June 9, 2023

# Virtual optimization of suspension parameters using Adams Car and pSeven

*Industry: Automotive | Product: pSeven | Company: MSC Software*

## Abstract

Benefits of integration between Adams Car by MSC Software, part of Hexagon's Manufacturing Intelligence division, and pSeven are demonstrated via an industrial use case involving the optimization of an automotive suspension damper.

In modern adaptive suspension systems, damping properties can be adjusted on the fly, depending on road conditions and driver preferences. An optimal balance between ride comfort and vehicle handling however needs to be maintained. Integration of Adams Car with pSeven allows the setting of multiple optimization criteria and an automated study of the model responses through a wide range of varying input parameters and under the influence of various design constraints.

Advanced algorithms of pSeven combined with SmartSelection technology allow engineers without specialized knowledge in Design Space Exploration to quickly switch between different problem statements and obtain optimal combinations of design parameters.

## Objective

The goal of this use case is creating an automated workflow for parametric analysis and optimization of car suspension dampers in order to ensure passenger vibration comfort and smooth vehicle behavior on the road.

## Challenges

- Different vehicle driving modes should be simulated.
- Dampers for various road profiles need to be optimized, so it addresses to single- and multi-objectives optimization.
- Heavy calculations but should be performed “overnight”.

## Solution

### I. Automation and problem statement

The ride simulation of the complete vehicle is implemented in Adams/Car. The model has the following features:

- Basic model - ‘sedan_rwd’ from a standard database <acar_concept>
- Tire-rode interaction model: PAC 2002
- Total mass: 1835 kg
- Unsprang mass: 313 kg

Automated simulation and analysis workflows were created for investigation of two main scenarios: basic study of a damper in fixed conditions and average performance for representative road profile.

**Rough road**

- ‘Full Vehicle Analysis’ → ‘File Driven Events’
- Driving speed is constant: ~ 90 km/h
- Simulation time: 7 s
- Number of integration steps: 700
- Duration of a single computation: ~ 1.5 min

*Figure 1. Driving on a rough road*

**Single obstacle**

- ‘Full Vehicle Analysis’ → ’Course events’ → ‘3D road’
- Short, medium and long obstacles
- Driving speed is constant

*Figure 2. Driving through a single obstacle*

These scenarios can be enriched or used in different combinations during optimization studies. Criteria for passenger vibration comfort can be formulated in various way. Automated workflow allows to introduce formulation of any complexity basing on the values of displacements, accelerations, forces, distributions of the energy density in the frequency domain, etc.

*Figure 3. 3D model of a full vehicle assembly in Adams/Car*

In this study, the following characteristics are explored:

- Angular movement and acceleration of the chassis around the lateral axis
*Y*(pitch) - Angular displacement and acceleration of the chassis around the longitudinal axis
*X*(roll). - Vertical chassis movement and acceleration along the
*Z*axis.

The metric for comparing responses is the root mean square (RMS) of the value series:

$$RMS=\sqrt{{\frac 1T}\int_{T}f^2(t)dt}$$

Vertical acceleration at the steering knuckle at the point of attachment of the suspension to the body, normal forces at the point of contact with the road are also considered.

*Figure 4. Acceleration around the longitudinal axis X (roll) in the time and frequency domains*

For each characteristic, the dependencies of the amplitude in the time *f(t)* and frequency *F(ω)* domains are extracted for visual check by the user.

Damper with a variable damping coefficient is described by a piece-wise linear curve of the dependency of the reaction force in compression / rebound from the stroke speed. So, the input parameters of the model are the parameters of the piece-wise linear dependence (**12 total parameters**):

- $z_1^x$ -
**slope**of the first segment of the piece-wise linear dependence**upon rebound**, - $d_1^x$ –
**slope**of the first segment of the piece-wise linear dependence**under compression**, - $v_z^x$ – the position of the inflection point of the line at
**rebound**, - $v_d^x$ – the position of the inflection point of the line during
**compression**, - $F_{zmax}^x$ – maximum
**rebound**damping force, - $F_{dmax}^x$ – maximum
**compression**damping force,

where the superscript is *x = r* in the case of the rear suspension and *x = f* in the case of the front suspension. Parameters are allowed to vary over a wide range to obtain a variety of suspension behavior.

*Figure 5. Parameterization of the dependence of the reaction force on the stroke speed for the suspension damper*

The curves of dependencies should stay within a special range due to technological characteristics of manufacturing of the damper.

- Constraints of avoided crossing of $f_1$ with $f_1^c$ and $f_3^c$. Constraints are met by setting a valid range of parameters variation.
- The line should not intersect with the second segment of the upper bounding line (Figure 6 a, b): $$f_1^c (v_i ) > f_1(v_i )$$
- The slope of the second line segment must not be negative: $f_2^{'} ≥ 0$ (Figure 6 c).

