Off-Road Vehicle Suspension System Design for Obstacle Traversal

A Classical Mechanics Engineering Problem (Quarter-Car Model)

Problem Statement

An automotive engineering firm is developing a heavy-duty off-road vehicle designed to traverse challenging terrain while maintaining passenger safety and comfort. The vehicle has the following specifications (representative of real-world UTVs and off-road vehicles):

Vehicle Specifications

• Total vehicle mass: $m = 900\ \text{kg}$
• Weight distribution: $60\%$ front / $40\%$ rear
• Front corner mass: $m_f = 270\ \text{kg}$ per wheel
• Rear corner mass: $m_r = 180\ \text{kg}$ per wheel
• Wheelbase: $L = 1.6\ \text{m}$
• Front track width: $1.26\ \text{m}$
• Rear track width: $1.39\ \text{m}$
• Suspension type: Double wishbone independent suspension (front and rear)

Consistency check (mass distribution):
Front axle mass $=0.6\cdot 900=540\ \text{kg}\Rightarrow 270\ \text{kg/wheel}$.
Rear axle mass $=0.4\cdot 900=360\ \text{kg}\Rightarrow 180\ \text{kg/wheel}$.
This matches the given corner masses.

Design Requirements

The vehicle must safely traverse a 100 mm ($0.1\ \text{m}$, 3.94 in) tall obstacle at various speeds while meeting the following criteria (typical of industry targets):

• Human comfort limit (ISO 2631-1): Maximum passenger vertical acceleration $$a_{\max} \le 1.5g = 14.7\ \text{m/s}^2$$
• Suspension travel requirement: Minimum 50 mm total suspension travel in each direction (compression and rebound)
• Ride frequency target (front): $1.5\text{–}2.0\ \text{Hz}$ (off-road applications)
• Damping ratio target: $\zeta = 0.3\text{–}0.5$ (comfort + control balance)

Questions

1. What is the maximum safe forward velocity at which the vehicle can traverse the $100\ \text{mm}$ obstacle without exceeding the $1.5g$ acceleration limit on passengers?
2. What spring stiffness (spring rate) is required for the front suspension to achieve a target natural frequency of $1.8\ \text{Hz}$, assuming a motion ratio of $\text{MR}=0.54$ (typical for double wishbone)?
3. What damping coefficient is needed for the shock absorbers to achieve a damping ratio of $\zeta = 0.4$?
4. What is the maximum force experienced by the suspension components when the vehicle strikes the obstacle at the calculated maximum safe velocity?
5. How much suspension compression travel is required to absorb the impact, and how does it compare to the 50 mm minimum travel requirement?

Analytical Reasoning

Before diving into mathematical formulation, we must understand the physical phenomena at play.

Energy Transformation

When the vehicle encounters a step obstacle, the wheel must rise by the obstacle height. This sudden vertical displacement injects energy into the suspension system, which must be handled via:

• Spring compression: elastic potential energy storage
• Damper dissipation: conversion of kinetic energy to heat via viscous damping

Two-Phase Impact Analysis

Phase 1 — Initial impact (road input):
The tire contacts the obstacle and the wheel center trajectory rapidly changes. A forward velocity $v$ generates an effective vertical velocity component $v_z$ associated with climbing the step geometry.

Phase 2 — Energy transfer to the body (sprung mass response):
The suspension transmits forces to the chassis. The peak acceleration experienced by passengers depends primarily on:

• Impact input severity (captured here by $v_z$)
• Effective stiffness at the wheel (wheel rate)
• Damping characteristics
• Corner (quarter-car) mass distribution

Critical Physics Considerations

• Geometric constraint: For a wheel of radius $R$ to climb a step of height $h$, the geometry sets a rapid change in direction at the contact point; higher $v$ increases the severity of the vertical input.
• Force amplification: Transient events can create forces substantially above static wheel loads; damper forces can dominate during rapid compression events.
• Natural frequency selection: The suspension natural frequency should avoid resonance with road input frequencies and human sensitivity bands.
• Motion ratio effect: With $\text{MR}=0.54$, the spring/damper sees reduced motion relative to the wheel; thus the hardware rates must be higher than the wheel-equivalent rates by approximately $1/\text{MR}^2$.

Mathematical Formulation and Resolution

Modeling Assumptions (Quarter-Car)

This analysis uses a single-degree-of-freedom (SDOF) quarter-car approximation at a front corner:

• Corner mass $m_f$ represents the effective sprung mass supported by one front wheel.
• The suspension is modeled as a linear spring and linear viscous damper at the wheel (wheel-equivalent), mapped to the shock via motion ratio.
• Tire stiffness, unsprung mass, and detailed tire–step contact dynamics are neglected (first-order design sizing).

Step 1 — Determine Required Front Spring Stiffness

The natural frequency of the suspension at the wheel is:

where $k_{\text{wheel}}$ is the wheel rate (effective stiffness at the wheel).

