UV6 d3ð Þ UL1 þ UL2 þ UL3 p2ð Þ ðUV<sup>2</sup> þ UV<sup>4</sup> þ UV6Þ � ð Þ UV<sup>1</sup> þ UV<sup>3</sup> þ UV<sup>5</sup> Loðð Þ� UV<sup>5</sup> þ UV<sup>6</sup> ð Þ UV<sup>1</sup> þ UV<sup>2</sup> ÞÞðh2UL<sup>1</sup> � UL3ÞÞ0 �ð Þ UL<sup>1</sup> ð Þ ð Þ UV<sup>1</sup> þ UV<sup>2</sup> d2UL<sup>1</sup> � p2ð Þ UV<sup>2</sup> � UV<sup>1</sup> <sup>000</sup>ð Þ �UL<sup>2</sup> ð�UV<sup>3</sup> þ UV4ÞÞ d2UL<sup>2</sup> � p2ð Þ UV<sup>4</sup> � UV<sup>3</sup> <sup>000</sup>ð Þ� �UL<sup>3</sup> <sup>ð</sup> UV<sup>5</sup> <sup>þ</sup> UV6ÞÞ d2UL<sup>3</sup> � p2ð Þ UV<sup>6</sup> � UV<sup>5</sup> <sup>00000000000000000000000000000000</sup>� T (14)

Vibration Analysis and Control in the Rail Car System Using PID Controls DOI: http://dx.doi.org/10.5772/intechopen.85654

Figure 3.

∑ð Þ¼ F<sup>1</sup> þ F<sup>2</sup> Mx€ (5)

δ ¼ K xð Þ <sup>1</sup> � x<sup>2</sup> (7)

∑ð Þ¼ F<sup>1</sup> þ F<sup>2</sup> ∑ð Þ Fd þ Fu (6)

∑ð Þ¼ Fd þ Fu Mx€ þ bx\_ þ Kx (8) ½ �þ Fd ½ �¼ Fu ½ � M x€ þ ½ � b x\_ þ ½ � K x (9)

Hence, x <sup>¼</sup> <sup>x</sup>:::::x<sup>1</sup> <sup>T</sup> (10)

dt <sup>¼</sup> f xt ð Þ ð Þ; u tð Þ; <sup>t</sup> (11)

<sup>T</sup> (13)

T

(14)

The spring deflection is expressed as Eq. (7):

Noise and Vibration Control - From Theory to Practice

(Figure 3).

122

<sup>x</sup>\_ <sup>¼</sup> dx

body as well as the deflection of the suspension systems.

where x tð Þ is the state vector and a set of variables representing the configurations of the system. The modeling of the rail car system and its suspension system as well as the track system is done on the MATLAB Simulink 2018 environment

The transfer function was used for representing the linear systems, and the inputs are the load changes, applied forces as well as the uneven track profiles, while the output of the system is the acceleration and displacement of the rail car

According to Sezer and Atalay [19], the vectors for the displacement, rail car disturbance and control forces are expressed as Eqs. (12)–(14), respectively:

x ¼ Xr,Lr,Vr, θr,∅r,ψr,Xb1,Lb1,Vb1, θb1∅b1,ψb1,Xb2,Lb2,Vb2, θb2∅b2,ψb2,

½ �¼ Fd ½000000000000000000000000ð Þ ksLYr<sup>1</sup> ð Þ ksVð Þ Vr<sup>1</sup> þ Vr<sup>2</sup> ð Þ ksLR1Lr<sup>1</sup> þ ksVð Þ Vr<sup>2</sup> � Vr<sup>1</sup> 00ð Þ ksLLr<sup>2</sup> ð Þ ksVð Þ Vr<sup>3</sup> þ Vr<sup>4</sup> ð Þ ksLR1Lr<sup>2</sup> þ ksVa Vð Þ <sup>r</sup><sup>4</sup> � Vr<sup>3</sup> 00ð Þ ksLLr<sup>3</sup> ð Þ ksVð Þ Vr<sup>5</sup> þ Vr<sup>6</sup> ð Þ ksLR1Lr<sup>3</sup> þ ksVa Vð Þ <sup>r</sup><sup>6</sup> � Vr<sup>5</sup> 00 ð Þ ksLLr<sup>4</sup> ð Þ ksVð Þ Vr<sup>7</sup> þ Vr<sup>8</sup> ð Þ ksLR1Lr<sup>4</sup> þ ksVa Vð Þ <sup>r</sup><sup>8</sup> � Vr<sup>7</sup> 00 ð Þ ksLLr<sup>5</sup> ð Þ ksVð Þ Vr<sup>9</sup> þ Vr<sup>10</sup> ð Þ ksLR1Lr<sup>5</sup> þ ksVa Vð Þ <sup>r</sup><sup>10</sup> � Vr<sup>9</sup> 00ð Þ ksLLr<sup>6</sup> ð Þ ksVð Þ Vr<sup>11</sup> þ Vr<sup>12</sup>

