Weight Estimation of Marine Propulsion and Power Generation Machinery

During the conceptual and preliminary design stage of a ship, designers need to ensure that the selected principal dimensions and parameters are good enough to deliver a stable ship (statically and dynamically) besides deadweight and speed. To support this, the initial intact stability of the proposed ship is required to be calculated, and in doing so, the lightship weight and its detailed breakdown are necessary to be known. After hull steel weight, machinery weight, mainly, marine propulsion and power generation machinery, play a vital role in the lightship weight estimate of a ship due to its robustness. The correct estimation of respective weights improves the accuracy of calculating a ship's initial stability typically to be designed and built. Hence, it would be advantageous for the designer to convince the shipowner. A total of 3006 marine propulsion (main marine diesel) engines and 348 power generation (auxiliary marine diesel) engines/generators of various power output (generators output for auxiliary engines), engine RPM and cylinder number of different engine makers are collected. These are analyzed and presented in both tabular and graphical forms to demonstrate the possible relationship between marine propulsion engine weight and power generation engine weight, and their respective power output, RPM, cylinder number, power-RPM ratio and power-RPM ratio per cylinder. In this article, the authors have attempted to investigate the behaviour of marine propulsion engine weight and power generation engine/generator weight regarding engine power output, generator power output, engine RPM and cylinder number (independent variables). Further attempts have been made to identify those independent variables that influence the weight of the marine propulsion engine and power generation engine/generator (dependent variables), and their interrelationships. A mathematical model has thus been developed and proposed, as a guiding tool, for the designer to estimate the weight of main and auxiliary engines more accurately during the conceptual and preliminary design stage.


Introduction
The conceptual and preliminary design stage is the beginning of a ship design process, before engaging in a shipbuilding contract between the shipowner and the shipyard. During this stage, designers select all necessary dimensions and parameters to fulfil the ship owner's requirements, the classification society, statutory regulations, etc. Sometimes, different issues, such as deadweight and speed, are required to be finalized during this design stage. Calculation of preliminary intact stability is a part of this process to ensure that the proposed dimensions and parameters are sufficient enough to deliver a stable ship (statically and dynamically). For this stability calculation, the weights of hull, machinery, and outfit, including their positions of the centre of gravity, are needed. Machinery weights are from marine propulsion (main marine diesel) engines (from now on the main engine), power generation engines (from now on the auxiliary engine), shaft generators, various deck machinery, main compressors, different pumps with motors, ballast water treatment plant, sewage treatment plant, hydraulic power unit, steering gear, etc. In short, every item onboard a ship is to be included in the calculation. In this article, only the weights of the main engine and auxiliary engine are included.
There are various ways to calculate the hull steel weight with an acceptable level of accuracy. Still, for various machinery/equipment mentioned earlier, there are no reliable and established formulas or procedures to calculate the appropriate weight. Typically, weights of various machinery/equipment are taken from a similar ship or a particular manufacturer's catalogue. Sometimes, during the machinery/equipment selection process, manufacturers and models are different from the initial selection, and hence the weight. This change in weight (particularly for heavy equipment, such as the main engine, auxiliary engine, deck machinery, etc.) significantly impacts the lightship weight, centre of gravity (cog), and stability. On the other hand, if the weight is calculated with the help of a reliable and established formula using machinery/equipment parameters, such as power output, RPM and number of cylinder, which are free from the influence of manufacturer and model, the issue of variation in weight can easily be avoided, if not entirely, at least partly.
Data analysis of the main engine's weight (from now on ME weight) is carried out against different independent variables. These are mainly main engine power output at 100% MCR (hereinafter power output), main engine RPM (hereinafter RPM), number of cylinders of the main engine (hereinafter cylinder number), power-RPM ratio output of the main engine (hereinafter power-RPM ratio) and power-RPM ratio output per cylinder of main engine (hereinafter power-RPM ratio per cylinder).
Power output, RPM, cylinder number, power-RPM ratio, and power-RPM ratio per cylinder (independent variables) significantly influence main and auxiliary engines' weight. However, the magnitudes of impact on ME weight vary from variable to variable. steam, turbo-electric and diesel-electric, nuclear power, and so on, were considered. It is highlighted that the proposed power equation yields an estimated engine weight 5% higher than presented the line through the data spots to allow for the fact that the graph ought to be a stepped line corresponding to cylinder numbers with approximately 10% weight steps for the addition of each cylinder. The article also suggested a procedure about the estimation of remaining components of machinery weight. Schneekluth and Bertram (1998) proposed 'weight -power'' linear relationship of a diesel engine under engine speed groups like slow speed (110 -140 rpm), medium speed (400 -500 rpm) and medium speed V-type engines (400 -500 rpm). Authors claimed that the proposed relationship would produce a reasonable estimate of engine weight (dry engine -without cooling water and lubricant). Roh and Lee (2018) studied the machinery weight from different angles. They considered the machinery (main engine) weight as a lightweight component together with a ship's structural weight and outfit weight to estimate the lightweight. Authors assumed that the main engine's weight is directly proportional to NMCR (Nominal Maximum Continuous Rating) of the engine. Using the admiralty equation, they finally proposed a non-linear equation to estimate the main engine weight involving length, breadth, draught, block coefficient and service speed. Papanikolaou (2014) mentioned the importance of the main machinery (main engine) and gearbox as part of the machinery weight. The importance of main machinery weight also took the effects of slowspeed, medium-speed and high-speed. While many approximate formulas and other approximations have been provided for the total machinery weight, very little has been mentioned about calculating the main machinery weight in the design stage.

