Mechatronics

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.

Scenario B

An electronics company is suffering a large number of circuit board failures in the field. All boards are tested before leaving the factory so the company knows that all boards shipped to site were sound. All failures happened at, or within seconds of, the first power up after installation. On inspection, all failures were found to be of the same microprocessor chip. The company suspects that the failures are being caused by static electric discharges to the microprocessor’s supply and drain (+ve and –ve power) terminals during installation. In its powered-off state these pins of the microprocessor behave as a fixed resistor R. Most of the circuitry around these pins is inactive until the system is powered up but there is one loop consisting of an inductor L and capacitor C linking the supply and drain that the company thinks may be vulnerable to static discharge. This loop can be considered to be simply R, L and C connected in series as shown below. Note that in this situation there is no supply voltage.

Static discharge events involve the movement of small quantities of charge from very high voltage sources and can be modelled to sufficient accuracy by assuming that a time-varying additional charge Q [equal to some function F(t)] is directly injected into C.

Questions :

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).

For Scenario B,

a) Produce a differential equation for the behaviour of the circuit linking the charge Q in C to some function F(t). Hence produce a Laplace transfer function linking the charge in C to the injected charge.

b) The inductance L was placed into this circuit to reduce the pick-up of ‘noise’ from nearly circuits. Noise has not been a problem in practice so, in an attempt to reduce the incidence of failure, the manufacturer is considering replacing the inductor with a wire link. What would the differential equation be in this case?

2. Comment on the similarities and differences between the equations for Scenario A part b) and Scenario B part a) above. To what extent are these scenarios analogous?

3. For Scenario A only:

· It is intended to investigate this scenario further by means of scale models. For such a model, suggest appropriate sensors to measure and capture the absolute positions of the ground and building, and the temperature of the rubber supports, over time.

· Propose two possible suitable actuators to move the ‘ground’ in this experiment.

· It is thought that some improvement in sound-deadening behaviour could be achieved by burying sound detectors deep beneath the building and using these, in the same way as is done in noise-cancelling headphones, to vibrate the building so that it moves exactly in anti-phase to the incoming sound. Propose sound detectors suitable for use in such a situation and recommend one actuator technology capable of producing the necessary counter-vibrations.