The double wishbone suspension is composed of control arms and elastic elements, and the road excitation is transmitted to the body through the wheel and suspension, as shown in Figure 1a. Path a1b1 consists of tire and is determined by tire radial stiffness and tire damping. Path cd consists of path c1d1, path c2d2, path c3d3 and path c4d4, and is affected by spring stiffness, shock absorber damping, upper control arm bushing stiffness and lower control arm bushing stiffness (Sharma, 2023). The vibration transfer path analysis and optimization can effectively attenuate the vibration transfer, which is of great significance to the vibration reduction performance of mechanical systems.

According to the equivalent force model of the suspension system (Figure 1b), taking (x0, x1, x2) as generalized coordinates, the vibration equation of the suspension system is established as, M 1 x 1 ¨ − c 11 x 0 ˙ − x 1 ˙ − k 11 x 0 − x 1 + c 21 x 1 ˙ − x 2 ˙ + k 21 + k 22 + k 23 x 1 − x 2 = 0 M 2 x 2 ¨ − c 21 x 1 ˙ − x 2 ˙ − k 21 + k 22 + k 23 x 1 − x 2 = 0 (1)

Where C₁₁ stands for tire damping, k₁₁ is tire radial stiffness, C₂₁ is suspension damping, k₂₁ is spring stiffness, k₂₂ is upper control arm bushing equivalent stiffness at the wheel center, k₂₃ stands for lower control arm bushing equivalent stiffness at the wheel center, M1 stands for the axle mass, M2 stands for the body mass.

The object of analysis in this paper is a double-swinging arm suspension system. In the impact process, the suspension stiffness and damping show nonlinear characteristics (as shown in Figure 2), so the double-swinging arm suspension system is a typical nonlinear system. The vibration transfer function analysis of nonlinear systems is usually linearized, and then the influence of nonlinear factors is analyzed.

The Laplace transform of Equation 1 is, M 1 s 2 x 1 s − c 11 s + k 11 · x 0 s − x 1 s + c 21 s + k 21 + k 22 + k 23 · x 1 s − x 2 s = 0 M 2 s 2 x 2 s − c 21 s + k 21 + k 22 + k 23 · x 1 s − x 2 s = 0 (2)

The transfer function of path a1b1, path cd and path a1b1cd is, H a 1 b 1 s = x 1 s x 0 s = c 11 s + k 11 M 1 s 2 + c 11 s + c 21 s + k 11 + k 21 + k 22 + k 23 − 1 M 2 s 2 + c 21 s + k 21 + k 22 + k 23 (3) H c d s = x 2 s x 1 s = c 21 s + k 21 + k 22 + k 23 M 2 s 2 + c 21 s + k 21 + k 22 + k 23 (4) H a 1 b 1 c d s = H a 1 b 1 s · H c d s = c 11 s + k 11 · c 21 s + k 21 + k 22 + k 23 M 1 s 2 + c 11 s + c 21 s + k 11 + k 21 + k 22 + k 23 · M 2 s 2 + c 21 s + k 21 + k 22 + k 23 − 1 (5)

Equation 5 shows that the suspension system transfer function is affected by spring loaded mass, n1on-spring loaded mass, shock absorber damping, spring stiffness, control arm bushing stiffness, tire radial stiffness and damping.

The transfer function of path cd is affected by the four paths, and the relation is H c d s = H c 1 d 1 s + H c 2 d 2 s + H c 3 d 4 s + H c 4 d 4 s (6)

There is coupling between the four paths in Equation 6, which makes it difficult to identify the main path. For the transfer function analysis of vehicle chassis system, classic TPA and working condition TPA are mostly used (Zhu et al., 2018; Sit et al., 2010). The classic TPA is used to obtain the transfer function by hammer method under static conditions, which is more suitable for linear systems. working condition TPA is mainly used in dynamic conditions and is more suitable for nonlinear systems, but there are path coupling problems. Under static and dynamic conditions, the elastic parameters of the suspension system are different. For example, under static conditions, the shock absorber does not work and c11 in Equation 2 is zero. In dynamic conditions, the equivalent stiffness of the suspension at the wheel center appears nonlinear due to the movement of the control arm, and the suspension transfer function also changes. Therefore, working condition TPA is more suitable for analyzing suspension transfer function. The parameters of the system are shown in Table 1.

In order to identify the main path in Equation 6, the sensitivity analysis method is used to analyze the influence of the parameters of the elastic element in each path on the transfer function Hcd (S).