Problem formulation
After hull (steel) weight, different machinery weight, mainly, main engines (ME) and auxiliary engines (AE) contribute the most to the lightship weight and centre of gravity (cog). The latter has the maximum impact on the initial intact stability, which is usually calculated at the conceptual and preliminary design stage of a ship. During this stage, in most cases, leading equipment makers and their models are not known. Designers assume a model from a particular maker which, later on, may not be the final one. Hence, the weight and cog calculations do not reflect the ship's performance on stability. As such, designers need to re-calculate the stability using the data of finally selected maker and model. This situation can be improved dramatically if a reliable guideline about the relationship between engine weight and other engine parameters is available. Keeping this problem in mind, this research focuses on the interrelationship among engine weight and various parameters of engines, such as power output, RPM, and cylinder number. Based on the relationship, an engine's weight can be estimated to solve the problem mentioned earlier, at least partly, if not entirely.

Engine power output
The engine power output refers to the engine's power at 100% maximum continuous rating (MCR) condition. The higher power output requires a bigger/longer engine dimensionally, and a bigger/longer engine means more weight irrespective of RPM and cylinder number. The power output can also be increased by adding additional cylinders, which results in extra weight. Therefore, the power output has a positive impact on the weight irrespective of RPM and cylinder number, and this is expected to be linearly associated. Hence, the first assumption is, ceteris paribus, the weight is a function of power output and assumed to be linearly associated.

Generator power output
The generator output refers to the electrical power delivered by a generator. In this research, the generator is driven by a diesel engine. The power output of the diesel engine is higher than that of a generator. Typically, diesel engine power output is about 4-5% higher than that of a generator output ( Fig.1). A higher generator output requires a higher engine power output and a more significant size generator, resulting in the generator and engine's higher weight irrespective of RPM and the diesel engine's cylinder number. Therefore, the generator power output has a positive impact on weight.
Hence, the second assumption is, ceteris paribus, the weight is a function of the power output and is expected to be linearly associated.

Engine RPM
The RPM of an engine refers to the number of revolutions of the crankshaft per minute (RPM).
Technically, keeping power output and cylinder number same for a particular engine, higher speed means lower power-RPM ratio. It causes smaller dimensions (maybe shorter stroke length, smaller 0 5,000 10,000 15,000 20,000 0 2,000 4,000 6,000 8,000 10,000 12,000 14,000 16,000 18,000 Engine power output (kW)

Generator power output (kW)
Observed Linear (Observed) cylinder bore diameter, and other scantlings) resulting in lesser weight. Therefore, RPM of an engine has a negative impact on the weight irrespective of power output and cylinder number and is expected to be linearly related. Hence, the third assumption is, ceteris paribus, weight is a function of RPM (negatively) and assumed to be linearly associated.

Cylinder number
Without any elaboration, it can be said that the cylinder number of an engine has a direct impact on the power output and hence weight. Therefore, cylinder number has a positive effect on weight irrespective of power output and RPM and is expected to be linearly associated. Accordingly, the fourth assumption is, ceteris paribus, weight is a function of cylinder number and assumed to be linearly associated.

Engine power-RPM ratio output
The power-RPM ratio output of an engine is defined by the ratio of power output to RPM, ( RPM ), at 100% MCR. Technically, an increase in power-RPM ratio means a reduction in RPM at constant power output. A decrease in RPM implies an increase in weight. Therefore, the weight has a positive impact on power-RPM ratio. Hence, the fifth assumption is, ceteris paribus, the weight is a power-RPM ratio function and is expected to be linearly associated.
Mathematically, the power output of an engine may be expressed in different units, like, HP (Horsepower), kW (kiloWatt), and PS (Metric horsepower). Torque may also be expressed in different units, like, lb-ft (Pound feet), N-M (Newton meter) and kg-M (Kilogram meter). An appropriate conversion factor is required for each combination of power to torque conversion and vice versa. To avoid this hassle, instead of torque, the ratio of power to RPM ( RPM ) is used throughout this article.
This is also convenient to use in a practical field to compare and validate the findings with other available results, which also used the same ratio.

