The advent of technology around the globe based on the Internet of Things, Cloud Computing, Big Data, Cyber-Physical Systems, and digitalisation. This advancement introduced industry 4.0. It is challenging to measure how enterprises adopt the new technologies. Industry 4.0 introduced Digital Twins, given that no specific terms or definitions are given to Digital Twins still challenging to define or conceptualise the Digital Twins. Many academics and industries still use old technologies, naming it Digital Twins. This young technology is in danger of reaching the plateau despite the immense benefit to sectors. This paper proposes a novel and unique definition for the Digital Twin based on mathematical modelling. The uniqueness of the meaning is that it is suitable and adaptable wherever it applies in industries or academics. This paper defines the Digital Twin as a “4-D Virtual replica that continuously simulates the entire mechanical behaviour of anything”. The novel concept of the DT for this study is based on the numerical analysis using Euler’s theory. A successful numerical Digital Twin model developed and validated the proposed idea of the Digital Twin based on Euler’s method. The numerical testing verified the accuracy and efficiency of the Digital Twin model throughout the exact representation of a vibrating system’s internal and external mechanical behaviours in all scenarios. The model still has some limitations and is open for further research; further research depends on the type of application. The contribution of this paper is summarised as follows: Proposing a novel concept for DT; Proposed DT applies to theory and practice; The concept is computationally fast, easy to implement and cost-effective; The proposed DT enhances condition monitoring in current real-time; The DT model continuously visualises and evaluates parts conditions; The proposed technique improves predictive and proactive maintenance.

The new industrial revolution considered industrial 4.0, where the systems used in manufacturing operations are integrated with the information and communication technologies (ICT). At this stage, the Cyber-Physical Systems (CPS) formed based on the advent of the materials and capabilities of the sensor. In other words, the IoT became the backbone of the fourth industrial revolution [

According to the Whitepaper, which was written by Grieves in the same year [

In the enthralling nexus of smart technologies, such as integrating AI with DTs emerges as a paradigm that exemplifies technological synergy, steering unprecedented advancements across myriad sectors as shown in

Throughout the lifecycle of any physical object in the real world, it is crucial to have its model in the digital world mirror each other continuously. The digital model consists of different models to visualize the physical object in 3D in the virtual world. Models are a critical part of DT as they are the initial step in developing or creating a twin for the physical object in the digital world. They provide a complete and detailed reflection of the physical object and insight into it through simulations. However, as the data have been gathered and combined through people, information systems, and sensors, the models themselves will have a specific function; these functions can be summarised as follows: (A) reproduction of rules, behaviors, and properties for the physical object (B) the autonomous operation in the virtual space (C) the prediction of issues before its occurrence (D) development of preventive strategies (E) performance validation.

Data is the brain that makes DT work, which provides intelligence to operate DT continuously. Data come from virtual and physical spaces; virtual space provides data from simulations and information systems. All the data combined to give the driver for the DT through big data analytics deals with a large and diverse set of data. Where valuable information can be minded and unnecessary data can be ignored. Data play a vital role as a full DT version of the physical object cannot work without it.

The ability to connect is essential for the creation of a digital twin. Using this technology can connect a physical element to its digital counterpart. By gathering, integrating, and transferring data through various integration methods, sensors make it possible to connect physical objects. Integrating the physical and virtual space is critical and completes the cycle between the model and the physical object. Connections are essential to bringing every element in every entity to live in the DT. However, connections can be divided into three categories as follows: (A) connection within the physical space, where the entities are interconnected for data exchange (B) connection within virtual space, where the model and information systems are interconnected (C) connection between the physical and virtual space, where the entities are linked with the corresponding virtual objects to provide a closed-loop for the sensing and control. The achievement of this comprehensive connection will provide an iterative optimization throughout the lifecycle of the products.

Services are the final form in which DT can be presented with, or is the user interfaces where the standard formats of functions can be visible to the user, such as input, output, and primary and standard information to communicate with the virtual and physical spaces. Services like the black boxes are used without knowing the internal mechanisms, all the users do, provide the input parameters and request the output. Services accessed with a bit of professional understanding simplify and ease DT usage and expand the DT applications to more user groups. Since many authors still assume that the DT is collected during product development, from digital artifacts to the collected data during product use; therefore, this paper proposes a novel concept for the DT. The idea or definition introduced by this paper is that the “DT is a Live virtual replica that continuously simulates the entire mechanical behavior of anything”, where the replication must mirror the entire internal and external mechanical behavior. This concept is suitable for both industries and academics. When used, for example, the entire representation of the thing in maintenance will represent the systems’ current real-time mechanical behavior. Still, it will also diagnose both electrical and mechanical faults and classify them. For this study, this paper, for the first time, validated the proposed concept using Euler’s method to replicate the entire mechanical behavior of a coil spring used in many systems through all industries and studies.

