Balancing of V-Engines

Balancing of V-Engines

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V-Engine

A V-engine is a two cylinder engine, which has a common crank and the axis of cylinder makes a "V" shape.

Since V-engines have a common crank and the crank revolves in one plane, there is no primary or secondary couple acting on the engine.

Consider a V-engine as shown in fig.1 having common crank OC and two connecting rods CP and CQ. The lines of stroke OP and OQ are inclined to vertical axis OY at an angle ‘α’.

fig_balancing-of-v-engines Fig-1

Let,

m=mass of reciprocating parts per cylinder, kgl=length of connecting rod, mr=radius of crank, mn=Obliquity ratio =l/rθ=Crank angle, measured from vertical axis OY, at any instantω=Angular velocity of crank, rad/s2α=V-angle i.e. angle between lines of strokes of two cylinders\begin{aligned} m &= \text{mass of reciprocating parts per cylinder, kg}\\ l &= \text{length of connecting rod, m}\\ r &= \text{radius of crank, m}\\ n &= \text{Obliquity ratio }= l/r\\ θ &= \text{Crank angle, measured from vertical axis OY, at any instant}\\ ω &= \text{Angular velocity of crank, rad/s}\\ 2 \alpha &= \text{V-angle i.e. angle between lines of strokes of two cylinders}\\ \end{aligned}

We know that,

  • Primary unbalanced force in single cylinder engine is,
FP=mrω2cosθF_P = mr\omega^2 \cos \theta
  • Secondary unbalanced force in single cylinder engine is,
FS=mrω2(cos2θn)F_S = mr\omega^2 \bigg({\cos2\theta \over n} \bigg)
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1. Primary forces (FPF_P)

(i) Primary forces in individual cylinders:

FP1=mrω2cos(αθ)FP2=mrω2cos(α+θ)F_{P1} = mr\omega^2 \cos (\alpha -\theta)\\ F_{P2} = mr\omega^2 \cos (\alpha +\theta)

(ii) Total primary force along Y axis (FPVF_{PV}):

As both components (FP1cosα)(F_{P1}\sdot \cos\alpha) and (FP2cosα)(F_{P2} \sdot \cos\alpha) are acting in same direction;

FPV=FP1cosα+FP2cosα=mrω2cos(αθ)cosα mrω2cos(α+θ)cosα=mrω2cosα[cos(αθ)+cos(α+θ)]=mrω2cosα(2cosθcosα)=2mrω2cos2αcosθ\begin{aligned} F_{PV} &= F_{P1} \cos\alpha + F_{P2} \cos\alpha\\ &= mr\omega^2 \cos (\alpha -\theta)\cos\alpha \ - mr\omega^2 \cos (\alpha +\theta)\cos\alpha\\ &= mr\omega^2 \cos\alpha [\cos (\alpha -\theta) + \cos (\alpha+\theta)]\\ &= mr\omega^2 \cos\alpha \sdot (2 \cos\theta \cos\alpha)\\ &= 2mr\omega^2 \cos^2\alpha \cos\theta \end{aligned}

(iii) Total primary force along X-axis (FPHF_{PH}):

As both components (FP1sinα)(F_{P1} \sdot \sin \alpha) and (FP2sinα)(F_{P2}\sdot \sin \alpha) are acting opposite to each other;

FPH=FP1sinα+FP2sinα=mrω2cos(αθ)sinα mrω2cos(α+θ)sinα=mrω2sinα[cos(αθ)cos(α+θ)]=mrω2sinα (2sinθsinα)=2mrω2sin2αsinθ\begin{aligned} F_{PH} &= F_{P1} \sin\alpha + F_{P2} \sin\alpha\\ &= mr\omega^2 \cos (\alpha -\theta)\sin\alpha \ - mr\omega^2 \cos (\alpha +\theta)\sin\alpha\\ &= mr\omega^2\sin\alpha [\cos (\alpha -\theta)-\cos (\alpha +\theta)]\\ &= mr\omega^2\sin\alpha \ \sdot (2\sin \theta \sin \alpha)\\ &= 2mr\omega^2\sin^2\alpha\sin \theta \end{aligned}

(iv) Resultant primary force (FpF_p):