By applying unit force to point a1 in Figure 1a according to Equations 3, 4, the transfer functions Ha1b1, Hcd and Ha1e can be obtained, as shown in Figure 3. The three curves have two peaks within the range of 0–100 Hz. The first peak frequency is 1.59 Hz, and the second peak frequency is 11.6 Hz, where 1.59 Hz is the natural frequency of suspension and 11.6 Hz is the natural frequency of tire. The second peak value of path Ha1b1 transfer function is larger than the first peak value, which is mainly affected by tire parameters. The first peak value of path Hcd transfer function is larger than the second peak value, which is mainly affected by suspension parameters. The trend of the Ha1e transfer function is consistent with that of the Hcd transfer function, and the coincidence of the two curves is 91.3%.

The transfer function Ha1e is the product of Ha1b1 and Hcd, and is affected by subsystem a1b1 and subsystem cd. Subsystem a1b1 has only one transfer path and only includes tire elastic components, and its transfer function is mainly affected by tire stiffness and damping. The subsystem cd has four transfer paths and contains multiple elastic elements. Therefore, the main transfer path of the whole system is mainly to analyze the main path of the main system cd.

In this paper, sensitivity analysis method (Strasser and Jiang, 2024) is used to analyze the influence of each elastic element in subsystem cd on its transfer function, so as to determine the main path. The characteristic equation of mechanical system can be expressed as, Δ = y ξ , x 1 , x 2 , · · · , x n (7)

Where ξ stands for the characteristic value,xi (i = 1,2,. , n) is the design variable.

The differential of Equation 7 is, δ Δ = ∂ y ∂ ξ δ ξ + ∑ i = 1 n ∂ y ∂ ξ k δ x k (8)

According to Equation 8, when the change of system design variable is δxk (i = 1,2,. n), then the change of the characteristic value is δξ and the change of the eigenequation is δ△.

Assuming that δxj ≠0, δxk = 0 (j≠k), and an eigenvalue of the system is ξt, Equation 8 can be expressed as δ Δ t = ∂ y ∂ ξ t δ ξ t + ∂ y ∂ x j δ x j (9)

Let δ△ t = 0 in Equation 9, then the system sensitivity is S t j = − ∂ y ∂ x j / ∂ y ∂ ξ t (10)

Parameters K21, C21, K22 and K23 in Figure 1 are taken as design variables, and according to Equation 10 the sensitivity analysis results to the transfer function are shown in Figure 4. At 1.59 Hz, the sensitivity of K21 is 18.43%, that of C21 is 11.96%, that of K22 is 1.85%, and that of K23 is 2.21%. Therefore, the impact of spring stiffness and shock absorber damping is obvious. At 11.6Hz, the sensitivity of K21 is 3.44%, the sensitivity of C21 is 1.75%, the sensitivity of K22 is 0.52%, and the sensitivity of K23 is 0.74%. Therefore, the above four design variables have little influence on the 11.6 Hz peak value.

Because tire parameters are not easy to change, the effective control of vibration transfer path is mainly achieved by optimizing suspension parameters. The sensitivity analysis shows that spring stiffness and shock absorber damping have significant influence on the transfer function in suspension system. For double control arm suspension, the suspension stiffness changes with the control arm position, and the shock absorber damping changes with the excitation amplitude, that is, the suspension stiffness and shock absorber damping show nonlinear characteristics. Therefore, it is necessary to fully consider the nonlinear effects of stiffness and damping when analyzing the vibration transfer path of the suspension system.

The variation range of K21 is 63 N/mm – 77 N/mm, and the variation range of C21 is 0.15–0.55 when the vehicle travels at different speed on different roughness. Figure 5 shows the suspension transfer function of different K21. K21 affects the first peak, but has little effect on the second peak. Near the first peak, the peak frequency moves to higher frequency with the increase of K21, and the peak value changes logarithmically with K21. Therefore, the K21 nonlinear characteristic mainly affects the result of the transfer function near the natural frequency of the suspension system.

Figure 6 is the suspension transfer function of different C21. The damping of shock absorber affects the amplitude, but does not affect the peak frequency. The effects of C21 on both peaks are significant. Compared with the low frequency region, C21 has a greater influence on the high frequency region of the transfer function. The peak values of S1 and S2 shows a logarithmic trend with the increase of C21, and the effect of C21 on S1 is greater than that on S2. Compared with K21, C21 has more obvious influence on suspension transfer function, which is mainly because C21 nonlinear characteristic is obviously larger than K21 nonlinear characteristic.

