Physical Modelling: 1.Out there world inside here 2.Modelling and Design Cycle 3.Practical example: Passive Dynamic Walkers 4.Base systems and concepts 5.Ideals, assumptions and real life 6.Similarities in systems and responses

Design Cycle:

Similarities in systems and responses: 1.Similarities in systems – why is a spring like a water tank like a capacitor??? 2.First derivative time response 3.Second derivative time responses (note there is more than one!) 4.Damping 5.Parallel verses Serial connection of components

Electronic components: Resistors: Voltage is proportional to current, Ohms law V = RI Voltage - Impedance Current ecen1400/labs/resistors.html

Electronic components: Resistors: Voltage is proportional to current, Ohms law V = RI Voltage - Current ecen1400/labs/resistors.html

Electronic components: Capacitors: Voltage is proportional to integral of current Voltage - Current /ecen1400/labs/capacitors.html

Electronic components: Capacitors: Voltage is proportional to integral of current Voltage - Impedance Current [think of s as the rate of change of output variable in an instant] /ecen1400/labs/capacitors.html

Electronic components: Inductors: Voltage is proportional to differential of current Voltage - Current

Electronic components: Inductors: Voltage is proportional to differential of current Voltage - Impedance Current

Force - Impedance Distance Spring Damper Mass Source Nise 2004 Mechanical: Electrical: Capacitor Resistor Inductor Source Nise 2004 Voltage - Impedance Current

Force - Impedance Velocity Spring Damper Mass Source Nise 2004 Mechanical Electrical: Capacitor Resistor Inductor Source Nise 2004 Voltage - Impedance Current

Modelling: Dynamic Systems

Consider dynamic systems: these change with time As an example consider water system with two tanks Water will flow from first tank to second [Assume I stays constant due to nature of Dams] Plot time response of system….? Modelling: Dynamic Systems I F L

Water flows because of pressure difference [Ignore atmospheric pressure – approx. equal at both ends of pipe] If have water at one end - what is its pressure? [Tanks with constant cross sectional area A] Pressure is force per unit area, = F / A, Force (F) is mass of water times gravity g Mass of water (M) is volume of water * density M = V *  Volume (V) is height of water, h, times its area A:V = h * A Combining: pressure is Dynamic Systems F Pressure

For first tank, pressure is p f = I *  * g For second tank, pressure is p s = L *  * g Thus flow is proportional to the difference in pressures: driving (effort) variable Flow ∝ p f – p s ∝ (I-L) *  * g as well as on the pipe (its restrictance, R) Here R is the constant of proportionality… Does flow Increase or decrease as the constant R is changed in different systems? Flow changes volume of tanks: Volume change = A * rate of change in height (L) = Flow

Flow changes volume of tanks: Volume change = A * rate of change in height (L) = Flow We write ‘rate of change in height L’ as dL/dt = Flow / A A tank has a capacitance - constantscollected together in C = Thus rate of change in height L: = Flow stops, and there is no change in height when I = L

Level change – not instantaneous Initially: Large height difference  Large flow  L up a lot Then: Height difference less  Less flow  L increases, but by less Later: Height difference ‘lesser’  Less flow  L up, but by less, etc Graphically we can thus argue the variation of level L and flow F is: Dynamic Flow F t T T L t I

Any system of the form: Rate of change of output variable in an instant = Input variable – Output variable Has a time response (depending on step input): Time Response of System Time Output Exponential

Time responses: Proportional components Or Ratio governed by constant of proportionality x f Time x t Input Output Time x t Input Output

Time responses: Proportional to derivative components (+ previous) Or Ratio governed by gain constant Time of response governed by time constant dx/dt f Time x t Input Output Time x t Input Output

Time responses: Proportional to second derivative components (+ previous) Or Ratio governed by gain constant Time of response governed by time constants Overshoot governed by damping constant. d 2 x/dt 2 f Time x t Input Output Time x t Input Output Time x t Input Output

Serial connection of components: Opposite to parallel connections What is equivalent spring? Draw a free body diagram of a spring Write down individual equations:F t = k t x t Consider laws to combine them:x t = x 1 + x 2 Consider what does not change:Force must be equal on each spring F t = F 1 = F 2 F t = k t x t = k t (x 1 + x 2 ) = k t (F 1 / k 1 + F 2 / k 2 ) cancel forces: k t = (1 / k / k 2 ) -1 Can extend method to any number of springs in series masteringphysics/13/springinseries.htm

Parallel connection of components: Opposite to serial connections Draw a free body diagram of a spring Write down individual equations:F t = k t x t Consider laws to combine them:F t = F 1 + F 2 Consider what does not change:x t = x 1 = x 2 F t = k t x t F 1 + F 2 = k t x t k t x t = k 1 x 1 + k 2 x 2 cancel distances: k t = k 1 + k 2 Can extend method to any number of springs in series

Remember to test: video-shows-nasas-flying-saucer-in-action- at-l /+jesusdiaz

Example test questions for PM 1.In the Researching phase of the engineering design cycle: state and describe at least five (5) steps when defining the problem. 2.Given a system has the following instantaneous (dynamic) relationships, sketch their characteristic graphs on appropriate axes: i) y ∝ x ii) y ∝ dx/dt iii) y ∝ d 2 x/dt 2 Initially a system starts with a component with the relationship given in (i) sketch its time response to a step change in the effort variable. Plot both the input and output on the same axes (Hint: time is the independent variable). An additional component with the relationship given in (ii) is added; add the new time response clearly labelling the graph. Finally, a component described by (iii) is added; plot and discuss the possible outputs. 3.Given two resistors are in parallel in a connected circuit with a unit voltage effort driving the current flow, draw the diagram labelling important components, variables and constants. Calculate the equivalent resistor value for the circuit. Help session available in Mon, Wed PM, with Howard