The following is mostly an academic problem.
The Response of an Undamped Two-degree-of-freedom System
Subjected to Base Displacement via Modal Analysis: twodof_base_disp_modal.pdf
The Response of an Undamped Two-degree-of-freedom System
Subjected to Base Displacement via Laplace Transforms: twodof_base_disp_Laplace.pdf
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See also: Two-degree-of-freedom Systems
– Tom Irvine