Positive Definite Matrices and a Mass-Spring system
In this guided exploration project you will explore how positive definite matrices arise naturally in detecting stability of structures in mechanics.
Some very rough guidelines are given below to help you start thinking about the project: more details will be provided during discussions.
- For simplicity, the first case is an one-dimensional mass-spring system. If this is completed, we can do two and three dimensional systems.
- Consider n masses m_1, m_2, \cdots, m_n arranged vertically in a straight line with n+1 springs as shown, with the top and bottom springs connected to rigid supports. We denote the displacement of each mass by y_i (considered positive downwards). We calculate the elongations of the springs e_1, e_2, \cdots, e_{n+1} is terms of the y_i (paying special attention to the topmost and bottommost springs) and create the displacement and elongation vectors satisfy \mathbf{e}=\mathbf{A}\mathbf{y}, where \mathbf{A} is the incidence matrix. Assuming the i-th spring to have (Hookean) stiffness c_i, we form the coefficient matrix \mathbf{C} and get the equation \mathbf{f}=\mathbf{C}\mathbf{e}, where \mathbf{f} is the internal force vector. We construct an external force (gravitational) to act on each mass m_i and form the force balance equation \mathbf{F}=\mathbf{A}^T \mathbf{f} (why is that?). Observe that this leads to the equation \mathbf{K} \mathbf{y}=\mathbf{F}, where \mathbf{K}=\mathbf{A^TCA} is known as the stiffness matrix of the system. Is this system statically determinate / indeterminate?
- Repeat the calculations with the bottommost spring free. Is this system statically determinate / indeterminate?
