A five cylinder in-line engine running at 750 rpm has successive cranks 144 degrees apart, the distance between the cylinder center lines being 375 mm. The piston stroke is 225 mm and ratio of the connecting rod to crank is 4. Examine the engine for the balance of primary and secondary forces and couples. Determine maximum values of these and the position with respect to the second crank/cylinder at which these values occur. The reciprocating mass for each cylinder is 10 kg.

Given a five-cylinder in-line engine running at 750 rpm with successive cranks 144 degrees apart, we need to examine the balance of primary and secondary forces and couples, and determine their maximum values and positions with respect to the second crank/cylinder.

Key Data:

• Engine speed: 750 rpm (≈78.54 rad/s)

• Distance between cylinders: 375 mm (0.375 m)

• Piston stroke: 225 mm (0.225 m), crank radius $R=0.1125$ m

• Connecting rod to crank ratio: 4, connecting rod length $L=0.45$ m

• Reciprocating mass per cylinder: 10 kg

Primary and Secondary Forces:

• Primary force for each cylinder: $F_{\text{primary}}=m{\omega }^{2}R\cos {\theta }_i$

• Secondary force for each cylinder: $F_{\text{secondary}}=\frac{m{\omega }^{2}R}{4}\cos 2{\theta }_i$

Crank Angles:

• Cylinder 1: 0°, Cylinder 2: 144°, Cylinder 3: 288°, Cylinder 4: 72°, Cylinder 5: 216°

Sum of Forces:

• Primary Forces: Sum of $\cos {\theta }_i$ ≈ 0, hence balanced.

• Secondary Forces: Sum of $\cos 2{\theta }_i$ ≈ 0, hence balanced.

Primary and Secondary Couples:

• Primary Couple: Sum of $\cos {\theta }_i\times {\text{distance}}_i$

  • Complex sum calculation: $S_p\approx -0.9375-j0.3051$

  • Magnitude: $\mathrm{\mid }{S}_p\mathrm{\mid }\approx 0.986$

  • Maximum primary couple: $10\times 6168.53\times 0.1125\times 0.986\approx 6842\text{ Nm}$

  • Position: 54° before TDC of the second crank.

• Secondary Couple: Sum of $\cos 2{\theta }_i\times {\text{distance}}_i$

  • Complex sum calculation: $S_s\approx -0.9375+j1.2904$

  • Magnitude: $\mathrm{\mid }{S}_s\mathrm{\mid }\approx 1.595$

  • Maximum secondary couple: $\frac{10\times 6168.53\times 0.1125}{4}\times 1.595\approx 2767\text{ Nm}$

  • Position: 81° after TDC of the second crank.

Final Answer:

• The maximum primary and secondary forces are balanced (zero).

• The maximum primary couple is \boxed{6842\ \text{Nm}} occurring at \boxed{54^\circ\ \text{before TDC}} of the second crank.

• The maximum secondary couple is \boxed{2767\ \text{Nm}} occurring at \boxed{81^\circ\ \text{after TDC}} of the second crank.

5-Cylinder Engine Balance Analysis

Problem Statement

A five cylinder in-line engine running at 750 rpm has successive cranks 144 degrees apart, the distance between the cylinder center lines being 375 mm. The piston stroke is 225 mm and ratio of the connecting rod to crank is 4. Examine the engine for the balance of primary and secondary forces and couples. Determine maximum values of these and the position with respect to the second crank/cylinder at which these values occur. The reciprocating mass for each cylinder is 10 kg.

Engine Specifications

• Number of cylinders: 5
• Configuration: In-line
• Crank angle spacing: 144°
• Cylinder spacing: 375 mm
• Piston stroke: 225 mm
• Connecting rod length: 450 mm (L/R = 4)

Operating Parameters

• Engine speed: 750 rpm
• Reciprocating mass per cylinder: 10 kg
• Crank angles: 0°, 144°, 288°, 72°, 216°
• Angular velocity: 78.54 rad/s

Engine Visualization

Crank Angle: 0°

Animation Speed: 5x

Solution

Primary and Secondary Forces

The primary and secondary forces for each cylinder are given by:

Primary force: \( F_{primary} = m \omega^2 R \cos \theta \)

Secondary force: \( F_{secondary} = \frac{m \omega^2 R}{4} \cos 2\theta \)

For this 5-cylinder engine with 144° crank spacing:

• Primary forces sum to zero (balanced)
• Secondary forces sum to zero (balanced)

Primary and Secondary Couples

The couples are calculated by multiplying each force by its distance from the reference plane (cylinder 2):

Primary couple: \( C_{primary} = \sum (F_{primary} \times x_i) \)

Secondary couple: \( C_{secondary} = \sum (F_{secondary} \times x_i) \)

After calculation, we find:

• Maximum primary couple: 6842 Nm at 54° before TDC of cylinder 2
• Maximum secondary couple: 2767 Nm at 81° after TDC of cylinder 2

Force Analysis

Primary Forces

Secondary Forces

Couple Analysis

Primary Couples

Secondary Couples

Numerical Results

Parameter	Value	Position Relative to Cylinder 2
Primary Forces	Balanced (Sum = 0)	N/A
Secondary Forces	Balanced (Sum = 0)	N/A
Maximum Primary Couple	6842 Nm	54° before TDC
Maximum Secondary Couple	2767 Nm	81° after TDC

Conclusion

The five-cylinder engine configuration provides excellent primary and secondary force balance due to its 144° crank spacing symmetry. However, unbalanced primary and secondary couples exist.

The maximum primary couple of 6842 Nm occurs when the second crank is 54° before top dead center (TDC), while the maximum secondary couple of 2767 Nm occurs when the second crank is 81° after TDC.

In practice, this couple imbalance can cause a rocking motion in the engine and may require balancing shafts or careful mounting design to minimize vibration transmission to the vehicle structure.