# The Simulation of an Automotive Air Spring Suspension Using a Pseudo-Dynamic Procedure

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## Abstract

**:**

## 1. Introduction

^{®}for the purpose. The objective was to obtain the natural frequencies of a suspension for several internal pressures, verify the non-linear behavior, and implement linearization methods to solve the problem of the evaluation of natural frequencies.

^{®}air spring suspension.

## 2. The Air Spring Stiffness

#### 2.1. The Displacement Field of the Toroidal Shell

- r is the meridian radius
- θ is the meridian angle
- R is the longitudinal (or circumferential) radius
- w is the transverse (or normal) displacement
- v is the meridian (membrane) displacement. This displacement does not participate in the simplified model to calculate the work done by the internal pressure in the shell deformation.

**e**must be calculated. Then, the volume of the shell is obtained with the theorem of Pappus–Guldin (in the book of Tom M. Apostol [6]) and integrating the following function for the meridian angular variable θ in interval [0, π]:

**e**. Volume increment is due only to transverse displacement w obtained after subtracting from Equation (1) a similar expression but with w = 0:

#### 2.2. The Pneumatic Stiffness

- (a)
- The first model assumes that only the compressed air participates as an elastic body in the evaluation of the spring stiffness. The shell torus enclosing the air does not contribute to the suspension stiffness;
- (b)
- The second model consists in superimposing the structural stiffness of the shell torus to the one of the air under compression, as two springs working in parallel.

_{a}in Equation (7) changes accordingly with a constant value of the stiffness in Equation (12):

_{a}depends on the air pressure, it is assumed to be constant during the displacement increment referred above. Equation (13) is thus a linearized form to update the structure configuration past an iteration Δa. The next section describes the linearization process of the iterative algorithm to obtain the solution for the structural deformation.

_{shell}and associated tangent stiffness K

_{T-shell}due to a displacement of magnitude a (the only unknown in Equations (3) and (4)) of the axisymmetric shell are:

#### 2.3. The Equilibrium Equation: The Iterative Newton–Raphson Method

- A continuous function F(a) can be written for an increment of its independent variable a, as F(a + Δa) ≅ F(a)+Δt × F’(a) from the Taylor’s formula up to a first-order expansion. Naming F(a + Δa) = F
_{n+}_{1}and F(a) = F_{n}; a + Δa = a_{n+}_{1}and a = a_{n}, then, the previous approach is equivalent to: F_{n+}_{1}− F_{n}≅ (a_{n+}_{1}− a_{n}) × F’(a). This formula models a linearized evolution of the structure constitutive relation for displacements inside the interval [a, a + Δa]. Graphically, this equation corresponds to a tangent line at a point of the curve that effectively represents the constitutive behavior of a structure, eventually with elastic, yet also represents the non-linear behavior of the internal reaction force vector from a prescribed displacements vector, as sketched in Figure 4. As point F(a + Δa) here obtained does not correspond to the effective reaction of the structure (this one given by Equation (7)), there is a residual difference to the real value, which is a parameter suitable to control the iteration pitch, as depicted in Figure 4. - The points labeled with “L” result from the linear approach, while points from initial internal reactions are labeled with “R” (Figure 4). Past the iteration, the linearly obtained next value is compared with the effective one, where the difference of effective and linear points is the residual value for the precision control of the iteration. This process is schematically represented by arrows in Figure 4, and is concluded when the final linear value of the internal reaction vector is close to the real one (the external force) by an error smaller than the prescribed tolerance.

#### 2.4. Example of Loading a Single Cell Bellows of a Pneumatic Suspension

- Radius of the meridian line r = 50 mm
- Radius of the circumference line (passing through the center of the transverse meridian circumference) R = 200 mm
- Initial pressure P = 2 bars (≅0.2 MPa)

## 3. Pseudo-Dynamic Simulation of the Automotive Suspension

#### 3.1. Applications to the Dynamic Analysis of Automotive Suspensions

_{0}(t) to the wheel axle (assuming here that the elasticity of the tire is much larger than the spring one), while the equivalent mass M of the vehicle body is displaced U

_{1}(t). The dynamic equilibrium of the spring/mass model of Figure 6b is:

_{spring}and F

_{damper}are the spring and the damper reaction forces due to the relative displacement and velocity between the road profile and the wheel hub. Supposing that the spring has a linear elastic behavior and the damper reaction force is also linear-dependent on the relative velocity, then Equation (18) can be written as:

_{t}is the spring stiffness (in N/m), and C is the damping constant (in Ns/m), which is a parameter assigned to viscous-type dampers. Constant C has a stable value when the damper is in good condition and is operated at a stabilized temperature. In the present study, a constant value of C is assumed.

