Editing Compound steam engine (section)

== The Yarrow-Schlick-Tweedy system ==
The presentation follows Sommerfeld's textbook, which contains a diagram (Figure 17) that is not reproduced for copyright reasons.<ref>{{Cite book |last=Sommerfeld |first=Arnold |url=https://books.google.com/books?id=pbs3BQAAQBAJ |title=Mechanics: Lectures on Theoretical Physics |date=1950 |publisher=Academic Press |isbn=978-1-4832-2028-4 |pages=76–98 |language=en}}</ref>

Consider a 4-cylinder engine on a ship. Let x be the vertical direction, z be the fore-aft direction, and y be the port-starboard direction. Let the 4 cylinders be mounted in a row along the z-axis, so that their pistons are pointed downwards. The pistons are connected to the same crankshaft via long vertical rods. Now, we set up the fundamental quantities of the engine:

* Let <math>M_1, M_2, M_3, M_4</math> be the effective masses of the compounded piston-rod system of each cylinder.
* Let cylinder 2 to be separated from cylinder 1 with a distance of <math>a_2</math> along the z-axis, and similarly for <math>a_3, a_4</math>.
* Let <math>l_1, l_2, l_3, l_4</math> be the length of each rod of the cylinder.
* Let <math>r_1, r_2, r_3, r_4</math> be the radii of the crankshaft connector of each cylinder.
* Let <math>\phi_1, \phi_2, \phi_3, \phi_4</math> be the angle of the crankshaft connector of each cylinder.
* Since the crankshaft is turned in tandem by all cylinders, <math>\phi_i - \phi_1</math> is a constant <math>\alpha_i</math> for each of <math>i = 2, 3, 4</math>.

Now, as the engine operates, the vertical position of cylinder <math>i</math> is equal to <math>x_i</math>. By trigonometry, we have

<math display="block">x_i =r_i \cos\phi_i + \sqrt{l_i^2(r_i\sin\phi_i)^2} = l_1 + r_i\cos\phi_i - \frac{r_i^2}{l_i} (1-\cos(2\phi_i))/2 + O(r_i^3/l^2)</math>

As each cylinder moves up and down, it exerts a vertical force on its mounting frame equaling <math>M_i\ddot x_i</math>. The YST system aims to make sure that the total of all 4 forces cancels out as exactly as possible. Specifically, it aims to make sure that the total force (along the x-axis) and the total torque (around the y-axis) are both zero:

<math display="block">\sum_{i=1}^4 M_i \ddot x_i = 0; \quad \sum_{i=2}^4 M_i a_i\ddot x_i = 0</math>

This can be achieved if

<math display="block">\sum_{i=1}^4 M_i x_i = Const; \quad \sum_{i=2}^4 M_i a_i x_i = Const</math>

Now, plugging in the equations, we find that it means (up to second-order)

<math display="block">\sum_{i=1}^4 M_i (r_i \cos\phi_i - \frac{r_i^2}{2l_i} \cos(2\phi_i))= 0; \quad \sum_{i=2}^4 M_i a_i (r_i \cos\phi_i - \frac{r_i^2}{2l_i} \cos(2\phi_i)) = 0</math>

Plugging in <math>\phi_i = \phi_1 + \alpha_i</math>, and expand the cosine functions, we see that with <math>\phi_1</math> arbitrary, the factors of <math>\sin(\phi_1), \cos(\phi_1), \sin(2\phi_1), \cos(2\phi_1)</math> must vanish separately. This gives us 8 equations to solve, which is in general possible if there are at least 8 variables of the system that we can vary.

Of the variables of the system, <math>M_i, r_i</math> are fixed by the design of the cylinders. Also, the absolute values of <math>a_2, a_3, a_4</math> do not matter, only their ratios matter. Together, this gives us 9 variables to vary: <math>l_1, l_2, l_3, l_4, \frac{a_3}
{a_2}, \frac{a_4}{a_2}, \alpha_2, \alpha_3, \alpha_4</math>.

The YST system requires at least 4 cylinders. With 3 cylinders, the same derivation gives us only 6 variables to vary, which is insufficient to solve all 8 equations.

The YST system is used on ships such as the [[SS Kaiser Wilhelm der Grosse]] and [[SS Deutschland (1900)]].<ref>{{Cite journal |last=Sommerfeld |first=A. |date=May 1904 |title=The Scientific Results and Aims of Modern Applied Mechanics |url=https://www.cambridge.org/core/journals/mathematical-gazette/article/abs/scientific-results-and-aims-of-modern-applied-mechanics/38680C8304708A68D86F82659971595B |journal=The Mathematical Gazette |language=en |volume=3 |issue=45 |pages=26–31 |doi=10.2307/3603435 |jstor=3603435 |s2cid=125314831 |issn=0025-5572|url-access=subscription }}</ref>