ð Þ ksLR1Lr<sup>6</sup> þ ksVa Vð Þ <sup>r</sup><sup>12</sup> � Vr<sup>11</sup> 0�

ð�UV<sup>3</sup> þ UV4ÞÞ d2UL<sup>2</sup> � p2ð Þ UV<sup>4</sup> � UV<sup>3</sup>

d2UL<sup>3</sup> � p2ð Þ UV<sup>6</sup> � UV<sup>5</sup>

½ �¼ Fu ½0 Uð Þ L1 þ UL2 þ UL3 ð Þ UV1 þ UV2 þ UV3 þ UV4 þ UV5 þ UV6

Loðð Þ� UV<sup>5</sup> þ UV<sup>6</sup> ð Þ UV<sup>1</sup> þ UV<sup>2</sup> ÞÞðh2UL<sup>1</sup> � UL3ÞÞ0 �ð Þ UL<sup>1</sup> ð Þ ð Þ UV<sup>1</sup> þ UV<sup>2</sup> d2UL<sup>1</sup> � p2ð Þ UV<sup>2</sup> � UV<sup>1</sup>

Xb3,Lb3,Vb3, θb3∅b3,ψb3,Xw1,Lw1,Vw1, θw1,ψw1,Xw2,Lw2,Vw2, θw2, ψw2,Xw3,Lw3,Vw3, θw3,ψw3,Xw4,Lw4,Vw4, θw4,ψw4,Xw5,Lw5,Vw5,

θ<sup>w</sup>5,ψ<sup>w</sup>5,Xw6,Lw6,Vw6, θ<sup>w</sup>6,ψ<sup>w</sup>6�T (12)

d3ð Þ UL1 þ UL2 þ UL3 p2ð Þ ðUV<sup>2</sup> þ UV<sup>4</sup> þ UV6Þ � ð Þ UV<sup>1</sup> þ UV<sup>3</sup> þ UV<sup>5</sup> 

<sup>00000000000000000000000000000000</sup>�

<sup>000</sup>ð Þ �UL<sup>2</sup>

<sup>000</sup>ð Þ� �UL<sup>3</sup> <sup>ð</sup> UV<sup>5</sup> <sup>þ</sup> UV6ÞÞ

Modeling of the rail car and its suspension system.

where.

Lr1,Lr2,Lr3,Lr4,Lr5,Lr<sup>6</sup> are the lateral rail disturbance functions; Vr1,Vr3,Vr5,Vr7,Vr9,Vr<sup>11</sup> are the vertical right rail disturbance functions; Vr2,Vr4,Vr6,Vr8,Vr10,Vr<sup>12</sup> are the vertical left rail disturbance functions; Xr,Lr,Vr are the longitudinal, lateral and vertical displacements of the rail car body; Xbi,Lbi,Vbi are the longitudinal, lateral and vertical displacements of the rail car bogie; Xwj,Lwj,Vwj are the longitudinal, lateral and vertical displacements of the rail car wheel sets; θr,∅r,ψr, are the roll, pitch and yaw displacements of the rail car body; θbi,∅bi,ψbi, are the roll, pitch and yaw displacements of the rail car bogie; θwj, ∅wj, ψwj, are the roll, pitch and yaw displacements of the rail car wheel set (i ¼ 1…::3; and j ¼ 1…::6Þ; p<sup>1</sup> is the lateral distance between the vertical primary suspensions; p<sup>2</sup> is the lateral distance between the vertical secondary suspensions; d<sup>1</sup> is the vertical distance between the wheel set and the bogie mass centre; d<sup>2</sup> is the vertical distance between the bogie mass centre and lateral secondary suspension; d<sup>3</sup> is the vertical distance between the lateral secondary suspension and railcar body mass centre; Lo is the longitudinal distance between the bogies; and a is the lateral distance between the contact points of the wheel-rail.

