Effect of passenger uncertainty on the inertial properties of a railway coach

: The rotation transformation matrix and translation transformation matrix are derived. They are combined to study the variation of inertial properties of the loaded coach with seating and standing passengers. After that, a CRH2 (China Railway Highspeed) motor coach and Chinese adults in statistical terms are illustrated for precise modelling. It is indicated that CG (Center of Gravity) positions and moments of inertia are all close to linear varying with passenger numbers but at different slopes before and after full-load. It is also found that yaw moment of inertia and pitch moment of inertia are highly correlated. The mass has larger correlation on CG z than CG x and CG y, and larger correlation on roll moment of inertia than yaw and pitch moment of inertia. It may offer some instructions and reference for more realistic simulation of railway vehicle dynamics and measure experiments.


Introduction
The advantages of large carrying capacity, high running speed, low energy consumption and low transportation cost of railway transportation make it always be a fascinating topic for railway engineers and researchers. Railway vehicle dynamics is one of the important aspect of studying dynamic performance of vehicles. For this the vehicle models are usually formulated as a multi-body system including many rigid bodies such as car body, bogie frame, axle-box and wheel-set, and still including many elastic components such as primary suspensions and secondary suspensions etc. It is usually assumed that the characteristic parameters of all rigid bodies and suspension components are determined in earlier studies [1,2]. However, there are some uncertainties [3][4][5] of many parameters due to manufacturing, installation, use or other reasons. Meanwhile, the values of some parameters may be varied with time and space during the operational period.
Many researchers have noticed these uncertainties and done some works during this decade. Lu et al. [6] established parametric finite element model and used Monte-Carlo (MC) method to study reliability and parameter sensitivities of the plate thickness, service load and material constants of the bogie frame. Shen [7] did sensitivity analysis of suspension parameters of a high-speed Electric Multiple Units (EMU) based on orthogonal experiments, and then presented an improved niche genetic algorithm to optimize the dynamic performance of the EMU. Mazzola et al. [8] adopted three methods, one-at-time method, MC simulation method based on Latin Hypercube Sampling (LHS) and MC method based on Design of Experiments (DOE), to investigate the propagation of suspension uncertainties to the critical speed of the vehicle. Suarez et al. [9] took the mass of rigid bodies and the height of CG as random parameters, and did 216 dynamics simulations to perform sensitivity analysis. It was concluded that the mass and the vertical moment of inertia are most sensitive parameters to dynamic behaviour. Bigoni et al. [10] used a stochastic Cooperrider truck model with two degrees of freedom to investigate the computational performance and convergence of the advanced Uncertainty Quantification (UQ) methods. The generalized polynomial chaos in the stochastic collocation form is highlighted and compared with MC and quasi-Monte-Carlo (QMC) methods. They [11] also applied high-dimensional model representation to global sensitivity analysis (GSA) of the Cooperrider bogie running on curved track with normal distribution of suspension parameters. It was found that the steering suspension components account for most of the variance of the critical speed. Luo et al. [12] used stochastic dynamics method to predict dynamics performance and optimize suspension parameters of a high-speed train where the suspension and wheel-rail parameters etc. are considered as random distribution. They found that the optimized suspension parameters have a wide range of line operation adaptability compared with conventional dynamic analysis. Bideleh et al. [13] studied the effects of the bogie suspension components on the wear, safety, and ride comfort. The multiplicative dimensional reduction method is used for GSA of the bogie with symmetric/asymmetric suspensions and straight/curved track. Gao et al. [14] conduct local sensitivity analysis (LSA), GSA and regional sensitivity analysis (RSA) of a railway bogie with independent and normal distribution of left-right suspension components. They found RSA is a good choice for importance evaluation and stability control due to its good efficiency and trusty precision. They [15] also found that different front-back symmetrical suspension components usually have different sensitivity indices to the critical speed of the bogie.
In order to make the vehicle dynamics model more realistic, model parameters should be carefully and accurately determined with good method and effective test equipment. However, even if we don't talk about parameter uncertainties from manufacturing, installation or experiment. There are still some uncertainties hard to quantify in railway vehicle system, such as passenger uncertainties for passenger train, cargo uncertainties for freight train etc. These uncertainties make the study of vehicle dynamics be more complex, and thus the result is not quite convincing to some extent. Therefore, it should be clear about the effects of these uncertainties on the vehicle performance.
In this paper, effect of passenger uncertainty on the inertial properties of a railway coach is studied. The expressions of inertial tensor using the rotation transformation matrix and translation transformation matrix are derived. A CRH2 motor coach and Chinese adults in statistical terms are illustrated to discuss how passengers, including passenger numbers and the proportion of female, affect CG positions and moments of inertia of the coach.
The correlation analysis is also conducted between different inertial parameters to provide comprehensive conclusions. The results provide a kind of methodology for similar problems and a validation method for some measure experiments.

