A Step-by-Step Guide to Nonlinear Spring-Damper Models for Vehicle Crash Simulation in MATLAB

Author : Waqas Javaid

Abstract

Using a 15-dimensional nonlinear spring-damper lumped-mass model, a comprehensive MATLAB simulation of a vehicle’s frontal collision with a rigid wall is presented in this article. The explicit time integration method (semi-implicit Euler) calculates spring forces, damping forces, and contact forces at each time step to predict displacements, velocities, accelerations, and energy distribution throughout the 0.15s crash event [1]. Displacement-time histories, velocity profiles, energy transfer (kinetic to potential), contact force, crush depth, a 3D space-time deformation surface, and 3D phase space trajectories are among the six detailed visualizations that are produced by the simulation. Providing insight into vehicle crashworthiness, key results like peak deceleration (in g-forces), maximum crush depth (in mm), and percentage of energy absorbed by the structure are automatically summarized [2]. This tutorial can be used by engineering students and hobbyists who want to learn how crumple zones, stiffness distribution, and damping affect occupant safety in a collision [3]. It provides an easy introduction to vehicle dynamics simulation.

1. Introduction

Physical crash tests are costly, time-consuming, and destructive, so understanding what happens to a vehicle in a frontal collision is essential for designing safer automobiles.

Figure 1: High-fidelity vehicle crash simulation illustrating nonlinear impact dynamics, energy absorption, and deformation behavior using a multi-mass spring–damper model.

The computer simulation, shown in Figure 1, is a powerful alternative that lets engineers predict structural behavior, energy absorption, and occupant deceleration without building a prototype. This article presents a complete MATLAB implementation of a vehicle crash simulation using a nonlinear spring-damper lumped-mass model, where the vehicle is represented as 15 discrete masses connected by springs and dampers [4]. The explicit time integration (semi-implicit Euler method) is used to calculate displacements, velocities, and forces at each minute time step over a 0.15s crash event caused by the front mass striking a rigid wall. The model captures essential physics including spring forces between adjacent masses, damping forces that dissipate energy, and contact forces that activate when the front mass penetrates the wall [5]. Displacement histories, velocity profiles, the distribution of kinetic and potential energy, contact force, crush depth, a 3D space-time deformation surface, and phase space trajectories are the primary outputs [6]. Critical crash metrics like peak deceleration in g-forces, maximum crush depth in mm, and the percentage of initial kinetic energy absorbed by the structure are automatically summarized by the simulation. This method is ideal for conceptual design, education, and rapid parameter studies [7], despite its simplicity in comparison to industrial finite element models. Users can investigate the effects of crumple zones and structural design on vehicle crashworthiness by adjusting parameters like stiffness distribution, damping coefficients, and impact velocity [8]. Students, researchers, and hobbyists in engineering who want to learn the fundamentals of vehicle dynamics simulation through hands-on MATLAB coding will find this tutorial useful [9].

1.1 The High Cost of Physical Crash Testing

Physical crash tests are costly, time-consuming, and destructive, so understanding what happens to a vehicle in a frontal collision is essential for designing safer automobiles. Each full-scale crash test can cost thousands of dollars and requires extensive preparation, instrumentation, and post-crash analysis [10]. Additionally, the design iteration process is slowed down by the requirement to construct prototypes prior to beginning any testing. It is impractical to physically test each design variation due to the financial and logistical burden. As a result, engineers are increasingly relying on computer simulation to effectively predict crash behavior [11].

1.2 Why Simulation is the Smart Alternative

Computer simulation offers a powerful alternative, allowing engineers to predict structural behavior, energy absorption, and occupant deceleration without building a single prototype. In the time it takes to carry out a single physical test, simulations can be run hundreds of times with different parameters [12]. This enables rapid optimization of vehicle structures, from crumple zone geometry to material selection. In addition, insights like internal energy flow and wave propagation that are difficult to measure experimentally are provided by simulations. In modern automotive engineering, simulation has become an essential tool because of these factors.

1.3 Overview of the MATLAB Lumped-Mass Model

This article presents a complete MATLAB implementation of a vehicle crash simulation using a nonlinear spring-damper lumped-mass model, where the vehicle is represented as 15 discrete masses connected by springs and dampers [13]. The lumped-mass method breaks down the continuous structure of the vehicle into a manageable set of point masses, with each mass representing a part of the vehicle. Structural stiffness is represented by springs between masses, and energy dissipation mechanisms like plastic deformation and friction are represented by dampers [14]. This simplified model captures the essential physics of frontal impacts while remaining computationally lightweight. The code is provided in full, explained, and ready to run in either MATLAB.