This study aims to conceptualize the Digital Twin (DT) technology. The idea introduced in this study is cost-effective and straightforward to implement in both industry and academia. While the aim is only to conceptualize the DT, the mechanical behavior of the coil spring is used to validate the proposed concept based on Euler’s method.

This study aims to conceptualize the DTs and prove that the DT mode can provide more accurate results similar to the experimental ones by analyzing the mechanical structure of coil springs used in suspension systems with the current time data flow, which eventually provides safety within automated mechanical systems by diagnosing faults and their classification.

The rest of this paper is organized as follows:

The workshop of “Dartmouth Summer Research Project”, which was organized by John McCarthy in 1956 on Artificial Intelligence (AI), now is considered by many to be the official declaration of as a research field in the AI [

In 1970, NASA was the first to generate the concept of the (physical twin) through system engineering and condition monitoring, in Apollo 13, two identical spaceships were created, one of which was launched in space to perform the mission and remained on Earth. NASA used the one that remained on the ground to analyze what was happening in the area [

Since 2016, there has been an overwhelming number of publications on DTs, however, none comprehensively review DTs. Some of the most important ones are summarised below. Following this, institutions and scholars submitted their definitions of DTs, including extensive and varied precise descriptions. It depends on the research as long as the report covers the three critical aspects of Grieves’ framework [

Ref. | Concepts and definitions of DT |
---|---|

[ |
It is an integrated Multiphysics, multiscale simulation of a vehicle or a system that uses the best available physical models, sensor updates, etc., to mirror its corresponding flying twin |

[ |
A digital replica of actual physical installation which can check the consistency for monitoring data, perform data mining to detect existing and forecast upcoming problems, and use AI knowledge engine to support effective business decisions |

[ |
Replication of the actual physical system, and he stated three dimensions for this replication (A) Physical entity (B) Digital counterpart (C) Connection between physical and digital. |

[ |
A product equivalent digital counterpart that exists along the product lifecycle from conception and design to usage and servicing knows the product past, current and future state and facilitates the development of product-related intelligent services |

[ |
A digital avatar encompassing CPS data and intelligence, representing structure, semantics, and behaviour of the associated CPS, and providing services to mesh the virtual and physical worlds |

[ |
A virtual dynamic model that is completely consistent with its physical counterpart in the real world and can accurately imitate its physical counterpart's attributes, behaviour, life, and performance. |

[ |
A virtual and computerised version of a physical system that can take use of real-time synchronisation of sensed data from the field and is closely tied to Industry 4.0. |

[ |
Having a high semantic content and considering both virtual product models as well as feedback data from the physical product along its whole lifecycle |

DT technology is becoming one of the most promising technologies in all industries. The railway industry is not included in this study simply because it is one of the few industries which did not take full advantage of the DT technology. As shown in

Ref. | Tool | Description |
---|---|---|

[ |
CAE | ALTAIR-Offers a CAE tool capable of predicting, optimising, tracking, and measuring product performance throughout the product life cycle. |

[ |
AWS | AMAZON EC2-Cloud-based environment for deploying a web application to the cloud. |

[ |
CAE | ANSYS-Combining all of an organisation’s digital information about a specific product and combining physics-based understanding with analytics are two examples of doing this. |

[ |
Autodesk | AUTODESK-Using augmented reality technologies borrowed from the media and entertainment software line, visualises the control capabilities, and IoT cloud services platform provider, the project spans both the factory and the product. |

[ |
Bsquare IoT | BSQUARE-Real-time configuration and state information of physical devices are represented digitally in this format. |

[ |
Dassault | Dassault-offers a virtual counterpart to a physical product that can improve manufacturing excellence by enabling people across the enterprise to collaborate and achieve continuous process improvement. |

[ |
IoT solution | DELOITTE-An evolving digital profile of physical objects’ and processes’ historical and current behaviour that aids in the optimisation of business performance. |

[ |
Docker | DOCKER-the product that uses operating system level virtualisation to develop and deliver software in containers |