Rotation transformation matrix
One case, suppose that there is an inertial reference coordinate system (OXYZ), a body with coordinate system  (1) In above equation, the denotations of c i =cos(θ i ), s i =sin(θ i ) for i=1-3 are used for simplicity. Similarly, the transformation matrix about X'-axis and Y'-axis are Therefore, the rotation transformation matrix from the body coordinate system (O'X'Y'Z') to the inertial reference system (OXYZ) is The transformation matrix A rot is an identity orthogonal matrix, that is, its transposed matrix is equal to its inverse matrix. The inertial tensor matrix of the body about the reference coordinate system (OXYZ) can be obtained by the following expression [16] through the rotation transformation matrix A rot It can be easily proved that the square matrix J 1 is similar with the matrix J c due to the orthogonality characteristic of the transformation matrix A rot . That is, they have same eigenvalues and characteristic polynomial, and moreover the following expressions are always true where rank(·), det(·) and tr(·) are operators of rank, determinant value and trace of a matrix.

Translation transformation matrix
Another case, the definitions of two coordinate systems are same as Fig. 1 except that two coordinate systems have different coordinate origins and the corresponding axes are parallel with each other which shows in Fig. 2. Suppose that the mass of the body is m and the values of CG position O' are (x, y, z) in the inertial reference coordinate system (OXYZ). Then the parallel-axis theorem can be directly applied to compute the inertial tensor of the body in the inertial reference system.
In above equation, the translation transformation matrix [17] A tra is If three mass centre axes of the body are originally not parallel to those of the inertial coordinate system, the rotation transformation matrix should firstly be used to make them parallel. After that the translation transformation can still be conducted to compute the inertial tensor of the body in the inertial reference system. It is in the form of (9) 3 Description of the model

The railway coach
A CRH2 motor coach of second class [18] is used as the research object to study the effect of passenger uncertainties on the inertial properties of the coach. The schematic diagram is shown in Fig In the analysis, it is assumed that the z-axis is upwards, the y-axis points to the regions of three seats and the x-axis is orthorhombic to the y-z plane and satisfying the right-hand screw rule. The floor plane is choose as x-y plane. Moreover, it is also assumed that the coordinate origin is located in the geometric centre in the x-y plane when the coach is in the condition of curb weight and no passenger. That is to say, half of 24500 millimetre in the length direction and 3095 millimetre in the width is the coordinate origin. The coordinate axes x, y, z and coordinate origin O are also illustrated in Fig. 3. In the context of railway vehicle dynamics [19], there are more specific terms about moments of inertia of rigid bodies. The moment of inertia around x-axis is usually called roll moment of inertia, pitch moment of inertia is around y-axis and yaw moment of inertia is around z-axis.

Inertial properties of Chinese adult
Different peoples in different countries often have different inertial properties. In the meantime, a same person in different ages still has different inertial properties. To a Chinese people, it often includes some big parts such as head and neck, upper torso, lower torso, thigh, calves, foot, upper arm, fore arm and hand in standard [20] which shows in Fig. 4. Each part has its own mass, CG position and moments of inertia. All parts are organically composed a person with general inertial properties. For simplicity, a body coordinate system (O'X'Y'Z') is also established in Here not a specific people is considered and the statistical inertial properties of Chinese adults are used.
According to the sampling survey and statistical analysis, the CG z position from the head point, moments of inertia J xx , J yy and J zz for a Chinese adult male can be regressed as a linear equation with two parameters [21] as follows  The CG z position from the foot bottom is For Chinese adult females, the above methods still hold effective only with different coefficients. The coefficients of CG z and moments of inertia for Chinese adults are listed in detail in Table 1. It should be noted that the coefficients for the moments of inertia of J xx and J yy are exchanged with each other since the X'-axis and Y'-axis are switched with each other compared with the reference [20]. For general Chinese adults, the value of J xx is the largest, J yy is the second and J zz is the lowest among three moments of inertia.

Numerical results and discussions
A detailed CRH2 coach is considered for numerical analysis. The CG z position of the coach in curb weight state is 1520 millimetre above the rail top -namely it is 220 millimetre in the z-axis of inertial reference frame of Fig. 3. The inertial parameters, including CG position, mass, roll moment of inertia J cx , pitch moment of inertia J cy and yaw moment of inertia J cz of the empty coach, are all invariable. In Table 2, parameters and their values for the coach and Chinese standing adults are presented.
For Chinese standing adults, their masses and heights are supposed to be independent and normally distributed around their nominal values. The nominal values and the standard deviation [22] are also given in Table 2. In the meanwhile, the rotation angles abound Z'-axis are also considered and randomly produced from -180 to 180 degree.
These data for standing adults can be used to produce the masses and heights of different passengers, and then they are substituted into equation (11) for CG z position computation and equation (10) for moments of inertia computation. For the CG x and CG y position of standing passengers, they are consistent with the CG x and CG y of the passenger themselves. For Chinese seating adults, the CG z position and moments of inertia are related to those of standing passengers. In the analysis, the rate of the male or female is also researched for a more realistic simulation of the actual situation. According to the regulation [24], the overcrowding rate of twenty percent is permitted for short trips of CRH coach of second class. However, it may exceed the standard at some lines in some cases, for example, Spring Festival peak travel season. Here the maximum overcrowding rate of fifty percent is studied.

