1. Introduction

There is a common problem in the readiness of the Main Defense Equipment of Weapon System (Alutsista) of Navy Indonesia Armed Forces which is often faced with one of them, it is Corrosion. Corrosion on KRI occurs in underwater ship buildings which are exposed to sea water conditions. Corrosion at the bottom of a ship's water line can result in personnel and material safety risks. There are 2 (two) ways to protect against corrosion, they are passive protection (by painting) and active protection (by cathodic protection method). In the KRI with KCR-40 type, the design of the bottom line of the ship's waterline protection has been carried out with ICCP, but the value of its failure risk and reliability is unknown, both functional and designs, so that the design of the tool cannot be used maximally.

This research aimed to determine the factors of failure and reliability value of the design-based ICCP (Reliability by Design) with the FTA and FMEA approach, the FTA aimed to identify the risks that contribute to the failure. This method was carried out with a top down approach, which began with the assumption of failure or loss from a top event then detailed the causes of a top event to a root cause, whereas FMEA was a tool used to analyze the reliability of a system and the causes of its failure, by providing basic information about the system reliability prediction, design, and process.

This research was expected to be used as a reference in planning the development of a forced current method protection system (ICCP). This research was limited to several aspects, it was testing the function of the ICCP in a static state, the analysis used was limited to failure factors and nonparametric reliability calculations. There were several previous researches as supporting research, namely FTA and FMEA models are used as a risk analysis method in safety system [1]. The paper presents about FMEA and Fishbone diagrams for possible failures that occur in the engine [2]. FTA and FMEA as a model for evaluating failures or important parts of the system [3]. FTA and FMEA as a model for analyzing the reliability of BPRS field connection configurations [4]. FTA and FMEA as a model to identify the potential impact of failures in the Requirements Phase [5].

HAZOP and FTA to provide risk analysis in the chemical industry [6]. FTA and FMEA as a model to improve quality in the cleaning process [7]. FTA and FMEA as a method for identifying critical function level failures in metal printing additive manufacturing systems [8]. FTA and FMEA as a method to improve the quality of large industrial products at the end consumer level [9]. FTA and FMEA as a method to identify modes, components, and sub-systems of critical failures in CNC systems [10]. FTA and FMEA as a method for analyzing failure modes and effects on power plant boilers.

The systematically of the paper is organized as follows. Section 2 presents the theory of Fault Tree Analysis (FTA), Failure Mode Effect Analysis (FMEA), Corrosion, Reliability, Flowchart of research. Section 3 and Section 4 explain the result and discussion of research. Section 5 describes conclusion of the research.

2. Material/Methods

2.1. Corrosion

Corrosion can generally be defined as a deterioration in the quality or destruction of a metal by reacting with its environment [11]. Reactions in this corrosion produce metal oxides, metal sulfides or other reaction products. Corrosion is divided into two according to the type of reaction, they are chemical corrosion or dry corrosion and electrochemical corrosion or wet corrosion [12]. Dry or chemical corrosion is a process of corrosion that occurs through pure chemical reactions without electrolytes, usually occurring at high temperatures without the presence of water vapor. Electrochemical corrosion or wet corrosion occurs when the reaction takes place in an electrolyte and there is a transfer of electrons between the materials concerned, this reaction is what happens a lot in the corrosion process [13]. Corrosion process also occurs due to the tendency of a metal to turn into a more stable state through an oxidation reaction, where the oxidation tendency of a metal varies depending on the reduction potential. Basically the process of corrosion is the dissolution of metals by the reaction of metal surfaces with chemicals contained in the metal environment. If the metal is exposed to seawater or freshwater medium which has a pH close to neutral then the metal will spontaneously oxidize according to the reaction [11]: Cathodic reactions that occur in this condition are [14]: Because chloride, nitrate and other anion ions are not involved in this reaction, the whole reaction can be written as follows: The resulting \(Fe(OH)_{2}\) compound will settle to the metal surface to form a layer. The formed layer is porous so that oxygen can diffuse into the layer, which in turn oxidation process will take place slowly. \(Fe(OH)_{2}\) is a compound that is less stable in the presence of oxygen, the compound \(Fe(OH)_{2}\) will further oxidize to form a more stable iron oxidation or also called iron rust [11]. There are four main elements that must be fulfilled so that corrosion can occur, if one of them disappears, then corrosion cannot occur. The four elements, among others [12]:

1. Anode;
2. Cathode;
3. Electrolytes;
4. Metal Joints.

2.2. Impressed current cathodic protection (ICCP) method

In planning the cathodic protection system with the ICCP method on ships, there are some things that are absolutely considered so that planning can be carried out on target, it is minimizing the occurrence of corrosion on the surface of the protected ship, including [15]:

1. The condition of the protected environmental structure;
2. Operational environmental conditions;
3. Equipment used.

The purpose of this ICCP is to minimize the corrosion or corrosion that occurs in metals immersed in water. Corrosion protection in open hulls and hidden areas such as sea-chests are generally carried out at a certain level except when making special installation plans [16]. Protection of the hull by the ICCP method, which is a method of protection by providing electrons to the hull with the help of an external source of electric current. The electrons given to the material come from permanent anodes made of metal [17].

The basic principle of the ICCP system is to provide more negative potential, so that the metal will move into the immune zone. The difference between the corrosion zone and the immune zone is based solely on definition, but in reality even though the metal is in the immune zone, corrosion continues at a slower rate [18]. The more negative potential given will slow down the anodic reaction, conversely the cathodic reaction will be faster, as a result the metal will become more cathodic [19]. Hexagonal tiling under small cell complex is given in Figure 1.

2.3. Reliability and probability of underwater ship building

Mode of failure in underwater ship hull, among others, is the loss of service capability (loss of serviceability), loss of human function either partially or as a whole. The consequences of a failure vary greatly, including the decline in service life, increased maintenance costs, user costs, the impact on the environment and this will all be seen clearly that some consequences cannot be compared and cannot be evaluated in monetary terms [20]. In underwater hull, a vital failure criterion is the loss of structural strength which influences the reduction in underwater hull plate thickness [21]. The thickness reduction of the underwater ship hull one of which comes from corrosion. One of the influencing factors is the corrosion attack on the outside of the hull plate (external corrosion) which is in direct contact with the sea water environment.

2.4. Fault tree analysis (FTA) method

Fault Tree Analysis is an analytical tool that graphically translates combinations of errors that cause system failure. This technique is useful to describe and assess events in the system [22]. This Fault Tree Analysis method is effective in finding the core problem because it ensures that an undesirable event or loss caused does not originate at a single point of failure. Fault Tree Analysis identifies the relationship between causal factors and is displayed in the form of an error tree that involves a simple logic gate. There are 5 stages to carry out analysis with Fault Tree Analysis (FTA), which are as follows [23]:

1. Define the problem and boundary conditions of a system under review;
2. A depiction of the Fault Tree graphic model;
3. Finding the minimum cut set from the Fault Tree analysis;
4. Perform a qualitative analysis of the Fault Tree;
5. Perform a quantitative analysis of the Fault Tree.

Logic gate describes the conditions that trigger failure, both single and a set of various conditions. The construction of fault tree analysis includes logic gates, AND gates and OR gates. Each failure that occurs can be described in a form of the failure analysis tree by transferring or moving failure components into symbolic (Logic Transfer Components) and Fault Tree Analysis forms [24]. The symbols in the Fault Tree Analysis used in describing an event are presented in Table 1.

2.5. Failure mode effect analysis (FMEA) method

Failure Mode and Effect Analysis (FMEA) is a systematic approach that applies a labeling method to help the thought process used by engineers to identify potential failure modes and their effects [25]. FMEA is a technique of evaluating the reliability of a system to determine the effects of the failure of the system. Failures are classified according to the impact they have on the success of a system's mission. The purpose of FMEA is to take action to minimize failure, starting with the highest consequences. There are five types of FMEA that can be applied in a manufacturing industry, they are:

1. Systems, focusing on functions [26]. System globally;
2. Design, focusing on product design;
3. Process, focusing on the production and assembly processes;
4. Service, focusing on service functions;
5. Software, focusing on software functions.

In the FMEA research conducted to see the risks that may occur in the maintenance operations and operational activities of the company. There are a number of things that help determine the interference, including [25]:

1. Frequency (occurrence).
   In determining this occurrence can be determined how much disruption can cause a failure in maintenance operations and plant operations.
2. Severity.
   In determining the level of damage (severity), it can be determined how serious the damage is resulting from the failure of the process in terms of maintenance operations and plant operations.
3. Detection rate.
   In determining the level of detection it can be determined how the failure can be known before it occurs. The detection rate can also be influenced by the number of controls that govern the process netting. The more controls and procedures that govern the netting system for handling maintenance operations and plant operations, it is expected that the detection rate of failure can be even higher.
4. Risk Priority Number (RPN).
   This value is a product of the results of the multiplication of severity, incidence, and detection rates. RPN determines the priority of failure. RPN has no value or meaning. This value is used to rank potential process failures. The RPN value can be shown by the following equation:

A protected structure

In designing this forced current cathodic protection system, the structure to be protected is the hull below the waterline KRI Terapang - 658. Following are the data on the KRI Terapang - 658 hull:

1. Materials: High Tension steel AH-36
2. Length over all: 46.23 meter
3. Length of water line: 41.4 meter
4. Length between Perpendicular (Lbp): 40.8
5. Length Perpendicular (Lpp): 40.8
6. Length (B): 7.9
7. Draft: 1.9
8. High :4.25
9. Coefisien blok (Cb): 0.39

The Indonesian warship KCR 40 type is given in Figure 2.