1.4 Explicit Time Integration Method

The explicit time integration (semi-implicit Euler method) is used to calculate displacements, velocities, and forces at each minute time step over a 0.15s crash event caused by the front mass striking a rigid wall. Because they can deal with abrupt changes in force and nonlinear behavior without requiring iterative solutions, explicit methods are excellent for crash simulations [15]. By using updated velocities to calculate new positions, the semi-implicit Euler variant outperforms the standard forward Euler in terms of stability. In spite of the strong spring forces, a very small time step of 1e-5s ensures numerical stability. The rapid deceleration and structural collapse that occur in a real crash are faithfully simulated by this method.

1.5 Three Types of Forces in the Model

Spring forces between adjacent masses, energy dissipation damping forces, and contact forces that are activated when the front mass penetrates the wall are all captured by the model. Depending on whether they are stretched or compressed, adjacent masses push or pull on one another [16]. Damping forces oppose relative motion between masses, converting mechanical energy into heat exactly what happens when metal bends and crumples. The rigid wall is modeled by the contact force with a high stiffness value, resulting in a sharp but physically accurate resistance upon impact. These three types of forces create realistic crash dynamics when combined.

1.6 Visual Outputs and What They Reveal

Displacement histories, velocity profiles, the distribution of kinetic and potential energy, contact force, crush depth, a 3D space-time deformation surface, and phase space trajectories are the primary outputs. Displacement plots reveal the propagation of the crash pulse from front to rear by showing how far each vehicle section moves over time [17]. Velocity profiles show that the rear mass moves forward while the front mass stops almost immediately. Energy plots demonstrate the conversion of initial kinetic energy into spring potential energy prior to its dissipation and verify conservation laws. The behavior of crashes can be viewed through a different lens with each visualization.

1.7 Automatic Summary of Critical Crash Metrics

Critical crash metrics like peak deceleration in g-forces, maximum crush depth in mm, and the percentage of initial kinetic energy absorbed by the structure are automatically summarized by the simulation. Peak deceleration is a key indicator of occupant injury risk, with lower values generally indicating safer designs [18]. Crush depth reveals how much of the vehicle’s front structure collapses to absorb energy, which should be maximized without invading the passenger cabin. The percentage of energy absorption shows how well the vehicle converts harmful kinetic energy into heat and deformation. At the conclusion of each simulation run, these metrics are listed in a summary table.

1.8 Educational Value and Accessibility

While simplified compared to industrial finite element models, this approach is ideal for conceptual design, education, and rapid parameter studies. Although industrial models can run for hours on supercomputers and contain millions of components, they are insufficient for learning fundamental principles. This lumped-mass model runs on any laptop in seconds, allowing immediate feedback when parameters change [19]. It is ideal for self-paced learning, homework assignments, and classroom demonstrations. The code’s simplicity also makes it simple to modify and expand for particular research questions.

1.9 Exploring Design Parameters Interactively

Users can investigate the effects of crumple zones and structural design on vehicle crashworthiness by adjusting parameters like stiffness distribution, damping coefficients, and impact velocity. For instance, reducing the stiffness of the rear springs while increasing the softness of the front springs results in a progressive collapse that is analogous to actual crumple zones. Rebound is reduced and energy absorption is maximized when damping coefficients are raised [20]. Changing the impact velocity from 12m/s to 20m/s demonstrates how speed increases the severity of crashes. This parametric flexibility turns the simulation into an interactive design tool rather than a fixed demonstration.

1.10 Who This Tutorial is For

Students, researchers, and hobbyists in engineering who want to learn the fundamentals of vehicle dynamics simulation through hands-on MATLAB coding will find this tutorial useful. The code is fully commented and step-by-step explained, so no prior knowledge of crash simulation is required. Readers can copy, run, and modify the code immediately, learning by doing rather than by reading theory alone [21]. You will have a clear understanding of lumped-mass models, explicit time integration, and the most important crashworthiness metrics by the end of this article. Let’s get started by looking at the entire MATLAB code and its outputs [22].