Engine power-RPM ratio output per cylinder
Like the power-RPM ratio, the power-RPM ratio per cylinder ( RPM * ) has a positive impact on engine weight. Hence, the sixth assumption is, ceteris paribus, the weight is a function of power-RPM ratio per cylinder and is expected to be linearly associated.
So far, it has been highlighted theoretically that engine power output, RPM, cylinder number, power-RPM ratio, and power-RPM ratio per cylinder are linearly associated with the corresponding weight of an engine. In other words, engine weight is a linear function of power output, RPM, cylinder number, power-RPM ratio, and power-RPM ratio per cylinder. Mathematically, the above assumptions can be expressed in equations (1-5) as follows. engine = (Power output) = a 1 + b 1 * (1) engine = (Cylinder number) = a 2 + b 2 * (2) engine = ( ) = a 3 + b 3 * (3) engine = (Power − ratio) = a 4 + b 5 * (4) engine = (Power − ratio per cylinder) = a 5 + b 5 * / Where engine is the weight of the engine P is the power output (kW) at 100% MCR for the main engine (ME) and electrical power output (kW) for the auxiliary engine (AE) N is the cylinder number of engine RPM is speed (revolution per minute) T is the power-RPM ratio (P/RPM)

Data collection and methodology
Studying the behavior/pattern of the weight of main engines and auxiliary engines, the detailed information of main engines, and auxiliary engines must be known. For this purpose, details of main engines and auxiliary engines of various manufacturers with different power output, RPM, cylinder number were collected and compiled.

Marine propulsion machinery weight
To develop the weight function and verify the assumptions, a total of 3,306 marine diesel engines' data of I-and V-type were compiled from different manufacturers' product list. All these engines are in production from time to time.
A general picture of sample engines is presented in Tables 1-2 and Figs.2-8. Table 1 summarizes several engines, average RPM, power output, and weight against the cylinder number for V-and I-type engines. One can easily understand the average RPM, power output, and weights of V-and I-type engines of same cylinder number. It also appears that for main engines of a particular cylinder number, these values are far apart for V-and I-type.  1  4  17  141  8,551  283  2  5  567  117  7,521  241  3  6  787  306  8,679  269  4  7  647  134 11,971  368  5  8  15 1,507  682  5 685  174 11034  389  6  9  179  263 21,286  666   7  10  40  147 45,345  1,520   8  11  36  82 54,740  1,838   9  12  151 1,478 3,066  31 38  119 56,855  1,892  10  13  4  97 61,360  2,160  11  14  12  599 13,159  167  4  97 66,080  2,300  13  16  99 1,267 5,593  58  2  875  6,740  59  15  18  14  861 13,543  158  17  20  4  876 10,775  78   Table 2 is constructed using Table 1. It shows a comparison of some selected characteristics, such as weight per unit power output, weight per unit power-RPM ratio output and power output per unit cylinder, of V-type and I-type main engines against cylinder number.  Table 2 that the magnitudes of the characteristics mentioned above for V-type and I-type engines having the same cylinder number are far apart from each other. This is probably because of inherent differences between V-type and I-type concerning geometrical configurations and design considerations. As such, it is not appropriate to combine both types in the same platform for analysis. Therefore, V-type engines are excluded from the analyses in this investigation. Only I-type engines are considered to ensure consistency, uniformity, and reliability in the analyses' results.  Figure 3 demonstrates the distribution of cylinder number with an average value of 7 and 67% of sample main engines are within 7 cylinders range. Figure 4 illustrates RPM's distribution with an average value of 193 RPM and 87% of sample main engines are within 193 RPM range. Figure 5 demonstrates the distribution of power output with an average value of 12,705 kW and 71% of sample main engines are within 12,705 kW power output range. Figure 6 demonstrates the distribution of power-RPM ratio with an average value of 1,347 and 79% of sample main engines are within the 1,347 power-RPM ratio range. Figure 7 displays the distribution of power-RPM ratio per cylinder with an average value of 184 and 71% of sample main engines are within 184 power-RPM ratios per cylinder range. Figure 8 demonstrates the weight distribution with an average value of 397 tonnes (kg x 10 3 ) and 71% of sample main engines are within 397 tonnes (kg x 10 3 ) weight range. -0.5 1-1.5 2-2.5 3-3.5 4-4.5 5-5.5 6-6.5 7-7.5 8-8.5 9-9.5 10-10.5 11-11.5 12-12.5 13-13.5 14-14.