Fp=(FPV)2+(FPH)2=(2mrω2cos2αcosθ)2+(2mrω2sin2αsinθ)2=2mrω2(cos2αcosθ)2+(sin2αsinθ)2\begin{aligned} F_p &= \sqrt{(F_{PV})^2 +(F_{PH})^2 }\\ &= \sqrt{(2mr\omega^2 \cos^2\alpha \cos\theta)^2 + (2mr\omega^2\sin^2\alpha\sin \theta)^2}\\ &= 2mr\omega^2\sqrt{(\cos^2\alpha \cos\theta)^2 + (\sin^2\alpha\sin \theta)^2} \end{aligned}

The angle made by resultant primary force with vertical axis is;

βP=tan1(FPHFPV)=tan1(2mrω2sin2αsinθ2mrω2cos2αcosθ)=tan1(tan2α  tanθ)\begin{aligned} \beta_P &= \tan^{-1} \bigg ( {F_{PH} \over F_{PV}}\bigg)\\ &= \tan^{-1} \bigg ( {2mr\omega^2\sin^2\alpha\sin \theta \over 2mr\omega^2 \cos^2\alpha \cos\theta} \bigg)\\ &= \tan^{-1} (\tan^2\alpha \ \sdot \ \tan\theta) \end{aligned}

2. Secondary forces (FSF_S)

(i) Secondary forces in individual cylinder:

FS1=mrω2cos2(αθ)nFS2=mrω2cos2(α+θ)nF_{S1} = mr \omega^2 {\cos2(\alpha-\theta) \over n}\\ F_{S2} = mr \omega^2 {\cos2(\alpha+\theta) \over n}\\

(ii) Total secondary force along Y axis (FSVF_{SV}):

As both components (FS1cosα)(F_{S1}\sdot\cos \alpha) and (FS2cosα)(F_{S2}\sdot \cos \alpha) are acting in same direction;

FSV=FS1cosα+FS2cosα=mrω2cos2(αθ)ncosα+mrω2cos2(α+θ)ncosα=mrω2cosα[cos2(αθ)n+cos2(α+θ)n]=1nmrω2cosα[2cos2θcos2α]=2nmrω2cosαcos2αcos2θ\begin{aligned} F_{SV} &= F_{S1} \cos \alpha + F_{S2} \cos \alpha\\ &= mr \omega^2 {\cos2(\alpha-\theta) \over n}\cos\alpha + mr \omega^2 {\cos2(\alpha+\theta) \over n}\cos\alpha\\ &= mr \omega^2\cos\alpha \bigg[{\cos2(\alpha-\theta) \over n} + {\cos2(\alpha+\theta) \over n}\bigg]\\ &={1\over n} mr \omega^2\cos\alpha \sdot[2\sdot\cos2\theta\sdot\cos2\alpha]\\ &= {2 \over n}mr \omega^2\cos\alpha\sdot\cos2\alpha\sdot\cos2\theta \end{aligned}

(iii) Total secondary force along X-axis (FSHF_{SH}):

As both components (FS1sinα)(F_{S1}\sdot\sin\alpha) and (FS2sinα)(F_{S2}\sdot\sin\alpha) are acting opposite to each other;

FSH=FS1sinαFS2sinα=mrω2cos2(αθ)nsinαmrω2cos2(α+θ)nsinα=mrω2sinα[cos2(αθ)ncos2(α+θ)n]=1nmrω2sinα[2sin2θsin2α]=2nmrω2sinαsin2αsin2θ\begin{aligned} F_{SH} &= F_{S1} \sin \alpha - F_{S2} \sin \alpha\\ &= mr \omega^2 {\cos2(\alpha-\theta) \over n}\sin\alpha - mr \omega^2 {\cos2(\alpha+\theta) \over n}\sin\alpha\\ &= mr \omega^2\sin\alpha \bigg[{\cos2(\alpha-\theta) \over n} - {\cos2(\alpha+\theta) \over n}\bigg]\\ &= {1\over n} mr \omega^2\sin\alpha \sdot[2\sdot\sin2\theta\sdot\sin2\alpha]\\ &= {2 \over n}mr \omega^2\sin\alpha\sdot\sin2\alpha\sdot\sin2\theta \end{aligned}