2.7. Goal

The steps of the FMEA method in this research are:

1. Determining the potential failure modes, causes of failure and the effects of failure, obtained from the results of discussions and interviews with the designer.
2. Furthermore, the weighting of Severity, Occurrence and Detection is weighted for each failure mode, weighting is obtained from the results of discussions with the designer.
3. After obtaining the results of Severity, Occurrence and Detection for each failure mode, the Risk Priority Number (RPN) can be calculated, so that the greatest risk of process failure is obtained. Based on the results of this high failure mode RPN, the FTA method is then used to see the root causes of the high failure mode. The steps of the FTA method in this research are:
4. Building or creating a fault tree with the highest failure mode is used as the top event.
5. Determining the minimum cut set of each basic event failure mode that is highest.
6. After the results of the minimum cut set are known, then a qualitative FTA analysis and Quantitative FTA analysis are performed to determine the probability and reliability values of each basic event.

The last stage of the output stage is evaluating the calculated data by providing conclusions and suggestions as well as solutions to proposed improvements, which are expected to reduce the level of defects so that the quality of future products is more optimal.

2.8. Flowchart

The flowchart of this research is given in Figure 3.

3. Result

3.1. Severity determination

The severity of the effects caused by failure in each ICCP tool trial process (Table 2).

Table 2. Result of Severity Score for Failure in ICCP.

Parts	Failure Mode	Potential Failure  Effect	S
Microcontroller	Error	ICCP tool does not work	8
Steel potential indicator and rectifier indicator	Do not designate  numbers correctly	Data displayed is inaccurate	8
Fan	Motor on fire	All Components in the box are at risk of damage	7
Transformer	Burnt out	ICCP tool does not work	9
Sensor Ag-Ag/Cl	Broken	Cannot provide data input of Corrosion protection limit	2

The highest component severity value is transformers with a value of 9, a value of 9 indicates the effect of "danger with signs" and the event criteria "Very High Failure that can frustrate the system and endanger the machine operator, in the presence of signs of previous damage". The failure mode on the transformer is "burning" with the potential failure effect "the ICCP device is not working". Followed by other component severity values including microcontroller and indicator with each value of 8, fan with a value of 7, and the lowest value with a value of 2 it is Ag - Ag / Cl sensor.

3.2. Determination of Rating Detection

The level of occurrence of how often the effects of these failures appear in each ICCP tool trial process. The result of occurrence scores for failure in ICCP is given in Table 3.

Table 3. Result of Occurrence scores for Failure in ICCP.

Parts	Failure Mode	Potential Failure Effect	O
Microcontroller	Error	Main power source is in trouble (voltage fluctuates)	1
Steel potential indicator  and rectifier indicator	Do not designate numbers correctly		5
Fan	Motor on fire		4
Transformer	Burnt out		2
Sensor Ag-Ag/Cl	Broken	Impact, natural factors	1

The highest component occurrence value is steel potential indicator and rectifier indicator with value 5 with failure mode "not pointing to the correct number" and the cause of potential failure "temperature in the heat box and voltage is up and down". A value of 5 indicates a probability of failure at the "Medium" level with a probability of failure of 4 per 600 hours, and a percentage of 0.83\%. Followed by the occurrence of other components including fans with a value of 4, Transformer with a value of 2, and the lowest value with a value of 1 it is the microcontroller and Ag -Ag / Cl sensor. The result of detection scores for failure in ICCP is discussed in Table 4.

Table 4. Result of Detection scores for Failure in ICCP.

Parts	Failure Mode	Potential Failure Effect	D
Microcontroller	Error	Main power source  is in trouble (voltage fluctuates)	1
Steel potential indicator  and rectifier indicator	Do not designate numbers correctly		6
Fan	Motor on fire		4
Transformer	Burnt out		2
Sensor Ag-Ag/Cl	Broken	Impact, natural factors	1

The highest component detection value is steel potential indicator and rectifier indicator with value of 6 with failure mode "not pointing to the correct number" and the cause of potential failure "temperature in the box is hot and voltage is up and down". A value of 6 indicates a detection rate of "Low" with the statement "Low the possibility of design controls will find a potential cause of mechanical damage or a cause of subsequent failure". Other component detection values include fans with a value of 4, Transformer with a value of 2, and the lowest value with a value of 1 it is the microcontroller and the Ag-Ag/Cl sensor.

3.3. Determination of risk priority number (RPN) value

After determining the rating of Severity, Occurance, and Detection, the next step is to calculate the Risk Priority Number (RPN) value. Failure modes which later have the highest RPN values will be viewed using the Fault Tree Analysis method so that the root of the problem is known and corrective action can be taken. The result of risk priority number (RPN) scores for failure in ICCP is given in Table 5 and the risk map of risk priority number (RPN) for ICCP is given in Figure 4.

Table 5. Result of Risk Priority Number (RPN) scores for Failure in ICCP.

Part	Failure Mode	(S)	(O)	(D)	RPN =   S*O*D
Microcontroller	Error	8	1	1	8
Steel potential indicator and rectifier indicator	Do not designate  numbers correctly	8	5	6	240
Fan	Motor on fire	7	4	4	112
Transformer	Burnt out	9	2	2	36
Sensor Ag-Ag/Cl	Broken	2	1	1	2

The highest RPN value occurs in the component "Steel potential indicator and rectifier indicator" with a value of 240, followed by the RPN value of the "fan" component with a value of 112, the RPN value of the "transformer" component with a value of 36, the RPN value of the "microcontroller" component with a value of 8 and the lowest RPN value is on the Ag-Ag/Cl Sensor component with a value of 2. The higher the RPN value of a component, the greater the degree of criticality failure on the ICCP tool.

Some treatments to reduce the risk of these components include:

1. Steel potential indicator and rectifier indicator:
   In this component the potential failure mode that occurs is "not pointing to the correct number" and the cause of the potential failure is "Temperature in the box is hot and the Voltage is Up and Down" then the actions that must be taken include:
1. Redesigning the laying of some components of the tool compilers to pay attention to the circulation of the freezing in the box, so that the tool works more optimally.
2. Controlling is carried out periodically while the tool is operating.
3. Ensuring that the electrical power used is stable so that there are no problems with the ICCP device during the ICCP device.
In this component the potential failure mode that occurs is "the motor is on fire" and the cause of the potential failure is "poor quality of wire and motor windings" then the actions that must be taken include:
1. Making sure to use a fan with a good quality motor winding.
2. Having control Periodically and scheduled while the tool is operating.

For parts that are at medium value, the risk value must be minimized, components that are at medium value, they are "Transformer" and "Microcontroller", steps are carried out for components that are in the yellow zone or are of moderate value among other things:

1. Transformer.
   In this component, the potential failure mode that occurs is "burning" and the cause of the potential failure is "The quality of the wire coil and transformer core is poor and the temperature in the box is hot", so actions that should be taken to reduce the risk value include:
   1. Using a transformer that is coiled and the core of the transformer is in good quality in accordance with the requirements of the tool specifications.
   2. Making sure the cooling fans are always in good condition if you need to install a backup fan with a parallel system automatically so that the temperature in the box is maintained.
2. Microcontroller.
   In this component the potential failure mode that occurs is "error" and the cause of the potential failure is "Main power source is in trouble (voltage is up and down)", then the actions that should be taken to minimize the value of risk include: Implementing controls on the main electric power source scheduled to ensure a stable supply of electricity.

Analysis of FTA

After the results of calculations with the FMEA method are known to be the highest RPN value of each failure mode, then the next step will look for the root causes of the failure mode in order to find the next stage of improvement recommendations. In determining the root of the problem, the Fault Tree Analysis (FTA) method is used.

From the previous step in the FMEA method obtained two failure modes that have high RPN values, it is in the "Steel potential indicator and rectifier indicator" section and the "Fan" component with failure mode in the steel potential indicator and rectifier indicator "cannot show the correct number" while on the fan with the fan motor failure mode on fire. Then the next step is to define the top event component that has a high RPN value by creating an error tree/FTA. The influence of the auxiliary parameter on the local velocity distribution with the property \(Η=0.2\) is given in Figure 5.

The minimum cut set from the structural image above is P1, P2, P3, P4.

1. Qualitative Analysis of FTA.
   Qualitative analysis is to get a combination of failures that cause the top event in a system or the minimum cut set itself. The results of a qualitative analysis of the failure modes above occur if the following events occur. These events are:
   1. P1: There is an overload on the electricity provider.
   2. P2: Overloaded cooling fan.
   3. P3: Bad quality of fan motor.
   4. P4: The circulation design in the box is not good.
2. Quantitative Analysis of FTA.
   From the results of this analysis, the probability of occurrence of each basic event can be searched with data of some damage as much as 7 events during the ICCP tool experiment, so that the probability of each event is (Table 6):

   Table 6. Result of Risk Priority Number (RPN) scores for Failure in ICCP.