Power generation machinery weight
To develop the weight function and verify the assumptions, the data of inline and V-type of a total of 348 auxiliary engines were collected from the product list of different manufacturers. All these engines are in production from time to time.
A general picture of sample auxiliary engines is presented in Tables 3-4 and Figs.9-13. Table 3 summarizes several engines, average RPM, power output, and weight against the cylinder number for V-and I-type engines. One can easily understand the average RPM, power output and weight of Vand I-type auxiliary engines of same cylinder number. It also appears that these values are far apart for auxiliary engines of a particular cylinder number for V-and I-type.  Table 4 is constructed using Table 3. It shows a comparison of some selected characteristics, such as the weight per unit power output, weight per unit power-RPM ratio and power output per unit cylinder, of V-and I-type auxiliary engines against the cylinder number. In Table 4, weight per unit power output for V-type engine varies from 12-19 and for I-type engine varies from 12-33. Weight per unit power-RPM ratio for V-type engines varies from 10,007-21,373 and for I-type engine varies from 9,814-59,328. Power output per unit cylinder for V-type engines varies from 92-582 and for I-type engines varies from 4-640. It is very clear from the table that the magnitudes of the characteristics mentioned above for V-type and I-type engines having the same cylinder number are far apart from each other. This is probably because of inherent differences between V-and I-type concerning geometrical configuration and design considerations. As such, it is not appropriate to combine both types in the same platform for analysis. Therefore, V-type engines are excluded from analysis in this investigation. Only I-type engines are considered to ensure consistency, uniformity, and reliability in the study results.  Figure 11 reflects the allocation of engine RPM with an average value of 1001 RPM and 71% of sample auxiliary engines are within this RPM range. Figure 12 demonstrates the distribution of power output with an average value of 2,036 kW and 65% of sample auxiliary engines are within this power output range. Figure 13 reflects the distribution of AE weight with an average value of 47 tonnes and 68% of sample auxiliary engines are within this AE weight range. To establish the relationship between the main engine weight and auxiliary engine weight and its dependent variables, the following functions of the linear equations 6 and 7 are chosen because individually, the independent variables are linearly associated with the dependent variable as per initial assumptions.
Where ME is the main engine weight (tonne = kg x 10 3 ) AE is the auxiliary engine weight (tonne = kg x 10 3 ) P is the power output (kW) of the main engine (ME) at 100% MCR and  Tables 5-9.

Engine power output
Main engine weights are analyzed against the power output to determine the ME weight's behaviour from different viewpoints and presented in Figs.14-15. Figure 14 illustrates the ME weight trend (observed values) against the power output, and it seems to be linearly associated. Figure 15 represents the behaviour of average ME weight (average by engine power output group) against average power output. It also shows the sign of linearity among variables. This depicts that with the rise in power output, the ME weight of the said engine increases linearly. Therefore, it is expected that with the increase of power output, the ME weight would be increased too. This is in line with the first assumption.
Figures are investigated for other forms of relationships, and the results are presented in Table 5.

Engine RPM
Main engine weights are analyzed against the RPM to determine the ME weight's behavior from different viewpoints, and the results are presented in Figs.16-17. Figure 16 shows the trend of the ME weight (observed values) against RPM. Figure 17 shows the behavior of average ME weight (average by RPM group) against average RPM. It also shows the sign of power form of relationship among variables. This demonstrates that with the increase of RPM, the ME weight decreases. Therefore, it is expected that with the rise in RPM, the ME weight would be reduced. This is in line with the third assumption, but only in the power form, instead of a linear arrangement of an equation.  Table 6. Table   6 shows the correlation coefficients under different equation forms and suggest that power form yields high correlation coefficients for Figs.16 and 17.

Fig.17
Average ME weight versus average RPM of I-type ME

Average
(average by cylinder number) against the cylinder number. It also shows the sign of linearity among variables. Although the observed values show a full range of ME weight against the cylinder number, this demonstrates that with the rise in the engine's cylinder number, the ME weight of the said engine increases linearly. Therefore, it is expected that with the increase in the cylinder number, the ME weight would be increased too. This is in line with the fourth assumption. Careful investigation of Fig.19 would suggest that the average rate of increase in ME weight per cylinder group are different.
A remarkable increase in ME weight is observed from 9 to 10 cylinders and may be considered the transition from the first group to the second group.
Figures are investigated for other forms of relationships, and the results are presented in Table 7.

Engine power-RPM ratio
Main engine weights are analyzed against the power-RPM ratio to determine the ME weight's behavior from different viewpoints and presented in Figs.20-21. Figure 20 demonstrates the trend of the ME weight (observed values) against the power-RPM ratio. Figures 21 illustrates the behavior of average ME weight (average by power-RPM ratio group) against average power-RPM ratio. It also shows the sign of linearity among variables. This demonstrates that with the rise in power-RPM ratio, the ME weight increases linearly. Therefore, it is expected that with the increase of power-RPM ratio, the ME weight would be increased too.
Figures are investigated for other forms of relationships, and the results are presented in Table 8.

