1. For Scenario A, produce the following mathematical models for this system to allow its response to a range of rumblings to be studied in simulations: a) In order to prevent movement during construction, the building is erected standing on steel supports that will be removed suddenly when the concrete has set. In these circumstances, at and shortly after the moment of release the rubber acts mainly as a damper and the value of k can be neglected. Develop a differential equation for the velocity v of the building modelling the behaviour of the building immediately after release.

1. For Scenario A, produce the following mathematical models for this system to allow its response to a range of rumblings to be studied in simulations:
a) In order to prevent movement during construction, the building is erected standing on steel supports that will be removed suddenly when the concrete has set. In these circumstances, at and shortly after the moment of release the rubber acts mainly as a damper and the value of k can be neglected. Develop a differential equation for the velocity v of the building modelling the behaviour of the building immediately after release.
b) Produce a differential equation for the behaviour of the building after construction linking its absolute position (i.e. its position relative to its original position) in the event that the ground moves according to some function F(t). Hence produce a Laplace transfer function linking the absolute position of the building (output) to the absolute position of the ground (input).
Scenario A
You are part of a team that is being consulted over the design of a new concert hall over an underground railway (tube) line. Your civil engineering colleagues have completed an outline design in which the main body of the office block is to be a reinforced concrete building. In an attempt to provide some barrier transmitting underground rumbling into the concert space, the building is to stand on rubber buffers (see below) sited in a concrete basement cast into the local bedrock. The building is restrained from moving horizontally because of its proximity to nearby buildings and has freedom to move in the vertical plane only.
Rubber has internal properties that combine springiness and a measure of damping so for all practical purposes the building can be modelled as a fixed mass M supported by a spring system of combined stiffness k and subject to a damping force equal to a constant c times its instantaneous velocity. Variations in the composition of the rubber can vary the effective values of k and c over a very wide range.
The underground rumblings involve small movements of thousands of tons of rock, so the mass of the building has virtually no effect on the displacement of its base. Consequently, in the event of a train passing, the base of the rubber mount can be assumed to move a distance x which is a function of time. Please note that this is not the same as subjecting the base to a force which is a function of time.