Engine dynamic characteristics research based on experiment and computer numerical simulation.
With the development of the computer simulation technology and testing technology, the dynamic characteristics of mechanical system is getting more and more attention (Xu and Tan, 2016). Especially for high-speed, high-precision, and complex mechanical system, dynamic characteristics significantly affect its working performance and product quality. Large-scale complex mechanical structures, such as engine systems and vehicle transmission systems, are composed of a number of components assembled together according to a certain functional requirement and the connection part among the components is known as the "joint surface" (Liao et al., 2004; Zhao et al., 2016). A variety of contact forms exist on the joint surface of these mechanical structures, such as elastic contact, plastic contact, coupled thermal-elasticity contact, and the contact between the oil film and mechanical parts (Tian et al., 2013). When the structure vibrates, damped vibrations having multiple degrees of freedom occur between joint surfaces and energy accumulation and consumption occurs at the joint surface, thus leading to the complex elasticity and damping characteristics of the joint surface (Wang et al., 2011). The characteristics of joint surface significantly affect the dynamic performance of the mechanical structures by reducing integral stiffness and increasing damping of the mechanical structure, which results in the decrease in natural frequencies and the complex vibration modes. Therefore, the contact problems of joint surface should be considered in the analysis of the dynamic characteristics of complex mechanical structures (Huang and Jin, 2012). It is necessary to further study the dynamic characteristics of composite structures, especially large and complex composite structures.
The remainder of this paper is organized as follows. In Section 2, the study progresses on dynamic characteristics of the combination structures are summarized and a novel method is proposed to simulate the dynamic characteristics of engine combination structures. In Section 3, the fundamental theory of the new method is introduced. Section 4 describes the application of this new method in the engine combination structures and presents the comparison of numerical simulation results with the experimental ones. Conclusions are summarized in Section 5.
2. Study Progresses on Dynamic Characteristics of Composite Structures
Several study methods of the dynamic characteristics of single or simple mechanical structures, such as experimental modal analysis, finite element analysis method, and boundary element method, are widely used. Engine modeling has achieved significant results. The group in Madison University established finite element model for each diesel engine part and performed the engine simulation. Yuehui Liu made a modal test of diesel engine body and finite element method (FEM) calculation (Sun, 2012). At present, the FEM has been applied to study dynamic characteristics of some structures successfully. However, the finite element analysis cannot provide precise calculation results for the complex assembled structures. Machine tools are composed of many substructures connected together by joints such as bolts, rivets, glue, and welds. Among these joints, the bolted joint is mainly employed in the fixed connection of parts for the convenience of repair and replacement. Characteristics of the joints determine the dynamic behaviors of machine tools and most vibration energy loss occurs at the joint between two joint surfaces. The contact problems of joint surface had been extensively studied since the 1960s, and the studies on the dynamic characteristics of joint surface had obtained great achievements (Li, 2002; Jalali, 2016; Mao et al., 2010; Mehrpouya et al., 2016; Bachmayer and Whitcomb, 2003; Jia et al., 2013), especially in the dynamics of bolt joint surfaces. Zhao presented a non-linear virtual material method based on surface contact stress to describe the bolted joint for accurate dynamic performance analysis of the bolted assembly and used the fractal geometry theory to describe the surface topography(Zhao et al., 2016). Hassan Jalali presented an identification approach in which the dynamic characteristic equation of linear interface parameters was adopted and the approach was applicable to both analytic and numerical problems (Jalali, 2016).
The scope of previous studies on joint surface was limited to two relatively simple mechanical members. However, the dynamic characteristics of large, complex composite structures were seldom reported at home and abroad. In the studies on large complex mechanical structures such as engine structures, complex model requires the longer computation time and allows the higher computation precision. In order to reduce the computational scale and guarantee the calculation accuracy, a novel method was presented to simulate the dynamic characteristics of engine combination structures in this paper. We comprehensively utilized the joint surface dynamic characteristics simulation method, experimental modal analysis method, and substructure modal synthesis method to deeply study the dynamic characteristics of large complex combination structures.
This method includes several main steps. The first step is to divide the engine into many substructures according to the dynamic substructure method, namely, the substructure modal synthesis method. The second step is to introduce virtual elements to simulate the dynamic characteristics of the joint surface of two substructures and identify the stiffness and damping characteristics of the virtual elements by the combined identification method of the theoretical modeling and dynamic experimental method (experimental modal analysis method). The third step is to assemble all the substructures in accordance with substructures modal synthesis and get dynamic characteristics of the engine by numerical simulation of the whole structure dynamic model (Cui and Yang, 2008). Finally, an experiment is conducted to measure the similarity between dynamic characteristics of the actual engine and the simulation results.