[ |
Predix | GE-Providing software representation of a physical asset based on the Predix platform and enabling companies to understand better, predict, and optimise the performance of each unique asset |

[ |
Nia TM | INFOSYS-Virtual replication of physical products, systems, and processes that are indistinguishable from their tangible counterparts |

[ |
Intellect-soft | INTELLECT-SOFT-As the environment changes, the digital representation of a physical object continuously updates its status, reporting it in measurements and pictures. |

[ |
SAP | IBM-A virtual representation of a physical object or system throughout its lifecycle uses real-time data to facilitate understanding, learning and deducing conclusions. |

[ |
Azure | MICROSOFT-Visualising the physical world, being intelligent, collaborative, interactive, and immersive, and providing a method to simulate electronic, mechanical, and combined system outcomes |

[ |
Oracle | ORACLE-An important concept that is going to be strategic to business operation as IoT deployments increase through the organisation |

[ |
Data V | PACCAR-A virtual version of an engine based on sensor data from the real-world versions to manage the maintenance and repair of engines |

[ |
Thing-works | PTC-A digital depiction of a field asset’s current and previous configuration states, taking serialised parts, software versions, options, and variants into account. |

[ |
PLM | SIEMENS-production of digital twins for manufacturing and production planning, and performance digital twins for performance, and acting on operational data |

[ |
SM | SM-Offering sets of analytical models that mirror the entire production process, encompassing machines, lines, plants, or supply chains |

[ |
DT | SIM-CI-A digital copy of a city allows us to mimic its vital infrastructures accurately |

[ |
SAP | SAP-A live digital representation or software model of a connected physical object |

[ |
TIBCO | TIBCO-A software representation of a device that can create efficiencies across the entire product lifecycle |

[ |
SS | TWIN THREAD-A digital representation of any physical asset, including all the information about the asset current and historical running conditions |

This section covers the analytical and numerical solutions (Euler method) for the main parameters that characterize the behavior for the different constructed vibration systems. The analysis is subject to the other operating conditions, incorporating the output responses, strains, normal, maximum induced shear, and shear Von-mises stress within the oscillating spring, the spring’s properties are shown in

LOAD PROPERTIES | |
---|---|

Mass/kg | 1 |

STEADY APPLIED FORCE (N) | 100 |

TRANSIENT APPLIED FORCE | 500 |

FREE LENGTH (Lf) | 100 |

HOLE DIAMETER (D) | 70 |

WIRE DIAMETER (d) | 6 |

OUTER DIAMETER (D3) | 66.5 |

MEAN DIAMETER (D2) | 60.5 |

SPRING INDEX (C) | 10.08 |

COIL DIAMETER (D1) | 54.5 |

NUMBER OF ACTIVE COILS | 9 |

NUMBER OF TOTAL COILS | 11 |

SOLID LENGTH (Ls) | 66 |

TO DETERMINE THE EXACT INDEX | 5.1 |

STRESS CORRECTION FACTOR (KW1) | 1.14 |

MAXIMUM LOAD (Fs) | 198.98 |

MAXIMUM TORSIONAL STRESS (Ss) | 162.52 |

CLEARANCE | 3.025 |

PITCH HEIGH | 9.11 |

SOLID DEFLECTION (delta x) | 34 |

YOUNG'S MODULUS (E) MPa | 190000 |

SHEAR MODULUS MPa | 72000 |

SPRING RATE (K) N/mm | 5.85 |

SPRING COMPLIANCE m/N | 0.17 |

Density Kg/mm^{3} |
0.0000079 |

Poisson Ratio | 0.3 |

TENSILE STRENGHT MPa | 1500 |

TORSIONAL STRESS Mpa | 750 |

yield strength (Mpa) | 1230 |

kc | 313186.81 |

Wahl Factor | 1.14 |

Alpha (coefficient of initial relative velocity) | 0.08 |

This section investigates the analytical solutions for the output response, normal strain, normal and maximum induced shear stress within each of the three different linear-vibration operating conditions as a function of time in detail, where the damping effect is ignored.

Therefore, the displacement y(t) based on the above equation obtained as follows:

The corresponding expression for the displacement of

Since

This section examines the analytical solutions for the output response, normal strain, normal and maximum induced shear stress within each of the three different linear-vibration operating conditions as a function of time in detail, and the damping effect.

Based on the initial conditions where at

By using the initial conditions at

Initial conditions of the steady-forced vibration system are applied to obtain the following expression of the object’s displacement.