(iv) Resultant secondary force (FSF_S):

Fs=(FSV)2+(FSH)2=(2nmrω2cosαcos2αcos2θ)+(2nmrω2sinαsin2αsin2θ)2=2nmrω2(cosαcos2αcos2θ)2+(sinαsin2αsin2θ)2\begin{aligned} F_s &= \sqrt{(F_{SV})^2 + (F_{SH})^2}\\ &= \sqrt{( {2 \over n}mr \omega^2\cos\alpha\sdot\cos2\alpha\sdot\cos2\theta)^ + ({2 \over n}mr \omega^2\sin\alpha\sdot\sin2\alpha\sdot\sin2\theta)^2}\\ &= {2 \over n} mr\omega^2 \sqrt{(\cos\alpha\sdot\cos2\alpha\sdot\cos2\theta)^2 + (\sin\alpha\sdot\sin2\alpha\sdot\sin2\theta)^2} \end{aligned}

The angle made by resultant secondary force with vertical axis is;

βS=tan1(FSHFSV)=tan1(2nmrω2sinαsin2αsin2θ2nmrω2cosαcos2αcos2θ)=tan1 (tanαtan2αtan2θ)\begin{aligned} \beta_S &= \tan^{-1} \bigg( {F_{SH} \over F_{SV}}\bigg)\\ &= \tan^{-1} \bigg( {{2 \over n}mr \omega^2\sin\alpha\sdot\sin2\alpha\sdot\sin2\theta \over {2 \over n}mr \omega^2\cos\alpha\sdot\cos2\alpha\sdot\cos2\theta} \bigg)\\ &= \tan^{-1}\ (\tan\alpha \sdot \tan2\alpha \sdot \tan2\theta) \end{aligned}

Advantages of V-engines

  • The design of V-engine is compact compared to multi-cylinder inline engines. So they consume less space.
  • Power output of V-engine is more compared to single cylinder engines because the crank receives power from both the cylinders.
  • The operation of V-engine is smoother for high speed performance.

Applications of V-engines

  • Because of its compactness, smoother operations and high power output, V-engines are used in sports cars.

V-Engine

A V-engine is a two cylinder engine, which has a common crank and the axis of cylinder makes a "V" shape.

Since V-engines have a common crank and the crank revolves in one plane, there is no primary or secondary couple acting on the engine.

Consider a V-engine as shown in fig.1 having common crank OC and two connecting rods CP and CQ. The lines of stroke OP and OQ are inclined to vertical axis OY at an angle ‘α’.

fig_balancing-of-v-engines Fig-1

Let,

m=mass of reciprocating parts per cylinder, kgl=length of connecting rod, mr=radius of crank, mn=Obliquity ratio =l/rθ=Crank angle, measured from vertical axis OY, at any instantω=Angular velocity of crank, rad/s2α=V-angle i.e. angle between lines of strokes of two cylinders\begin{aligned} m &= \text{mass of reciprocating parts per cylinder, kg}\\ l &= \text{length of connecting rod, m}\\ r &= \text{radius of crank, m}\\ n &= \text{Obliquity ratio }= l/r\\ θ &= \text{Crank angle, measured from vertical axis OY, at any instant}\\ ω &= \text{Angular velocity of crank, rad/s}\\ 2 \alpha &= \text{V-angle i.e. angle between lines of strokes of two cylinders}\\ \end{aligned}

We know that,

  • Primary unbalanced force in single cylinder engine is,
FP=mrω2cosθF_P = mr\omega^2 \cos \theta
  • Secondary unbalanced force in single cylinder engine is,
FS=mrω2(cos2θn)F_S = mr\omega^2 \bigg({\cos2\theta \over n} \bigg)
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1. Primary forces (FPF_P)

(i) Primary forces in individual cylinders:

FP1=mrω2cos(αθ)FP2=mrω2cos(α+θ)F_{P1} = mr\omega^2 \cos (\alpha -\theta)\\ F_{P2} = mr\omega^2 \cos (\alpha +\theta)

(ii) Total primary force along Y axis (FPVF_{PV}):

As both components (FP1cosα)(F_{P1}\sdot \cos\alpha) and (FP2cosα)(F_{P2} \sdot \cos\alpha) are acting in same direction;