3.1. Experimental Modal Analysis of Engine Combination Structure
In order to carry out experimental modal analysis, the pulse hammer excitation method was adopted under the condition of free suspension state to individually hang engine block, crankshaft, cylinder head and their combination structures with elastic rope (Yasuda et al., 1997). The modal test system is shown in Fig. 1.
In order to obtain modal parameters precisely, the multi-position exciting technology was used and the excitation frequency of the test system was set as 1000 Hz (Fu et al., 2000). Because single point measure method is prone to omit some modes, the multi-position measure method is chosen for each structure to avoid mode omission. Through the test, the natural frequencies and vibration modes (eigenvalues and eigenvectors) of engine block, crankshaft, cylinder head, and their combination structures can be obtained.
3.2. Sub-Structure Modal Synthesis Method
Substructure modal synthesis method is also called dynamic substructure method and mainly used in the dynamic characteristics analysis of complex combination structures, such as engine combination structures. In the dynamic analysis and design of some large complex structures, degree of freedom can reach ten thousand and even hundred thousand, thus leading to the unimaginable computation load. Therefore, degree of freedom should be reduced effectively. The dynamic substructure method, namely, the substructures modal synthesis method, can decrease the calculation load caused by the large degree of freedom. The method is described as follows:
Generally, in the substructure modal synthesis method, the vibration system's kinematics equation is converted into modal coordinate space equation and the higher modes of the substructures are removed. Therefore, it is required to convert the original system's physical coordinate into modal coordinate for all modal synthesis methods. Detailed steps are introduced as follows (Zhao et al., 2016; Cui and Yang, 2008).
Substructure kinematics equation can be established by dividing the whole structure into several substructures. Then, each one's kinematics equation can be established according to any model or method. Supposing that the r -th kinematics equation of substructures is expressed as:
[m.sub.r] [[??].sub.r] + [k.sub.r] [x.sub.r] = [f.sub.r] (l)
where [m.sub.r], [k.sub.r] and [f.sub.r] are the mass matrix, stiffness matrix, and exciting force vector based on generalized coordinate [x.sub.r].
Then, the substructure kinematics equation is converted into the modal coordinate equation. Supposing that the right-hand side of Eq. (1) is zero, its eigenvalues are solved. The main modes of substructures are calculated and the other modes, such as rigid body modes and constrained modes, are determined. These modes constitutes modal matrix [[phi]].sub.r]. Modal matrix [[phi]].sub.r] consists of substructures' lower order modes under simple conditions. The first coordinate conversion of Eq. (1) can be implemented by taking [[phi]].sub.r] as transformation matrix.
[x.sub.r] =[[phi]].sub.r][p.sub.r] (2)
Substructures kinematics equation can be expressed with the other equation containing modal coordinate [p.sub.r] :
[[bar.m].sub.r] [[??].sub.r] + [[bar.k].sub.r] [p.sub.r] = [[bar.f].sub.r] (3)
where [[bar.m].sub.r], [[bar.k].sub.r] and [[bar.f].sub.r] are the mass matrix, stiffness matrix, and exciting force array based on modal coordinate [p.sub.r]. In addition, they are defined as follows:
[[bar.m].sub.r] = [[phi]].sup.T.sub.r] [m.sub.r][[phi].sub.r],[[bar.k].sub.r] = [[phi]].sup.T.sub.r][k.sub.r][[phi].sub.r], [bar.f] = [[phi]].sup.T.sub.r][f.sub.r] (4)
The non-connected kinematics equation of the whole structure can be obtained if all the substructures' kinematics equations with modal coordinate are combined together.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
It can be simplified as follows:
[bar.M][??] + [bar.K]p = [bar.F] (5)
where both and are block diagonal matrices.
When the second coordinate conversion is completed, the kinematics equation of the whole connected structure expressed with generalized modal coordinate can be obtained according to the constraint conditions of the substructures' joint surfaces. Because the integration of the substructures is implemented in the collection p consisting of substructure modal coordinate [p.sub.r], the constraint conditions should be expressed with the collection p. Obviously, the collection p is related to modal matrix [phi] and the constraint condition expression varies with [phi].