The mathematical model can be obtained in either the state space or using the transfer function for necessary control actions. The control action is to improve the speed of response, system's balance and stability and to reduce the steady-state error as well as the amplitude of oscillations. Eqs. (15) and (16) express the form of the second-order systems:

$$
\dot{y}(t) + \xi \alpha\_n \dot{y}(t) + \alpha\_n^2 y(t) = k\_d \alpha\_n^2 u(t) \tag{15}
$$

$$G(s) = \frac{X(s)}{F(s)} = \frac{1}{ms^2 + bs + k} = \frac{k\_d o o\_n^2}{s^2 + 2\xi o o\_n s + o\_n^2} = \frac{k\_d o o\_n^2}{\left(s + \sigma\right)^2 + o\_d^2} \tag{16}$$

where.

kd is the dc gain which is the steady-state step response to the magnitude of the step input expressed as Eq. (17)

$$k\_d = \frac{1}{k} \tag{17}$$

Po ¼ e

Eq. (26) relates the percent overshoot to the damping ratio:

Vibration Analysis and Control in the Rail Car System Using PID Controls

DOI: http://dx.doi.org/10.5772/intechopen.85654

The input parameters of the system are presented in Table 1.

2.1 The proportional-integral-derivative (PID) control

10 Distance between the centre of gravity and the front position of the rail car

11 Distance between the centre of gravity and the middle position of the rail car

12 Distance between the centre of gravity and the rear position of the rail car

Source: [20, 21].

Input parameter for rail car system modeling.

Table 1.

125

�<sup>1</sup> ζπ ffiffiffiffiffiffiffiffiffiffiffiffi

<sup>ξ</sup> <sup>¼</sup> �lnPo ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>π</sup><sup>2</sup> <sup>þ</sup> ln<sup>2</sup>

The PID control represents the proportional, integral and derivative controls. It is a form of classic control designed to automatically reduce rise and settling times, steady-state errors and percent overshoot. The block diagram of the PID controller is illustrated in Figure 4. The threshold value otherwise referred to as the reference point is pre-set on the controller while real-time measurement using sensors is taken. The error generated, which is the difference between the threshold value and the actual measurement, represents the deviation from the ideal process; thereafter, the actuator effects real-time control to adjust process variables. The output signal of the PID controllers often responds to changes over time with respect to the actual

S/N Parameter Notation Value Unit Average mass of the rail car M<sup>1</sup> 50,500 kg Average mass of bogie M<sup>2</sup> 2410 kg Mass of primary suspension system Mp 30,000 kg Mass of secondary suspension system Ms 30,000 kg Moments of inertia Ii 56,900 kgm<sup>2</sup> Rail car roll inertia Ir 68,200 kgm<sup>2</sup> Rail car pitch inertia Ip 71,000 kgm<sup>2</sup> Average mass of first wheel set and axle m<sup>1</sup> 1300 kg Average mass of second wheel m<sup>2</sup> 1300 kg

 Spring constant of the primary suspension system <sup>k</sup><sup>1</sup> 2.4 � 106 N/m Spring constant of the secondary suspension system <sup>k</sup><sup>2</sup> 5.6 � <sup>10</sup><sup>5</sup> N/m Spring constant of the wheel <sup>k</sup><sup>3</sup> 4.0 � <sup>10</sup><sup>5</sup> N/m 16 Damping constant of the primary suspension system <sup>b</sup><sup>1</sup> 1.2 � 103 Ns/m 17 Damping constant of the secondary suspension system <sup>b</sup><sup>2</sup> 2.95 � <sup>10</sup><sup>4</sup> Ns/m Damping constant of the wheel <sup>b</sup><sup>3</sup> 5.0 � <sup>10</sup><sup>4</sup> Ns/m

