Combining substructure and power flow theory, in this paper an external program is written to control MSC. Nastran solution process and the substructure frequency response are also formulated accordingly. Based on a simple vehicle model, characteristics of vibration, noise, and power flow are studied, respectively. After being compared with the result of conventional FEM (finite element method), the new method is confirmed to be feasible. When it comes to a vehicle with the problem of lowfrequency noise, finite element models of substructures for vehicle body and chassis are established, respectively. In addition, substructure power flow method is also employed to examine the transfer characteristics of multidimensional vibration energy for the whole vehicle system. By virtue of the adjustment stiffness of drive shaft support and bushes at rear suspension lower arm, the vehicle interior noise is decreased by about 3 dB when the engine speed is near 1050 rpm and 1650 rpm in experiment. At the same time, this method can increase the computation efficiency by 78%, 38%, and 98% when it comes to the optimization of chassis structure, body structure, and vibration isolation components, respectively.
With the continuous enhancement of life quality of human beings, consumers’ requirements of vehicle NVH (noise, vibration, and harshness) performance become accordingly more stringent. The NVH performance is a key factor which determines if a vehicle can stand on the market. The NVH performance is mainly measured based on two indicators, body vibration and interior noise. Powertrain and vehicle driving road are considered as the main excitation sources. Through chassis components and isolation components, the energy is transferred in multiple directions and eventually input to the body structure, leading to vibration of thin plate parts. Besides, coupled with interior acoustic cavity, vibrations will generate lowfrequency noise peaks, which may influence the comfort of passengers. As a result, reducing the vibration energy input to vehicle body and controlling vibration of thin plate parts are two effective ways to make the enhancement of vehicle NVH performance [
Power flow takes into account vibration speed, vibration transfer force, and their phase relations at the same time and reflects the characteristics of structure vibration response from nature. Combined with the power flow theory and FEM, the substructure method is developed in this work [
The power flow describes the energy transfer at each structure point, and it can guide the control of structure and noise efficiently. The power flow is defined as follows [
In this formula,
It is difficult to characterize the structure vibration with transient power flow, so the average power flow is taken as the evaluation index to describe the energy characteristic of structure response. Vibration power flow is expressed as follows:
The time averaged vibration power is given by [
As the schematic showed in Figure
Schematic of substructure.
Steadystate frequency response equation is
In this equation,
The above equation for displacement response is solved as follows:
The equation of motion for each subsystem in the frequency domain for the complex displacement
The subscript
Note that the number
Subsystem
Subsystem
Let the superscript
The FRFs of the system can be represented as
The connection DOFs from the partitioned equation (
Now let
Note that for the fully coupled system
Substitute these transmitted forces into (
Set (
Substituting (
Now let us obtain coupled FRF equation
Recall from equation the coupled system (
Recall from (
For the coupled system, (
Substitute (
Substitute (
Note that at the system level
Thus,
Divide each side of (
An individual admittance function
By now, the entire FRF matrix of the coupled structure
The acoustic analysis is based on inviscid flow with linear pressuredensity relation as
Combining the above equations, the governing equation of the fluid domain is
After finite element discretization, the assembly of equations for the fluid domain is
The matrix
The structural equation assembly can be written as
The matrix
Therefore, the combined fluidstructure interface equation is
The above equations are solved simultaneously for unknowns in structure and fluid domains, either by direct frequency response or by modal frequency response [
A 3D vehicle model is built and is simplified to three parts: body, acoustic cavity, and chassis. The body structure is simulated as shell elements CQUAD4, whose thickness is 2 mm with a total of about 12 thousand of elements. The acoustic cavity model can be built with the closed body model and solid element CHEXA with acoustic properties, and there are about 66 thousand of elements in acoustic cavity model. According to (
Simple vehicle model for conventional method.
Unit sinusoidal excitation torque was input at the mass center of powertrain in rotation direction of crankshaft at a frequency ranging from 20 Hz to 200 Hz with increment of 1 Hz. The vibration and noise characteristics using conventional and substructure methods are, respectively, examined. The isolator can reduce the vibration transmission energy. Its master side is connected to chassis and slave side is connected to body structure. The excitation power is transferred to the body though 4 isolators, and the radiated noise from the vehicle is generated by body structure vibrations.
From Figure
Simple vehicle model for substructure method.
Taking the velocity characteristic of the master side and slave side of
Velocity characteristic of
Acoustic characteristic at point
With the acoustic characteristic at an interior point
Directed at the low and middle frequency noise problem existing in FR cars, the full vehicle finite element model was established for NVH analysis [
In terms of NVH analysis, body structure is a significant transmission path. To a great extent, the response of body structure determines the interior noise level [
Schematic of ACM welding spot.
2layer welding
3layer welding
Based on the 3D geometry model of body in white, small structures which nearly have no effects on body response were simplified, including pipe bundle, wire harness, and bolts. In accordance with the experience, the uniform structural and fluid damping coefficient of this type of vehicle are chosen as 0.04 and 0.12, respectively. All thin plates are modeled with shell elements. The average size is 10 mm × 10 mm, and there is a total of about 600,000 shell elements. To ensure the credibility of the analysis results, element quality was inspected according to Table
Criterion of element check.
Aspect  Length  Skew/°  Warpage/°  Trias angle/°  Quads angle/°  Jacobian  