*Figure 6. a) – Line segment symbols; b), c) – Constraints violation*

Constraints of type 2 and 3 can be represented as an analytical quadratic function, so there are **8 quadratic constraints** in total.

To solve the problem firstly an automation of simulation is needed. The workflow created in pSeven uses the capabilities of Adams Car command language for the automated execution of pre-processing commands, solver calls and results post-processing.

*Figure 7. Workflow for automation of model updating in pSeven*

On Fig. 7:

- Updating the characteristics of the damper
*F(v)*. - Updating simulation parameters in the script file of Adams Car (cmd), selection of simulation type, solver settings (the number of integration steps and simulation time).
- Execution of Adams Car command script, call to postprocessor.
- Reading results.
- Calculation of output parameters; RMS, FFT, etc.

This workflow is prepared as a model for parametric and optimization studies on top.

### II. Parametric study

This step is performing a basic parametric study to ensure if the model is stable enough, and investigating the basic dependencies that helps to finalize an optimization problem statement. For Design Space Exploration automated workflow is presented as a Composite block which allows to include a complex workflow as a single block into an external cycle, execute several parallel computations and save intermediate results (data caching) to be able to restart in case the simulation is interrupted.

*Figure 8. Workflow for parametric study in pSeven*

Design Space Exploration block (DSE) is a tool for Design of Experiments (DoE) and optimization studies. Embedded SmartSelection technology applies the most suitable algorithm for the current task.

*Figure 9. DSE block configuration for parametric study*

From the correlation analysis it is obvious that there is one pair of parameters which are competitive (**sigma_acc_disp_vert and sigma_acc_roll**). This means these parameters could be targets for optimization because they can’t be improved together.

*Figure 10. Correlations analysis in pSeven*

Sensitivity analysis in pSeven is a powerful tool helping to understand which parameters can be removed from the consideration.

*Figure 11. Sensitivity analysis in pSeven*

### III. Optimization study

After everything is ready, the last step is an optimization study with **12 parameters, 8 quadratic constraints** and **2 targets** to improve for the best configuration.

*Figure 12. Feasible domain for optimization study*

There are two independent objectives for optimization therefore a Pareto frontier is obtained as a result for further choosing of the most appropriate combination of parameters.

- Minimizing acceleration fluctuations in the vertical direction (
**sigma_acc_disp_vert**)

$$\min_{X_{param} ∈R^{12}}\sqrt{{\frac 1T}\int_{T}a^2_{disp\_vert}(t)dt}$$

- Minimization of acceleration fluctuations around the longitudinal axis (
**sigma_acc_roll**)

$$\min_{X_{param} ∈R^{12}}\sqrt{{\frac 1T}\int_{T}a^2_{roll}(t)dt}$$

The workflow remains the same except DSE setup where two outputs should be switched to Minimization type.

*Figure 13. DSE block configuration for optimization study*

Thanks to the fact that we placed all the outputs to the list of responses, we can play around with these list settings to study different problems with a couple of clicks.

## Results

After the two-objectives optimization study is finished a Pareto frontier is obtained as a set of optimal solutions.

*Figure 14. Pareto frontier as a result of the two-objectives optimization*

When comparing the values of objective functions at these points, it is impossible to get the optimal value for all criteria at once: when the RMS of vertical acceleration decreases, the RMS of lateral roll acceleration increases, and vice versa.

pSeven Surrogate-based optimization (SBO) algorithm found 36 Pareto-frontier points in 600 iterations which means ~ 20 iterations per point. For reference, classical methods would require ~ 3,600 model calls (~ 100 per point). So, thanks to SBO it is possible to finish the solution “overnight” and estimate the result just by updating a report in the morning. pSeven visualization tools allow to draw the dependence corresponded to each point of the result.

*Figure 15. Quality change in the characteristics of dampers along the Pareto frontier*

Comparing initial configuration to the optimal points of Pareto frontier we see that all optimal points obtained during the optimization study are better than initial one. In addition, if we will consider the particular configuration in the middle of the frontier, we can say that the vertical overload spread is **reduced by 10%** and angular accelerations on the lateral roll are **reduced by 2%** in comparison with the initial configuration.

## Conclusion

This use case demonstrates integration capabilities of Adams Car and pSeven. Automated workflow allows to easily extend simulation scenarios for various driving scenarios and setups. Domain experts can thus carry out many parametric studies using a unified methodology while developing adaptive suspension systems.

A two-criteria optimization of characteristics of the damper of the front and rear suspensions is performed: minimization of fluctuations in acceleration around the transverse axis of the vehicle and minimization of fluctuations in vertical acceleration.

Obtained Pareto frontier allows evaluating various suspension modifications and choosing the best trade-off solution among the optimal solutions.

Thanks to the SBO optimization technique, optimal solution obtained with a minimum number of model calls. This approach allows user to quickly solve similar problems with both single- and multi-criteria statements.