   No	Classification of Basic event	Effect	Probability (pf)
   1	Overload occurred at the electricityprovider	Steel potential indicator and rectifier indicator are broken	0.28
   		Weak fan motor
   2	Steel potential indicator and rectifier indicator are broken	Steel potential indicator and rectifier indicator are broken	0.28
   		The temperature in the box is hot so that other components are disrupted
   3	Bad quality of fan motor	Fan work is not optimal	0.42
   		Hot box temperature
   		Fan is burning
   4.	Circulation design in the box is not good	Steel potential indicator and rectifier indicator are broken	0.14

The probability of a top event with the Boolean algebraic approach "direct manual approach" is as follows: \begin{eqnarray*} P(T) &=& P (P_1\cup P_2 \cup P_3 \cup P_4)\\ &=& P (P_1) + P (P_2) + P (P_3) + P (P_4) - P (P_1 \cup P_2) - P (P_1 \cup P_3) - P (P_1 \cup P_4) - P (P_2 \cup P_3)\\ &&- P (P_2 \cup P_4) - P (P_3 \cup P_4)+ P (P_1 \cup P_2 \cup P_3 \cup P_4)\\ P(T) &=& 0,28 + 0,28 + 0,42 + 0,14 - (0,28 . 0,28) - (0,28 . 0,42) - (0,28 . 0,14) - (0,28 . 0,42)\\ && - (0,28 . 0,14) - (0,42 . 0,14) + (0,28 . 0,28 . 0,42 . 0,14)\\ &=& 1,12 - 0,0784 - 0,1176 - 0,0392 - 0,1176 - 0,0392 - 0,0588 + 0,00461\\ P(T) &=& 0,67 \end{eqnarray*} For the probability value of the occurrence of the top event or the possibility of failure in the steel potential indicator and the rectifier indicator "that is equal to 0.67 or 67\% during the trial period, while for the reliability value of the Top event is $$R=1-Pf\longrightarrow 1 - 0,67 = 0,33 \;\;\textrm{atau}\;\;33\%.$$

Discussion

From the ICCP tool test results showed that there were several problems at the initial stage of the trial, with a total of 7 problems, then using the FMEA method to identify the problem, from the initial identification conducted there were 5 components to be assessed and calculated the risk value. The 5 components included: Microcontroller, Voltage and Ampere Indicator, Fan, Transformer, and Ag-Ag/Cl Sensor.

The next FMEA step is determining the rating/weighting of Severity, occurrence, and Detection failure mode, determining the rating/weighting is through an interview with the tool designer. From the results of data processing the determination of SOD rating obtained RPN results with the highest value, it is the component of "steel potential indicator and rectifier indicator" and the "fan" component. Furthermore, based on the value (Occurrence) and severity of the effect (Severity) that has been weighted, it is necessary to know the risks that must be minimized by using the Risk Map matrix, from risk mapping components that have the highest value/are in the red zone carried out several mitigations. After mitigating the components in the red zone, data processing is continued using the FTA method.

In the FTA method the two components that have a high value are mapped to a failure tree to find out the basic event of the problem. Data processing by the FTA method is done qualitatively and quantitatively, if the data qualitatively produce a minimum cut set of the top events of the two components that have high RPN values. From the results of FMEA and FTA analyzes that have been obtained, the results of data processing according to the purpose of this research were known to be "factors causing failure and reliability of the design of ICCP tool designs" and "mitigation of the tool compiler components that have a high level of risk".

5. Conclusion

Based on the results of the research that has been carried out, it can be concluded that:

1. The main factors causing failure in the design of ICCP tools occur in the component of "Steel potential indicator and rectifier indicator" with a failure mode "not pointing to the correct number", this will result in corrosion control which is expected to be uncontrolled properly and correctly due to incorrect data input. After analyzing the FTA, the reliability value was 33\%.
2. Mitigation of tool components that have a high level of risk among other things in the "Indicator of steel potential and rectifier indicators": the first was to redesign the laying of some components of the tool compilers to pay attention to the circulating circulation in the box so that the tool works more optimally, the second was to carry out periodic control while the device was operating, and third was to ensure that the electrical power used was stable so there were no problems with the ICCP device while the ICCP device was operating.

Acknowledgements

This study was supported by Studies Program of Defense Technology in Indonesia Defense University. We thank you to the 2nd Fleet Command in Surabaya, East Java for this research data. We also thank you to Kukuh Susilo, M. Eng. for smart discussion in corrosion field.

Competing Interests

The authors declare no competing interest.

1. Introduction

The science of how fuel spray penetration affect physical and engineering systems has been an interesting subject of interest. These systems whose operations are influenced by the aforementioned penetration processes include combustion chambers, gas turbines, thrust engines, agricultural spray systems, vapor diffusion and distribution processes and general gas turbulence operations. Adequate monitoring and prediction of the flow dynamics associated with fuel spray penetration thus require great attention. As a result, different studies have been flagged up from experimental, analytical and numerical point of views. This include the tremendous work by Szchin et al. , [1]. In their work, they presented a penetration model for fuel spray and performed different parametric studies by basically focusing on the impact of density, pressure and semi spray cone angle on spray penetration. They went further to perform an experiment for the validation of some of their results. They concluded by establishing mathematical relations between the fuel spray term and some vital parameters. In a different study, the penetration of liquid in diesel spray focusing on two fluid flow model has been critically considered [2, 3].

The systems were studied under steady condition with the impact of the injection droplet size cell investigated. Loth et al. , [4] applied numerical methods for motion of dispersed particles, droplets and bubbles to properly investigate the dispersion and spray penetration phenomenon while Pozorski et al. , [5] further did a study by considering turbulent spray penetration fluid flow. It has also been established that structure and break up properties of sprays are attributes that should be considered when performing analysis on the dynamics of fuel sprays [6]. Due to the aforementioned effects, Ganji et al. , [7] did an extension on the modelling work of spray penetration by studying the impact of nonlinear turbulence model on the spray performance in a two-phase flow. Ebrahimian et al. , [8] presented an approximate analytical solution by employing Variational iteration method (VIM) to examine the initial stage of fuel spray penetration. Bararnia et al. , [9] presented a solution of the Falkner-Skan wedge flow by HPM Padé method. Mebine et al. , [10] generated a unified model for fuel spray penetration, Ghosh et al. , [11] did a study on induced air velocity within droplet driven sprays and Jalilpour et al. , [12] obtained an approximate analytical solution of analytical model for fuel spray penetration using Pade - homotopy perturbation method (HPM) by considering both initial and two phase fluid. They also verified the Pade HPM with an efficient numerical scheme. In [13, 14, 15, 16], authors applied the Adomian decomposition method, DJ method and Variation of parameter method (VPM) to study the thermal and environmental behaviour of oxidizing micro-dust that is susceptible to combustion. Furthermore, Yinusa and Sobamowo [17] presented the analysis of dynamic behaviour of a tensioned Carbon nanotube in thermal and pressurized environments. They employed integral transform to solve the governing model and obtain the dynamic behaviour and stability criteria of the carbon nanotube. Hikmah et al., et al. [18] did an elementary study on the impact of tea dust geometry and size on its explosion using some food particles. Furthermore, the classical DTM, Multi-step DTM (MsDTM) as well as other approximate and total analytical schemes have been widely employed to solve different governing equations as applied to different research areas [19, 20, 21, 22, 23, 24].

Motivated by the previous works, this work introduces a generalized analytical solution for generating and monitoring two phase fuel spray dynamics using differential transform method (DTM). The method is verified, compared and validated in order to ascertain the efficiency and appropriateness of the scheme.