Threshold  ⩽5 

⩽40  ⩽15 



As shown in Figure
Benchmark of body in white modal.
Modal order  Test results  FEM results  Relative error  Vibration mode 

1  28.78  28.82  +0.01%  Deformation of back door 
2  32.99  35.35  +7.15%  First order of roof 
3  38.99  38.91  −0.21%  Second order of roof 
4  42.96  42.54  −0.98%  Global twist 
Body in white.
Finite element model
Modal test
In Table
Modal shape.
1st order 28.82 Hz
2nd order 35.35 Hz
3rd order 38.9 Hz
4th order 42.54 Hz
According to the criterion of element quality in the table, the full vehicle finite element model for NVH analysis is established as shown in Figure
Finite element model of full vehicle.
Based on the finite element model in Figure
Substructure 1 consists of trimmed body and acoustic cavity, and substructure 2 is composed of chassis assembly. The two substructures are connected through the vibration isolators at the points, as is shown in Figure
Substructures and connection point.
Based on the above NVH finite element model, the substructure power flow analysis method was used to obtain the velocity response at the connection point between the transmission shaft support bushing and the body. Besides, comparisons with the vibration response result of conventional FEM are presented in Figure
Velocity response.
Translation
Rotation
The curves in Figure
Based on the NVH finite element model, the substructure power flow analysis method is employed to obtain the interior acoustic response, which is compared with the result obtained through conventional method. Besides, the curves are plotted in Figure
Two sets of curves are of high consistency. In 20–100 Hz full frequency range, the acoustic characteristic curve to a great extent retains peak frequency features of conventional FEM frequency response result. There is little difference between noise peak values of two results, which proves that the result of substructure power flow method is sufficiently accurate.
32 Hz, 48 Hz, and 56 Hz are potential frequencies corresponding to noise peak. The interior noise response curves peak at around 32 Hz, 48 Hz, and 56 Hz, and it becomes most distinct at 32 Hz. At the location of driver’s right ear, the noise reaches 68 dB; besides, at the middle of the second and back row seats, the noise reaches 71 dB.
Characteristics of interior noise response.
Driver’s right ear
Middle of second seats
Middle of rear seats
The substructure FEM of frequency response is employed to analyze the structure vibration velocity and vibration transmission force. Combined with the basic theory of power flow, the characteristics of vibration transmission energy between substructures are investigated. Due to the scalar property of power flow, the risky transmission path can be ranked and identified, which can also be applied in conducting the adjustment of vibration isolation system and body structure. The vibration excitation is generated by powertrain. Through the powertrain mounts, front suspension bushing, rear suspension bushing, exhaust pipe hook, transmission shaft support bushing, and chassis structure, the excitation is transferred to the body structure. When it comes to the vehicle vibration system, the inputted multidimensional energy to response substructure, which is the body structure, should be highlighted.