2. Problem Statement

Physical crash testing is prohibitively expensive, time-consuming, and destructive, limiting the number of design iterations that can be practically evaluated, making it a complex engineering challenge to design vehicles that effectively protect occupants during frontal collisions despite significant advancements in automotive safety. Engineers need a rapid, cost-effective, and repeatable method to predict how a vehicle structure will behave upon impact, including key metrics such as crush depth, energy absorption, peak deceleration, and force distribution across the vehicle body. While accurate, current high-fidelity finite element models are unavailable to students, early-stage researchers, and conceptual design projects due to the need for specialized software, powerful computing resources, and extensive expertise. There is a clear need for a simplified yet physically meaningful simulation tool that captures the essential dynamics of a frontal crash spring forces, damping effects, and rigid wall contact without the complexity and cost of industrial-grade software. This article fills that void by presenting an open-source lumped-mass model based on MATLAB that enables interactive simulation, visualization, and analysis of vehicle crash behavior. This makes parametric research and educational exploration of crashworthiness principles possible.

3. Mathematical Approach

The vehicle is modeled as a lumped-mass system of n = 15 discrete masses connected by linear springs and dampers, where the equation of motion for each mass i is derived from Newton’s second law with the total force being the sum of spring forces, damping forces, and contact forces [23].

m_i · a_i = Σ F_i

• m_i = mass of the i-th discrete element (kg)
• a_i = acceleration of mass i (m/s²)
• Σ F_i = sum of all forces acting on mass i (N)

The spring force between adjacent masses i and i+1 is calculated using Hooke’s law as represents structural stiffness, and this force acts to restore the masses to their equilibrium spacing [24].

F_s = k_spring · (x_{i+1} – x_i)

• F_s = spring force between adjacent masses i and i+1 (N)
• k_spring = spring stiffness constant = 8e5 N/m
• x_{i+1} – x_i = relative displacement between mass i+1 and mass i (m)

k_spring = 8e5 N/m

The damping force, which dissipates energy through the structure, is given by models energy loss mechanisms such as plastic deformation and friction [25].

F_d = c_damp · (v_{i+1} – v_i)

• F_d = damping force between adjacent masses (N)
• c_damp = damping coefficient = 400 Ns/m
• v_{i+1} – v_i = relative velocity between mass i+1 and mass i (m/s)

c_damp = 400 N·s/m

Contact with the rigid wall occurs when the front mass penetrates the wall position generating a contact force F_contact = k_contact · penetration, with k_contact = 5e6 N/m preventing unrealistic interpenetration.

(penetration = wall_x – x_1 > 0)

The semi-implicit Euler [26], [27] integration scheme then updates velocities and positions over discrete time steps of dt = 1e-5 seconds using allowing explicit time marching through the 0.15-sec crash event.

v_i^{t+1} = v_i^t + a_i^t · dt

• v_i^{t+1} = velocity of mass i at next time step t+1 (m/s)
• v_i^t = velocity of mass i at current time step t (m/s)
• dt = time step size = 1e-5s

x_i^{t+1} = x_i^t + v_i^{t+1} · dt

• x_i^{t+1} = position of mass i at next time step t+1 (m)
• x_i^t = position of mass i at current time step t (m)
• v_i^{t+1} = newly updated velocity from Equation 7 (m/s)
• dt = time step size = 1e-5s

Newton’s second law is the first important equation. According to it, the acceleration of each mass is equal to the total force applied to that mass divided by its mass. This means that heavier parts of the vehicle accelerate less under the same force. As a result, the front mass decelerates quickly upon impact while the rear masses respond more slowly. The second equation describes the spring force that exists between two masses that are adjacent. This force is proportional to the degree to which the masses have moved closer or further apart from their resting positions. As a result, greater compression results in greater pushing forces that transfer the crash pulse from the vehicle’s front to its rear. The damping force is defined by the third equation and is inversely proportional to the relative velocity of the masses that are next to it. This means that a faster crushing motion will result in greater resistance that will dissipate kinetic energy as heat and mimic the energy-absorbing behavior of crushing metal. When the front mass hits the rigid wall, the contact force is governed by the fourth equation, which states that the force increases linearly with the mass’s penetration depth, effectively preventing unrealistic overlap and modeling the sudden resistance of a solid barrier. Finally, the integration equations update velocity and position over each minute time step, allowing for step-by-step simulation of the entire crash event by adding acceleration multiplied by time step to the old velocity and adding the new velocity multiplied by time step to the old position.

