fig. 1
Basic knowledge about Firing Order
The firing order is a sequence of giving spark/firing to the cylinders of multicylinder inline engines. Firing order is provided in multicylinder inline engines in order to avoid vibrations (If firing order is not provided in multicylinder inline engines, then all the pistons will move in upward direction at the same time and same way they move in the downward direction at the same time. This creates a lot of vibration.) For avoiding these vibrations, firing order is provided in multicylinder inline engines.
Numerical
A four stroke five cylinder inline engine has a firing order of 145321. The centre lines of cylinders are spaced at an equal interval of 15 cm, the reciprocating parts per cylinder have a mass of 1.5 kg, the piston stroke is 10 cm and the connecting rods are 17.5 cm long. The engine rotates at 600 rpm. Determine the value of maximum primary and secondary unbalanced forces and couples about the central plane.
Solution:
Given data:
$\begin{aligned}
\text {mass of reciprocating parts } m &= 1.5 \ kg \\
\text {Stroke } S &= 10 \ cm = 0.10 \ m \\
\text {Radius of crank } r &= {S \over 2} = {0.10 \over 2} = 0.05 \ m \\
\text {Length of connecting rod } L &= 17.5 \ cm = 0.175 \ m \\
\text {Obliquity ratio } n &= {L \over r} = {0.175 \over 0.05} = 3.5 \\
\text {Speed of engine } N &= 600 \ rpm \\
\omega &= {2 \pi N \over 60} = {2 \pi \times 600 \over 60 } = 62.85 { rad \over s}
\end{aligned}$
The force and couple data is given in table:
Plane 
Mass (m), kg 
Radius (r), m 
Centrifugal Force (m x r), kg.m 
Distance From R.P.(l), m 
Couple (m x r x l), kg.m
^{
2
} 
Primary Crank Position '
θ
' 
Secondary Crank Position '2
θ
' 
1 
1.5 
0.05 
0.075 
0.3 
0.022 
0° 
0° 
2 
1.5 
0.05 
0.075 
0.15 
0.011 
576° i.e. 216° 
1152° i.e. 72° 
3 (R.P.) 
1.5 
0.05 
0.075 
0 
0 
432° i.e. 72° 
864° i.e. 144° 
4 
1.5 
0.05 
0.075 
0.15 
0.011 
144° 
288° 
5 
1.5 
0.05 
0.075 
0.3 
0.022 
288° 
576° i.e. 216° 
fig.2 (a) Position of planes
fig.2 (b) Primary crank positions
 Assuming the engine to be vertical, the position of cylinders are shown in fig.2(a).
 The angles are measured from cylinder 1.
 Cylinder 3 is considered as a reference plane (R.P.)
 The engine is working on a four stroke cycle, the angle between two crank $= {720 \over 5} = 144\degree$
 Considering firing order : 145321
$\begin{aligned}
\therefore \ &\theta_1 = 0\degree \\
&\theta_4 = 144\degree \\
&\theta_5 = 144\degree + 144\degree = 288\degree \\
&\theta_3 = 288\degree + 144\degree = 432\degree \\
&\theta_2 = 432\degree + 144\degree = 576\degree
\end{aligned}$
1. Primary force polygon
For drawing a primary force polygon, take a suitable scale of 1 cm = 0.075 kg.m. From fig.2(c) it is seen that, primary force polygon is closed and hence there is no unbalanced primary force.
fig.2 (c) Primary force polygon
2. Primary couple polygon
fig.2 (d) Primary couple polygon

Draw the primary couple polygon by taking a suitable scale of $1\ cm = 0.011 \ kg \ m^2$. The vector $\overrightarrow {od'}$ represents the unbalanced primary couple $C_p$.

Now from fig.2 (d), measure $\overrightarrow {od'}$;
$\overrightarrow {od'} = 1.5 \ cm$
 Hence, the unbalanced primary couple is,
$\begin{aligned}
C_p &= \overrightarrow {od'} \times scale \times \omega^2 \\
&= 1.5 \times 0.011 \times (62.85)^2 \\
&= 65.18 \ N \sdot m
\end{aligned}$
3. Secondary force polygon
fig.2 (e) Secondary crank positions
fig.2 (f) Secondary force polygon
Draw the secondary force polygon by taking a suitable scale of $1 \ cm = 0.075 \ kg.m$.
From fig.2 (f), it is seen that the secondary force polygon is closed, hence there is no unbalanced secondary force.
4. Secondary couple polygon
fig.2 (g) Secondary couple polygon

Draw the secondary couple polygon by taking suitable scale of $1 \ cm = 0.011 \ kg m^2$. The vector $\overrightarrow {od'}$ represents the unbalanced secondary couple ${C_s}$

