Suppose a space station is built in the shape of a regular tetrahedron with all sides of unit length. Answer all questions in Exercise 6.3.12.
(a) Sketch the space station and find its incidence matrix A.
(b) Show that ker A is six-dimensional, and find a basis.
(c) Explain which three basis vectors correspond to rigid translations.
(d) Find three basis vectors that correspond to linear approximations to rotations around the three coordinate axes.
(e) Suppose the bars all have unit stiffness. Compute the full stiffness matrix for the space station.
(f) What constraints on external forces at the four nodes are required to maintain equilibrium? Can you interpret them physically?
(g) How many nodes do you need to fix to stabilize the structure?
(h) Suppose you fix the three nodes in the xy plane. How much internal force does each bar experience under a unit vertical force on the upper vertex?