Fig.21
Average ME weight versus average power-RPM ratio of I-type ME

Engine power-RPM ratio per cylinder
Main engine weights are analyzed against the power-RPM ratio per cylinder to determine the ME weight's behaviour from different viewpoints and presented in Figs.22-23. Figure 22 demonstrates the ME weight (observed values) trend against the power-RPM ratio per cylinder. Figures 23 illustrates the behaviour of average ME weight (average by power-RPM ratio per cylinder group) against average power-RPM ratio per cylinder. It also shows the sign of linearity among variables. This demonstrates that with the rise in power-RPM ratio per cylinder, the ME weight increases linearly. Therefore, it is expected that with the increase of power-RPM ratio per cylinder, the ME weight would be increased too.
Figures are investigated for other forms of relationships, and the results are presented in Table 9.    (Fig.14), ME weight versus cylinder number (Fig.18), and average ME weight versus cylinder number (Fig.19) follow the polynomial form (second-order) of an equation. Rest

Weight analysis by speed groups
In Sub-Sub-Section 3.2.3, it is explained how the RPM of an engine affects the engine weight. In real life, all high-power output engines are of low-speed group engines, and they are cumbersome in weight (for example, 80,080 kW, 102 RPM and 2,300 tonnes, Model -RT-fles96C, version B, Wartsila).
Also, all high-speed engines are of low power output, and they are comparatively very light in weight (for example, 599 kW, 2100 RPM and 2 tonnes, Model -C18ACERT, Caterpillar). In between, there is another group called medium-speed (for example, 6,600 kW, 600 RPM and 80 tonnes, Model -12DKM-36e, Daihatsu). Because of the full range of operating speed and hence the torque, the design criterion is also different, and it prompted the analysis of the engine weight under various speed groups. In the following sections, engine weights are examined under three speed groups -low, medium, and high.
Usually, from an operational point of view, main engines are classified according to their design speed (revolution per minute, RPM). They are low-, medium-and high-speed. Wartsila Diesel, one of the leading engine manufacturers, defines low-, medium-and high-speed as RPM 400 and below, 401 to 1000 and more than 1000. In this research, the same specification is followed.

Low-speed group
Main engine weights under low-speed group (speed ≤ 400 RPM) are analyzed against the power output, RPM, cylinder number, power-speed ratio and power-speed ratio per cylinder to determine the ME weight's behaviour from different viewpoints and presented in Figs.24-33. Their trends are almost identical to that of Figs.14-23, but with different responses and correlation coefficients. Figures are also investigated for other types of relationships, and the results are presented in Table 11.

Generator power output
Auxiliary engine weights are analyzed against the power output to determine the AE weight's behaviour from different viewpoints and presented in Figs.54-55. Figure 54 shows the AE weight (observed values) trend against the power output, and it seems to be linearly associated. Figure 55 shows the behaviour of average AE weight (average by power output group) against average power output. It also shows the sign of linearity among variables. This shows that with the rise in power output, the AE weight increases linearly. Therefore, it is expected that with the increase of power output, the AE weight would be increased too. This is in line with the second assumption.

Engine RPM
Auxiliary engine weights are analyzed against the RPM to determine the behaviour of the AE weight from different viewpoints, and the results are presented in Figs.56-57. Figure 56 illustrates the trend of the AE weight (observed values) against RPM. Figure 57 demonstrates the behaviour of average AE weight (average by RPM group) against average RPM. It also shows the sign of power form of relationship among variables. This shows that with the rise in RPM, the AE weight decreases.
Therefore, it is expected that with the increase in RPM, the AE weight would be reduced. This is in line with the third assumption, but in the power form, instead of a linear form of an equation.
Figures are additionally investigated for other forms of relationships, and the results are presented in Table 16.

Cylinder number
Auxiliary engine weights are analyzed against the cylinder number to determine the AE weight's behaviour from different viewpoints and presented in Figs.58-59. Figure  cylinder number. Therefore, it is expected that with the rise in the cylinder number, the AE weight would be increased too. This is in line with the fourth assumption.
Figures are additionally investigated for other forms of relationships, and the results are presented in Table 17.