CG position
Firstly, the CG position of the coach with different passengers are studied. The probability of the adult female is set to p=0.5. Fig. 6 shows error bar of CG x of the coach with passenger numbers. The abscissa is the passenger numbers from 10 to 150, and the ordinate is the x coordinate value of CG. For each working condition, the 50000 data points with normal distribution are computed for statistical analysis of their mean values, standard deviation, minimum and maximum values. In the diagram, the mean value is denoted by a small circle, the standard deviation is showed by half of the height of a narrow rectangle, and the short horizontal lines located at lower and upper positions are minimum and maximum values in this case. It is seen from the diagram that the mean of x coordinate values are all negative since the passenger zones are slightly to the back of the coach in Fig. 3. The mean value is -8.88 millimetre with 10 passengers to -75.97 millimetre with 100 passengers where the limit value is reached and all passengers are in seat. After that, with increasing numbers of standing passengers, the mean value is slowly down to -70.56 millimetre with 150 passengers. Furthermore, it is also found that the standard deviation, the range of minimum and maximum values at 100 passengers point are the minimum since all passengers are in seat and there is no empty seat. It greatly decreases the deviation. The largest standard deviation and the range are both near the point of 150 passengers where the value are 63.85 and 563.90 millimetre. Fig. 6 Error bar of CG x of the coach with passenger numbers (p=0.5) Fig. 7 Error bar of CG y of the coach with passenger numbers (p=0.5) Fig. 7 shows error bar of CG y of the coach with passenger numbers. The coordinate representations are the same as Fig. 6. The mean values of the y coordinate are always increasing and positive when the seated passengers are increased from 10 to 100 numbers because the y-axis of the inertial coordinate system points to the direction of three seats region. After that, the standing passengers begin to increase, the mean values of the y coordinate start to decrease toward the negative direction since the centre of aisle region is partial to the two seats regions which is in the negative y coordinate direction. There is a minimum mean y coordinate in absolute values when all passenger numbers are close to 130. It can also be seen that the standard deviation, the range of minimum and maximum values of the y coordinate are all small when there is no empty seat, i.e. passenger numbers are from 100 to 150 compared with other conditions. Fig. 8 shows error bar of CG z of the coach with passenger numbers. The ordinate axis with number of 220 millimetre is the CG z position of the empty coach in the inertial reference system. Since the CG z position of the empty coach is lower than the bottom of the seat which is 325 millimetre above the floor, in other words, the CG z position of adult passengers, in spite of male or female and seating or standing, are all higher than that of the empty coach, thus the CG z positions are all increased along. With increasing of passenger numbers, the CG z is linearly increased with different slopes. The slope is bigger when the passenger numbers exceed 100 because the standing passengers make more contribution to the CG z. This is also true for the range of minimum and maximum values of CG z since standing passengers have much bigger variation of CG z than that of seating passengers. Meanwhile, it is found that the CG z is increased 63.26 millimetre when the coach is in full load of 100 passenger numbers compared with empty coach. The values are 81.20 millimetre and 106.23 millimetre if the passenger numbers increased to 120 and 150 separately.

Mass and moments of inertia
Secondly, the mass and moments of inertia of the coach are discussed. Fig. 9 shows increase rate of mass and moments of inertia of the coach versus passenger numbers. The mean value is only illustrated for each working condition. The symbols J x , J y and J z are moments of inertia around x-axis, y-axis and z-axis with new CG position but all coordinate axes are still parallel to original inertial reference coordinate axes. It is seen from the diagram that the increase rate of mass is completely increased by linear way with increasing numbers of passengers. It can be regarded as part of validity and effectiveness of the method and program. In the meantime, the increase rate of moments of inertia J x , J y and J z are all linearly increased with different slopes before and after the point of 100 passenger numbers. The slopes of moments of inertia J y and J z are smaller before the point but larger after the point, while moment of inertia J x is reversed.  Table 3 where the slopes of CG coordinates are also given. The point of 100 passengers is regarded as a break point of the slope concluded from Fig. 6 to Fig. 9.  In above analysis, the probability of the adult female are all set to p=0.5. Here the effect of probability is further researched. Fig. 10 shows the increase rate of mass and moments of inertia of the full load coach with probability of the adult female. The linear decreasing tendency is obvious for the four parameters with increasing of the female probability. That is to say, the higher of the proportion of female is, the more of the female numbers are when the total number is constant. It decreases total mass and thus decrease the moments of inertia to some extent.

