Vibration and noise analysis of heavy-duty trucks based on powertrain lightweighting

4.2.1. Genetic algorithm (GA) method [19, 20]

The Genetic Algorithm (GA) is a probabilistic global search method which mimics the process of natural biological evolution. Usually, the evolution begins with an initial population of randomly generated individuals. This evolution is an iterative process with the population in each iteration called a generation. In each generation, a fitness value solved from the objective function in the optimization problem is associated with each individual to evaluate the performance of the related solution. Each candidate individual (called solution) has a set of properties which can be mutated and altered. A standard representation of each candidate solution is as an array of bits. Once the genetic representation and the fitness function are defined, a GA proceeds to initialize a population of solutions and then to improve it through repetitive application of the selection, crossover, and mutation operators until a globally optimal solution is achieved. The process of GA is illustrated in Fig. 5.

Fig. 5Flow chart of genetic algorithm(GA)

4.2.2. Object function

One combination of the main vibration energy distribution of powertrain mount system was selected as the objective function of optimization design, denoted as:

Here, $F\left(x\right)$ is the object function with the weighted sum of the decoupling index $F_i\left(x\right)$ in each coordinate direction of the mounting system, ${\alpha }_i$ is the weighting coefficient.

4.2.3. Design variables

Optimization of a Powertrain Mounting System (PMS) for vibration control includes the selection of stiffness and shape of individual mount and adjustment of placement and orientations in order to minimize the transfer of vibration from the PowerTrain System (PTS). Considering the installation position of PMS, limitations of package space, cost and available time, the mounting stiffness were chosen as the design variables. Due to the approximate symmetry around the $xz$-plane, only the stiffness of front and rear mount (i.e., $x_i=K_{uf}$, $K_{vf}$, $K_{wf}$, $K_{ur}$, $K_{vr}$, $K_{wr}$) is considered.

4.2.4. Constraint equation

The objective of optimization in this paper is to tune all the natural frequency of the PMS to the desired range (i.e., 5 Hz-21.2 Hz is shown in Section 1) to avoid resonance and to decouple each modes of vibration through dynamic analysis and optimization.

Subject to:

4.2.5. Optimization process

Combined with the content of our research, a detailed process is given as follows:

Step 1. Initialization: Considering the design variable $0<\mathrm{m}\mathrm{a}\mathrm{x}\left(K_{ij}\right)<2^{11}$, each design variable $x_i$ of integer string is converted into an 11-bit binary number before these processes are performed. Then the binary substrings of six design variables are composed to one 66-bit candidate solution.

Step 2. Generation of initial populations: The initial population (called individual) is randomly generated. And then, N populations forms one generation “S” as the initial iteration.

Step 3. Calculation of fitness values: The fitness function represents the quality of the individual. The fitness function for the problem listed in this paper can be defined as:

here $C_{\mathrm{m}\mathrm{a}\mathrm{x}}$ is the maximum estimated value of the objective function $F\left(x\right)$.

Step 4. Selection: Apply the selection operator to the generation to select a relative better “parent” solutions breeding.

Step 5. Crossover: Apply the crossover operator to the generation to transform part chromosome of each individual randomly with certain crossover probability.

Step 6. Mutation: Apply the mutation operator to the generation to select some genes in an individual randomly and changing them within the respective acceptable range.

Step 7. Termination condition: when the value of genetic generation less than the value of the maximum generation, turn to step 2 repeated the following process. The process is terminated when the value of genetic generation larger than the value of the maximum generation. The individual with the maximum fitness is considered as the optimal solution.

The control parameters of GA refer to the population size, probability of crossover, and probability of mutation. These control parameters have been studied and obtained by varying them in the GA convergence process and are given in Table 6.

