Optimizing damper placement in automotive mechanical suspension using Bayesian Optimization (BO)

Repository

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Overview

The project aims to optimize the placement of dampers in an automotive suspension system to minimize an objective function that balances passenger comfort, vehicle vibration, handling, and suspension stroke constraints.

The system is modeled as a half-car with a sprung mass (vehicle body) and two unsprung masses (wheels), driven over a random road profile. Bayesian optimization is used to tune parameters, particularly damper placement parameters: motion ratio and inclination angle.

Given:

• Sprung mass: \( m_s = 1200 \, \text{kg} \)
• Unsprung mass (per wheel): \( m_u = 50 \, \text{kg} \)
• Tire stiffness: \( k_t = 200,000 \, \text{N/m} \)
• Time step: \( \Delta t = 0.002 \, \text{s} \)
• Simulation steps: \( N = 5000 \)
• Road profile: Random, ISO 8608-like, with \( \Delta t \)
• Parameters (partial list):
  • Compression damping: \( C_c \in [500, 5000] \, \text{Ns/m} \)
  • Rebound damping: \( C_r \in [1000, 8000] \, \text{Ns/m} \)
  • Motion ratio: \( \text{MR} \in [0.5, 1.5] \)
  • Inclination angle: \( \theta \in [0, 30]^\circ \)
• Objective: Minimize \( J = 0.35 J_{\text{comfort}} + 0.25 J_{\text{vibration}} + 0.25 J_{\text{handling}} + 0.15 J_{\text{constraints}} \)

Functions:

1. Simulates the half-car model and evaluates the objective function for a given set of suspension parameters.
2. Compression and rebound damping coefficients (Ns/m) controls damper force.
3. Motion ratio scales the damper's effect based on its mechanical advantage in the suspension linkage.
4. Inclination angle affects the effective damping force via cosine projection while knee point, blowoff, hysteresis, temp coeff. define nonlinear damping behavior.
5. Modelling the vehicle's response to road inputs using a 10-state half-car model (sprung/unsprung masses, pitch, roll, etc.).
6. Damping forces are computed incorporating nonlinear effects like blow-off thresholds and temperature-dependent damping.
7. Outputs (acceleration, displacement, pitch, roll, tire forces, max stroke) are used to compute the objective function
8. Damper placement affects comfort (via acceleration) and handling (via pitch/roll).
9. Bayesian Optimization is utilized and it optimizes the suspension parameters by modeling the objective function with a Gaussian Process (GP) and maximizing the Expected Improvement (EI) acquisition function.
10. Initialization uses Latin Hypercube Sampling to generate initial parameter sets and optimization loop fits the GP to observed data, predicts mean/variance for new points, and selects the next parameters by maximizing EI over 1000 random samples.
11. All 16 parameters are optimized within bounds.
12. The GP indirectly learns how motion ratio and inclination angle impact the objective, optimizing their values to improve damping effectiveness.
13. A random road profile is generated to simulate realistic road inputs.
14. An ISO 8608-like roughness model with smoothed noise, providing displacement and velocity inputs to the half-car model is used.
15. The road profile drives the suspension dynamics, testing the damper’s ability to absorb disturbances.
16. Power Spectral Density (PSD) of chassis displacement to evaluate vibration, which is influenced by damper settings is computed.

Solution

Half-Car Model

The half-car model has 10 states:

• Sprung mass displacement \( z_s \), velocity \( v_s \)
• Pitch angle \( \theta \), angular velocity \( \omega_y \)
• Roll angle \( \phi \), angular velocity \( \omega_x \)
• Front unsprung displacement \( z_{uf} \), velocity \( v_{uf} \)
• Rear unsprung displacement \( z_{ur} \), velocity \( v_{ur} \)

Vehicle parameters:

• Wheelbase: \( L \in [2.4, 2.6] \, \text{m} \)
• Track width: \( T \in [1.4, 1.6] \, \text{m} \)
• Pitch inertia: \( I_y = m_s L^2 / 12 \)
• Roll inertia: \( I_x = m_s T^2 / 12 \)

Suspension Displacements:

\[ z_{sf} = z_s + \frac{L}{2} \theta - \frac{T}{2} \phi \quad (\text{front}) \]

\[ z_{sr} = z_s - \frac{L}{2} \theta + \frac{T}{2} \phi \quad (\text{rear}) \]

Relative Velocities:

\[ v_{\text{rel},f} = v_s + \frac{L}{2} \omega_y - \frac{T}{2} \omega_x - v_{uf} \quad (\text{front}) \]

\[ v_{\text{rel},r} = v_s - \frac{L}{2} \omega_y + \frac{T}{2} \omega_x - v_{ur} \quad (\text{rear}) \]

Forces:

• Spring force (stiffness \( k_s \), preload \( F_{\text{preload}} \)):

  \[ F_{\text{spring},f} = -k_s (z_{sf} - z_{uf}) - F_{\text{preload}} \]

  \[ F_{\text{spring},r} = -k_s (z_{sr} - z_{ur}) - F_{\text{preload}} \]

  Gas pressure: \( k_s' = k_s (1 + 0.01 p_{\text{gas}}) \).

