Engine

The mean piston speed (v_mps) is the average speed of the piston in a reciprocating engine which is related to the stroke (S) and the crankshaft angular displacement for one stroke (θ_s) at a given angular velocity (ω).

By design, most reciprocating engines have an angular displacement of 1/2 revolution (π rad) per stroke.

Mean piston speed is proportional to:

• Friction losses (higher fuel consumption)
• Volumetric flow rate (higher power)
• piston acceleration & deceleration (higher stress on mechanical components)

Mean piston speed is a much more important (and limiting) feature of an engine than its RPM.

Fuel consumption will be the limiting factor for most «working» engines, so their speeds are kept as low as possible. For racing engines, power is the goal, so we try to get the highest mean piston speed possible. The upper limit is being imposed by the area of the intake valve(s) with respect to the bore area. This is what restricts the flow and is usually fixed by the cylinder head design. The other limit is imposed by the strength of the mechanical components.

Wankel engine

The mean «piston» speed is a rather meaningless value in the case of a Wankel engine. As a trial, this site will consider the more meaningful mean rotor tip speed instead. The rotor tip motion against the housing is similar to the piston motion against the cylinder wall in a reciprocating engine.

By design, the rotor of a Wankel engine rotates 3 times slower than the output shaft. So the mean rotor tip speed ( v_mrts ) can be found with the rotor angular velocity ( ω_R ) and the rotor radius ( R ):

Where ω is the angular velocity of the output shaft.

Connecting rod length (more)

Let us analyze the position of the piston (L_p) inside a traditional engine with respect to the crankshaft angular position (θ_ck).

If we assume that L_p = 0 and θ_ck = 0 when the piston is at TDC, then the position of the piston with respect to the stroke (S) is:

(more)

Lp + Lcrcosθcr + S 2 cosθck = Lcr + S 2

Lp = Lcr + S 2 ( 1 − cosθck ) − Lcrcosθcr

Dividing by S :

Lp S = Lcr S + 1 2 ( 1 − cosθck ) − Lcr S cosθcr

(2a)

The only term unknown is the last one, related to θ_cr. But , looking at the preceding figure and with the help of the Pythagorean theorem:

( Lcrcosθcr ) 2 + ( S 2 sinθck ) 2 = L 2 cr

Dividing by S ² :

( Lcr S cosθcr ) 2 + ( 1 2 sinθck ) 2 = ( Lcr S ) 2

( Lcr S cosθcr ) 2 = ( Lcr S ) 2 − ( 1 2 sinθck ) 2

Lcr S cosθcr = √ ( Lcr S  ) 2 − 1 4 sin2θck

And replacing in equation (2a):

Lp S  = Lcr S  + 1 2 ( 1 − cosθck ) − √ ( Lcr S  ) 2 − 1 4 sin2θck

(2)

What this equation means is that

Lp

⁄

S

  = 0 when θ_ck = 0 (at TDC); after half a revolution (θ_ck = 180°, at BDC),

Lp

⁄

S

  = 1; finally, after a complete revolution (θ_ck = 360°, back at TDC),

Lp

⁄

S

  = 0. the position between TDC and BDC will be slightly different depending on the value of

Lcr

⁄

S

. The following graph shows piston position according to different values of this ratio:

When

Lcr

⁄

S

  = ∞, we have a perfect sine wave, i.e the piston spends as much time near TDC as it does near BDC. As the ratio decreases (min.: 0.5), more time is spent near BDC and less near TDC. This is the opposite of what we want: we prefer that the piston stays at TDC as long as possible (i.e. when combustion happens) because we prefer a combustion at constant volume and the only way is to make sure volume changes as slowly as possible at this moment.

Moreover, when we decrease that ratio, the maximum angle between the connecting rod and the cylinder axis (θ_{cr max}) increases constantly. We can evaluate this angle the following way:

(more)

From the previous drawing:

Lcrsinθcr = S 2 sinθck

sinθcr = 0.5sinθck Lcr ⁄ S

The maximum value of sinθ_ck is 1 (when θ_ck = 90°), so the maximum value of θ_cr is:

sinθcr max = 0.5 Lcr ⁄ S

(3)

As we can see on the previous graph, this angle increases rapidly from 0° (at

Lcr

⁄

S

  = ∞) to 90° (at

Lcr

⁄

S

  = 0.5); An unacceptable situation because the lateral forces against the cylinder wall will be too large and might induce piston wobbling and increased wear. The acceptable limit is usually around 17°.

Typical values

Typical values for

Lcr

⁄

S

are between 1.60 and 2.00; a good compromise being 1.80. Smaller, the connecting rod angle is too large and longer, the space required by the engine becomes too large. Within that range, variations are almost unnoticeable, and shouldn't affect power output. Although there might be some minor tuning adjustments (for example, the valve events).

Wankel engine

The Wankel «up-and-down» rotor motion is a perfect sinusoidal motion, just like a traditional piston engine with an infinitely long connecting rod.