## 3 Kinematic Conceptual Design Process
### Kinematic Conceptual Design Process of Conventional Engine Mechanisms  
The kinematic design of mechanisms requires type synthesis, dimensional synthesis, and analysis. Regarding the kinematic design of the driving system for a conventional engine mechanism, since its kinematic structure is the slider-crank mechanism, the conceptual kinematic design is subject to the determination of the lengths of the crank and connecting rod, the piston offset, and the compression height of the piston to achieve a target performance, and then is subject to kinematic and dynamic analysis. The guidelines for the main dimensions [1,19] and the balancing of the shaking force and shaking moment for engine mechanisms are well established [15, 16, 17, 18, 19]. Kinematic Conceptual Design Process of In-Line Four-Cylinder VCR Engine Mechanisms Considering Vertical Second Harmonic Acceleration  
In general, since a VCR engine mechanism requires more links and joints than the conventional engine to vary the compression ratio, there can be many types of candidate mechanisms. Hence, the first step of the conceptual design process of a VCR engine is the type synthesis. For this, graph theory can be applied to enumerate feasible kinematic structures [20], or appropriate VCR engine mechanisms can be chosen from the atlas of kinematic chains [14,20,21]. Considering the number of graphs to be enumerated and the complexity of the dimensional synthesis of two-degree of freedom variable mechanisms, it would be enough to enumerate or choose basic one-degree of freedom candidate mechanisms excluding their control function [22]: the variable action can be achieved by moving one of the ground pivots. After selecting candidate mechanisms, the next step is the dimensional synthesis to determine the joint positions and link lengths that satisfy desired specifications such as the TDC position and stroke. For this, the displacement equations of the selected mechanisms can be used to determine the dimensions. The next step of the design process would be the kinematic and dynamic analysis of the initially designed candidate mechanisms. For the dynamic analysis, however, the mass properties of links and gas pressure are required, which are not available at the conceptual design stage. Instead, the harmonic acceleration analysis of candidates is the proper next step, since the distributed mass of a link can be substituted by dynamically equivalent point masses located at neighboring joints [23] and the shaking forces in the in-line four-cylinder engine depend upon the second harmonic acceleration. For illustration purpose, only the vertical harmonic acceleration of each joint will be considered in this paper. The same procedure can be applied to the horizontal acceleration. The vertical displacement of  
Figure 2: Harmonic acceleration components at the piston pin of slider-crank mechanism: (**a**) Cylinder 1 and (**b**) Cylinder 2.  
the \(i\)th joint of a candidate VCR engine mechanism, \(f_{P_{y}}(\theta)\), which is a periodic function of period \(2\pi\) of crank angle \(\theta\), can be represented by a Fourier series as  
\[f_{P_{y}}(\theta)=a_{0}+\sum_{n=1}^{\infty}(a_{n}\cos n\theta+b_{n}\sin n \theta), \tag{6}\]  
where  
\[ \begin{array}{l}a_{0}=\frac{1}{2\pi}\int_{-\pi}^{\pi}f_{P_{y}}(\theta)d\theta,\\ a_{n}=\frac{1}{\pi}\int_{-\pi}^{\pi}f_{P_{y}}(\theta)\cos n\theta d\theta\,, \qquad n=1,\,2,\,\cdots\,,\\ b_{n}=\frac{1}{\pi}\int_{-\pi}^{\pi}f_{P_{y}}(\theta)\sin n\theta d\theta\,, \qquad n=1,\,2,\,\cdots\,.\end{array}  \tag{7}\]  
Differentiating Equation (6) with respect to \(\theta\) twice gives the Fourier series expression of the vertical acceleration (length/rad\({}^{2}\)) of the \(i\)th joint. The next step of the design process is the selection of feasible VCR engine mechanisms among candidates on the basis of the second harmonic acceleration analysis when they are used in the in-line four-cylinder engine.