2. Description of problem and governing the equation

Consider the spherical shaped fuel droplets with entrained air as shown in Figure 1. These two components form air-fuel mixtures in proportions usually well specified. The study of the influence of air jet on fuel spray generates strong nonlinear model that is difficult to solve generally using an analytical scheme for a specified air jet velocity. Also, an augmentation in the nozzle distance causes the velocity of the fuel droplets to be approximately equal to the velocity of the air entrained. This equality in velocity can be used to model the dynamical process as a two phase flow involving the spray droplets and the air jet. In order to derive the flow and penetration model for the process, the continuity and Navier stoke equations are employed as shown below [12]:

The continuity equation is: \begin{align*} (Continuity of air - fuel mixture) = (Continuity of entrained air) + (Continuity of fuel spray droplet) \end{align*} which when simplifies becomes The momentum equation is: By incorporating the geometrical assumptions as described by Jalilpour et al. , [12], the nonlinear governing equation becomes where \(k_1 = 4(1 - \alpha_d) \rho + 1,\) \( k_2 = \frac{4(1 - \alpha_d) \rho \ tan \Theta}{D_0} \) and \( k_3 = \frac{4(1 - \alpha_d) \rho \tan^2 \Theta}{D_0^2}. \) Then the generalized model becomes subject to the initial condition \( \zeta(0) = 0.\) Recall that the nonlinear governing equation as shown in Equation (4) may be expressed as Since it is part of the aim of the present study to establish a generalized solution to the model, then the index \(n\) should be well taken care of. This will be obtained following the procedures below. Let \begin{align*} f = \bigg( (4 (1 - \alpha_d) \rho + 1) + \bigg( \frac{4(1 - \alpha_d) \rho \ tan \Theta}{D_0} \bigg) \zeta + \bigg( \frac{4(1 - \alpha_d) \rho \tan^2 \Theta}{D_0^2} \bigg) \zeta^2 \bigg)^n \end{align*} Equation (6) becomes; We need to generate a differential equation for the function f in Equation (7). This is achieved by employing the chain rule as shown below: \begin{align*} \frac{df}{d\tau} = \frac{df}{d \tau} \frac{d \zeta}{d \tau} \end{align*} Performing the differentiation, we get
\( \frac{df}{d \tau} = n\bigg( (4 (1 - \alpha_d) \rho + 1) + \bigg( \frac{4(1 - \alpha_d) \rho \tan \Theta}{D_0} \bigg) \zeta + \bigg( \frac{4(1 - \alpha_d) \rho \tan^2 \Theta}{D_0^2} \bigg) \zeta^2 \bigg)^{n-1} \times \bigg(\bigg( \frac{4(1 - \alpha_d) \rho \tan \Theta}{D_0} \bigg) \\+ 2 \bigg( \frac{4(1 - \alpha_d) \rho \tan^2 \Theta}{D_0^2} \bigg) \zeta \bigg) \frac{d \zeta}{d \tau} \)
simplifying the resulting equation, we get
\( \frac{df}{d \tau} = n \frac{\bigg( (4 (1 - \alpha_d) \rho + 1) + \bigg( \frac{4(1 - \alpha_d) \rho \tan \Theta}{D_0} \bigg) \zeta + \bigg( \frac{4(1 - \alpha_d) \rho \tan^2 \Theta}{D_0^2} \bigg) \zeta^2 \bigg)^n}{\bigg( (4 (1 - \alpha_d) \rho + 1) + \bigg( \frac{4(1 - \alpha_d) \rho \tan \Theta}{D_0} \bigg) \zeta + \bigg( \frac{4(1 - \alpha_d) \rho \tan^2 \Theta}{D_0^2} \bigg) \zeta^2 \bigg)} \times \bigg(\bigg( \frac{4(1 - \alpha_d) \rho \tan \Theta}{D_0} \bigg) + 2 \bigg( \frac{4(1 - \alpha_d) \rho \tan^2 \Theta}{D_0^2} \bigg) \zeta \bigg) \frac{d \zeta}{d \tau} \)
The two coupled differential equations shown will then be solved simultaneously in order to obtain an analytical solution for \(\zeta(\tau)\) considering any value of the exponent \(n\).
\( \bigg( (4 (1 - \alpha_d) \rho + 1) + \bigg( \frac{4(1 - \alpha_d) \rho \tan \Theta}{D_0} \bigg) \zeta + \bigg( \frac{4(1 - \alpha_d) \rho \tan^2 \Theta}{D_0^2} \bigg) \zeta^2 \bigg) \frac{df}{d \tau} - n \times \bigg(\bigg( \frac{4(1 - \alpha_d) \rho \tan \Theta}{D_0} \bigg) \\+ 2 \bigg( \frac{4(1 - \alpha_d) \rho \tan^2 \Theta}{D_0^2} \bigg) \zeta \bigg) f \frac{d \zeta}{d \tau} = 0 \)

3. Methods of solution

As a result of the presence of the nonlinear terms in Equation (5), obtaining an exact solution becomes uneasy. The usual practice is to employ numerical schemes. However, recently, several semi- or approximate analytical methods have been developed to handle nonlinear equations exactly. In this study, the developed nonlinear model in Equation (5) is be solved analytically using Differential transform Method (DTM)

3.1. Basic principle: differential transform method (DTM)

The basic concept and the procedure of DTM for solving differential equations are outlined in previous works [22]. However, the DTM recursive relation for transforming models into another domain in an algebraic form is presented below:

Table 1. DTM recursive relations.

1.	\(Z(t)= U(t) \pm V(t), \ \ Z(k) = U(k) \pm V(k)  \)
2.	\(Z(t) = \propto U(t), \ \  Z(k) = \propto U(k)    \)
3.	\(Z(t) = \dfrac{dU(t)}{dt},  \ \  Z(k) = (k+ 1) U[k+ 1]   \)
4.	\(Z(t) = \dfrac{d^2U(t)}{dt^2},  \ \ Z(t) = (k+ 1) (k+ 2) U[k+ 2]\)
5.	\(Z(t) = \dfrac{d^m U(t)}{dt^m},  \ \ Z(k) = (k+ 1) (k+ 2) \dots (k+ m) U[k+ m],  \ \ or \ \dfrac{(k+m)!}{k!} U[k+m]\)
6.	\(Z(t) = U(t) * V(t),  \ \ Z(k) = \sum_{L= 0}^{K} V[l] U[K - l]\)
7.	\(Z(t) = t^m,  \ \ Z(k) =  \delta (k - m )  = \tau_{ij} =

The Table 1 above shows the recursive relations for different operators. These relations help map a given function into another domain in an algebraic form. The solution will then be obtain by inverting the obtain series solution.