The balance conditions of joint surface force have been satisfied in modal coordinate system and only the displacement compatible condition should be considered. Supposing that the displacement of joint surface point's compatible equation in physical coordinate is:
HX = o (6)
where H represents the constraint matrix; X represents the displacement vector. According to Eq. (2), we get the compatible equation expressed with the coordinate p
H[phi]p = 0 or Bp = 0 (7)
Combining Eq. (5) with Eq. (7), the kinematics equation of the whole structure assembled with all the substructures is obtained. The number of the degrees of freedom of the whole structure is smaller than the sum of that of all the substructures because of the mutual constraints among the substructures. If the collection p is divided into dependent coordinate [p.sub.d] and independent [p.sub.i] =q, the matrix B is treated in the same way. Then Eq. (7) can be rewritten as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)
where S is transformation matrix of the second coordinate conversion. After the coordinate conversion of Eq. (5) according to Eq. (8), the kinematics equation of the connected whole structure expressed by generalized coordinate q is obtained as
M[??] + Kq = F (9)
M = [S.sup.T][bar.M]S, K = [S.sup.T] [bar.K]S,F = [S.sup.T][bar.F] (10)
After solving the Eq. (9), the natural frequencies of the whole structure are obtained and the main modals, frequency responses, and kinematics status expressed by generalized coordinate q, are also obtained.
Finally, after the coordinate conversion of Eqs. (8) and Eq.(2) to physical coordinate, the main modes of each substructures and each kinematics status expressed by physics coordinate are obtained.
3.3. Simulation of Dynamic Characteristics of Joint Surface With Virtual Elements
The dynamic characteristics of the joint surfaces should be considered in the establishment of the dynamics model of complex assembled structures (Zhao et al., 2016; Yasuda et al., 1997). Moreover, in the application of the modal synthesis theory, the dynamic characteristics of joint surfaces should be identified or simulated according to the dynamic substructure method (Bachmayer and Whitcomb, 2003).
A simulation method of dynamic characteristics of the joint surface with the concept of virtual elements was applied in engine combination structures. The method is described as follows:
Supposing that there are two substructures: l and y. Mass matrix and stiffness matrix of substructures l are [M.sub.l] and [K.sub.l] , respectively. Mass matrix and stiffness matrix of substructures y are [M.sub.y] and [K.sub.y] , respectively. Virtual elements are introduced to simulate the connection characteristics between two substructures. This virtual element could be defined by two characteristics: elasticity and damping. The schematic diagram of the simulation of the dynamic characteristics of joint surface by using virtual elements is shown in Fig. 2.
Virtual elements are adopted to simulate the features of joint surfaces between engine block and cylinder head, the outer surface of crankshaft, and the inner surface of sliding bearing.
3.4. Parameter Identification Method of Virtual Elements of Joint Surface of Combination Structures
To study the dynamic characteristics of the joint surface, the stiffness and damping characteristics of the virtual elements should be identified according to the combination identification method of theoretical modeling and dynamic experimental method described in Subsection 3.1. Specific identification process is introduced as follows:
The equations of a free vibration system can be written as
M[??] + C[??] + Kx = 0 (11)
where M, C, and K are respectively the mass matrix, damping matrix and stiffness matrix of the system and can be obtained by analysis methods. The characteristic equation of the system is
([[lambda].sup.2.sub.i] M + [[lambda].sub.i] C + K)[[phi].sub.i] = o (12)
Considering stiffness and damping of the joint surface, we get
[[[lambda].sup.2.sub.i] M + [[lambda].sub.i](C + [C.sup.J]) + K + [K.sup.J]] [[phi].sub.i]= 0 (13)
where C is zero because structural damping is generally ignored and [C.sup.J] and [K.sup.J] are unknown stiffness and damping parameters of the joint surface and should be identified. By dynamic experiment and modal parameter identification, eigenvalues [[lambda].sub.i] and eigenvectors [[phi].sub.i] of the system can be obtained. After substituting each eigenvalue and eigenvector into Eq. (13), linear equations with unknown joint surface parameters can be obtained as
AZ = B (14)
where Z is a vector of unknown parameters of joint surface; A and B are known coefficient matrix and constant matrix involving the system mass, stiffness, damping and the measured frequency response function. Therefore, the parameter identification problem of the joint surface is converted into the solution problem of linear equations. Multiplying [A.sup.T] on both sides of Eq. (14) gives
[A.sup.T] AZ = [A.sup.T]B (15)
Thus, the least squares solution of Z is obtained as:
Z = [([A.sup.T] A).sup.-1] [A.sup.T]B (16)
where Z is the parameter of joint surface obtained by solving Eq. (16).
4. Result Analysis And Discussion
The application of the novel study method of dynamic characteristics in the engine is described below.
4.1. Model Establishment of Engine Block+Cylinder-Head+Crankshaft
In the dynamic analysis and design of some large complex structures such as engine block, cylinder-head, crankshaft and their combination structures, degree of freedom may reach hundreds of thousands and even one million, thus leading to the unimaginable computation load. Therefore, the degree of freedom should be reduced effectively. The method of dynamic substructure described in Subsection 3.2, namely, substructures modal synthesis method, can decrease the computation load. In the method, the whole structure is firstly divided into a lot of substructures or components and then each structure is subjected to finite element analysis based on the comparison of modal tests to obtain dynamic characteristics of every substructure. The engine substructures include engine block, cylinder head, and crankshaft. The solid model of these substructures are shown in Fig. 3. On the basis of the reliable finite element models of each substructure, the finite element model of the whole structure is obtained.