ð Þ Po

<sup>1</sup> � <sup>ξ</sup><sup>2</sup> <sup>p</sup> (25)

<sup>q</sup> (26)

di 6 m

d<sup>2</sup> 6 m

d<sup>3</sup> 6 m

ω<sup>d</sup> is the damped natural frequency expressed as Eq. (18)

$$
\rho\_d = \rho\_n \sqrt{1 - \xi^2} \tag{18}
$$

σ is the real part of the pole expressed as Eq. (19)

$$
\sigma = \xi o\_{\mathfrak{n}} \tag{19}
$$

ω<sup>n</sup> is the undamped natural frequency at which the system oscillates expressed as Eq. (20)

$$
\rho\_n = \sqrt{\frac{k}{m}}\tag{20}
$$

ξ is the damping ratio which defines the rate or nature of amplitude of oscillation (Eq. (21))

$$\xi = \frac{b}{2\sqrt{km}}\tag{21}$$

The performance of the control system is measured by the settling time, delay time rise time, percent overshoot and the steady-state error.

The settling time ts is the time it takes the system to fall within a certain percent (mostly 2%) of the steady-state value for a step input response expressed as Eq. (22):

$$t\_s = \frac{-\ln T\_f}{\xi a\_n} = 4\zeta = \frac{4}{\sigma} \tag{22}$$

On the other hand, the rise time is the time it takes the signal to change from a low value to a high value (say 10–90% or 0–100%). The time it takes the peak value to occur known as the time is expressed as Eq. (23):

$$t\_p = \frac{\pi}{o\nu\_d} \tag{23}$$

The steady-state error E sð Þ is the difference between the input reference signal R sð Þ and the output signal Y sð Þ expressed as Eq. (24):

$$E(\mathfrak{s}) = R(\mathfrak{s}) - Y(\mathfrak{s}) \tag{24}$$

Similarly, the delay time is the time required for the response to reach half the final value for the first time, while the percent overshoot is the percent by which the step response of the system exceeds the final steady-state value. It is a parameter that defines the instability of a system (Eq. (25)):

Vibration Analysis and Control in the Rail Car System Using PID Controls DOI: http://dx.doi.org/10.5772/intechopen.85654

$$P\_o = e^{-1} \frac{\zeta \pi}{\sqrt{\mathbf{1} - \xi^2}} \tag{25}$$

Eq. (26) relates the percent overshoot to the damping ratio:

$$\xi = \frac{-\ln P\_o}{\sqrt{\pi^2 + \ln^2(P\_o)}}\tag{26}$$

The input parameters of the system are presented in Table 1.

#### 2.1 The proportional-integral-derivative (PID) control

The PID control represents the proportional, integral and derivative controls. It is a form of classic control designed to automatically reduce rise and settling times, steady-state errors and percent overshoot. The block diagram of the PID controller is illustrated in Figure 4. The threshold value otherwise referred to as the reference point is pre-set on the controller while real-time measurement using sensors is taken. The error generated, which is the difference between the threshold value and the actual measurement, represents the deviation from the ideal process; thereafter, the actuator effects real-time control to adjust process variables. The output signal of the PID controllers often responds to changes over time with respect to the actual


#### Table 1.

where.

as Eq. (20)

(Eq. (21))

Eq. (22):

124

step input expressed as Eq. (17)

Noise and Vibration Control - From Theory to Practice

kd is the dc gain which is the steady-state step response to the magnitude of the

kd <sup>¼</sup> <sup>1</sup>

ω<sup>n</sup> is the undamped natural frequency at which the system oscillates expressed

ξ is the damping ratio which defines the rate or nature of amplitude of oscillation

ffiffiffiffi k m r

ω<sup>n</sup> ¼

<sup>ξ</sup> <sup>¼</sup> <sup>b</sup> 2 ffiffiffiffiffiffi km

(mostly 2%) of the steady-state value for a step input response expressed as

ts <sup>¼</sup> �lnTf ξω<sup>n</sup>

The performance of the control system is measured by the settling time, delay

The settling time ts is the time it takes the system to fall within a certain percent