_{equiv}(t) on assuming a virtual profile for a road where the wheel/suspension set travels at a prescribed horizontal velocity. This equation must be integrated numerically with a time integration algorithm as explained below, but for now, it is important to note that the following term on the right-hand side of Equation (20) corresponds to the spring internal reaction force to a prescribed displacement U

_{1}(t) at any instant t:

_{t}can be measured with a load cell inserted between one end of the spring and the ram of the displacement actuator mounted in the test rig. The evaluation of the spring stiffness K

_{t}at each time step t is carried out upon recording the consecutive values of the internal reaction force, respectively, at instants t and +Δt calculating the tangent stiffness K

_{t}

_{+Δt}by Equation (22):

#### 3.2. The Direct Time Integration Algorithm: The Newark Constant Acceleration Method. Application to Pseudo-Dynamic Methods

**R**} is the structure internal reaction vector, and [C] and [M] are, respectively, the damping (viscous type) and mass matrices, which are assumed constant at time step t and during the time increment Δt, while on the right-hand side, {

**F**(t)} is the external force vector. In linear systems, the internal reaction vector is the result of the matrix and vector product:

_{t}(from (11-d)) can be assumed to be constant during a deformation increment associated with the time step Δt, in order to develop Newmark’s [9] method, where the stiffness matrix in Equation (25) corresponds to a tangent stiffness during the time increment Δt.

#### 3.3. Description of the Method and Application to the Dynamic Analysis

#### 3.3.1. Operation Principles

#### 3.3.2. Engineering Design of the Rig and Applications

- The number of degrees of freedom to obtain accurate results may be predictably high, a less favorable attribute in the design of pseudo-dynamic test rigs;
- The size and engineering complexity of the equipment for pseudo-dynamic testing are not remarkable advantages considering that a real-time dynamic analysis test rig records internal reactions and damping forces. In turn, with pseudo-dynamic methods, viscous damping forces have to be numerically inserted in the time integration algorithm, as referred above.
- The dynamic analysis of some components in the automotive industry cannot be approached only with a set of time-dependent displacements, since the inclusion of nodal rotations in discretized models is necessary. This requirement makes the adequate design of the test rig more expensive, with an additional need of implementing angular transducers, more elaborate test-specimen fixtures, and specific load cells, which is an option that may not guarantee accurate results.

- Sprung mass Ms = 340 kg;
- Damping Constant Cs = 2.25 kNs/m
- Spring Stiffness Ks = 31.2 kN/m.

_{0}= 8 ms

^{−2}, at t = 0. Malmedahl [13] probably chose the trigger instant for acceleration time start not necessarily at t = 0, but rather slightly anticipated, as shown in Figure 8. The external impulsive force was computed from the initial acceleration as F

_{0}= M

_{s}× a

_{0}(t = 0) = 340 × 8 = 2720 N.

#### 3.3.3. Testing a Bellows Air Spring under Dynamic Load Using a Pseudo-Dynamic Procedure

^{®}(from Kyoto, Japan) test machine (Figure 10) with 50 KN of capacity force. For safety reasons and to avoid exceeding the machine capacity, the spring displacement was restricted to an amplitude of 20 mm.

- Torus radius r = 50 mm
- Radius of circle passing through the transverse sections radius: R = 200 mm
- Initial pressure (preload pressure) P
_{0}= 2 bar (≅0.2 MPa) - External force F(t) = 10 KN (suddenly applied as a step load)
- Virtual mass of the SDOF system M = 50 Kg

_{max}= 0.0066 s). The spring stiffness ranges from about K

_{min}= 191280 N/m (first time step) up to K

_{max}= 473911 N/m (time step for maximum displacement). It is noted that the spring stiffness here calculated refers to the effect of the axial displacement (equal to the double of the ovalization amplitude a) on the internal reaction force due to the air pressure as mentioned above:

_{mean}= 332595 N/m.

_{max}= 2 × U

_{static}= 2 × (10,000/332,595) = 0.0601 m, which is a result close to the amplitude vibration calculated with the Newark algorithm as observed in the graphical results of Figure 11. The results show the accuracy of the method, where the amplitude of the vibrating system is double that of the static deflection. Also, the natural period T

_{0}of this SDOF model (calculated with the averaged stiffness of the spring) is T

_{0}= 0.077 s, which is also in good agreement with the numerical result in Figure 11.

^{®}(SIMULIA-DASSAUL SYSTÈMES

^{®}—from Vélizy-Villacoublay, France). The compression was carried out in an isothermal regime with an initial pressure of 4 bars.

^{®}simulation, the shell container was modeled using three-node quadratic axisymmetric shell elements (SAX2). Three integration points across the thickness were used for all of the layers i.e., the 0.40-mm thick outer rubber layers and the inner 0.20-mm thick steel layer. Rubber and steel were assumed to be elastic and isotropic with moduli equal to 0.1 GPa and 200 GPa, respectively. The Poisson’s ratios were 0.48 for the rubber and 0.30 for the steel. Geometrically non-linear analyses were conducted for the imposed internal pressure values. As observed, there is a good agreement of both results, despite a very small difference obtained with ABAQUS

^{®}.