Power-RPM ratio
Auxiliary engine weights are analyzed against the power-RPM ratio to determine the AE weight's behaviour from different viewpoints and presented in Figs.60-61. Figure 60 demonstrates the trend of the AE weight (observed values) against the power-RPM ratio. Figure 61 illustrates the behaviour of average AE weight (average by power-RPM ratio group) against average power-RPM ratio. It also shows the sign of linearity among variables. This shows that with the rise in power-RPM ratio, the AE weight increases linearly. Therefore, it is expected that with the increase of power-RPM ratio, the AE weight would be increased too.
Figures are additionally investigated for other forms of relationships, and the results are presented in Table 18. Table 18 shows the correlation coefficients under different equation forms, and it suggests that the power form yields the highest correlation coefficients for Fig.60 and 61, but yields from polynomial (second-order) type are not very far from the highest values.

Power-RPM ratio per cylinder
Auxiliary engine weights are analyzed against the power-RPM ratio per cylinder to determine the AE weight's behaviour from different viewpoints and presented in Figs.62-63. Figure 62 demonstrates the AE weight (observed values) trend against the power-RPM ratio per cylinder. Figure 63 illustrates the behaviour of average AE weight (average by power-RPM ratio per cylinder group) against average power-RPM ratio per cylinder. It also shows the sign of linearity among variables. This shows that with the rise in power-RPM ratio per cylinder, the AE weight increases linearly. Therefore, it is expected that with the increase of power-RPM ratio per cylinder, the AE weight would be increased too.
Figures are additionally investigated for other forms of relationships, and the results are presented in Table 19.

Multiple linear regression analysis
Multiple linear regression analysis is a mathematical method used to determine the mathematical relationship involving more than one independent variable, unlike a single independent variable in simple linear regression analysis. This method uses the past data of both dependent and independent variables to establish a relationship to predict the dependent variable against a set of independent variables. A general but comprehensive description of the multiple linear regression analysis is available in Saha, M. (2015, 2016).
For this research, the following multiple linear regression model is chosen to represent the relationship expressed in equations 6 and 7. WME = b0 + b1*P + b2*N + b3*RPM (8) Where ME is the main engine weight (tonne = kg x 10 3 ) AE is the auxiliary engine weight (tonne = kg x 10 3 ) P is the power output (kW) of the main engine (ME) at 100% MCR and electrical power output (kW) of the auxiliary engine (AE) N is the cylinder number of engine RPM is the speed of the engine (revolution per minute) By the least-squares method (Walpole, R.E. and Myers, R.H., 1978) the required simultaneous equations with unknown regression coefficients are obtained Saha, M. 2015, 2016).
Using the observed data of WME, WAE, P, RPM, N, the values of the required statistical notations in simultaneous equations are calculated and inserted into the simultaneous equations. The solution of These simultaneous equations' solution yields the regression coefficients' estimate, and the required regression equation is formed. Subsequently, the statistical testing parameters such as i) standard deviation, ii) coefficient of multiple determination and iii) F statistic and its critical value at 5% significance level, are calculated to demonstrate the model's adequacy.
The same method was applied for the analysis of main engine weight under various speed groups (low-, medium-and high-speed engines)