The front wheels of the vehicle are fixed on the vibration test bench, as shown in Figure 7a. Class A pavement, Class B pavement and Class C pavement unevenness excitation are calculated, and the displacement excitation is applied to the test bench to drive the suspension work. The Z-direction load of the tire is collected by the six-direction force testing equipment, as shown in Figure 7b, and the fluctuation of tire load is calculated. The six-direction force sensor is mounted on the tire through a special wheel hub, its brand is ME with accuracy 0.2%, resolution 0.2% and response time 1.0 m. The accelerometers are mounted on the upper and lower ends of the shock absorber as shown in Figure 7c. The acceleration sensor is glued to the structure by strong adhesive, the brand is PCB, and the sensitivity is 2.385mv/m/s². The acquisition device is LMS SCADAS with resolution 0.001 Hz and frequency range 0–100 Hz. The relative velocity of the shock absorber is calculated from the acceleration data. The vibration acceleration of the body is collected by LMS equipment, and the sensor is installed on the body floor, as shown in Figure 7d.

In the small deformation condition, the main path of the suspension system is c2d2 and c1d1 respectively. Under large deformation conditions, C21 has a great influence on the system dynamics. In order to further analyze the vibration transfer characteristics of suspension system under different excitation amplitudes, it is necessary to consider the C21 nonlinear factor. Under the action of forced excitation f(s), Equation 1 can be expressed as M 1 x 1 ¨ − c 11 x 0 ˙ − x 1 ˙ − k 11 x 0 − x 1 + c 21 x 1 ˙ − x 2 ˙ + k 21 + k 22 + k 23 x 1 − x 2 = 0 M 2 x 2 ¨ − c 21 x 1 ˙ − x 2 ˙ − k 21 + k 22 + k 23 x 1 − x 2 = f s (11)

The spectral density of road displacement excitation f (s) in Equation 11 is (Chiu et al., 2024; Ma et al., 2022) G d n = G d n 0 n n 0 − w (12)

Where n is the spatial frequency, N₀ is the reference frequency, G_d (n) is the roughness coefficient and w is the frequency index.

In the range of (n₁, n₂), the variance of road roughness is σ = ∫ n 1 n 2 G d n d n (13)

According to the sine wave superposition principle, Equations 12, 13 the random excitation of the displacement of uneven pavement is f x = ∑ i = 1 n 2 G d n m i · Δ n i · ⁡ sin 2 π n m i x + ς i (14)

Where n is the spatial frequency, n_mi is the central frequency of △ n_i interval, ζ is the random number, and X is the vehicle displacement.

The displacement excitation of road roughness can be calculated by Equation 14 and applied to the test-bed to carry out the vibration test of suspension system.

The relative velocity data of the shock absorber is shown in Figure 8. The displacement excitation at 0.0–1.0 s is Class A road surface excitation, the relative velocity range of the shock absorber is −0.2 m/s - 0.2 m/s, and the damping coefficient of the shock absorber is 0.15; The displacement excitation at 1.0 s −2.0 s is Class B road surface excitation, the relative velocity range of the shock absorber is −0.4 m/s - 0.4 m/s, and the damping coefficient of the shock absorber is 0.35; The displacement excitation at 2.0 s −3.0 s is Class C road surface excitation, the relative velocity range of the shock absorber is −0.6 m/s - 0.6 m/s, and the damping coefficient of the shock absorber is 0.55. Tire load fluctuation is shown in Figure 9, the rougher the road surface, the greater the tire load fluctuation.

The body response can be calculated from the tire load FZ and Transfer Function, and the expression is a z = F z H a 1 e s (15)

Because the damping coefficient of shock absorber varies with the amplitude of road excitation, the nonlinear characteristics of shock absorber should be considered in order to accurately fit the acceleration response of vehicle body. For Class A pavement, the damping coefficient of the shock absorber is 0.15, the corresponding transfer function is the C21 = 0.15 curve in Figure 5, and the working load is the load in the range of 0.0–1.0 s in Figure 8. According to Equation 15, the body M2 response results is shown in Figure 10, and the fitting result agree with the test result by 90.3%.

For Class B pavement, the damping coefficient of the shock absorber is 0.35, the corresponding transfer function is the C21 = 0.35 curve in Figure 5, and the working load is the load in the range of 1.0 s–2.0 s in Figure 8. The M2 response results is shown in Figure 11, and the fitting result agree with the test result by 93.5%.

For Class C pavement, the damping coefficient of the shock absorber is 0.55, the corresponding transfer function is the C21 = 0.55 curve in Figure 5, and the working load is the load in the range of 2.0 s–3.0 s in Figure 8. The M2 response results is shown in Figure 12, and the fitting result agree with the test result by 92.6%.

The agreement between the fitting results and the test results of Class A pavement, Class B pavement and Class C pavement is more than 90%, which indicates the correctness of the analysis method in this paper. In other words, the nonlinear effect of damper damping should be considered in the analysis of double wishbone suspension dynamic response.