The relationship between spring rate and wheel rate through motion ratio $\text{MR}$ is:

Therefore:

Given:

• $f_n = 1.8\ \text{Hz}$
• $m_f = 270\ \text{kg}$
• $\text{MR}=0.54$

Compute:

Unit conversion:

Wheel rate:

Step 2 — Calculate Required Damping Coefficient for $\zeta = 0.4$

Critical damping for an SDOF system is:

Using $k_{\text{wheel}}\approx 34536\ \text{N/m}$ and $m_f=270\ \text{kg}$:

For target damping ratio $\zeta=0.4$:

Motion ratio mapping to the shock hardware (same $\text{MR}^2$ scaling):

Step 3 — Obstacle Impact Kinematics (Effective Vertical Velocity)

For a wheel of radius $R$ climbing a step of height $h$, a simple geometric approximation for the instantaneous vertical velocity component is:

Assume a typical off-road tire radius:

Step 4 — Maximum Acceleration Constraint and Maximum Safe Velocity

Peak compression estimate (underdamped SDOF with initial velocity input)

With damping ratio $\zeta$, the damped natural frequency factor is:

Using $\omega_n=11.3097\ \text{rad/s}$ and $\zeta=0.4$:

Approximate maximum compression from an initial velocity $v_z$:

Peak force at the wheel

At peak response (approximation), wheel force is the sum of spring and damper contributions:

Substitute $x_{\max}=v_z/10.3655$:

Compute the coefficient:

So:

Acceleration limit

Passenger vertical acceleration is constrained by:

Convert to forward velocity using $v_z\approx 0.571v$:

Step 5 — Suspension Travel Required (Compression)

Comparison to the minimum travel requirement (50 mm compression):

• Required compression at the safe speed: $\approx 66\ \text{mm}$
• Minimum requirement stated: $50\ \text{mm}$ compression

A suspension designed with only $50\ \text{mm}$ compression travel would bottom out before meeting the $1.5g$ objective at that speed (so either travel must increase or speed must decrease).

Step 6 — Maximum Forces on Components at the Safe Speed

At $v_{z,\max}=0.6873\ \text{m/s}$ and $x_{\max}=0.0663\ \text{m}$:

Spring force

Damper force (wheel-equivalent)

Total peak force

Factor above static load

Static load at the front corner:

Load factor:

Conclusions

Based on a classical mechanics quarter-car model (spring–damper SDOF mapped through motion ratio), the results are:

1. Maximum Safe Velocity (1.5g limit):

   $$\boxed{v_{\max}\approx 1.20\ \text{m/s} \approx 4.33\ \text{km/h} \approx 2.69\ \text{mph}}$$

   This is consistent with typical crawling speeds for technical off-road obstacles.

2. Required Spring Stiffness (front, $f_n=1.8\ \text{Hz}$, MR=0.54):

   $$\boxed{k_{\text{spring}} \approx 118.4\ \text{kN/m} \approx 676\ \text{lb/in}}$$ $$\boxed{k_{\text{wheel}} \approx 34.5\ \text{kN/m}}$$

3. Required Damping Coefficient ($\zeta=0.4$):
   Wheel-equivalent:

   $$\boxed{c_{\text{wheel}} \approx 2.44\times 10^3\ \mathrm{N \cdot s/m}}$$

   Shock hardware (mapped through MR):

   $$\boxed{c_{\text{shock}} \approx 8.38\times 10^3\ \mathrm{N \cdot s/m}}$$

4. Maximum Component Forces at Safe Speed:

   $$\boxed{F_{\text{total}} \approx 3.97\ \text{kN} \approx 892\ \text{lbf}}$$

   This corresponds to roughly $1.5\times$ the static corner load.

5. Suspension Compression Travel Required:

   $$\boxed{x_{\max}\approx 66\ \text{mm}}$$

   Therefore, to meet the $1.5g$ requirement at $v_{\max}$, the design must provide at least $\sim 66\ \text{mm}$ compression travel. The stated $50\ \text{mm}$ minimum is not sufficient at that speed (unless the speed is reduced further or additional compliance is introduced).

Engineering Implications (Design Takeaways)

• The low safe speed for a 100 mm step illustrates why off-road vehicles must crawl over sharp obstacles when passenger acceleration limits are strict.
• Motion ratio strongly drives hardware sizing: spring/damper hardware rates scale approximately with $1/\text{MR}^2$.
• To increase obstacle-crossing speed while keeping acceleration limits, typical options are:
  • Increased available travel (and/or progressive-rate springs)
  • Tire compliance modeling and tuning (effective first-stage "spring")
  • Semi-active/active damping strategies
  • Accepting higher peak accelerations beyond comfort thresholds

This problem demonstrates how classical mechanics—Newton's laws, harmonic oscillator dynamics, and motion ratio mapping—directly informs practical suspension design tradeoffs between performance, comfort, and constraints.