This section studies the analytical solutions for the output response, normal strain, normal and maximum induced shear stress for the non-linear vibration system, consisting of four springs under three different operating conditions, excluding the damping effects throughout all the segments.

Determining the natural frequency for the system is shown in

The equations of motion at node 1 within the system would be:

The equation of motion at node 2

The equation of motion at node 3

The equation of motion at node 4

Since no damping is taking place throughout the entire system, an expression that associates the displacements

Using the relationships given in

Using the initial conditions at

Using the

Taking the determinant of the square matrix

Based on the further derivation from the above determinant matrix, the expression for the

Therefore, the natural frequency of the system would be:

Since it is a free vibration, therefore the initial displacement of node one can be decided or assumed, and by knowing this displacement at node 1 (

The equations of motion at nodes 2, 3, and 4 will be the same as expressed in the previous system from

The equations of motion at nodes 2, 3, and 4 will be the same as expressed in the previous system from

This section studies the numerical solutions for the output response for the non-linear vibration system, consisting of three active springs under three different operating conditions, including the damping effect within the first spring only.

The equation of motion at a junction point (node 2) between spring one and spring two within the system would be defined as follows:

The equation of motion at a junction point between spring two and spring 3 (node 3) within the system is described as follows:

The equation of motion at a junction point between spring three and spring four (node 4) within the system is described as follows:

Analytical solutions for

To find the acceleration at node 1 (

Assuming the acceleration between the two-time steps is constant, the velocities in these two-time steps are known. Therefore, taking the average velocities and multiplying it by the change of time (time-step) will result in the change in displacement for node 1. Adding this change of displacement to the initial displacement will result in the displacement at the end of the given time-step. Therefore, the displacement

By rearranging

Making

Since the velocity at node two is known, then using the same technique as for node 1 to obtain the displacement at node two at a given time (t), which results in the following equation:

By rearranging

Making

Since the velocity at node three is known, then using the same technique as for node 1 to obtain the displacement at node three at a given time (t), which results in the following equation:

By rearranging

Making

Since the velocity at node four is known, then using the same technique as for node 1 to obtain the displacement at node four at a given time (t), which results in the following equation:

The equation of motion at nodes 2, 3 and 4 will be the same as mentioned in

Considering the transient forced-vibration system will lead to the following equation of motion of a steady forced oscillating object within the system would be:

The equation of motion at nodes 2, 3 and 4 will be the same as mentioned in

This section investigates the analytical solutions for the output response, normal strain, normal and maximum induced shear stress within each of the three different linear-vibration operating conditions as a function of time in detail, where the damping effect is ignored, as shown in

It is noticed in

According to

The damping effect into the system mentioned in the undamped linear transient-forced vibration system leads the output response slightly distorted compared to the rest of the spring’s motion within the first few seconds of the spring’s movement observed. The spring’s movement observation is interpreted as the transient response throughout this duration, which has gotten damped quite rapidly. The entire system’s overall response is the same as the steady-state response. Incorporating the damping effect into the system leads to the simple harmonic motion response. Using the same transient force, the undamped linear transient-forced vibration system has been mentioned. They obtained the maximum normal and induced shear stress values of about −14.78 and −433.90 MPa. Once again, this signifies that failure would occur due to shearing mode since the obtained shear stress exceeds the shear yield stress, as shown in

The construction of this system would be similar to that mentioned within the section of the un-damped non-linear steady-forced vibration systems; the only difference is throughout the analysis of springs’ motions within the system, which is attached to the base neither gets extended nor gets compressed. From the theoretical studies section, where the numerical solutions using the Euler method for the displacements of

Spring 1: −50.62 MPa (Maximum Normal Stress) and −1485.7 MPa (Maximum induced shear stress). Spring 2: −33.80 MPa (Maximum Normal Stress) and −992.15 MPa (Maximum induced shear stress). Spring 3: −25.35 MPa (Maximum Normal Stress) and −744.11 MPa (Maximum induced shear stress). These values show that the shear failure would occur among all of these springs since their shear stresses exceeded the yielding shear stress.