4. Method of solution: differential transform method (DTM)

Recall the coupled ODE which is to be transformed using the Table 1 above; with initial conditions \( \zeta[0] = 0\), \( F[0] = a.\) Applying the principle of DTM transformation, the recursive relations for Equation (9) and Equation (10) are: \( k_1 (k+ 1) F_{k+ 1} + k_2 \sum_{ l = 0}^{k } (l + 1) F_{l + 1} \zeta_{k - l} + k_2 \sum_{m = 0}^{k} \bigg( \sum_{l = 0}^{m } (l + 1) F_{l + 1}\zeta_{m - l} \zeta_{k - m} \bigg) - n \bigg( k_2 \sum_{l = 0}^{k} ( l + 1) \zeta_{l+ 1} F_{k - l} \\ + 2k_3 \sum_{m = 0}^{k} \bigg( \sum_{l = 0}^{m} (l+ 1) \zeta_{l + 1} \zeta_{m- l} F_{k -m } \bigg) \bigg)= 0 \) After iterating with the highest counter, the term by term DTM solution becomes;
\( \zeta[0 ] = 0,\)
\( \zeta[1] = 2 \dfrac{v_i}{a + 1}, \)
\( \zeta[2] = v_i \frac{ \biggl[ -8n \rho \ tan (\Theta) v_i a \alpha_d + 4 D_0 a^2 \rho \alpha_d +8 n \rho \ tan (\Theta) v_i a - 4 D_0 a^2 \rho + 8 D_0 a \rho \alpha_d - D_0 a^2 - 8 D_0 a \rho + 4 D_0 \rho \alpha_d - 2 D_0 a - 4D_0 \rho - D_0 \biggl] }{(a+ 1)^3 D_0 (4 \rho \alpha_d -4 \rho - 1) }, \)
\( \zeta[3] = \frac{2}{3} \frac{v_i}{(a+ 1)^5 D_0^2 (4 \rho \alpha_d - 4 \rho - 1)^2} \biggl[ -48 D_0 \tan (\Theta) a^3 n \rho^2 v_i + 16 (\tan (\Theta))^2 a^2 n \rho v_i^2 \alpha_d + 16 D_0^2 \rho^2 \alpha_d^2 - 32 D_0^2 \rho^2 \alpha_d - 8 D_0^2 \rho \alpha_d + \\64 (\tan (\Theta))^2 a n \rho^2 v_i^2 \alpha_d - 12 D_0 \tan (\Theta) a^3 n \rho v_i + 8 D_0^2 a^4 \rho + 64 D_0^2 a^3 \rho^2 + 16 D_0^2 a^4 \rho^2 + \\16 (\tan (\Theta))^2 a n \rho v_i^2 \alpha_d - 96 D_0 \tan (\Theta) a^2 n \rho^2 v_i - 48 D_0 \tan (\Theta) a n \rho^2 v_i - 24 D_0 \tan (\Theta) a^2 n \rho v_i - \\32 (\tan (\Theta))^2 a n \rho^2 v_i^2 - 32 (\tan (\Theta))^2 a^2 n \rho^2 v_i^2 - 32 (\tan (\Theta))^2 a n^2 \rho^2 v_i^2 - 16 (\tan (\Theta))^2 a^2 n \rho v_i^2 - \\16 (\tan (\Theta))^2 a n \rho v_i^2 + 64 (\tan (\Theta))^2 a^2 n^2 \rho^2 v_i^2 - 128 D_0^2 a \rho^2 \alpha_d - 32 D_0^2 a \rho \alpha_d - 64 (\tan (\Theta))^2 a n^2 \rho^2 v_i^2 \alpha_d- \\32 D_0^2 a^3 \rho \alpha_d - 192 D_0^2 a^2 \rho^2 \alpha_d + 64 D_0^2 a \rho^2 \alpha_d^2 - 48 D_0^2 a^2 \rho \alpha_d + 16 D_0^2 a^4 \rho^2 \alpha_d^2 - 32 D_0^2 a^4 \rho^2 \alpha_d + \\64 D_0^2 a^3 \rho^2 \alpha_d^2 - 8 D_0^2 a^4 \rho \alpha_d - 128 D_0^2 a^3 \rho^2 \alpha_d + 96 D_0^2 a^2 \rho^2 \alpha_d^2 - 12 D_0 \tan (\Theta) a n \rho v_i + 96 D_0^2 a^2 \rho^2 \\32 D_0^2 a^3 \rho^2 + 32 D_0^2 a \rho + 64 D_0^2 a \rho^2 + 48 D_0^2 a^2 \rho - 32 (\tan (\Theta))^2 a n \rho^2 v_i^2 \alpha_d^2 + 64 (\tan (\Theta))^2 a^2 n \rho^2 v_i^2 \alpha_d + \\32 (\tan (\Theta))^2 a^2 n \rho^2 v_i^2 \alpha_d^2 - 32 (\tan (\Theta))^2 a n^2 \rho^2 v_i^2 \alpha_d^2 + 64 (\tan (\Theta))^2 a^2 n^2 \rho^2 v_i^2 \alpha_d^2 - 128 (\tan (\Theta))^2 a^2 n^2 \rho^2 v_i^2 \alpha_d + \\4 D_0^2 a + 8 D_0^2 \rho + D_0^2 a^4 + 4 D_0^2 a^3 + 6 D_0^2 a^2 + 16 D_0^2 \rho^2 + D_0^2 + D_0^2 + 12 D_0 \tan (\Theta) a n \rho v_i \alpha_d+ \\96 D_0 \tan (\Theta) a n \rho^2 v_i \alpha_d + 24 D_0 \tan (\Theta) a^2 n \rho v_i \alpha_d - 48 D_0 \tan (\Theta) a n \rho^2 v_i \alpha_d^2 + 192 D_0 \tan (\Theta) a^2 n \rho^2 v_i \alpha_d + \\12 D_0 \tan (\Theta) a^3 n \rho v_i \alpha_d - 96 D_0 \tan (\Theta) a^2 n \rho^2 v_i \alpha_d^2 + 96 D_0 \tan (\Theta) a^3 n \rho^2 v_i \alpha_d - 48 D_0 \tan (\Theta) a^3 n \rho^2 v_i \alpha_d^2 \biggr]. \) Applying the principle of DTM inversion to the obtained term by term solution, Making necessary substitutions, the desired approximate analytical solution for the spray penetration history becomes:
\( \zeta(\tau) = \Biggl[ 2 \frac{v_i \tau}{a + 1} + v_i \frac{\biggl[ -8n \rho \ \tan (\Theta) v_i a \alpha_d + 4 D_0 a^2 \rho \alpha_d +8 n \rho \ \tan (\Theta) v_i a - 4 D_0 a^2 \rho + 8 D_0 a \rho \alpha_d - D_0 a^2 - 8 D_0 a \rho + 4 D_0 \rho \alpha_d - 2 D_0 a - 4D_0 \rho - D_0 \biggl] }{(a+ 1)^3 D_0 (4 \rho \alpha_d -4 \rho - 1) }\\\tau^2 + \frac{2}{3} \frac{v_i \tau^3}{(a+ 1)^5 D_0^2 (4 \rho \alpha_d - 4 \rho - 1)^2} \bigg[-48 D_0 \tan (\Theta) a^3 n \rho^2 v_i + 16 (\tan (\Theta))^2 a^2 n \rho v_i^2 \alpha_d + 16 D_0^2 \rho^2 \alpha_d^2 - 32 D_0^2 \rho^2 \alpha_d - 8 D_0^2 \rho \alpha_d + \\64 (\tan (\Theta))^2 a n \rho^2 v_i^2 \alpha_d - 12 D_0 \tan (\Theta) a^3 n \rho v_i + 8 D_0^2 a^4 \rho + 64 D_0^2 a^3 \rho^2 + 16 D_0^2 a^4 \rho^2 + \\16 (\tan (\Theta))^2 a n \rho v_i^2 \alpha_d - 96 D_0 \tan (\Theta) a^2 n \rho^2 v_i - 48 D_0 \tan (\Theta) a n \rho^2 v_i - 24 D_0 \tan (\Theta) a^2 n \rho v_i - \\32 (\tan (\Theta))^2 a n \rho^2 v_i^2 - 32 (\tan (\Theta))^2 a^2 n \rho^2 v_i^2 - 32 (\tan (\Theta))^2 a n^2 \rho^2 v_i^2 - 16 (\tan (\Theta))^2 a^2 n \rho v_i^2 - \\16 (\tan (\Theta))^2 a n \rho v_i^2 + 64 (\tan (\Theta))^2 a^2 n^2 \rho^2 v_i^2 - 128 D_0^2 a \rho^2 \alpha_d - 32 D_0^2 a \rho \alpha_d - 64 (\tan (\Theta))^2 a n^2 \rho^2 v_i^2 \alpha_d - \\32 D_0^2 a^3 \rho \alpha_d - 192 D_0^2 a^2 \rho^2 \alpha_d + 64 D_0^2 a \rho^2 \alpha_d^2 - 48 D_0^2 a^2 \rho \alpha_d + 16 D_0^2 a^4 \rho^2 \alpha_d^2 - 32 D_0^2 a^4 \rho^2 \alpha_d + \\64 D_0^2 a^3 \rho^2 \alpha_d^2 - 8 D_0^2 a^4 \rho \alpha_d - 128 D_0^2 a^3 \rho^2 \alpha_d + 96 D_0^2 a^2 \rho^2 \alpha_d^2 - 12 D_0 \tan (\Theta) a n \rho v_i + 96 D_0^2 a^2 \rho^2 + \\32 D_0^2 a^3 \rho^2 + 32 D_0^2 a \rho + 64 D_0^2 a \rho^2 + 48 D_0^2 a^2 \rho - 32 (\tan (\Theta))^2 a n \rho^2 v_i^2 \alpha_d^2 + 64 (\tan (\Theta))^2 a^2 n \rho^2 v_i^2 \alpha_d + \\32 (\tan (\Theta))^2 a^2 n \rho^2 v_i^2 \alpha_d^2 - 32 (\tan (\Theta))^2 a n^2 \rho^2 v_i^2 \alpha_d^2 + 64 (\tan (\Theta))^2 a^2 n^2 \rho^2 v_i^2 \alpha_d^2 - \\128 (\tan (\Theta))^2 a^2 n^2 \rho^2 v_i^2 \alpha_d + 4 D_0^2 a + 8 D_0^2 \rho + D_0^2 a^4 + 4 D_0^2 a^3 + 6 D_0^2 a^2 + 16 D_0^2 \rho^2 + D_0^2 + D_0^2 + 12 D_0 \tan (\Theta) a n \rho v_i \alpha_d+ \\96 D_0 \tan (\Theta) a n \rho^2 v_i \alpha_d + 24 D_0 \tan (\Theta) a^2 n \rho v_i \alpha_d - 48 D_0 \tan (\Theta) a n \rho^2 v_i \alpha_d^2 + 192 D_0 \tan (\Theta) a^2 n \rho^2 v_i \alpha_d + \\12 D_0 \tan (\Theta) a^3 n \rho v_i \alpha_d - 96 D_0 \tan (\Theta) a^2 n \rho^2 v_i \alpha_d^2 + 96 D_0 \tan (\Theta) a^3 n \rho^2 v_i \alpha_d - 48 D_0 \tan (\Theta) a^3 n \rho^2 v_i \alpha_d^2\bigg] \Biggr]. \)

5. Results and Discussion

5.1. Verification

The verification of the analytical method employed in the present study is shown in Figure 2 with the numerical values displayed in Table \ref{table2}. The graph and table confirm the efficiency of the DTM used as excellent agreements are noticed.

5.2. Effect of Droplet initial velocity on spray penetration

Figure 3 depicts the influence of droplet initial velocity on spray penetration for the two phase fluid flow considered. An increase in the initial velocity increases the spray penetration as a result of the increase in the diffusion capability of the air-fuel mixture. This initial spray velocity is important and can be used to monitor the penetration history of the spray when in used.

5.3. Effect of droplet volume fraction on spray penetration

Figure 4 and Figure 5 show the influence of droplet volume fraction on spray penetration. The impact illustrates negligible effect of the volume fraction term on the spray penetration for small values and close values. However, for increased and large space values, noticeable influences begin to spring up. This constant parameter has a proportional relationship with spray penetration.

5.4. Effect of half cone angle and density on spray penetration

Figure 6 and Figure 7 describe the impact of half cone angle and density on spray penetration. When these parameters are augmented, the penetration capability of the two phase spray increases. These parameters can be used to control processed where high spray penetration is necessary in two phase fluid flow involving fuel droplet and entrained air

5.5. Effect of orifice diameter on spray penetration

Figure 8 explains the influence of orifice diameter on spray penetration. An increase in the orifice diameter causes the spray penetration to increase. This is due to the easy admittance of the spray with minimal restriction. However, when larger diameters are required with specified penetration values, the initial injection velocities may be controlled with the orifice diameter since the continuity equation must always be satisfied for the two phase fluid flow spray penetration process

5.6. Dynamic impact of half cone angle and density on spray penetration

Figure 9 and Figure 10 display the dynamics of spray penetration considering the half cone angle and the droplet initial velocity. The dynamic plot can be used to monitor more than a variable at a time.