Now from fig.2 (g), measure $\overrightarrow {od'}$;
$\overrightarrow {od'} = 4.7 \ cm$
 Hence, the unbalanced secondary couple is,
$\begin{aligned}
C_s &= \overrightarrow {od'} \times scale \times {\omega^2 \over n} \\
&= 4.7 \times 0.011 \times {(62.85)^2 \over 3.5} \\
&= 58.3 \ N \sdot m
\end{aligned}$
fig. 1
Basic knowledge about Firing Order
The firing order is a sequence of giving spark/firing to the cylinders of multicylinder inline engines. Firing order is provided in multicylinder inline engines in order to avoid vibrations (If firing order is not provided in multicylinder inline engines, then all the pistons will move in upward direction at the same time and same way they move in the downward direction at the same time. This creates a lot of vibration.) For avoiding these vibrations, firing order is provided in multicylinder inline engines.
Numerical
A four stroke five cylinder inline engine has a firing order of 145321. The centre lines of cylinders are spaced at an equal interval of 15 cm, the reciprocating parts per cylinder have a mass of 1.5 kg, the piston stroke is 10 cm and the connecting rods are 17.5 cm long. The engine rotates at 600 rpm. Determine the value of maximum primary and secondary unbalanced forces and couples about the central plane.
Solution:
Given data:
$\begin{aligned}
\text {mass of reciprocating parts } m &= 1.5 \ kg \\
\text {Stroke } S &= 10 \ cm = 0.10 \ m \\
\text {Radius of crank } r &= {S \over 2} = {0.10 \over 2} = 0.05 \ m \\
\text {Length of connecting rod } L &= 17.5 \ cm = 0.175 \ m \\
\text {Obliquity ratio } n &= {L \over r} = {0.175 \over 0.05} = 3.5 \\
\text {Speed of engine } N &= 600 \ rpm \\
\omega &= {2 \pi N \over 60} = {2 \pi \times 600 \over 60 } = 62.85 { rad \over s}
\end{aligned}$
The force and couple data is given in table:
Plane 
Mass (m), kg 
Radius (r), m 
Centrifugal Force (m x r), kg.m 
Distance From R.P.(l), m 
Couple (m x r x l), kg.m
^{
2
} 
Primary Crank Position '
θ
' 
Secondary Crank Position '2
θ
' 
1 
1.5 
0.05 
0.075 
0.3 
0.022 
0° 
0° 
2 
1.5 
0.05 
0.075 
0.15 
0.011 
576° i.e. 216° 
1152° i.e. 72° 
3 (R.P.) 
1.5 
0.05 
0.075 
0 
0 
432° i.e. 72° 
864° i.e. 144° 
4 
1.5 
0.05 
0.075 
0.15 
0.011 
144° 
288° 
5 
1.5 
0.05 
0.075 
0.3 
0.022 
288° 
576° i.e. 216° 
fig.2 (a) Position of planes
fig.2 (b) Primary crank positions
 Assuming the engine to be vertical, the position of cylinders are shown in fig.2(a).
 The angles are measured from cylinder 1.
 Cylinder 3 is considered as a reference plane (R.P.)
 The engine is working on a four stroke cycle, the angle between two crank $= {720 \over 5} = 144\degree$
 Considering firing order : 145321
$\begin{aligned}
\therefore \ &\theta_1 = 0\degree \\
&\theta_4 = 144\degree \\
&\theta_5 = 144\degree + 144\degree = 288\degree \\
&\theta_3 = 288\degree + 144\degree = 432\degree \\
&\theta_2 = 432\degree + 144\degree = 576\degree
\end{aligned}$
1. Primary force polygon
For drawing a primary force polygon, take a suitable scale of 1 cm = 0.075 kg.m. From fig.2(c) it is seen that, primary force polygon is closed and hence there is no unbalanced primary force.
fig.2 (c) Primary force polygon
2. Primary couple polygon
fig.2 (d) Primary couple polygon

Draw the primary couple polygon by taking a suitable scale of $1\ cm = 0.011 \ kg \ m^2$. The vector $\overrightarrow {od'}$ represents the unbalanced primary couple $C_p$.

Now from fig.2 (d), measure $\overrightarrow {od'}$;
$\overrightarrow {od'} = 1.5 \ cm$
 Hence, the unbalanced primary couple is,
$\begin{aligned}
C_p &= \overrightarrow {od'} \times scale \times \omega^2 \\
&= 1.5 \times 0.011 \times (62.85)^2 \\
&= 65.18 \ N \sdot m
\end{aligned}$
3. Secondary force polygon
fig.2 (e) Secondary crank positions
fig.2 (f) Secondary force polygon
Draw the secondary force polygon by taking a suitable scale of $1 \ cm = 0.075 \ kg.m$.
From fig.2 (f), it is seen that the secondary force polygon is closed, hence there is no unbalanced secondary force.
4. Secondary couple polygon
fig.2 (g) Secondary couple polygon

Draw the secondary couple polygon by taking suitable scale of $1 \ cm = 0.011 \ kg m^2$. The vector $\overrightarrow {od'}$ represents the unbalanced secondary couple ${C_s}$

Now from fig.2 (g), measure $\overrightarrow {od'}$;
$\overrightarrow {od'} = 4.7 \ cm$
 Hence, the unbalanced secondary couple is,
$\begin{aligned}
C_s &= \overrightarrow {od'} \times scale \times {\omega^2 \over n} \\
&= 4.7 \times 0.011 \times {(62.85)^2 \over 3.5} \\
&= 58.3 \ N \sdot m
\end{aligned}$
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