• Tire force:

  \[ F_{\text{tire},f} = -k_t (z_{uf} - z_{rf}) - C_t (v_{uf} - \dot{z}_{rf}) \]

  \[ F_{\text{tire},r} = -k_t (z_{ur} - z_{rr}) - C_t (v_{ur} - \dot{z}_{rr}) \]

• Anti-roll force:

  \[ F_{\text{roll}} = -k_{\text{roll}} \phi \]

Equations of Motion:

1. Sprung mass (vertical):

   \[ m_s \ddot{z}_s = F_{\text{spring},f} + F_{\text{spring},r} + F_{\text{damper},f} + F_{\text{damper},r} \]

   \[ \ddot{z}_s = \frac{F_{\text{spring},f} + F_{\text{spring},r} + F_{\text{damper},f} + F_{\text{damper},r}}{m_s} \]

2. Pitch:

   \[ I_y \ddot{\theta} = \left( F_{\text{spring},f} + F_{\text{damper},f} \right) \frac{L}{2} - \left( F_{\text{spring},r} + F_{\text{damper},r} \right) \frac{L}{2} \]

   \[ \ddot{\theta} = \frac{\left( F_{\text{spring},f} + F_{\text{damper},f} \right) \frac{L}{2} - \left( F_{\text{spring},r} + F_{\text{damper},r} \right) \frac{L}{2}}{I_y} \]

3. Roll:

   \[ I_x \ddot{\phi} = \left( F_{\text{spring},r} - F_{\text{spring},f} \right) \frac{T}{2} + F_{\text{roll}} \]

   \[ \ddot{\phi} = \frac{\left( F_{\text{spring},r} - F_{\text{spring},f} \right) \frac{T}{2} - k_{\text{roll}} \phi}{I_x} \]

4. Unsprung masses:

   \[ m_u \ddot{z}_{uf} = -F_{\text{spring},f} - F_{\text{damper},f} + F_{\text{tire},f} \]

   \[ m_u \ddot{z}_{ur} = -F_{\text{spring},r} - F_{\text{damper},r} + F_{\text{tire},r} \]

States updated via Euler integration:

\[ v_s(t + \Delta t) = v_s(t) + \ddot{z}_s \Delta t, \quad z_s(t + \Delta t) = z_s(t) + v_s(t) \Delta t \]

Damping Force Calculation

Damping force:

\[ F_{\text{damper}} = -C_{\text{base}} v_{\text{rel}} - F_{\text{hysteresis}} \]

• \( C_{\text{base}} = C_r \) if \( v_{\text{rel}} \geq 0 \), else \( C_c \).
• If \( |v_{\text{rel}}| > v_{\text{blowoff}} \), \( C_{\text{base}} \leftarrow 0.5 C_{\text{base}} \).
• If \( |v_{\text{rel}}| > v_{\text{knee}} \), \( C_{\text{base}} \leftarrow C_{\text{base}} \cdot \frac{v_{\text{knee}}}{|v_{\text{rel}}|} \).
• Temperature: \( C_{\text{base}} \leftarrow C_{\text{base}} (1 - \alpha (T - 20)) \), \( T = 20 + 0.001 N \).
• Hysteresis: \( F_{\text{hysteresis}} = h |v_{\text{rel}}| \).

Damper Placement:

• Motion ratio (\( \text{MR} \)):

  \[ F_{\text{damper}} \leftarrow F_{\text{damper}} \cdot \text{MR} \]

• Inclination angle (\( \theta \)):

  \[ C_c' = C_c \cdot \text{MR} \cdot \cos(\theta), \quad C_r' = C_r \cdot \text{MR} \cdot \cos(\theta) \]

Objective Function

\[ J = 0.35 J_{\text{comfort}} + 0.25 J_{\text{vibration}} + 0.25 J_{\text{handling}} + 0.15 J_{\text{constraints}} \]

• Comfort:

  \[ J_{\text{comfort}} = \sqrt{\frac{1}{N} \sum_{t=1}^N a_s^2 \cdot w(a_s)}, \quad w(a_s) = 1 \text{ if } |a_s| < 1, \text{ else } 1.4 \]

• Vibration:

  \[ J_{\text{vibration}} = \frac{1}{M} \sum_{k=1}^M \text{PSD}(z_s, k) \quad (0.5–20 \, \text{Hz}) \]

• Handling:

  \[ J_{\text{handling}} = 0.4 |\theta|_{\text{max}} + 0.4 |\phi|_{\text{max}} + 0.2 \max(0, -F_{\text{tire,min}}) \]

• Constraints:

  \[ J_{\text{constraints}} = 1000 \cdot \max(0, s_{\text{max}} - s_{\text{limit}}) \]

  \( s_{\text{max}} = \max(|z_{sf} - z_{uf}|, |z_{sr} - z_{ur}|) \).