General
There are some significant findings of various V-type and I-type marine propulsion engines' parameters, and auxiliary engines produced globally based on collected data presented in Tables 21 and 22, respectively. Parameters are cylinder number, RPM, power output, and weight. From Table   21, one can understand that the range of mentioned parameters are significantly different, and it is unfair to compare those by specified parameters. Similarly, from Table 22, one can understand that the range of mentioned parameters are significantly different except RPM, and it is unfair to compare those by specified parameters.
Mathematically, this means that providing the example for the combined group, 93% of the variation in the dependent variable (ME weight) is contributed due to the difference in the independent variables and the remaining (7%) is called the error of estimation. Statistically, it is called an error of the sum of squares (SSE) or unexplained variation. This variation behaves randomly or unpredictably (Murray, R.S. 1992). It also reflects the difference in the regression line. However, this estimation error is due to the absence of one or more critical independent variables responsible for the change in the dependent variable. The coefficient of multiple determination and the error of estimation are inversely related, meaning that for the higher value of the coefficient of multiple determination, the error of estimation would be low and vice versa. The lower value of the coefficient may also occur if the collected data suffers inconsistency, meaning that some of the independent variables have low values (Dev, A.K. and Saha, M. 2015).
Validation of a postulated mathematical model is an alternative way to demonstrate the model's adequacy of fitness to the system in question. To carry out the mathematical model's validation (Equations 10, 11, 12 and 13), each model is applied to estimate WME for each main engine and compared with actual data of main engine weight regarding deviation (% error). The summary of the result is presented in Table 24. The table shows the outline of the variation of model values from the actual values. The table shows that the range of error is very high for all models, more on the negative side, except for the high-speed group. This means that models yield a wide range of error, which is not acceptable. This also suggests that the distribution of error has a backward skew. Due to this wide range of error, the proposed multiple linear regression model does not appropriately explain the data.
The possible reasons are described as follows.
The Moreover, detailed investigations reveal that excessive negative deviations occur in the case of highspeed main engines. Mathematically, this is very much expected, due to the negative contribution of speed in final regression equations of all groups (equations 10-13). It is clear from Table 24 that with high R 2 value, the model could yield an extensive range of error. With this outcome, it can be concluded that the multiple linear regression model, combined or separate speed group, does not provide sufficient adequacy to fit the data. Furthermore, based on the mean value of power output, cylinder number and RPM of sample engines, low-speed, medium-speed, and high-speed engines, their contributions to the mean ME weight are calculated and presented in Table 25. If any combinations of independent variables deviate from this order, the model will yield the different model value depending on the combination's equivalent mean value. Referring to the statement made in point (a) above, various equations, such as power form, polynomial form and transformed form of existing independent variables, such as power-RPM ratio and power-RPM ratio per cylinder are tried, including validation. The results are presented in Table   26. One can identify that the power form and transformed independent variables (power-RPM ratio and power-RPM ratio per cylinder) yield the comparatively better fit to data regarding deviation from an actual value.   Table 27 shows a summary of the validation results of different relationships under a different form of equations and speed groups. According to the table and based on the range of error, W = a * T n (power form) provides the best fit for all engines, low and medium-speed engines, and W = a * T 2 + b * T + c (polynomial-second-order) provides an excellent fit to high-speed engines.  shows that the independent variables' values lower than the sample data's equivalent mean value increased the estimated engine weight giving a higher positive deviation. Inversely, the independent variables' values more elevated than the equivalent mean value of the sample data resulted in a decrease in the estimated engine weight giving a higher negative deviation. This phenomenon is due to RPM's negative contribution in final regression equations, and it is very much expected. However, someone may alternatively choose to use the individual relationship of the independent variables with the dependent variable.  (Table 1), the model will provide a reasonable estimate of engine weight. The assessment could be high and low (+3% to -4%) accordingly. Alternatively, one may choose to use the individual relationship under different equation form and other speed groups as appropriate.

Power generation machinery
Weights    (14), which passes the F statistic test, the multiple determination coefficient value is 0.88, which can be considered high. Mathematically, 88% of the dependent variable (AE weight) variation is contributed due to the difference in the independent variables, and the remaining (12%) is called the error of estimation. Statistically, it is called an error of the sum of squares (SSE) or unexplained variation. This variation behaves randomly or unpredictably (Murray, R.S. 1992). It also reflects the interpretation of the regression line. However, this estimation error is due to the absence of one or more influential independent variables responsible for the dependent variable's change. The coefficient of multiple determination and the error of estimation are inversely related, meaning that for the higher value of the coefficient of considerable determination, the assessment error would be low and vice versa. The lower value of the coefficient may also occur if the collected data suffers inconsistency, meaning that some of the independent variables have low values (Dev, A.K. and Saha, M. 2015  This suggests that the distribution of error has a forward skew. Due to this wide range of error, the proposed multiple linear regression model does not explain the data. The possible reasons are precisely similar to those, as explained in Sub-Section 6.2, for ME weight. Moreover, detailed investigation reveals that the excessive positive deviations occur in low power output with high-speed auxiliary engines (13 kW and 1800 RPM). Mathematically, it is expected due to the high positive intercept (b0 = 57) in the final regression equation (14). This situation suggests that the mentioned model (equation 14) cannot explain auxiliary engine weight data.
Furthermore, based on the mean value of power output, cylinder number, and RPM of sample auxiliary engines, their contributions to the mean AE weight are calculated and presented in Table 30.
If any combinations of independent variables deviate from this order, the model will yield the different model value depending on the combination's equivalent mean value. Referring to the statement made in point (a), Sub Section 6.2, various models, such as power form, polynomial form and transformed form of existing independent variables, such as power-RPM ratio and power-RPM ratio per cylinder are tried, including validation. The results are presented in Table   31. One can identify that the power form and transformed independent variables (power-RPM ratio and power-RPM ratio per cylinder) yield the comparatively better fit to data regarding deviation from an actual value. Linear: y = m*x + c, Power: y = m*x n , Polynomial: y = a*x 2 + b*x + c As explained in the ME weight case, when independent variables are closer to the sample's mean value, the model would yield a reasonable estimate with negligible error. This is equally applicable for AE weight and is demonstrated in Fig.65 (μ refers to the equivalent mean value of various independent variables of sample data). Finally, the proposed mathematical model may estimate auxiliary engine weight for initial stability calculation during the preliminary and conceptual design stage. Using the above model as a guide, designers may estimate the expected engine weight against expected auxiliary engine parameters. If the desired independent variables are close to the mean value (Table 2), then the model would provide a reasonable estimate of engine weight. However, for low and high values, the calculations could be high and low, respectively.  Table 10 suggests that the weight-power relationship (Fig.15), weight-power-RPM ratio relationship ( Fig.21) and weight-power-RPM ratio per cylinder relationship (Fig.23)  During the preliminary and conceptual design stage, the designers have only two options to estimate the engine weight, i) based on engine power (kW), ii) based on engine power (kW) and engine speed (RPM). The cylinder number is not useful at that stage. As such, weight-power (Fig.15) and weightpower-RPM ratio (Fig.21) relationships are the most appropriate to estimate the engine weight. Accordingly, Fig.66 is developed using the relationship of weight-power output of main engines under the power equation (Fig.15). Figure 66 shows the full range of estimated weight against the full range of estimated power output calculated during the conceptual and preliminary design stage. It shows the relationships between the estimated ME weight and estimated power output of propulsion engine under power equation. Figure 67 is developed using the relationship of ME weight and a power-RPM ratio under power equations (Fig.21) for various constant RPMs. It is clear from the Fig.99 that the ME weight decreases with RPM's increase for particular power output. This is in line with the initial assumption.      Table 20 suggests that the weight-power relationship (Fig.55), weight-power-RPM ratio relationship ( Fig.61) and weight-power-RPM ratio per cylinder relationship (Fig.63) yield the highest correlation coefficients under power equation form. It is evident from the validation results (Table 29) (Table 31) dictate that the power equation yields the lower range of error (%) and a smaller standard deviation (%). Table 33 displays a summary of the above findings. It is very clear from the Table that the above mentioned three relationships under power equation form yield the optimum result in terms of mentioned statistical parameters. During the preliminary and conceptual design stage, the designers have only two options to estimate the engine weight, i) based on generator power (kW), ii) based on generator power (kW) and engine speed (RPM). The cylinder number is not useful at that stage. As such, weight-power (Fig.55) and weightpower-RPM ratio (Fig.61) relationships are the most appropriate to estimate the engine weight. Accordingly, Fig.74 is developed using the relationship of weight-power output of auxiliary engines under power equations (Fig.55). It shows the relationships between the auxiliary engine's estimated weight and the generator's calculated power output under the power equation. This is in line with the initial assumption.