With reference to NVH analysis power flow FEM model, the FRF matrix of chassis substructure and body structure are calculated, respectively, with MSC.NASTRAN. Subsequently, the frequency response analysis is carried out. Later, vibration velocities and vibration transmission forces at 20 elastic connecting points between body substructure and chassis substructure are obtained, and the total power of every connecting point in the body structure can be figured out according to (
Total power.
Transmission shaft support bushing
Rear suspension lower arm (right)
Rear suspension lower arm (left)
Furthermore, the power flow in different direction of motion is investigated, by which the composition of total power flow can be determined. Subsequently, the isolator parameters in key direction can be adjusted, and the power flow contribution of this direction can be reduced. Figure
Power of transmission shaft support bushing in 6 directions.
Figure
Power of rear suspension lower arm in 6 directions.
Right
Left
The total powers in 20 transmission paths are calculated, respectively, and ranked, and 3 risky transmission paths are identified, respectively, transmission shaft support bushing, rear suspension lower arm bushing (right), and rear suspension lower arm bushing (left). Furthermore, the power flow characteristics in each direction are analyzed, by which the main motion directions of transmission shaft support bushing and rear suspension lower arm bushing are identified, which are the rotation along
There is a total of about 2 million nodes in the NVH finite element model of the full vehicle, and about 9.5 million DOFs need to be solved.
In order to overcome the practical limitations of long computation time, extensive work has been performed, and AMLS (automated multilevel substructuring) and FastFRS (fast frequency response solver) methods are currently commonly used to solve large FE modes.
In AMLS (automated multilevel substructuring), a finite element model of a structure is automatically divided into two substructures, each of which is then subdivided into its own substructures. This subdivision is repeated recursively until thousands of substructures have been defined in a tree topology [
With the aim to compare substructure method with the conventional method, the AMLS and fastFRS method are not used in this paper. However, it is worth mentioning that the efficiency will be improved if AMLS and fastFRS were adopted. Besides, comparing with using AMLS and fastFRS methods only, if the substructure method was adopted, the required computer storage and computational time will significantly reduce when a single substructure was changed.
The model is submitted to a Dell workstation, equipped with double 3.46 GHz Intel Xeon X5690 CPUs and 48 GB memory. The computing time is about 5 hours for the vibration and noise response with conventional FE method.
The chassis substructure model contains about 680 thousand nodes, and about 3.5 million DOFs need to be solved. FRF solution takes about 1 hour. Meanwhile, the body substructure model contains about 1.32 million nodes, and about 6 million DOFs need to be solved. Its FRF solution also takes about 1 hour. It only takes nearly 4 minutes to invoke the substructure FRF matrix and carry out the full vehicle substructure frequency response analysis, which is equivalent to 0.07 hours. The analyzing time of the conventional method and the substructure method is displayed in Table
Analysis time of finite element model.
Computation method  Analysis project  DOFs/million  Time consumed/hours  Total time consumed/hours 