Table 1. Comparism and Validation of the analytical scheme.

\(\tau\)	\(\stigma(\tau)\)
	RKF45	Sazhin <i>et al.</i> [1]1} (Experiment)	Jalipour <i>et al.</i> [1]12} (HPM)	Present study (DTM
0.0	0.0000000000	0.0000000000	0.0000000000	0.0000000000
0.1	13.4494000600	13.4494000621	13.4494000611	13.4494000600
0.2	19.0203239700	19.0203239700	19.0203239700	19.0203239700
0.3	23.2950442300	23.2950442303	23.2950442301	23.2950442300
0.4	26.8988001100	26.8988001100	26.8988001100	26.8988001100
0.5	30.0737728000	30.0737728055	30.0737728042	30.0737728000
0.6	32.9441674900	32.9441674900	32.9441674900	32.9441674900
0.7	35.5837678400	35.5837678400	35.5837678400	35.5837678400
0.8	38.0406479300	38.0406479300	38.0406479300	38.0406479300
0.9	40.3482001800	40.3482001811	40.3482001801	40.3482001800
1.0	42.5307373500	42.5307373500	42.5307373500	42.5307373500
1.1	44.6066136500	44.6066136500	44.6066136500	44.6066136500
1.2	46.5900884700	46.5900884700	46.5900884700	46.5900884700
1.3	48.4925015400	48.4925015403	48.4925015402	48.4925015400
1.4	50.3230470900	50.3230470900	50.3230470900	50.3230470900
1.5	52.0893024300	52.0893024300	52.0893024300	52.0893024300
1.6	53.7976002300	53.7976002301	53.7976002302	53.7976002300
1.7	55.4532970300	55.4532970300	55.4532970300	55.4532970300
1.8	57.0609718900	57.0609718900	57.0609718900	57.0609718900
1.9	58.6245757000	58.6245757005	58.6245757007	58.6245757000
2.0	60.1475455600	60.1475455600	60.1475455600	60.1475455600

6. Conclusion

In this work, analysis has been performed on the generalized dynamics of two phase fuel spray penetration using differential transform method. The results of the DTM solution were verified, compared and validated. It was confirmed that DTM is an efficient method for the problem studied. Also, the parametric studies performed helps with the understanding of how the two phase spray penetration process can be properly monitored and controlled. The results show that an increase in the initial velocity and orifice diameter causes a corresponding increase in spray penetration while an antonymous effect is noticed for an increased semi cone angle and density. This work will find vital applications in the optimization of systems whose operation are influence by the aforementioned spray penetration process.

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of Interest:

The authors declare no conflict of interest.

1. Introduction

Research into vibration analysis of thin isotropic rectangular plate resting on elastic nonlinear foundation is vast gaining significant awareness among researchers due to its wide applications and important in the field of engineering. Geotechnics engineers need to understand the behaviour of plates when embedded in soil for their design, structural engineers requires same information for the design of the structural foundations likewise highway engineers rely on the information for the highway pavement design. In the design of elastic soil foundation, the adoption of two- parameter foundations gives better results than the use of Winkler foundation alone, which is associated with limitation of shear interaction among the spring elements. In the study of dynamic behavior of plates, Jain et al. [1] worked on free vibration of rectangular plate. In another work, natural frequency of rectangular plate was determined by Bhat [2] using Rayleight method. Few years later, Balkaya [3] investigated the dynamic response of rectangular plate using differential transform method (DTM). Thereafter, Gupta et al. [4] analyzed forced vibration of rectangular plate with varying thickness. In a further study, some other researchers [5, 6, 7, 8, 9, 10] studied buckling and vibration of plates and beams.

Several authors already applied different method of solutions for analysis of thin rectangular plate. However, in numerical analysis [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], it is very important to carry out convergence and stability study which increases the computational time and cost otherwise the solution will diverge. Furthermore, exact method [24, 25, 26] are having the limitation of handling nonlinear problem due to the complex mathematics involved. These limitations had led to the introduction of semi-analytical methods. Ozturk and Coskun [27] used Homotopy perturbation method (HPM) in the study of plate dynamic behaviour. However, despite the effectiveness, there is setback of finding embedded parameters. In another study, Galerkin method of solution was adopted by Njoku [28] for vibration analysis of thin isotropic rectangular plate. The method suffers the limitation of extension of the series solution to provide precise result. In a later work, Pirbodaghi et al.[29] utilized Homotopy analysis method (HAM) for investigation of vibration analysis of beam. HAM suffers from limitation of assumption of solution for the expression. Variation of iteration method (VIM), was first proposed by He [30,31, 32, 33, 34, 35, 36], has been applied to investigate many nonlinear partial differential equation. The approach uses Lagrange multiplier to find the analytical solution with very fast convergence. This present study adopts the use of exact method to handle the linear part of the system governing equation and solving the rest of equation with very effective method of VIM. The advantage of this method over other hybrid method calls for its application in this research.

Despite the effectiveness of the method and high prediction of results, the author realized that, with several researches on dynamic analysis of plate, Laplace transform and VIM has not been used to determine analytical solution of thin rectangular isotropic plate resting on two-parameter foundations. Therefore, the present study is on determination of analytical solution of free vibration of thin isotropic rectangular plate resting on nonlinear foundation. The analytical solution obtained is used for investigation of the controlling parameters.

2. Problem formulation and mathematical analysis

Considering homogenous rectangular plate of uniform thickness resting on Winkler and Pasternak foundations as shown in Figure 2. The two opposite edge \(y=0\) and \(y=b\) are regarded as simply supported.

The domain are \(0 \leq x \leq a\), \(0 \leq y \leq b\) where \(a\) and \(b\) represents the length and breadth of the rectangular plate as shown in Figure 2. The following assumptions are made for the development of the governing equation [37]:

1. Normal stresses in the direction transverse to the plate are considered small.
2. Thickness of plate is smaller compared to the other dimensions.
3. Plate is of constant thickness.
4. Normal to the undeformed middle surface remains straight and unstretched in length and still normal to the deformed middle surface.

The governing equation for thin isotropic rectangular plate as reported by Leissa [38] is;

where, \(w(x,y,t)\) represents the transverse deflection, \(D\) is the flexural rigidity \(\frac{Eh^{3}}{12(1-v^{2})}\), represents modulus of elasticity \(h\), represents the plate thickness, \(v\) represents the Poisson ratio of plate material, \(\rho\) represents the mass density of the plate, \(\omega\) represents the radial frequency \((rad/s), k_{w}\), and \(k_{p}\) are Winkler foundation and Pasternak foundation parameter respectively.

Using the following dimensionless variables: According to Kantorovich type approximation, the free vibration of Equation (1) can be written as: Assuming the two opposite edges of Figure 1, \(Y = 0\) and \(Y = 1\) to be simply supported, deflection function can be represented as follows: Substituting the derivative of Equation (6) into governing differential equation where \(\lambda \bigg( \frac{a}{b}\bigg)\)represents the aspect ratio, \(m\) is an integer, \(\Omega\) is the frequency parameter, \(a\) represents side length along \(x-axis\).

2.1. Boundary conditions

Three boundary conditions are considered at \(X = 0\) and \(X = l\) namely, Simply supported and clamped edge (SC), Simply supported and simply supported edge (SS) and Simply supported and free edge conditions (SF).

3. Method of Solution: Laplace transform and variation iteration method

3.1. Basic ideal of Laplace transform

If \(f(t)\) is a function of a variable \(t\). \(\mathscr{L}{F(t)}\) and is defined by the integral: Some of the properties used in this study include: where \(F^{n}(t)\) represents the \(n-th\) derivative of \(F(t)\) and \(\mathscr{L}\{F(t)\} = f(s)\). If Laplace transform of \(F(t)\) is \(f(s)\), then the inverse Laplace transform of \(f(s)\) is expressed as \(F(t) = \mathscr{L^{-1}}\{f(s)\}\), where \(\mathscr{L^{-1}}\) is called inverse Laplace operator. The inverse Laplace of Equations (12) and (13) are:

3.2. Laplace and variation iteration method

Assuming the following nonlinear differential equation: \(L\) represents the linear operator, \(N\) is nonlinear operator, \(f\) is the source or analytical function. VariatiFon iteration method use the correction function for Equation (17) as: where \(\lambda\) is general Lagrange multiplier identified through variational theory. The subscript \(n\) represents the \(nth\) term and \(\tilde{w_{n}}\) is a constrained variation \((\delta \tilde{w_{n}} = 0) \). Laplace transform of both sides of Equation (12) gives: Apply convolution to Equation (19), we get Optimal value of \(\bar{\lambda}(x - \zeta)\) is obtained taking variation with respect to \(w_{n}(x)\) given as; Applying variation with respect to \(w_{n}(x)\) gives: Assume \(L\) is linear differential operator with constant coefficients as; where \(a_{i}'s\) are constants. The coefficient contains non-constant terms of the form \(x^{k}\). The Laplace transform of initial operator term is given as: The variation with respect to \(w\) is given as: Extremum condition \(w_{n+1}\) needs that \(\delta w_{n+1}\). Meaning the right hand side of Equation (25) should be set to zero. Hence stationary condition is;

3.3. Application of LVIM to the governing equation

Following the basic principle of LVIM , the governing equation is now analyzed as: Taking variation with respect to \(w_{n}(x)\) on both sides of Equation (27), we get Simplifying Equation (28) gives, Extremum condition \(w_{n+1}\) needs that \(\delta w_{n+1} = 0\) . Meaning the right hand side of Equation (29) should be set to zero. For simplicity we adopt, Substituting Equation (31) into Equation (27) results; Assuming Applying condition (9) at \(x = 0\) on Equation (34) gives Then Inverse Laplace gives the first iteration: Inverse Laplace gives the second iteration: The same approach is continued till frequency parameter \(\Omega\) obtained converges. Substituting boundary condition at \(x = 1\) to find the unknowns introduced results into simultaneous equation.