Correlation analysis
Finally, the correlation analysis is conducted and the probability of the adult female is set to p=0.5 for general condition. Pearson's method [25] is used to compute linear correlation coefficient since the data of the considered parameters is near normal and independent distribution with one peak through data validation. It computes the correlation coefficient ρ X,Y of the vector X and Y by the equation (12) where cov(· , ·) denotes the covariance of the two vectors and σ is the standard deviation. Fig. 11 shows correlation coefficient map of the full-load coach. The parameters are the mass, CG x, CG y, CG z and moments of inertia J x , J y and J z . Numbers in squares are the corresponding correlation coefficients between horizontal and vertical parameters. It is found that the maximum correlation coefficient for different parameters is 0.9998 between moments of inertia J y and J z , which means they are highly correlated. The strong correlation exists between the mass and CG z whose value is 0.7303. There is slight correlation between the parameter groups (CG x, J y ), (CG x, J z ) and (CG y, J x ). The rest of parameter groups have negligible correlation since the correlation values are less than 0.1. It should be noted that the correlation relation between two parameters is changed with different passenger numbers. Fig. 11 Correlation coefficient map of the full-load coach (p=0.5) In railway vehicle engineering, the correlation effects of mass on other parameters are desirable to some extent since the mass is relatively convenient to measure. In Fig. 12, the correlation coefficients of CG with the mass at different passenger numbers is given. It can be seen from the diagram that passenger numbers have great effect on the correlation coefficient of CG z and the positive correlation are all along. With increasing numbers of passengers, the correlation coefficient becomes smaller and smaller, particular after 100 passenger numbers where it undergoes a sudden drop from 0.7303 to 0.2050. After that, the value is slowly decreased to 0.1100 with 150 passengers. However, there are both positive and negative correlation coefficients between the mass and CG y or CG z. The correlation coefficients are all lower than 0.01 within the considered passenger numbers range.  Fig. 13 shows the correlation coefficients of moments of inertia with the mass at different passenger numbers. The correlation effect of the mass on moment of inertia J x is a bit larger than J y and J z whose coefficients are close to the same in the whole range. When the coach is full-loaded with 100 passengers, the maximum and positive correlation coefficients exist as 0.0906, 0.0508 and 0.0520 for J x , J y and J z respectively.

Conclusions
In the paper, effect of passenger uncertainties, including different passenger numbers and different probabilities of adult female, on the inertial properties such as CG position and moments of inertia of a railway coach is investigated in detail.
The rotation transformation matrix of a body rotating with its CG at any angle is derived, and furthermore the translation transformation matrix of the body parallel moving some distance from CG to the inertial reference system is also given. They are combined to study different passengers, including seating passengers at any position and standing passengers at any position and facing toward any angle, on the moments of inertia of the loaded coach.
After that, a CRH2 motor coach of second class with 100 standard seating capacity is considered as the research object. The outline, distribution of functional regions and size of the coach are briefly illustrated. In the meanwhile, Chinese adults are considered as passengers, and the CG z position and moments of inertia are regressed as linear expressions with their masses and heights according to relevant specification.
It is indicated in the results that inertial parameters are nearly varied in linear way with different slopes before and after 100 passenger numbers. The mean values of CG x are slightly to the back of the coach, while CG z with passengers are higher than the empty coach within the considered passenger numbers range. For the mean values of CG y, they are deviated to three seats region when the passenger numbers are smaller than 130. Furthermore, it is also found that the increase rate of moments of inertia of J y and J z are larger than J x with increasing of passengers numbers. There are different ranges of minimum and maximum values for different inertial parameters with different passenger numbers. The mass and inertia moments are all decreased with increasing proportion of female.
It is also concluded through correlation analysis that moments of inertia of J y and J z are highly correlated, the mass and CG z are strong correlated, and the correlation coefficients of other parameter groups are lower in fullloaded state of the coach. The correlation coefficients for different parameter groups are varied at different passenger numbers. There is strong correlation between the mass and CG z when the passenger numbers are lower than 100.
The correlation coefficients between the mass and other inertial parameters are all lower than 0.1. That is, they have slight or negligible correlation relation.
The conclusions can be directly used to investigate the effect of passenger uncertainties on vehicle system dynamics. It can also be used to study cargo uncertainties, including the mass and the position of the cargo, on vehicle performance of the freight train. Moreover, it provides a kind of idea for similar problems and validation method for some measure experiments.

Availability of data and materials
The datasets supporting the conclusions of this article are included within the article.