Marine propulsion machinery weight
For estimating the propulsion engine weight, the following options may be followed.

Option -I
Use the power output at 100% MCR (calculated value during the conceptual design stage) to estimate the engine weight ( Fig.66) irrespective of speed

Option -II
Use the power output at 100% MCR (calculated value during the conceptual design stage) and designated engine RPM (designers may have some reservations about RPM) to estimate the main engine weight (Fig.67).

Option III
Use the power output at 100% MCR (calculated value during the conceptual design stage) and designated engine RPM (designers may have some reservations about RPM) to estimate the main engine weight under speed groups as appropriate (Fig.68-73).

Power generation machinery weight
For estimating the diesel generator weight, the following options may be followed.

Option -I
Use the generator's power output at 100% MCR (calculated value during the conceptual design stage) to estimate the auxiliary engine weight (Fig.74) irrespective of speed.

Option -II
Use the power output of the generator at 100% MCR (calculated value during the conceptual design stage) and designated engine RPM (designers may have some reservations about RPM) to estimate the auxiliary engine weight (Fig.75).

Option III
Use the regression equation 14

Conclusion
This article investigates the trends of the main propulsion engines' weight and auxiliary engines regarding power output, cylinder number, RPM, power-RPM ratio, and power-RPM ratio per cylinder.
Based on the analyses' results, the engine's weight (main engine and auxiliary engine) significantly depends positively on power output, cylinder number, power-RPM ratio, and power-RPM ratio per cylinder and negatively on RPM but at different degrees of responses. All independent variables are linearly (linear equation form), exponentially (power equation form) and polynomial form (secondorder) associated with the dependent variable. Hence, it can be concluded with sufficient confidence level, that power output, cylinder number, RPM, power-RPM ratio, and power-RPM ratio per cylinder jointly or independently have a significant impact on the weight but at different degrees of response.
It is important to note that only the main engine and generator's required power output is available during the conceptual and preliminary design stage. In some cases, the owner may choose the RPM range of the main engines. Figures 66-73 and 74-75 may help estimate the main engine and auxiliary engine weights.
Despite the limitations of sample size, this research's various outcomes may be useful to ship designers as a guiding tool. Using these findings, ship designers may estimate the weights of a propulsion engine and an auxiliary engine for initial stability calculations during the conceptual and preliminary design of a ship. These works can be fine-tuned further with a larger sample size. For future study, artificial intelligence, neural networking may be considered to develop and validate the model.