A hybrid modeling approach for automotive vibration isolation mounts and shock absorbers

Abstract

This paper presents a novel modeling approach to predict the nonlinear dynamic characteristics of automotive mounts and shock absorbers. Firstly, the concept of the hybrid ANN (artificial neural network)–mechanical modeling approach is presented, which consists of an equivalent mechanical model to characterize the trend of the dynamic characteristic and a neural network model to compensate for the errors introduced by uncaptured nonlinearities. Then experiments are carried out on a rubber mount, a hydraulic mount and a shock absorber to measure their static and dynamic characteristics. Parameters of the equivalent mechanical model are identified, and the neural network model is trained. The hybrid models are validated under harmonic and random excitations with a relative error of less than 8%. Finally, the developed models for the rubber mounts, hydraulic mounts and shock absorbers are integrated into a vehicle model to evaluate the impact of nonlinearity on vehicle ride comfort. The results show that ignoring the nonlinearity of the hydraulic mounts will introduce the largest calculation errors in the analysis of vehicle ride comfort, followed by the shock absorber, and the rubber mount is the smallest.

Appendix

The working principle of the test benches in this paper is shown in Fig. 

Fig. 20

Schematic diagrams of the test bench for vehicle vibration isolators

20.

1.1 Formula derivation

The fractional operator in Eq. (5) is obtained by using the definition of the Grünwald fractional derivative [22]:

$$ \frac{{{\text{d}}^{\beta } }}{{{\text{d}}t^{\beta } }}x{(}t{)} \approx (\Delta t)^{ - \beta } \sum\nolimits_{j = 0}^{N} {B_{j + 1} x} {(}t - j\Delta t{)} $$

(1)

where b is the damping coefficient. β is the fractional derivative order (0 < β < 1). Δt is the constant time step size. t is the current time. N is the truncated calculation step. B_j+1 is the integral coefficient:$B_{j + 1} = \frac{ j - 1 - \beta }{j}B_{j}$, B₀ = 1.

Similarly, the fractional operators in Eq. (7) can be expressed as:

$$ \frac{{{\text{d}}^{\alpha } }}{{{\text{d}}t^{\alpha } }}x{(}t{)} \approx (\Delta t)^{ - \alpha } \sum\nolimits_{j = 0}^{N} {A_{j + 1} x} {(}t - j\Delta t{)} $$

(2a)

$$ \frac{{{\text{d}}^{\beta } }}{{{\text{d}}t^{\beta } }}x{(}t{)} \approx (\Delta t)^{ - \beta } \sum\nolimits_{j = 0}^{N} {B_{j + 1} x} {(}t - j\Delta t{)} $$

(2b)

$$ \frac{{{\text{d}}^{\beta } }}{{{\text{d}}t^{\beta } }}f_{c} {(}t{)} \approx (\Delta t)^{ - \beta } \sum\nolimits_{j = 0}^{N} {B_{j + 1} f_{c} } {(}t - j\Delta t{)} $$

(2c)

where α and β are the time derivative orders. τ is the relaxation time. K₀ and K_∞ are the storage stiffness in static and high frequency, respectively. The A_j+1 and B_j+1 are integral coefficients: $A_{j + 1} = \frac{ j - 1 - \alpha }{j}A_{j}$,$B_{j + 1} = \frac{ j - 1 - \beta }{j}B_{j}$, A₀ = B₀ = 1.

The five-parameter fractional derivative viscoelastic force can be rewritten as:

$$ f_{c} {(}t{) = }\frac{{K_{0} x(t) + K_{0} \tau^{\beta } (\Delta t)^{ - \beta } \sum\nolimits_{j = 0}^{N} {B_{j + 1} x} {(}t - j\Delta t{)} + (K_{\infty } - K_{0} )\tau^{\alpha } (\Delta t)^{ - \alpha } \sum\nolimits_{j = 0}^{N} {A_{j + 1} x} {(}t - j\Delta t{)} - \tau^{\beta } (\Delta t)^{ - \beta } \sum\nolimits_{j = 1}^{N} {B_{j + 1} f_{c} } {(}t - j\Delta t{)}}}{{1 + \tau^{\beta } (\Delta t)^{ - \beta } }} $$

(3)