## 2 Understanding of Harmonic Acceleration in the In-Line Four-Cylinder Engine  
A slider-crank mechanism used in the conventional fixed compression ratio internal combustion engine is shown in Figure 1. The displacement of the piston pin, \(s\), can be derived as  
\[s=r\left(\sin\theta+\frac{l}{r}\sqrt{1-\left(\frac{e-r\cos\theta}{l}\right)^{2 }}\right), \tag{1}\]  
where \(\theta\) is the crank angle, \(r\) is the crank length, \(l\) is the connecting rod length, and \(e\) is the piston offset.  
Figure 1: Slider-crank mechanism.  
Equation (1) and its time derivatives can be directly used to determine the motion of the piston, however, in order to find the effects of changes in the design variables \(r\) and \(l\) on the acceleration thus on the inertia forces, an approximate form of Equation (1) is used [17; 18; 19]. When \(e=0\), for example, the binomial series expansion of the square root term in Equation (1) yields  
\[\sqrt{1-\left(\frac{r}{l}\right)^{2}\cos^{2}\theta}=1-\frac{1}{2}\left(\frac{r }{l}\right)^{2}\cos^{2}\theta-\frac{1}{8}\left(\frac{r}{l}\right)^{4}\cos^{4} \theta-\frac{1}{16}\left(\frac{r}{l}\right)^{6}\cos^{6}\theta-\cdots. \tag{2}\]  
In most engines, since the crank-connecting rod ratio \(r/l\) is less than 1/3, Equation (2) can be estimated quite closely by the first two terms. Hence, Equation (1) can now be approximated as  
\[s\cong r\left\{\sin\theta+\frac{l}{r}-\frac{1}{2}\left(\frac{r}{l}\right)\cos ^{2}\theta\right\}. \tag{3}\]  
Substituting the trigonometric identity \(\cos^{2}\theta=(1+\cos 2\theta)/2\) into Equation (3) gives the displacement of the piston pin as  
\[s\cong r\left\{\sin\theta+\frac{l}{r}-\frac{1}{4}\left(\frac{r}{l}\right)(1+ \cos 2\theta)\right\}. \tag{4}\]  
Since the position of the piston pin is a periodic function of the crank angle, the Fourier series approximation of Equation (1) would yield the same as Equation (4).  
Assuming the angular velocity of the crank, \(\omega\) is constant, the second time derivative of Equation (4) gives the acceleration of the piston pin as  
\[\ddot{s}\cong r\omega^{2}\left(-\sin\theta+\frac{r}{l}\cos 2\theta\right), \tag{5}\]  
which consists of the harmonics of the crank angle \(\theta=\omega t\) where \(t\) is the time.  
Let \(m_{eq}\) be the equivalent concentrated mass of the piston and connecting rod at the piston pin, then the inertia force at the piston pin, \(F=m_{eq}\ddot{s}\), has its primary force component \(-m_{eq}r\omega^{2}\sin\theta\) due to the first harmonic acceleration and the secondary force component \(m_{eq}\omega^{2}\left(r^{2}/l\right)\cos 2\theta\) induced by the second harmonic acceleration.  
When the slider-crank mechanism is used in an in-line four-cylinder engine with crank throws at \(90^{\circ},270^{\circ},270^{\circ}\), and \(90^{\circ}\) phase angles, the harmonic acceleration components up to the fourth harmonic at the piston pins in cylinder 1 and cylinder 2 are calculated as shown in Figure 2. The \(n\)th harmonic acceleration components expressed in terms of \(\cos(n\theta)\) and \(\sin(n\theta)\) in cylinders 1 and 4 appear as \(\cos\{n(\theta+\pi)\}\) and \(\sin\{n(\theta+\pi)\}\) in cylinders 2 and 3, each of which becomes \(-\cos(n\theta)\) and \(-\sin(n\theta)\) for odd \(n;\cos(n\theta)\) and \(\sin(n\theta)\) for even \(n\). Hence, the inertia forces due to the odd harmonic acceleration components in cylinders 1 and 2, and in cylinders 3 and 4 would cancel out and sum to zero, however those induced by the even harmonic acceleration components do not cancel out but add up and result in unwanted vibration. Hence, the minimization of the second harmonic acceleration is a good guideline for the kinematic conceptual design of a new in-line four-cylinder VCR engine.