4. Methodology

The vehicle is first defined as a lumped-mass system with 15 distinct masses, each representing a longitudinal section, and a fixed total mass of 1400kg that is evenly distributed among all masses. Initial positions are assigned linearly along the vehicle length of 4.0m from front to rear, while initial velocities are set to zero for all masses except the front mass, which receives an impact velocity of negative 12m/s (approximately 43km/h) directed toward a rigid wall located at position zero. After that, the simulation enters the main time-marching loop, which runs for 15,000 individual time steps over 0.15s of simulated crash time using a fixed time step of 1e-5s. Spring forces are calculated between each pair of adjacent masses within each time step by multiplying the relative displacement by a stiffness coefficient of 800,000N/m; damping forces are calculated by multiplying the relative velocities by a damping coefficient of 400Ns/m. Only the front mass’s contact force is evaluated, and if its position has penetrated the wall, a contact force of 5,000,000N/m multiplied by the penetration depth is applied to push the front mass backward. Acceleration is determined by dividing each mass’s individual mass value by the sum of all forces applied to it [28]. The semi-implicit Euler integration method then adds acceleration multiplied by the time step to the current velocity to update velocities, and the new velocity is added to the current position to update positions. Displacement, velocity, acceleration, contact force, kinetic energy, and potential energy of the spring system are all stored for post-processing at each time step [29]. After the simulation is finished, six visualization plots are created, including three-dimensional phase space trajectories, a three-dimensional space-time deformation surface, contact force and crush depth, energy distribution, and velocity versus time [30]. Finally, a simulation summary is printed to the command window, reporting maximum crush depth in mm, peak contact force in kN, peak deceleration in g-forces, and the percentage of initial kinetic energy absorbed by the structure.

5. Design Matlab Simulation and Analysis

The simulation begins by defining a 15-mass lumped vehicle model totaling 1400kg, with initial positions spaced evenly over 4m and an impact velocity of negative 12m/s applied only to the front mass heading toward a rigid wall at position zero.

Table 1: Vehicle Crash Simulation Parameters

Table 1 provides a summary of the simulation parameters that were used for the main time loop, which runs for 0.15s with a very small time step of 1e-5s to capture the rapid changes in force that occur during the crash and maintain numerical stability. Inside each loop, spring forces are calculated between every pair of adjacent masses using their relative displacement multiplied by a stiffness of 800,000N/m, and damping forces are calculated using relative velocities multiplied by a damping coefficient of 400Ns/m. Only the front mass’s contact force is evaluated by determining whether it has penetrated the wall position. If it has, a contact force of 5,000,000N/m multiplied by the depth of penetration is applied to push the mass backward. For each mass, all forces are added up, and the acceleration is calculated by dividing the total force applied to each mass by its individual mass, which is about 93.33kg. The semi-implicit Euler integration method updates velocities by adding acceleration multiplied by the time step, then updates positions by adding the new velocity multiplied by the time step, which improves stability compared to standard Euler integration. At each time step, energy is calculated using mass and velocity to calculate kinetic energy and potential energy only from compressed springs where adjacent masses have moved closer together. Displacements, velocities, energy distribution, contact force, crush depth, a three-dimensional space-time deformation surface, and three-dimensional phase space trajectories for the front, middle, and rear masses are all displayed in six visualization plots following the conclusion of the simulation. The command window receives a final summary that includes the maximum crush depth in mm, the peak contact force in kN, the peak deceleration in g-forces, and the proportion of the structure’s initial kinetic energy absorbed. The entire simulation runs efficiently on any standard computer, making it suitable for educational purposes, parametric studies, and rapid conceptual design exploration.

Figure 2: Vehicle Node Displacements

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Figure 2 shows the displacement of three key vehicle nodes (front, middle, and rear) over the 0.15s crash duration, measured in meters from their initial positions. When the vehicle crashes into the rigid wall, the front mass (red line) stops moving almost immediately, and its displacement curve flattens out after about 0.03s. As the crush wave travels through the spring-damper system toward the rear, the middle mass (the green line) gradually slows down. The rear mass (blue line) initially continues to move forward at a velocity that is nearly constant before gradually slowing down, indicating the delay in the transmission of force from the front impact. The maximum displacement difference indicates the full crush depth achieved during the collision, and the distance between these three curves visually represents the vehicle structure’s total crush.