Based on the system’s construction that comprises three active springs (as mentioned for damped non-linear steady-forced vibration system) and as shown in

5.852466 | −5.85247 | 0 | 0 | 0 |

5.852466 | 19.02051 | −13.1680478 | 0 | 0 |

0 | −13.168 | 23.70248611 | −10.5344 | 0 |

0 | 0 | −10.5344383 | 17.11846 | −6.58402 |

0 | 0 | 0 | −6.58402 | 6.584024 |

0.085434 | 0.085434 | 0.085434076 | 0.085434 | 0.085434 |

−0.08543 | 0.085434 | 0.085434076 | 0.085434 | 0.085434 |

−0.08543 | 0.085434 | 0.161375477 | 0.161375 | 0.161375 |

−0.08543 | 0.085434 | 0.161375477 | 0.256302 | 0.256302 |

−0.08543 | 0.085434 | 0.161375477 | 0.256302 | 0.408185 |

The DT model shows the current real-time visual representation of the mechanical behaviour of the vibrating system (Coil Spring). This model is based entirely on only one input which is the load applied on the spring.

This paper proposed a novel numerical concept of the Digital Twin based on Euler’s method. The Digital Twin model successfully replicated the mechanical behaviour and virtually represented the mechanics of materials for the physical coiled spring. Successfully developed a numerical way to validate the proposed idea of the DT. The technique suggested still has limitations and is subject to further research. This paper defined the Digital Twin as “Digital Twin is a virtual replica of anything, where the replication must mirror the entire internal and external mechanical behaviour of the replicated thing.”. The DT model virtually represented all stresses acted internally on the spring. While the resulting strains and stresses are accurate based on Euler’s method, this paper proposed a novel concept for the DT. The DT model captured all the variations of the normal and maximum induced shear stress in current real-time. Additionally, the model showed the instant representation of the system’s behaviour and showed that in the case of free force, vibration behaved similarly concerning the time is irregular compared to the conduct of the linear-steady forced vibration system. The damping effect into the system mentioned in the section of the undamped linear transient-forced vibration system leads the output response slightly distorted compared to the rest of the spring’s motion within the first few seconds of the spring’s motion’s movement observed.

The non-linear steady and transient forced vibration used in the un-damped case illustrates all measurements’ output response (deflection, strain, maximum normal and induced shear stresses). The output for each of the four springs behaves harmonically with the same natural frequency, with different values concerning the time due to the other geometric properties of each spring. Non-Linear Steady-Forced Vibration Systems (Damped) is the same as the undamped system. The only difference is throughout the analysis of springs’ motions within the system, which is attached to the base neither gets extended nor extended gets compressed. The model shows the overall displacement of the coils and the displacement between each coil. The model still has some limitations and is open for further research; fatigue analysis is one of most types of failure accrues to mechanical systems. Since all the stresses shown in the model’s interface are in the current real-time, it is essential to improve the fatigue analysis further.

Maximum Amplitude in

Constant used in

Constant used in

Actual Damping

Critical Damping

Spring compliance (1/k)

Wire diameter

Hole Diameter

Inner Diameter

Mean Diameter

Outer Diameter

Young’s Modulus

Applied External Force

Shear Modulus

Spring Rate

Equivalent spring Rate

Wahl Stress factor

Solid length

Mass

Divided Segments

Number of Active coils

Number of Total coils

Pitch height

Time

Period

Tension Force

Tensile Strength

Poisson Ratio

Normal strain

Normal stress

Alternating stress

Damping Ratio

Phase Angle

Normal strain

Normal stress

Actual frequency

Natural frequency

Driving frequency

Actual frequency

Normal strain

Normal stress

Alternating stress

Damping Ratio

Constant in

Constant in

Constant in

Constant in

Deflection

Velocity

Acceleration

Coefficient of initial relative velocity

Time interval

Solid Deflection

Deflection

Velocity

Acceleration

The authors would like to thank the Society of Automotive Engineers (SAE), for making the data available on their website.

The authors did not receive funding for this research study.

The authors of this research paper have made equal contributions to the study from inception to completion. Each author has played a significant role in conceiving and designing the research, collecting and analysing data, interpreting results, and writing and revising the manuscript. Specifically, each author has collaboratively formulated the research objectives and hypotheses, participated equally in data collection, either through experiments, surveys, or data acquisition, conducted statistical analyses and interpreted the findings jointly, contributed equally to the drafting and revision of the manuscript, including the literature review and discussion sections, and reviewed and approved the final version of the manuscript for submission. We affirm that all authors have made substantial and equal contributions to this research and are in full agreement with the content of the manuscript.

The data and materials supporting the findings of this study are available at

The authors declare that they have no conflicts of interest to report regarding the present study.