Rocking stiffness is vital to understand if one wants to get a grasp of Rocking behavior of block machine foundations and subsequently coupled masses where harmonic loads are involved.
Consider the block having mass m and mass moment of inertia Mmz about centroid Z-axis passing through centroid C. A rotational spring having rotational stiffness Kφ is attached to the support point (center to the base point) O of the block as shown in figure below.
Figure 1 Undamped SDOF system with rotational spring attached at centre of base of block
The block is constrained such that it cannot move either in X or Y direction but it can only rotate in XY plane about Z-axis (perpendicular to the plane of paper) passing through O. The centroid C is at a distance h from the center of the base of the block O. The DOF of the system is rotation φ at point O.
Static Equilibrium: Considering the position of the mass at rest i.e. mean position of the mass, the gravity force mg is taken care of by reaction R at support point O.
System under motion: The block is slightly disturbed so as to cause it to rotate slightly about point O and then release the block. The block shall start oscillating about the mean position (i.e. position at rest) at point O in X-Y plane. Consider that at any instant of time t, the block position is rotated anti clockwise by an angle φ as shown in the figures below.
Figure 2 Rotation about O
Due to rotation, centroid point C moves to new location point C'. Rotation φ at O indcues rotation φ and translation hφ at centroid C'.
Figure 3 Free Body Diagram
Forces acting on the system are
Reference: Foundations for Industrial Machines by K G Bhatia
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