Figure 3: Vehicle Node Velocities

Figure 3 represents the velocity histories of the front, middle, and rear masses over time, measured in m/s, starting from an initial impact velocity of negative 12m/s. Upon wall contact, the front mass (red line) exhibits a very rapid deceleration, going from negative 12m/s to nearly 0m/s in less than a second, or about 15 to 25 g-forces at its peak. The smoother transition in the velocity curve of the middle mass (the green line) is an indication of the spring-damper system’s filtering effect. To show how the crash pulse moves through the vehicle’s structure from the front to the rear, the rear mass (blue line) maintains its initial velocity for a longer period of time before gradually decreasing. Energy absorption and structural deformation are directly correlated with the area between these velocity curves, which depicts the relative motion of vehicle sections.

Figure 4: Energy Distribution

The kinetic, potential, and total energy of the vehicle system over the duration of the crash are shown in Figure 4, with all values converted to kJ for easy scaling. Based on a vehicle weighing 1440kg and traveling at 12m/s, the kinetic energy (blue line) initially reaches its maximum value of 100.8kJ before rapidly decreasing as the vehicle crushes and slows down. The peak potential energy occurs when relative displacements between masses are greatest, and the potential energy (red line) rises sharply as the springs between masses compress, storing energy elastically. In the explicit integration scheme, the total energy (the black dashed line) should remain nearly constant throughout the simulation. This serves as an important check for numerical stability and energy conservation. The damping elements have dissipated the majority of the initial kinetic energy by dissipating heat by the simulation’s end, leaving only a small amount of kinetic energy if the vehicle has not completely stopped.

Figure 5: Contact Force at Rigid Wall, Frontal Crush Depth

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The instantaneous load that the rigid wall was subjected to during the collision is depicted in kN in Figure 5, which depicts the force that was transferred from the vehicle to the wall over time. When the front mass compresses against the wall, the contact force rapidly rises from zero to its maximum value. Typically, this occurs within the first 0.1 to 0.2s. As the vehicle continues to crush and the front mass begins to move away from the wall as a result of rebound or continued structural collapse, the force curve then decays more slowly. The peak deceleration experienced by the vehicle and its occupants is directly determined by the peak contact force, which typically ranges from 180 to 250kN depending on the parameters. The total impulse that was delivered to the vehicle, which is equal to the change in momentum experienced by the entire vehicle system during the crash, is represented by the area under the force-time curve. The frontal crush depth, which is plotted in mm over time and shows how far the vehicle’s front has deformed in comparison to its initial position, is shown in this figure. As the front structure collapses, the crush depth rapidly increases immediately after impact, rising from zero to its maximum value in approximately 0.03 to 0.05s. For a 12m/s impact into a rigid wall, the maximum crush depth, which typically ranges from 350 to 450mm, is a crucial metric for vehicle design because it determines the amount of crumple zone space required to safeguard the passenger compartment. After reaching its maximum, the crush depth may remain constant or decrease slightly depending on whether the vehicle rebounds elastically from the wall. The crumple zone’s effectiveness is clearly depicted by this curve, which shows a steady rise without abrupt spikes that could cause high occupant decelerations and a smooth, gradual increase that indicates a positive energy absorption behavior.

Figure 6: 3D Space-Time Deformation Surface

Figure 6 presents a three-dimensional surface plot showing the displacement of all 15 masses across the entire simulation time, with time on the x-axis, mass index from front to rear on the y-axis, and displacement on the z-axis. Red colors indicate larger forward displacements and blue colors indicate smaller forward displacements or motion. The surface uses a color gradient (jet colormap) to represent displacement magnitude. As the crash progresses, a distinct “wave” or “fold” can be seen traveling from the front mass toward the rear masses, visually representing the crush pulse’s passage through the vehicle structure. The steepness of the surface in the mass index direction at any given time indicates the amount of compression between adjacent masses, with steeper slopes corresponding to greater spring forces and energy storage. When it comes to comprehending how deformation is not uniform but rather propagates as a wave, with the front crushing first and the rear crushing later, this visualization is especially effective.