FPV=FP1cosα+FP2cosα=mrω2cos(αθ)cosα mrω2cos(α+θ)cosα=mrω2cosα[cos(αθ)+cos(α+θ)]=mrω2cosα(2cosθcosα)=2mrω2cos2αcosθ\begin{aligned} F_{PV} &= F_{P1} \cos\alpha + F_{P2} \cos\alpha\\ &= mr\omega^2 \cos (\alpha -\theta)\cos\alpha \ - mr\omega^2 \cos (\alpha +\theta)\cos\alpha\\ &= mr\omega^2 \cos\alpha [\cos (\alpha -\theta) + \cos (\alpha+\theta)]\\ &= mr\omega^2 \cos\alpha \sdot (2 \cos\theta \cos\alpha)\\ &= 2mr\omega^2 \cos^2\alpha \cos\theta \end{aligned}

(iii) Total primary force along X-axis (FPHF_{PH}):

As both components (FP1sinα)(F_{P1} \sdot \sin \alpha) and (FP2sinα)(F_{P2}\sdot \sin \alpha) are acting opposite to each other;

FPH=FP1sinα+FP2sinα=mrω2cos(αθ)sinα mrω2cos(α+θ)sinα=mrω2sinα[cos(αθ)cos(α+θ)]=mrω2sinα (2sinθsinα)=2mrω2sin2αsinθ\begin{aligned} F_{PH} &= F_{P1} \sin\alpha + F_{P2} \sin\alpha\\ &= mr\omega^2 \cos (\alpha -\theta)\sin\alpha \ - mr\omega^2 \cos (\alpha +\theta)\sin\alpha\\ &= mr\omega^2\sin\alpha [\cos (\alpha -\theta)-\cos (\alpha +\theta)]\\ &= mr\omega^2\sin\alpha \ \sdot (2\sin \theta \sin \alpha)\\ &= 2mr\omega^2\sin^2\alpha\sin \theta \end{aligned}

(iv) Resultant primary force (FpF_p):

Fp=(FPV)2+(FPH)2=(2mrω2cos2αcosθ)2+(2mrω2sin2αsinθ)2=2mrω2(cos2αcosθ)2+(sin2αsinθ)2\begin{aligned} F_p &= \sqrt{(F_{PV})^2 +(F_{PH})^2 }\\ &= \sqrt{(2mr\omega^2 \cos^2\alpha \cos\theta)^2 + (2mr\omega^2\sin^2\alpha\sin \theta)^2}\\ &= 2mr\omega^2\sqrt{(\cos^2\alpha \cos\theta)^2 + (\sin^2\alpha\sin \theta)^2} \end{aligned}

The angle made by resultant primary force with vertical axis is;

βP=tan1(FPHFPV)=tan1(2mrω2sin2αsinθ2mrω2cos2αcosθ)=tan1(tan2α  tanθ)\begin{aligned} \beta_P &= \tan^{-1} \bigg ( {F_{PH} \over F_{PV}}\bigg)\\ &= \tan^{-1} \bigg ( {2mr\omega^2\sin^2\alpha\sin \theta \over 2mr\omega^2 \cos^2\alpha \cos\theta} \bigg)\\ &= \tan^{-1} (\tan^2\alpha \ \sdot \ \tan\theta) \end{aligned}

2. Secondary forces (FSF_S)

(i) Secondary forces in individual cylinder:

FS1=mrω2cos2(αθ)nFS2=mrω2cos2(α+θ)nF_{S1} = mr \omega^2 {\cos2(\alpha-\theta) \over n}\\ F_{S2} = mr \omega^2 {\cos2(\alpha+\theta) \over n}\\

(ii) Total secondary force along Y axis (FSVF_{SV}):

As both components (FS1cosα)(F_{S1}\sdot\cos \alpha) and (FS2cosα)(F_{S2}\sdot \cos \alpha) are acting in same direction;