Table 1. Parameters for validation of the model.

Edge Condition/Dimensionless	Simply-supported (SS)		Simply supported -Clamped (SC)		Simply-supported -Free (SF)
Natural frequency	Bhat et al.;[2]	Present	Leissa [38]	Present	Leissa [38]	Present
\(\Omega 1\)	19.7392	19.7434	23.6463	23.6486	11.7195	11.7606

Table 1 contains parameters for validation of the approach to ascertain the correctness of the results. The polynomials are represented as \(\psi_{11}, \psi_{12}, \psi_{21}\) and \(\psi_{22}\). Equation (42) can be written in matrix form as: The following Characteristic determinant is obtained applying the non-trivial condition Solving Equation (44) gives the natural frequencies. Substitute the result obtained into Equation (43), we get Setting \(\alpha = 1\) and find \(\beta\) Same procedure is repeated for other modes. The following convergence criterion may be used where \(\varepsilon\) is the tolerance parameter taken to be \(0.0001\) for this study, \(\Omega_{j}\) represents the Eigenvalue. The iteration converges at third iteration for first mode frequency parameter.

4. Results and discussion

The solution of Laplace and Variation iteration method is presented here. Table 2 shows the comparison of present results to that of previously published work. It is realized from the Table 2 that, good agreements is achieved with that of the past results.The fundamental modal shape of the thin rectangular plate are shown in Figures 3, 4, 5 and it is observed that the shape obeys classical plate theory. Also, Table 3 shows different deflection values of transverse displacement for the first three mode frequency parameters of SC, SS and SF boundary condition considered. Table 4 shows the convergence study, it is observed that the fundamental natural frequency converges at the third iteration while higher modes are obtained by increasing the number of iterations. This phenomenon is peculiar to vibration problem.

Table 2. Showing validation of results.

Edge Condition/Dimensionless	Simply-supported (SS)		Simply supported -Clamped (SC)		Simply-supported -Free (SF)
Natural frequency	Bhat et al.[2]	Present	Leissa [38]	Present	Leissa [38]	Present
\(\Omega 1\)	19.7392	19.7434	23.6463	23.6486	11.7195	11.7606

It is also observed that, the presence of elastic foundation and aspect ratio has no significant changes on the mode shape of the rectangular plate. Since dimensionless analysis is carried out, the results are valid for all thin plates. Table 4 shows that the value of frequency parameters \(\Omega\) decreases in the order of \(SC \geq SS \geq SF\).

Table 3. Results of different deflection values.

Transverse displacement	SF	SS	SC	SF	SS	SC	SF	SS	SC
	\(\Omega 1\)			\(\Omega 2\)			\(\Omega 3\)
w[0]	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
w[0.05]	0.0500	0.0498	0.0497	0.0496	0.0492	0.0490	0.0489	0.0482	0.0479
w[0.10]	0.0998	0.0984	0.0978	0.0970	0.0935	0.0921	0.0916	0.0858	0.0837
w[0.15]	0.1492	0.1445	0.1425	0.1400	0.1288	0.1240	0.1224	0.1048	0.0983
w[0.20]	0.1981	0.1871	0.1825	0.1767	0.1514	0.1410	0.1376	0.1009	0.0881
w[0.25]	0.2464	0.2251	0.2163	0.2054	0.1592	0.1409	0.1351	0.0750	0.0556
w[0.30]	0.2938	0.2575	0.2429	0.2248	0.1514	0.1237	0.1153	0.0328	0.0091
w[0.35]	0.3403	0.2836	0.2615	0.2340	0.1288	0.0915	0.0807	-0.0166	-0.0397
w[0.40]	0.3857	0.3027	0.2715	0.2323	0.0935	0.0482	0.0358	-0.0623	-0.0784
w[0.45]	0.4300	0.3144	0.2727	0.2199	0.0492	-0.0011	-0.0134	-0.0945	-0.0973
w[0.50]	0.4732	0.3183	0.2652	0.1971	0.0000	-0.0506	-0.0607	-0.1061	-0.0915
w[0.55]	0.5153	0.3144	0.2496	0.1647	-0.0492	-0.0945	-0.0998	-0.0946	-0.0624
w[0.60]	0.5564	0.3027	0.2266	0.1239	-0.0935	-0.1275	-0.1254	-0.0624	-0.0173
w[0.65]	0.5967	0.2836	0.1976	0.0760	-0.1288	-0.1462	-0.1340	-0.0167	0.0326
w[0.70]	0.6363	0.2575	0.1641	0.0227	-0.1514	-0.1487	-0.1240	0.0327	0.0751
w[0.75]	0.6757	0.2251	0.1279	-0.0345	-0.1592	-0.1356	-0.0958	0.0750	0.0999
w[0.80]	0.7154	0.1871	0.0914	-0.0939	-0.1514	-0.1095	-0.0520	0.1009	0.1018
w[0.85]	0.7561	0.1445	0.0571	-0.1542	-0.1288	-0.0754	0.0036	0.1048	0.0821
w[0.90]	0.7987	0.0984	0.0281	-0.2148	-0.0935	-0.0400	0.0664	0.0859	0.0487
w[0.95]	0.8444	0.0498	0.0078	-0.2753	-0.0492	-0.0117	0.1325	0.0482	0.0155
w[1.00]	0.8948	-9.0000	-9.0000	-0.3364	0.0000	0.0000	0.1995	0.0000	0.0000

Table 4. Showing convergence study of the results.

Edge Condition /Dimensionless Natural frequency	Iteration	(SS)		(SC)
		Bhat et al. [39]	Present	Leissa [38]	Present	Leissa [38]	Present
\(\Omega 1\)	N3	19.7392	19.7434	23.6463	23.6486	11.7195	11.7606
\(\Omega 1\)	N4	19.7392	19.9574	23.6463	23.8905	11.7195	11.7445
\(\Omega 2\)		49.3481	49.3271	58.6465	58.6240	27.7563	27.7563
\(\Omega 1\)	N6	19.7392	19.7418	23.6463	23.6487	11.7195	11.6855
\(\Omega 2\)		49.3481	49.0637	58.6465	58.3220	27.7563	27.6965
\(\Omega 1\)	N7	19.7392	19.7394	23.6463	23.6465	11.7195	11.6846
\(\Omega 2\)		49.3481	49.3271	58.6465	58.6240	27.7563	27.7563

4.1. Effect of foundation parameter on natural frequency

Table 5 illustrates the impact of foundation parameter on natural frequency. It is clear from the Figures 6, 7 and 8 that the foundation parameter has impact on natural frequency, increasing values of the foundation parameter increases the natural frequency. This satisfies the principle of classical vibration. Stiffness increment results to natural frequency increment. This also corroborated with finding reported in [38]. The effect of increase in natural frequency is much significant in higher values of the elastic foundation.

Table 5.Variation elastic foundation coefficient on natural frequency.

Edge Condition	Natural frequency	kw=5	kw=15	kw=45	kw=120	kw=200	kw=250
SS	\(\Omega 1\)	19.865457	20.115575	20.847934	22.575127	24.282429	25.291033
	\(\Omega 2\)	49.398659	49.49977	49.801882	50.549258	51.334467	51.81918
SC	\(\Omega 1\)	23.751809	23.961395	24.579431	26.060476	27.552648	28.445534
	\(\Omega 2\)	58.688978	58.774102	59.02877	59.660676	60.327409	60.7404
SF	\(\Omega 1\)	11.896571	12.309687	13.473248	16.016504	18.34471	19.660326
	\(\Omega 2\)	27.846269	28.025251	28.555467	29.839817	31.151479	31.94393

4.2. Effect of variation of aspect ratio on natural frequency

The influence of aspect ratio on natural frequency are shown in Table 6 and Figures 9, 10, 11 respectively. It is shown that, the natural frequency increase with increases in aspect ratio. This is because, the plate becomes more stiff as the aspect ratio increases resulting in the natural frequency increases.

Table 6. Variation of aspect ratio on natural frequency.