#### 3.4. Pseudo-Dynamic Test of a Real Pneumatic Spring: Procedures and Results

^{®}(Emerson Industrial Automation, www.numatics.com), is performed with the Shimadzu test machine, as shown in Figure 10. The double bellows tangent stiffness is calculated from the ratio of an incremental internal reaction R

_{t}(measured with the load cell of the test machine) by the respective incremental relative displacement of both bellows flanges at each time step. The main problem data for the dynamic procedure were:

- A pneumatic double-bellows type spring (Numatics-Emerson
^{®}) having a rigid end flanges diameter of 160 mm. Internal pressure is p_{0}= 4 bar; - A virtual mass M = 100 Kg (no damping considered);
- An impulsive step force F(t) = 1500 N sustained after instant t = 0
^{+}s; - No damping was considered in the dynamic model.

_{mean}= 142,000 N/m. For the virtual mass M = 100 Kg, the natural period is T

_{0}= 0.166 s.

_{0}≅ 6.2 l (0.0062 m

^{3}). The dynamic response for this SDOF system was obtained with a total Newmark’s non-linear solution, where the spring tangent stiffness was calculated at each time step upon changing the internal pressure according to the adiabatic equation.

_{mean}= 142,000 N/m) showed that 21 mm was the maximum value. Although acceptably close, repeated cycles (with the pseudo-dynamic procedure) revealed a progressive degradation of the agreement between the experimental and theoretical results.

## 4. Conclusions

- A simple yet accurate shell model was developed by variational techniques based on the minimum deformation energy concept and essential thermodynamics of gases. For enhanced accuracy, the influence of the velocity of the deformation was included by selecting isothermal or adiabatic compression regimes for the air suspension.
- The present model predictions agreed very well with a finite element model constructed in a commercial code with axisymmetric high order shell finite elements.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Axisymmetric toroidal shell element under internal pressure: displacement field by axial loads.

**Figure 5.**Internal reaction vs axial displacement (in compression) of the pneumatic spring with isothermal and adiabatic (γ = 1.3) processes. Initial internal pressure p

_{0}= 2 bar (displacements indicated are half of each prescribed spring stroke for symmetry).

**Figure 6.**Sketch of a vehicle suspension and the equivalent dynamic model for a case of passive transmission: (

**a**) drawing; and (

**b**) symbology.

**Figure 8.**Pseudo-dynamic test rig for structural validation of bus passenger seats (figure not included in the work of Pinto et al. [12]).

**Figure 9.**Acceleration at the wheel of a vehicle tire and suspension set hitting the soil suddenly from a small height. Points were collected from Malmedahl [13] having shifted them of about 0.1 s so that the starting point could match approximately the corresponding results here evaluated.

**Figure 10.**Mounting of the bellows pneumatic accessory in a SHIMADZU

^{®}testing machine between the working table and upper crosshead with an adaptor ram and a pressure air control unit.

**Figure 11.**Internal reaction versus axial displacement of the pneumatic spring analyzed. (

**a**) Displacement versus time after application of an impulsive step load F

_{0}= 10 KN; (

**b**) Force/displacement relation of the pneumatic spring.

**Figure 12.**Spring response from prescribed axial displacements with the presented method and ABAQUS

^{®}(Axisymmetric shell elements with ABAQUS

^{®}, with geometry similar to the previous example but initial pressure = 4 bar).

**Figure 13.**Results for the vibration of a single degree of freedom (SDOF) system with the pneumatic spring in analysis: (

**a**) with Newmark’s method (natural period is ≅0.180 s); and (

**b**) with a pseudo-dynamic simulation (natural period is ≅0.175 s).

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

De Melo, F.J.M.Q.; Pereira, A.B.; Morais, A.B.
The Simulation of an Automotive Air Spring Suspension Using a Pseudo-Dynamic Procedure. *Appl. Sci.* **2018**, *8*, 1049.
https://doi.org/10.3390/app8071049

**AMA Style**

De Melo FJMQ, Pereira AB, Morais AB.
The Simulation of an Automotive Air Spring Suspension Using a Pseudo-Dynamic Procedure. *Applied Sciences*. 2018; 8(7):1049.
https://doi.org/10.3390/app8071049

**Chicago/Turabian Style**

De Melo, Francisco J. M. Q., António B. Pereira, and Alfredo B. Morais.
2018. "The Simulation of an Automotive Air Spring Suspension Using a Pseudo-Dynamic Procedure" *Applied Sciences* 8, no. 7: 1049.
https://doi.org/10.3390/app8071049