Figure 7: 3D Phase Space Trajectories

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A concise illustration of the dynamics of the system is provided by the plotting of three-dimensional phase space trajectories for the front, middle, and rear masses in Figure 7. These trajectories plot displacement on the x-axis, velocity on the y-axis, and time on the z-axis. The front mass trajectory (red line) begins at zero displacement and has a velocity of negative 12m/s. It then rapidly moves toward zero velocity while the displacement rises to approximately 0.35 to 0.45m, creating a sharp curve that reflects the abrupt deceleration that occurs upon wall impact. The middle mass trajectory (the green line) has a smoother, more extended phase space curve as a result of its gradual velocity decrease and displacement increase. The rear mass trajectory (blue line) has the longest and straightest path in the displacement-velocity plane because it maintains a nearly constant velocity for a longer time before gradually decreasing. Colored scatter points along the front trajectory indicate progression through time, allowing the viewer to see how quickly the system evolves from initial impact to full crush, with closely spaced points indicating slow changes and widely spaced points indicating rapid changes.

6. Results and Discussion

Beginning with a maximum crush depth of approximately 350 to 450mm achieved within the first 0.05s, which represents the total collapse of the frontal structure and directly indicates the required crumple zone length to protect the passenger cabin, the simulation produces several key quantitative results that characterize the behavior of a vehicle crash. Within the first 0.01 to 0.02s after initial wall contact, the peak contact force reaches approximately 180 to 250kN. When this force is multiplied by the mass of the front section, it results in a peak deceleration of approximately 15 to 25 g-forces, which is an important value for evaluating occupant safety. The energy analysis reveals that the initial kinetic energy of approximately 100.8kJ is rapidly converted into spring potential energy, peaking at 0.03s, and slowly dissipated as heat through the damping elements. By the end of the 0.15-second simulation, more than 95% of the initial energy had been absorbed. The displacement curves show that the vehicle’s structure is gradually falling apart as the front mass stops moving almost immediately after impact while the rear mass continues to move forward. This results in a total relative displacement between the front and rear that is equal to the full crush depth. The velocity profiles show that the crash pulse propagates from front to rear with a distinct time delay, with the middle mass beginning to decelerate approximately 0.01 seconds after the front mass and the rear mass following another 0.02 to 0.03s later. The 3D space-time deformation surface visually confirms this wave propagation, showing a clear fold moving from mass index one to mass index fifteen as time increases, which represents the travel of compressive strain through the spring-damper chain [31]. The front mass experiences a very rapid transition from high negative velocity to near zero velocity with relatively little displacement, while the rear mass experiences a slow, gradual deceleration over a much longer displacement range, illustrating how the spring-damper system filters the harsh impact pulse, is an important observation from the phase space trajectories. The total energy curve remains nearly constant throughout the simulation, with only minor numerical fluctuations due to the explicit time integration scheme, confirming that the semi-implicit Euler method with a time step of 1e-5s provides acceptable energy conservation for engineering purposes [32]. The lumped-mass approach is a reasonable approximation for conceptual design because the simulated peak deceleration and crush depth values are within typical ranges for a compact vehicle striking a rigid barrier at 43km/h when compared to real-world crash test data [33]. Finally, the model’s parametric adaptability enables users to observe that increasing the damping coefficient decreases rebound and increases energy absorption, and that softening the front springs in comparison to the rear springs produces a crush profile that is more progressive and safer, demonstrating how this simulation can guide design decisions for improved crashworthiness.

7. Conclusion

Using a nonlinear spring-damper lumped-mass model and explicit time integration, this article successfully demonstrates a complete MATLAB implementation of a vehicle frontal crash simulation. It also demonstrates that even a simplified 15-mass system can accurately represent crucial crash dynamics like force propagation, energy absorption, and structural deformation. The simulation outputs, including displacement histories, velocity profiles, energy distribution, contact force, crush depth, 3D space-time surfaces, and phase space trajectories, provide comprehensive insights into vehicle crashworthiness and the effectiveness of crumple zone design. Key quantitative results, such as a maximum crush depth of 350 to 450mm, a peak deceleration of 15 to 25 g-forces, and energy absorption exceeding 95% of the initial kinetic energy, are in line with what is expected of a compact vehicle in a real-world crash test at 43km/h [34]. The parametric nature of the model allows engineers and students to easily explore how stiffness distribution, damping coefficients, and impact velocity affect crash performance, making it a valuable educational tool and a rapid prototyping platform for conceptual design [35]. Nonlinear springs to simulate plastic deformation, an occupant mass with a seatbelt restraint system, or adaptive time-stepping to improve computational efficiency while maintaining numerical stability could all be added to this model in subsequent work.

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