Edge Condition	Natural frequency	\(\lambda=0.4\)	\(\lambda=0.7\)	\(\lambda=1.0\)	\(\lambda=1.5\)	\(\lambda=2.5\)	\(\lambda=3.0\)
SS	\(\Omega 1\)	11.448741	14.705711	19.739209	32.076214	71.554632	98.696055
	\(\Omega 2\)	41.057554	44.314525	49.348024	61.685024	101.16349	128.3048
SC	\(\Omega 1\)	16.627624	19.277232	23.64632	35.051125	73.438926	100.26994
	\(\Omega 2\)	51.326727	54.171092	58.646365	69.912813	107.42	133.79231
SF	\(\Omega 1\)	3.0081474	6.562375	11.684537	24.010127	63.28683	90.296225
	\(\Omega 2\)	17.636147	21.826148	27.756345	41.173975	81.606845	108.92412

5. Conclusion

In this study, the dynamic analysis of isotropic rectangular plates resting on Winkler and Pasternak foundations is analyzed. The governing equation is transform to nonlinear ordinary differential equation using Galerkin method of separation. The nonlinear ordinary differential equations have been solved using Laplace transform and variation of iteration method. The accuracies of the obtained analytical solutions were ascertained with the results obtained by earlier researcher. The obtained analytical solutions were used to examine the effects of foundation parameter, aspect ratio. The rate of convergence is increased with the introduction of exact method for analyzing the linear part of the governing equation while the remaining part are treated with variation of iteration method, practical applications of the study are base plate of tower, steel hinged steel column structures and culvert covers. From the parametric studies, the following observations were established:

1. Increase in elastic foundation parameter increases the natural frequency.
2. Increase in aspect ratio increases the natural frequency.
3. Increasing the combine elastic foundation parameters increases the natural frequency.
4. Accurate higher mode frequency can be obtained with increase in number of iterations.
5. SF boundary condition has the least value of frequency parameter followed by SS edge condition.
6. The effect of increase in natural frequency is much significant in higher value of the elastic foundation.

Abbreviations

Abbreviations	Nomenclature
\(a\)	Length of the plate
\(b\)	Width of the plate
\(C\)	Clamped edge plate
\(E\)	Young's modulus
\(F\)	Free edge support
\(S\)	Simply supported edge
\(d/dx\)	Differential operator
\(w\)	Dynamic deflection
\(X\)	space coordinate along the length of thin plate Symbol
\(h\)	plate thickness
\(\rho\)	Mass density
\(D\)	Modulus of elasticity
\(\Omega\)	natural frequency

Acknowledgments

The author expresses sincere appreciation to the management of University of Lagos, Nigeria, for providing material supports and good environment for this work.

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of Interest:

The authors declare no conflict of interest.

1. Introduction

The responsibility of the SQA engineer is to

• Assess the software development process.
• Evaluate the conformance of software processes and software product
• Evaluate the effectiveness of the software processes.

This paper brings out all these above activities carried out for CDEU software.

ISPD describes the software life cycle processes and identifies the essential documents as the outcome of various life cycle processes like System requirement specification document, software design document, software project management plan, configuration management plan etc. An appropriate development life cycle model can be chosen based on software category using the guideline for selection of software development life cycle. The processes, activities and tasks applicable for the software category are mapped to the chosen life cycle model.

With the above explained framework we are able to establish complete traceability. The SDLC model adopted helped in finding missing functionality and to accommodate new requirements which came at last stage of the project. This framework helped to find the anomaly in the early stages of the project.

The PTC has two compressors C1 and C2. It is mounted back to back to reach a cooling effect of 80K by removing 0.7W heat load. The CDEU delivers sinusoidal drive to both the compressors. The drive can be programmed for various parameters based on the heater load. There is various logic for CDEU operation. Safety logics are also built-in like, over voltage and over current protection etc. The software requirements and interface specifications for the cryo cooler is organized into following sections:

• Mission Requirements
• System Requirements
• Functional Requirements
• Memory organization and Hardware interface registers
• Software Requirement Specifications

Since CDEU being the new system, all the requirements were discussed and complete testing was carried out. If it is a repeat project then only changes will be discussed in the SRR. Testing will be carried out for the changes and regression testing will be carried out to see the effect of change on other modules.

2. Cryocooler drive electronics unit system overview

CDEU delivers sinusoidal drive to both compressor C1 and C2. It can be programmed for parameters like voltage, phase, offset and frequency. CDEU monitors voltage, current and frequency, temperature of the cold tip and pressure of the gas for both the compressors. This CDEU package is stack of three cards Power card, Instrumentation card and Controller card hosted on the mother board. It is interfaced with TCP for data commands and pulse commands through 1553 interface. TCP is the bus controller and CDEU is one of the remote terminals for it. It also has 1553 interface with telemetry package to monitor the health of cryo cooler and other parameters like temperature etc. Figure 1 shows the various modes of operation of CDEU.

The functionalities are broadly classified as:

• Different modes of operation bases on requirement
• Command reception through 1553 interface , command decoding and execution
• Control of linear motor drive parameters, Configuration of system parameters
• Acquisition of parameters from hardware and transmit to TM subsystem
• Fault detection and recovery logic

3. ISRO software control board adaption of IEEE 12207

ISRO has created a framework for developing software and ensuring its quality through the implementation of a software standard IEEE-12207 as defined in the ISRO Software Process Document. ISCB is responsible for the effective implementation of ISPD encompassing all classes of software pertaining to various centres and units of ISRO.

4. Software quality assurance for CDEU

4.1. Software requirement specifications

All the requirements were discussed in the SRS committee. Since this being a new system there is no heritage for it, hence all the requirements were reviewed. If it is a repeat project then only changes will be reviewed. The SRS committee recommendations for any change in requirements or to include any new module for the correctness of the system are taken for implementation. Minutes for the SRS and closeout will be released soon after the review. It is most important for the completion of process and also to establish traceability. After the review approved SRS document will be released.

4.2. Design phase

Approved SRS document after the SRR is the input to software design. The software design review is carried out with members from different groups like mission, controls, subsystem and SQA. The design review was carried out in steps. Initially with the designer peer review was carried out. Any change in the design, correctness of requirement implementation is verified by the quality engineer. After all the modules peer review is completed the minutes will be presented to the SDR committee. The CDEU design was verified against the approved SRS. Any changes or action on QA engineer or on designer will be included in the minutes. The action should be closed before the code is released to SQA engineer. Figure 2 shows the SDLC followed for onboard software.

Some of the recommendations like WDT flow chart has to be modified, Message ID can be displayed appropriately all these were taken care in the next version of SDD. This will be the approved design document.

4.3. Coding Phase

With the approved design document the coding will begin. The code is written in assembly language 8086. CPU86 IP core is qualified for on-board and to be used with the safe subset instructions. Once the code is available for software quality assurance engineer, codewalk through has to be carried out. The automated tool helps in verifying the safe subset. Figure 3 shows the utility which will generate the report file saying whether the instruction is in safe subset or not. Any observations during codewalk through will be presented to committee.

4.4. Testing Phase

Test cases were generated based on the system level requirements, negative cases were also generated to test the same. Test cases for all the modes of CDEU were tested. Also mode transition from one mode to other mode was also carried out. It will be going to halt mode automatically being in configuration mode or in the operation mode. Figure 4 shows the mode transition testing carried out as part of initial bench level testing [1]. Negative logic tests were tested in-depth. Even the mission scenario tests were carried out. All boundary conditions were thoroughly verified. All the telecommands and telemetry were verified. This is the major test to know that CDEU is functioning as per requirements [2].

One of the tests carried out for soft start and soft stop for CDEU by PWM enable and disable command is shown in Figure 5. The soft start voltage and the step size for soft start are commandable. Once PWM enable command is given till it reaches the commanded voltage it increases in steps [3]. Once PWM disable is given it decreases in steps and reaches zero volts.

5. Reviews and recommendations

All the observations and non-conformances found during codewalk through and testing will be presented to sub system review board. Based on committee recommendations change was carried out in the operation module to take care of safety logic. Regression testing was carried out for the same. Configuration management was also carried out. Initial version of the code was released once the designer level tests were carried out. Later after testing and with committee recommended change new version of code is released. Delta code walk through was also carried out. As part of database verification all the limits, register values were verified.

6. Conclusion

The software is developed based on ISRO Software Process Document and documents were generated as per IEEE template. Software development process is clearly stated along with linkage of activities with inputs, process, outputs, responsibility and linkage to ISPD processes in this document. Our existing practices are much more strengthened in terms of documentation, process followed with the application of ISPD. Many non conformances were detected at the early stages and implemented. Hence it has saved much time during testing. It has shown our ability to refrain our existing processes with international standards. SQA provides the pre-certificate to project which has the list of SQA activities complied for the project.

Acknowledgments

We wish to convey our gratefulness to deputy director,- space craft reliability and quality area, group head-reliability \& quality assurance software group, division head-mission software quality assurance division and all our colleagues in Mission Software Quality Assurance Division, Onboard Software Quality Assurance Division and sub system designers for their help and support.

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of Interest:

The authors declare no conflict of interest.