Because an engine is a structure assembled with different parts, there are all kinds of joint surfaces in the engine, such as the surface between block and cylinder head, the outer surface of crankshaft, and the inner surface of sliding bearing, thus leading to the difficulty in model establishment. Considering that dynamic characteristics of the joint surface of the engine combination structure cannot be carried out theoretically or experimentally, virtual elements were introduced and the stiffness and damping characteristics of virtual elements of the joint surface were identified through the combination method of theoretical modeling and dynamic experiment. The specific identification process was described in Subsection 3.4.
After obtaining the dynamic characteristics of substructures such as engine block, cylinder head, and crankshaft, all the substructures were assembled together according to the mutual coupling conditions among the substructures and the dynamic model of engine was obtained according to the substructure modal synthesis method. With the stiffness and damping parameters of the joint surface identified above as the boundary conditions, the dynamic characteristics of the engine combination structures can be obtained by numerical simulation of the engine dynamic model. The three-dimensional solid model of engine combination structure composed of engine block, crankshaft, and cylinder head was established (Fig. 4).
4.2.Finite Element Modal Analysis and Comparison
The engine combination structure's solid model was meshed by the tetrahedron fournode elements. Then the finite element model of the combination structure is obtained, as shown in Fig. 4. By substituting the identified stiffness and damping parameters of joint surface among engine block, crankshaft, and cylinder head into the dynamic model of the whole structure, the dynamic characteristics of the engine structures can be obtained by numerical simulation.
0When solving modals with the finite element analysis, Lanczos method, an efficient solving method of engineering eigenvalue, is adopted (Luo and Hu, 2015; Cai and Liang, 2014). At last, the natural frequency comparison of numerical simulation results of the engine combination structures and the experimental results are listed in Table 1.
In Table 1, the relative errors between numerical simulation results of natural frequency for the combination structures and the experimental ones are less than 7%, which satisfies engineering requirements. At the same time, the study found that testing and simulation vibration modes are also in good agreement. The comparison results proved that the results were consistent with actual situations.
In this paper, a novel method was proposed to simulate the dynamic characteristics of engine combination structures. We comprehensive utilized the joint surface dynamic characteristics simulation method, experimental modal analysis method, and substructure modal synthesis method to explore the dynamic characteristics of large complex combination structures. By comparing the dynamic characteristics results obtained from the novel method and the experimental ones, we found that the relative errors between the simulation results and the experimental results were less than 7%, which satisfied engineering requirements. The comparison results proved that the results were consistent with actual situations. The main conclusions are drawn as follows:
It is feasible to simulate the engine combination structures according to the substructure modal synthesis method. This method can also be applied to other large complex composite structures. The method extracts the lower-order modes and removes the higher-order modes of the substructures, thereby reducing solution time and improving computational efficiency.
The dynamic characteristics of joint surface can be effectively simulated with virtual elements. The simulation is also called the establishment of the equivalent dynamic model of joint surface. The equivalent dynamic model of joint surface facilitates the analysis and study of dynamic characteristics, dynamic tests and parameter identification.
The parameter identification method of joint surface by the combination of theoretical and experimental methods can also be feasible. The combination of the equivalent dynamic model of joint surface and dynamic test yields the effective and comprehensive identification method in engineering applications.
The novel method proposed in this paper is suitable for dynamic characteristics study of engine combination structures and can be used for other kinds of machinery combination structures. The method provides a solid foundation for improving the dynamic performance of the whole machine.
This work is supported by the National Natural Science Foundation of China (Grant No.51375462)
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Yinping Jing (1), Zhiqin Cui (1), Kun Huang (1), Zhen Liu (2), Wei Liu (1), Xinling Dong (3)
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(1) College of Mechatronic Engineering, North University of China, 030051, Taiyuan, China
(2) China North Advanced Technology Generalization Institute, 100089, Beijing, China
(3) China Institute of Defense Science and Technology, 101601, Beijing, China
Table 1 - Comparison of numerical simulation results with the experimental one (unit: Hz) mode experimental simulation errors order result result (%) 1 220.35 235.19 6.73 2 365-19 362.15 -0.83 3 408.17 401.46 -1.64 4 458.03 429.92 -6.14 5 541.48 531.62 -1.82 6 574.26 603.11 5.02