Bayesian Optimization

Bayesian optimization:

1. Initializes 10 parameter sets via Latin Hypercube Sampling.
2. Uses Gaussian Process (GP) with RBF kernel:

   \[ k(x_1, x_2) = \sigma_f^2 \exp\left( -\frac{\|x_1 - x_2\|^2}{2 l^2} \right) \]

3. Maximizes Expected Improvement (EI):

   \[ \text{EI}(x) = (J_{\text{best}} - \mu(x)) \Phi(z) + \sigma(x) \phi(z), \quad z = \frac{J_{\text{best}} - \mu(x)}{\sigma(x)} \]

4. Iterates 30 times, evaluating \( J \).

Sample Calculation

Parameters:

• \( C_c = 3000 \, \text{Ns/m} \)
• \( C_r = 5000 \, \text{Ns/m} \)
• \( \text{MR} = 1.2 \)
• \( \theta = 10^\circ \)
• \( v_{\text{blowoff}} = 1.5 \, \text{m/s} \)
• \( v_{\text{knee}} = 1.0 \, \text{m/s} \)
• \( h = 0.05 \)
• \( \alpha = 0.005 \, /\text{°C} \)
• \( k_s = 30,000 \, \text{N/m} \)
• \( F_{\text{preload}} = 500 \, \text{N} \)
• \( p_{\text{gas}} = 15 \, \text{bar} \)
• \( C_t = 200 \, \text{Ns/m} \)
• \( k_{\text{roll}} = 5000 \, \text{N/rad} \)
• \( L = 2.5 \, \text{m} \)
• \( T = 1.5 \, \text{m} \)
• \( s_{\text{limit}} = 0.08 \, \text{m} \)

Damping Force:

At \( t \), assume \( v_{\text{rel},f} = 0.8 \, \text{m/s} \), \( T = 25 \, \text{°C} \).

• \( C_{\text{base}} = C_r = 5000 \, \text{Ns/m} \).
• \( |v_{\text{rel},f}| = 0.8 < 1.5 \), no blow-off.
• \( |v_{\text{rel},f}| = 0.8 < 1.0 \), no knee adjustment.
• Temperature:

  \[ C_{\text{base}} = 5000 (1 - 0.005 (25 - 20)) = 5000 \cdot 0.975 = 4875 \, \text{Ns/m} \]

• Hysteresis:

  \[ F_{\text{hysteresis}} = 0.05 \cdot 0.8 = 0.04 \, \text{N} \]

• Base damping:

  \[ F_{\text{base}} = -4875 \cdot 0.8 = -3900 \, \text{N} \]

• Placement:

  \[ C_r' = 5000 \cdot 1.2 \cdot \cos(10^\circ) = 5000 \cdot 1.2 \cdot 0.9848 = 5908.8 \, \text{Ns/m} \]

  \[ F_{\text{damper},f} = (-3900 - 0.04) \cdot 1.2 \cdot 0.9848 \approx -4618.8 \, \text{N} \]

Spring Force:

Assume \( z_{sf} - z_{uf} = 0.02 \, \text{m} \).

\[ k_s' = 30,000 (1 + 0.01 \cdot 15) = 34,500 \, \text{N/m} \]

\[ F_{\text{spring},f} = -34,500 \cdot 0.02 - 500 = -690 - 500 = -1190 \, \text{N} \]

Tire Force:

Assume \( z_{uf} - z_{rf} = 0.01 \, \text{m} \), \( v_{uf} - \dot{z}_{rf} = 0.5 \, \text{m/s} \).

\[ F_{\text{tire},f} = -200,000 \cdot 0.01 - 200 \cdot 0.5 = -2000 - 100 = -2100 \, \text{N} \]

Acceleration:

Assume similar rear forces:

\[ \ddot{z}_s = \frac{(-1190 - 1190) + (-4618.8 - 4618.8)}{1200} \approx \frac{-11,617.6}{1200} \approx -9.68 \, \text{m/s}^2 \]

Objective (Single Step):

Assume \( \theta = 0.01 \, \text{rad} \), \( \phi = 0.005 \, \text{rad} \), \( F_{\text{tire,min}} = -2100 \, \text{N} \), \( s_{\text{max}} = 0.09 \, \text{m} \).

• Comfort:

  \[ J_{\text{comfort}} \approx \sqrt{9.68^2 \cdot 1.4} \approx \sqrt{131.3} \approx 11.46 \]

• Handling:

  \[ J_{\text{handling}} = 0.4 \cdot 0.01 + 0.4 \cdot 0.005 + 0.2 \cdot 2100 = 420.006 \]

• Constraints:

  \[ J_{\text{constraints}} = 1000 \cdot (0.09 - 0.08) = 10 \]

Results

Output:

• \( C_c = 3000 \, \text{Ns/m} \), \( C_r = 5000 \, \text{Ns/m} \)
• \( \text{MR} = 1.2 \), \( \theta = 10^\circ \)

Damper Placement:

• Motion ratio (1.2): Boosts damping by 20%.
• Inclination (10°): \( \cos(10^\circ) \approx 0.9848 \), near-full force transmission.

Conclusion

The half-car model and Bayesian optimization optimize damper placement effectively. Motion ratio and inclination angle adjust damping force transmission, balancing comfort and handling. The physics-based simulation is realistic, and the GP minimizes evaluations.