# Engineering

## ECM2111 - Mathematical Modelling of Engineering Systems (2010)

MODULE TITLE CREDIT VALUE Mathematical Modelling of Engineering Systems 15 ECM2111 Prof Mike Belmont (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS
DESCRIPTION - summary of the module content
AIMS - intentions of the module
To introduce students to mathematical models of engineering systems. To expose them to standard methods of systems analysis using transform methods in both continuous and discrete variable form.
INTENDED LEARNING OUTCOMES (ILOs) (see assessment section below for how ILOs will be assessed)
Note: List A comprises core outcomes that will be covered fully in lectures and must be achieved by all students to meet the minimum university requirement for progression. List B comprises outcomes that are EITHER more difficult to achieve OR are to be achieved by private study (or both). All outcomes will be assessed, and coverage of List B outcomes is essential for both BEng and MEng students. [b]A: THRESHOLD LEVEL[/b] 1) System modelling; motivational examples of real physical systems and their behaviour (sourced by student experience), spring mass damper, RLC networks, heated reservoir, liquid tanks with capillary pipes; input/output relationships, the lumped parameter element concept, linearity, constant and time varying, reciprocity, non-linearity; linear time-invariant reciprocal lumped parameter, (LT) approximations to real elements; storage elements, dissipative elements; approximations to real first and second order systems by building up connected networks of LT elements, need for extra system elements (signed summer, constant coefficient multiplier); description of network as single system block with single input, single output; derivation of system differential equation;system order and relationship to number of storage elements and physical link to number of initial conditions; coupled systems, 3rd order and above, multi-input, multi-output systems. (2) Time Domain Methods: Solution of 1st and 2nd order differential equations for IC’s only, zero source; time constants, natural frequencies and damping; forced response for steps and sinusoids, transient and steady state; the impulse function, impulse response and time domain convolution, causality. (3) Transform Methods: The Laplace transform, simple manipulative properties and the use of transform pair tables, emphasis on partial fractions; transfer functions, the s plane, polynomial ratio transfer functions poles and zeros; stability. Use of LPs’ to solve LT system equations. (4) System Synthesis: Block diagram algebra, feedback configurations. (5) Discretisation of linear differential equations; Z transform system responses. [b]B: GOOD TO EXCELLENT[/b] (1) System modelling; direct derivation of system differential equation from first principles; time-varying, non-reciprocal and non-linear elements and systems incorporating them, dependence of non-linear system behaviour on initial conditions; linearisation techniques; vector and matrix description of multi-input multi-output systems, introduction to distributed systems. (2) Time Domain Methods: Response to periodic inputs, introduction to state variables, deterministic versus stochastic signals. (3) Transform Methods: Fourier transforms, orthogonal transforms in general, Walsh as an example, transform methods for solving distributed systems. (4) System Synthesis: Minimal form synthesis, canonical forms. (5) Z transform skill set equivalent to Laplace; obtain discrete frequency responses.
SYLLABUS PLAN - summary of the structure and academic content of the module
Generic modelling of engineering systems as networks. Electromechanical, thermal and fluid systems examples. SISO and MIMO systems. The Laplace transform, partial fractions and the use of tables. The concepts of transfer function, stability, gain and phase shift. Continuous variable frequency response. Discretisation of differential equations, finite-difference equations, Z transforms. Discrete frequency response. Solving linear ordinary differential equations. Convolution, poles and zeros. Bode plots with first and second order systems examples. Block diagram algebra.
LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
DETAILS OF LEARNING ACTIVITIES AND TEACHING METHODS
ASSESSMENT
FORMATIVE ASSESSMENT - for feedback and development purposes; does not count towards module grade
SUMMATIVE ASSESSMENT (% of credit)
 Coursework Written Exams 15 85
DETAILS OF SUMMATIVE ASSESSMENT
DETAILS OF RE-ASSESSMENT (where required by referral or deferral)
RE-ASSESSMENT NOTES
RESOURCES
INDICATIVE LEARNING RESOURCES - The following list is offered as an indication of the type & level of
information that you are expected to consult. Further guidance will be provided by the Module Convener