4.2.6. Optimization results

Using the GA method and MATLAB program, the optimized results are summarized in Table 7 and Table 8. Table 7 shows the optimized mount stiffness in a Local Coordinate System (LCS). Table 8 shows the optimized results of natural frequency and mode energy distribution. It can be seen from Tab.8 that the maximum natural frequencies are 21.1 Hz (roll direction) which meet the target value range (5 Hz-21.2 Hz) presented in Section 1. Specially, the efficiency of decoupling at that frequency was reached 96.4 % in the roll direction. Most modes except the modes along $Z$ direction and pitch direction are well decoupled. Considering that the frequency in the $Z$ and pitch direction is lower and far from the engine excitation, the optimization result is accepted. Next, a series of experimental test are performed to evaluate the vehicle performance after optimization.

5.2.1. Vibration analysis of mount and steering wheel at idle

The test results of mount and steering wheel before and after optimization at idle are summarized in Fig. 7 and Table 10, respectively. Here, active in Table 10 represents the engine side and passive represents the frame side. It is seen from Fig. 7 that the vibration accelerations of steering wheel after optimization decreased significantly at all the directions.

As shown in Table 10, it can be observed that all the transmissibility decreased after optimization. Especially, before optimization, the transmissibility of mount front left and mount rear left at $y$-direction exceeds one, that is to say, at $y$-direction of those two mounts, the vibration shows a magnified effect. The value of the transmissibility of mount rear right at $y$-direction is very close to one. After optimization, those three values have an obvious reduction.

Fig. 7Accelerations of steering wheel with and without optimization at idle

5.2.2. Vibration of engine mount at 7th gear WOT

At the 7th gear WOT running condition, the mount accelerations at active and passive of front left, front right, rear left and rear right in $X$, $Y$ and $Z$ directions of VCS are measured, and here the accelerations in $X$, $Y$, and $Z$-direction for the RL mount are given for illustration in Fig. 8. It is seen that the accelerations at the active side (i.e., the dashed lines) increase significantly with the engine speed, but the accelerations at the passive side (i.e., the solid lines) increase slowly and relative smooth with the engine speed.

Fig. 8Mount accelerations at active and passive of Rear Left (RL)

a)$X$ direction

b)$Y$ direction

c)$Z$ direction

As the curves are very similar in both the left and right side, only the results of the right side were presented for illustration. Figs. 9-10 show the test results of the mount accelerations of before and after optimization at passive sides in three directions (i.e., $X$, $Y$, $Z$ direction). The dashed lines in figures represent the initial status without optimization. The solid lines in figures represent the status after optimization. From Fig. 9-10, it can be observed that nearly all the accelerations decreased to a different extent after optimization. It demonstrated that the optimization had significant improvement on vibration.

It is seen from Fig. 9 that the lines at the $X$ direction before and after optimization are very close to each other while the lines at $Z$ direction before and after optimization have obvious improvement. The reduction of acceleration at the $Z$ direction is much larger than other two directions with engine speed. The second one is at the $Y$ direction. The reduction of acceleration at the $X$ direction is the smallest one. That is to say, the optimization for the mount at the front side has a better effect on vibration in the $Z$ direction than other two directions.

As shown in Fig. 10, it is seen that the lines at all three directions before and after optimization have a quite difference. The reduction of acceleration at the $Z$ direction is relative larger than other two directions at all engine speeds. The second one is at the $Y$ direction. The reduction of acceleration at the $X$ direction is the smallest one. That is to say, the optimization for the mount at rear side has relative better improvement on vibration at all directions.

Fig. 9Mount accelerations before and after optimization at passive of Front Right (FR)

a)$X$ direction

b)$Y$ direction

c)$Z$ direction

Fig. 10Mount accelerations before and after optimization at passive of Rear Right (RR)

a)$X$ direction

b)$Y$ direction

c)$Z$ direction

5.2.3. Vibration of seat track and steering wheel at 7th gear WOT

When the engine speed is from 650 to 2000 rpm at 7th gear WOT, the accelerations at seat track before and after optimization in the $X$, $Y$, and $Z$ directions are shown in Fig. 11 while the accelerations at steering wheel before and after optimization in the $X$, $Y$, and $Z$ directions are shown in Fig. 12.