FSV=FS1cosα+FS2cosα=mrω2cos2(αθ)ncosα+mrω2cos2(α+θ)ncosα=mrω2cosα[cos2(αθ)n+cos2(α+θ)n]=1nmrω2cosα[2cos2θcos2α]=2nmrω2cosαcos2αcos2θ\begin{aligned} F_{SV} &= F_{S1} \cos \alpha + F_{S2} \cos \alpha\\ &= mr \omega^2 {\cos2(\alpha-\theta) \over n}\cos\alpha + mr \omega^2 {\cos2(\alpha+\theta) \over n}\cos\alpha\\ &= mr \omega^2\cos\alpha \bigg[{\cos2(\alpha-\theta) \over n} + {\cos2(\alpha+\theta) \over n}\bigg]\\ &={1\over n} mr \omega^2\cos\alpha \sdot[2\sdot\cos2\theta\sdot\cos2\alpha]\\ &= {2 \over n}mr \omega^2\cos\alpha\sdot\cos2\alpha\sdot\cos2\theta \end{aligned}

(iii) Total secondary force along X-axis (FSHF_{SH}):

As both components (FS1sinα)(F_{S1}\sdot\sin\alpha) and (FS2sinα)(F_{S2}\sdot\sin\alpha) are acting opposite to each other;

FSH=FS1sinαFS2sinα=mrω2cos2(αθ)nsinαmrω2cos2(α+θ)nsinα=mrω2sinα[cos2(αθ)ncos2(α+θ)n]=1nmrω2sinα[2sin2θsin2α]=2nmrω2sinαsin2αsin2θ\begin{aligned} F_{SH} &= F_{S1} \sin \alpha - F_{S2} \sin \alpha\\ &= mr \omega^2 {\cos2(\alpha-\theta) \over n}\sin\alpha - mr \omega^2 {\cos2(\alpha+\theta) \over n}\sin\alpha\\ &= mr \omega^2\sin\alpha \bigg[{\cos2(\alpha-\theta) \over n} - {\cos2(\alpha+\theta) \over n}\bigg]\\ &= {1\over n} mr \omega^2\sin\alpha \sdot[2\sdot\sin2\theta\sdot\sin2\alpha]\\ &= {2 \over n}mr \omega^2\sin\alpha\sdot\sin2\alpha\sdot\sin2\theta \end{aligned}

(iv) Resultant secondary force (FSF_S):

Fs=(FSV)2+(FSH)2=(2nmrω2cosαcos2αcos2θ)+(2nmrω2sinαsin2αsin2θ)2=2nmrω2(cosαcos2αcos2θ)2+(sinαsin2αsin2θ)2\begin{aligned} F_s &= \sqrt{(F_{SV})^2 + (F_{SH})^2}\\ &= \sqrt{( {2 \over n}mr \omega^2\cos\alpha\sdot\cos2\alpha\sdot\cos2\theta)^ + ({2 \over n}mr \omega^2\sin\alpha\sdot\sin2\alpha\sdot\sin2\theta)^2}\\ &= {2 \over n} mr\omega^2 \sqrt{(\cos\alpha\sdot\cos2\alpha\sdot\cos2\theta)^2 + (\sin\alpha\sdot\sin2\alpha\sdot\sin2\theta)^2} \end{aligned}

The angle made by resultant secondary force with vertical axis is;

βS=tan1(FSHFSV)=tan1(2nmrω2sinαsin2αsin2θ2nmrω2cosαcos2αcos2θ)=tan1 (tanαtan2αtan2θ)\begin{aligned} \beta_S &= \tan^{-1} \bigg( {F_{SH} \over F_{SV}}\bigg)\\ &= \tan^{-1} \bigg( {{2 \over n}mr \omega^2\sin\alpha\sdot\sin2\alpha\sdot\sin2\theta \over {2 \over n}mr \omega^2\cos\alpha\sdot\cos2\alpha\sdot\cos2\theta} \bigg)\\ &= \tan^{-1}\ (\tan\alpha \sdot \tan2\alpha \sdot \tan2\theta) \end{aligned}

Advantages of V-engines

  • The design of V-engine is compact compared to multi-cylinder inline engines. So they consume less space.
  • Power output of V-engine is more compared to single cylinder engines because the crank receives power from both the cylinders.
  • The operation of V-engine is smoother for high speed performance.

Applications of V-engines

  • Because of its compactness, smoother operations and high power output, V-engines are used in sports cars.

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