On the other hand, the rise time is the time it takes the signal to change from a low value to a high value (say 10–90% or 0–100%). The time it takes the peak value

> tp <sup>¼</sup> <sup>π</sup> ωd

The steady-state error E sð Þ is the difference between the input reference signal

Similarly, the delay time is the time required for the response to reach half the final value for the first time, while the percent overshoot is the percent by which the step response of the system exceeds the final steady-state value. It is a parameter

<sup>¼</sup> <sup>4</sup><sup>ζ</sup> <sup>¼</sup> <sup>4</sup>

ffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>ξ</sup><sup>2</sup> q

ω<sup>d</sup> ¼ ω<sup>n</sup>

ω<sup>d</sup> is the damped natural frequency expressed as Eq. (18)

σ is the real part of the pole expressed as Eq. (19)

time rise time, percent overshoot and the steady-state error.

to occur known as the time is expressed as Eq. (23):

R sð Þ and the output signal Y sð Þ expressed as Eq. (24):

that defines the instability of a system (Eq. (25)):

<sup>k</sup> (17)

σ ¼ ξω<sup>n</sup> (19)

p (21)

<sup>σ</sup> (22)

E sðÞ¼ R sðÞ� Y sð Þ (24)

(18)

(20)

(23)

Input parameter for rail car system modeling.

Figure 4. The block diagram of the PID controller.

measurements and set points. The PID variables are iteratively adjusted until the steady-state error from the output signal is eliminated. This control action is done by adjusting the controller gain with resulting decrease in the rise time and percent increase in the overshoot, which makes the system go unstable. This rise time is further reduced with the integral control action. Finally, the derivative action is introduced to compensate for the offset. This reduces the percent overshoot and settling time, thus making the system stable over time.

Eq. (27) gives the expression for the control action of the PID controller:

$$u\_c = K\_p e(t) + \frac{K\_i}{T\_i} \Big|\_{0}^{t} e(t)dt + K\_d T\_d \frac{de(t)}{dt} \tag{27}$$

system's transfer function and subsequent importation of the parameters obtained

Vibration Analysis and Control in the Rail Car System Using PID Controls

DOI: http://dx.doi.org/10.5772/intechopen.85654

The PID control system and its connection to the rail car system are shown in

The actively controlled suspension system can be activated via the use of solenoid, hydraulic, electromagnetic means or through a magnetorheological damper. This system is designed to use the solenoid actuators because of its lightweight, simplicity in structure, ease of installation and short response time, which makes it

Figure 7 shows the step response before the iterative adjustment of the PID control. The amplitude of oscillation, which is a function of the percent overshoot, is 2 mm, and the system could not return to the equilibrium position after 3 s. The shape of the plot represents a system that is underdamped, which signifies the need for damping to minimize unwanted motion. The system whose step response is depicted in Figure 7 is relatively unstable as vibration will reduce the system's and

Figure 8 shows the step response from the controlled system. When compared to Figure 1, the amplitude of oscillation has reduced to 1.15 mm and settling time 0.5 s under the effect of the PID control action. The system is relatively stable as the

The aim of the control design is to keep the system variables close to the reference in order to compensate for the effect of load and rail disturbances. The system requirement is to check unpleasant motion and ensure rail car stability by reaching a compromise between the stiff primary suspension and soft secondary

into the linear time-invariant system.

Figures 5 and 6, respectively.

The PID control and the rail car systems.

highly sensitive to disturbances.

3. Results and discussion

suspension system.

Figure 5.

Figure 6.

The PID control system.

ride performance.

127

The Nichols-Ziegler tuning rules employed for tuning the PID control as well as the summary of the effects of its control action on the PID are presented in Tables 2 and 3, respectively.

The signal (U) which passes through the controller computes the derivative and integral of error signal. The signal error is thereafter sent to the system in order to obtain the system's output (Y). The PID controller was designed in the MATLAB Simulink 2018 environment to generate a continuous time control. Using the Nichols-Ziegler rules, the tuning of PID controller was done by generating the

