The parameter identification of the rubber bushing dynamic model involves two steps. Firstly, the identification of the parameters of the elastic element and the friction element, and then the identification of the parameters of the viscoelastic element.
4.2. Identification of the Parameters of the Viscoelastic Elements
The identification of parameters for the rubber bushing’s viscoelastic elements was conducted using dynamic loading test curves. Due to the numerous parameters that need to be identified and the strong nonlinearity of the viscoelastic unit, the particle swarm optimization (PSO) algorithm was employed to search for the optimal solution based on fitness evaluation. The PSO algorithm updates the fitness, velocity, and position of particles iteratively to find the best parameters. The corresponding Equation (17) is as follows:
where
is the inertia weight; the velocity and position of the current particle are represented by
and
;
and
are random numbers ranging between 0 and 1;
and
are learning factors; the range of the velocity and position of the particles are [
] and [
], respectively;
is the best position (parameter values) of particle i found so far in the iterations;
is the best position among all particles in the swarm at iteration t.
To improve the optimization speed, a modification has been made in the particle swarm optimization (PSO) algorithm, where a random particle is selected to update the velocity during the velocity update process. This approach, known as “random particle updating”, helps accelerate the optimization process and prevents the algorithm from getting stuck in local optima. The velocity update formula with random particle updating is given by Equation (18).
is an additional coefficient introduced for the random particle updating, controlling the influence of the random particle’s position on the velocity update; is a random number ranging between 0 and 1; is the randomly selected particle from the current particle swarm.
To evaluate the results of each optimization, this study uses an optimization degree λ to assess the effectiveness of the search process. The optimization degree, denoted as “OD”, is a measure of how much the global fitness changes during the search. If the global fitness changes during the current search, the optimization degree is set to 1. However, if the global fitness remains unchanged during the current search, the optimization degree is calculated using the following Equation (19):
where
is the average fitness value of the
i-th optimization search population, and
is the best fitness value of the current population.
A larger inertia weight
is advantageous for global search, while a smaller inertia weight is advantageous for local search. By adjusting the inertia weight in a timely manner based on the optimization degree
of each optimization search, it is possible to search for the best particle [
21].
Equation (20) is used to update the inertia weight; D represents the population size of the particles, and G represents the maximum number of iterations.
The parameter identification of the rubber bushing model’s viscoelastic element is essentially an optimal parameter estimation problem. To ensure that the identified parameters closely match the experimental data, an appropriate fitness function needs to be established. This is achieved by setting up an optimization function that minimizes the error between the numerical and experimental values based on the designed parameterized dynamic model. The objective function, based on the errors in dynamic stiffness, is formulated as follows in Equation (21):
where
n is the number of identification conditions,
represents the dynamic stiffness test data, and
represents the dynamic stiffness of the
i-th condition calculated by the bushing dynamics model.
To ensure the accuracy of parameter identification for the rubber bushing model and improve the fitting precision of the model, it is important to enforce constraints during the parameter identification process to prevent significant discrepancies between the model’s fitted data and the experimental data. Therefore, the following constraints are established:
Calculate the bushing dynamic stiffness using Equation (23):
The objective function Equation (21) is used as the fitness function for the parameter identification of the rubber bushing model. Two different optimization algorithms, adaptive chaotic multi-particle swarm optimization (ACMPSO) and particle swarm optimization (PSO), are applied to identify the parameters of the viscoelastic element.
After debugging, the following algorithm parameters are selected:
Population size: 50, particle dimension: 7, position vector
, particle search space lower limit
, particle search space upper limit
, maximum inertia weight
, minimum inertia weight
, learning factors
,
,
, and maximum number of iterations of 300. The PSO algorithm uses linearly decreasing inertia weight. The identification process is carried out using 0.2 mm dynamic stiffness data, as shown in
Figure 8.
Figure 9 is a comparison of the optimization performance between the two algorithms, and it is clear that the PSO algorithm has fallen into premature convergence, while the ACMPSO algorithm has a stronger optimization ability.
Figure 10 illustrates the results of parameter identification and model fitting based on the rubber bushing dynamic model using experimental data obtained from a 0.2 mm amplitude test. The model is then used to fit dynamic stiffness data obtained from tests with amplitudes of 0.2 mm, 0.3 mm, 0.5 mm, and 1 mm, respectively.
As shown in
Figure 11, the dynamic model’s fitted data for bushing amplitudes of 0.2 mm, 0.3 mm, and 0.5 mm closely match the test data with little deviation. However, when comparing the fitted data with the experimental data for the large amplitude of 1 mm loading test condition, it is evident that the fitting accuracy has decreased. The identification parameter results are shown in
Table 2.