Fig. 11Acceleration before and after optimization at seat track (at 7th gear WOT)

a)$X$ direction

b)$Y$ direction

c)$Z$ direction

Fig. 12Acceleration before and after optimization at steering wheel (at 7th gear WOT)

$X$ direction

b)$Y$ direction

c)$Z$ direction

As compared with the lines before optimization (i.e., the dashed lines), Fig. 11 and Fig. 12 show that most of the accelerations decreased after the optimization at each engine speed. However, the variations of accelerations before and after optimization are very obvious. It demonstrated that the optimization has significant improvement on vibration at seat track and at steering wheel.

5.2.4. Vibration transmissibility analysis at 7th gear WOT

To evaluate the performance of vibration isolation, we defined the transmissibility as the ratio of active accelerations to the passive ones. Usually, the transmissibility is as low as possible. The value of transmissibility equals to 20 % and is set up as the target in the engineering application. It can be written as:

Here, $a_p$is the vibration of passive side, $a_a$ is the vibration of active side, $T_A$ is the transmissibility.

The relations between the transmissibility before and after optimization with the engine speed are plotted in Figs. 13-16 in three directions (i.e., $X$, $Y$, $Z$ direction). The dashed curve lines in figures represent the initial status without optimization. The solid curve lines in figures represent the status after optimization. The dashed straight lines in figures represent the requirement target. All the curves after optimization show that the transmissibility of each mount decreased significantly as compared with the curves before optimization. After optimization, the reduction of all the curves is relative flat which is good for the improvement of comfort. It demonstrates that the vibration isolation meets the requirement after optimization.

Fig. 13Mount transmissibility before and after optimization at Front Left (FL)

a)$X$ direction

b)$Y$ direction

c)$Z$ direction

Fig. 14Mount transmissibility before and after optimization at Front Right (FR)

a)$X$ direction

b)$Y$ direction

c)$Z$ direction

As shown in Fig. 13 and Fig. 14, the transmissibility of front mount in all the directions meets the targets before optimization, except in the $Y$ direction. With the increase of engine speed, the transmissibility has some improvements and the targets in all the directions are satisfied after optimization. It is can be observed that the front mount has better isolation performance in all the directions after optimization.

As shown in Fig. 15 and Fig. 16, the transmissibility of rear mount in the $Y$ and $Z$ directions do not meet the target before optimization. The transmissibility of rear mount has relative larger decrease in the $Y$ and $Z$ directions after optimization, and the targets in all the directions are satisfied. It can be observed that the optimization can improve the isolation performance of rear mount in any direction, especially the improvement in the $Y$ and $Z$ directions are relative significant.

Fig. 15Mount transmissibility before and after optimization at Rear Left (RL)

a)$X$ direction

b)$Y$ direction

c)$Z$ direction

Fig. 16Mount transmissibility before and after optimization at Rear Right (RR)

a)$X$ direction

b)$Y$ direction

c)$Z$ direction

5.2.5. Interior sound pressure level (SPL) at 7th gear WOT

The interior noise is mainly composed of structure vibration noise (i.e., engine vibration, etc.) and air noise (i.e., engine noise, exhaust noise, road noise, etc.) via different paths to reach the target location and superposition. In order to evaluate the effects of the optimization of the Powertrain Mounting System (PMS) on the interior noise, interior noise at the driver’s left ear was measured at 7th gear WOT acceleration condition. Both the noise test results with and without optimization are presented in Fig. 17 and Fig. 18.

Fig. 17Noise color map at driver’s left ear (at 7th gear WOT)

a) Initial status

b) Optimization status

As shown in Fig. 17(a), it can be seen that the third-order vibration of engine contributes more on the interior noise before optimization, referred to engine speed at 1100-1700 RPM and the frequency at 55-85 Hz. From Fig. 17(b), it can be seen that the energy distribution of the interior noise at the third-order vibration of engine was significantly decreased after optimization. And no obvious peak point appeared. From Fig. 18, it can be seen that interior sound pressure level (SPL) at the 7th gear WOT acceleration condition was obviously improved as compared with the initial status before optimization, especially the improvement at the peak point. Therefore, the optimization of the PMS can better improve the vehicle interior noise at a low-frequency band.

Fig. 18Interior noise at driver’s left ear (at 7th gear WOT)