Recent content by Ax_xiom

After some thinking, I don't think this is possible. For an arbitrary stage i, the expression for the mass ratio will be this $$N_i = \frac{m_i + \alpha_i p + U}{s m_i + \alpha_i p + U}$$ where U is the mass of all upper stages. And we (ideally) want to find ##\frac{m_i + \alpha_i p + U}{U}## in...

s will be around 0.1. These results make sense but this is a solved issue currently. This is the issue I am currently having

the ##c_i##'s and ##V_f## will be variable but the ##c_i##'s will typically be around ##2400-3000ms^{-1}## and ##V_f## will be anywhere from ##2600 - 9000ms^{-1}##

This is nice, but @pasmith pointed out a way to make the Jacobian upper triangular, which makes the computation of ##\Delta X_n## much easier to scale and makes everything much more convenient. The bigger problem I am currently facing is that the system of equations I initially came up with was...

So not related to this specific topic but related to this project. So turns out the equations I was working with are likely wrong. So the method I'm using starts like this: $$ \Delta V = c_1 \ln(\frac{m_1 + m_2 + (\alpha_1 + \alpha_2 + 1)p}{s m_1 + m_2 + (\alpha_1 + \alpha_2 + 1)p}) + c_2...

Do you mean it's possible to solve the linear system without using Excel's matrix operations each time? If that is correct, I believe these would be the solutions to the system: $$ \begin{align} \Delta\lambda = \frac{s\lambda f_6 - \sum_{i=1}^5 \frac{f_i}{N_i}}{\sum_{i=1}^5 c_i (s -...

There's 3 equations in my post, but that's the version for 2 stages. I want to be able to expand it up to 5 stages where the system of equations will be this: $$\begin{align} 1 - c_{1}\lambda + \frac{sN_{1}}{1 - sN_{1}} = 0 \\ 1 - c_{2}\lambda + \frac{sN_{2}}{1 - sN_{2}} = 0 \\ 1 -...

A few problems: I want to implement up to 5 stages, which would result in 6 equations and 6 variables that would need to be graphed which would require a 6D space to visualise I'm not sure how easy that would be to use or how accurate it would be Can Excel even draw 3D graphs?

So I'm working on a project where I am trying to work out what the ideal sizes of rocket stages are and I am using Excel to allow the user to interact with this quickly. The method I am using is derived from this video, and I end up with this system of equations: \begin{align} 1 - c_{1}\lambda +...

I'm assuming that this is computationally inefficient and it's much faster to just solve directly. This is good to know but I attempted to implement this in Excel, and I was only doing it with a 3x3 matrix so it was much easier to just use Excel's MINVERSE() and MMULT() functions

So what happens comptationally during the ##J(X_k) \Delta X_k = -F(X_k)## step? Do you express the problem as a series of linear equations and solve them? Edit: I think that might be the case. I was under the impression that using Gaussian manipulation to solve a system of equations would give...

So the formula used to solve non-linear equations using the Newton-Raphson method is this $$ X_{k+1} = X_k - J^{-1}(X_k)F(X_k) $$ and instead of finding ##J^{-1}(X_k)##, we solve for the change that will be applied to ##X_k## (##\Delta X_k##) using this relation $$J(X_k) \Delta X_k = -F(X_k)$$...

But not in the case of a coil rotating in a magnetic field (if so, are the underlying physics causing a current to be generated in a rotating wire in a magnetic field different to the physics of the same thing happening in a wire surrounded by a rotating magnet? If so, why?)

So in most cases, is the induction from a coil in a magnetic field not due to Faraday's Law of Induction? So the electric field is generated everywhere, but a current is only generated within the wire due to it having free electrons?

So Faraday's Law of induction states this: $$ \nabla \times E = - \frac {\partial B} {\partial t} $$ Or if we write it in it's integral form: $$ \int E \cdot dl = - \frac {d \Phi_B} {dt} $$ which (to my understanding) means that the magnitude of the EMF around a coil of wire will be equal to...