Conventional FEM  Frequency response of full vehicle  9.5  5  5 


Substructure FEM  Chassis FRF analysis  3.5  1  4.07 
Body FRF analysis  6  3  
Substructure frequency response  —  0.07 
As for substructure finite element analysis of frequency response, the chassis substructure FRF is firstly calculated and then the body substructure FRF. Besides, the total time needed to solve the full vehicle FRF is about 4.07 hours. Comparing with the conventional FEM analysis of frequency response, 56 minutes are saved, indicating that the efficiency is improved by 18%.
Obviously, the application of the substructure finite element analysis of frequency response in the NVH model greatly improves the efficiency, especially for models with a great number of DOFs, whose changed structure schemes require repeated calculations. The application of this method will substantially shorten the calculation time.
The chassis substructure requires to be optimized. Because the analysis model retains the substructure FRF matrix file of the body, with the application of substructure method, only the FRF of optimized chassis substructure needs to be recomputed. And then the FRF matrixes of each substructure are combined and frequency response analysis is carried out, taking about 1.07 hours in total. Compared with the conventional FEM, this method reduces the analysis time from 5 hours to 1.07 hours, and the efficiency is improved by 78%, as shown in Figure
The body substructure needs to be optimized. With the application of substructure method, only the FRF of the optimized body substructure needs to be recomputed. Subsequently, the FRF matrixes of each substructure are combined and frequency response analysis is carried out, which takes about 3.07 hours in total. Compared with the conventional FEM, which can be found in Figure
In order to optimize the parameter of vibration isolators between chassis and body substructure, only the DAMP code needs to be rewritten, which takes just about 0.07 hours in total. In comparison with the 5 hours required by the conventional method, this optimization scheme takes only 4 minutes, and the analysis efficiency is improved by 98%, as presented in Figure
Optimization of chassis.
Optimization of body.
Optimization of isolators.
Briefly, the application of substructure FEM analysis of frequency response greatly shortens the engineering computation time. As for chassis structure optimization scheme, the efficiency is improved by 78%; when it comes to body optimization scheme, the efficiency is improved by 38%; regarding vibration isolators’ optimization scheme, the efficiency is improved by 98%. Details are displayed in Table
Analysis time of optimization.
Conventional FEM/hours  Substructure FEM/hours  Efficiency improved  

Chassis optimization  5  1.07  78% 
Body optimization  5  3.07  38% 
Isolators optimization  5  0.07  98% 
The transmission shaft support bushing is made of soft rubber, and its function is to reduce the unbalanced vibration of the transmission shaft transferring to the body structure [
Samples of transmission shaft support bushing.
Installation
Original sample 50 HA
1# sample 40 HA
2# sample 70 HA
Figure
Interior noise response.
Driver’s right ear
Middle of second seats
Middle of rear seats
The bushing of rear suspension lower arm is made of rubber, whose main function is to attenuate the vibration transferred from suspension. The installation locations are points 8 and 9 in Figure
Samples of rear suspension lower arm bushing.
Installation
Original Sample
3# sample
The interior noise is tested when the vehicle rapidly speeds up at the 4th gear, and related data can be found in Figure
Interior noise response.
Driver’s right ear
Middle of second seats
Middle of rear seats
To conclude, this paper combined the substructure modeling and power flow theory and derived the function of vibration transmission force and vibration velocity at each interface. The finite element model based on substructure was developed and was used in the analysis of substructure power flow characteristics.
Simple vehicle model was selected as an example, and vibration and acoustic characters were analyzed based on MSC.NASTRAN. The accuracy of substructure frequency response analysis method was verified by comparing with the conventional FEM solution.
Combined with the NVH finite element model of full vehicle and substructure frequency response analysis method, the present study investigated the risky paths causing interior noise and the vibration modes in these paths from the perspective of energy. And then the stiffness of transmission shaft support bushing increased and the radial stiffness of rear suspension lower arm bushing decreased. Through conducting the experimental test, the interior noise was palpably improved.
The substructure frequency response analysis method retains the FRF matrix of each substructure, so the unchanged substructure does not need to be recomputed in the subsequent optimization analysis. This will significantly shorten the engineering computation, and the analysis efficiency will be improved. When it comes to the optimization schemes of chassis, body, and vibration isolators, the computation efficiency can be raised by 78%, 38%, and 98%, respectively.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This project is supported by National Natural Science Foundation of China (Grant no. 51575410).