I came across the following problem in "Introduction to Applied Mechanics" by Gilbert Strang, and am a little confused about the solution to this problem. The following figure shows the problem.
The following vectors are defined for the problem,
- $\mathbf{x} = [x_1, x_2, x_3]^T$ represents the displacements of the masses.
- $\mathbf{e} = [e_1, e_2, e_3, e_4$ represents the elongations of the springs.
- $\mathbf{y} = [y_1, y_2, y_3, y_4]$ represents the force in the springs.
- $\mathbf{f} = [f_1, f_2, f_3]$ represents the force on the masses.
Let us assume that upward direction is positive, i.e. a displacement in the upward displacement will result in a positive $x_i$.
Thus the elongations of the $i^{th}$ spring is given by, $e_i = x_{i-1} - x_{i}$. This means that an upward movement of the mass above the spring, and the downward movement of the mass below the spring will result in elongating the spring, and thus positive elongation. It must be noted that $x_0, x_4 = 0$ as these correspond to the boundary conditions represented by the walls on which the ends of springs 1 and 3 and clamped.
$$ \implies \mathbf{e} = \left( \begin{array}{ccc} -1 & 0 & 0\\ 1 & -1 & 0\\ 0 & 1 & -1\\ 0 & 0 & 1\\ \end{array} \right) \mathbf{x} = \mathbf{A}\mathbf{x} $$
The above equation makes sense, as a negative $x_1$ (movement downwards), will correspond to an elongation of spring 1. Similarly, a positive $x_3$ (movement upwards) will correspond to elongation of spring 4.
The forces in the springs (positive for tension, and negative for compression) is given by the following,
$$ \mathbf{y} = \left( \begin{array}{cccc} k_1 & 0 & 0 & 0\\ 0 & k_2 & 0 & 0\\ 0 & 0 & k_3 & 0\\ 0 & 0 & 0 & k_4\\ \end{array} \right) \mathbf{e} = \mathbf{C}\mathbf{e} $$
Now the force on the $i^{th}$ mass is $f_i = y_i - y_{i+1}$. If the spring above the $i^{th}$ mass is under tension then $y_i$ will be positive, and will tend to pull the mass upwards, and thus it has a positive contribution $y_i$ to the force on the mass. The opposite is true for the spring below the mass, thus has a negative contribution $y_{i+1}$.
$$ \mathbf{f} = \left( \begin{array}{cccc} 1 & -1 & 0 & 0\\ 0 & 1 & -1 & 0\\ 0 & 0 & 1 & -1\\ \end{array} \right) \mathbf{y} = \mathbf{B}\mathbf{e} $$
The problem here is that $\mathbf{B}$ must equal $\mathbf{A}^T$, but according the conventions I have followed it does not. In fact $\mathbf{B} = -\mathbf{A}^T$.
Can someone tell me where I am going wrong?