ES197 – Systems
Contents
Week 2 ................................................................................................................................................................................................ 2
Lecture 0 ......................................................................................................................................................................................... 2
Lecture 1: Translational Mechanical Systems ................................................................................................................................. 2
Element Laws .............................................................................................................................................................................. 2
Differential Equations (UNDERSTAND THIS BIT LATER) .............................................................................................................. 4
Lecture 2: Rotational Mechanical Systems ..................................................................................................................................... 4
Definitions and Classifications for Dynamics Systems ................................................................................................................ 4
Element Laws .............................................................................................................................................................................. 5
Interconnection Laws ...................................................................................................................................................................... 7
Differential Equations (FINISH OFF LATER) ................................................................................................................................. 7
Week 3: Electrical Systems ................................................................................................................................................................. 7
Applications for Electrical Systems ................................................................................................................................................. 7
Element Laws (Resistor, Capacitor, Inductor) ................................................................................................................................. 7
Interconnection Laws .................................................................................................................................................................. 8
Week 4: Thermal Systems ................................................................................................................................................................... 9
Applications for Thermal Systems .................................................................................................................................................. 9
Element Laws (Thermal Resistors and Thermal Capacitors) ........................................................................................................... 9
Interconnection Laws and Differential Equations ........................................................................................................................... 9
Week 5: Data Driven Models ............................................................................................................................................................ 10
Different Modelling Approaches ................................................................................................................................................... 10
Applications of Data-Driven Models ............................................................................................................................................. 10
Fitting Polynomial Models to data using MATLAB ........................................................................................................................ 11
Regression ................................................................................................................................................................................. 11
Appreciation of Potential Problems .............................................................................................................................................. 12
Week 6: 1st Order Step Response ..................................................................................................................................................... 12
1st Order and 2nd Order Differential Equations across domains ................................................................................................... 12
Difference between Free and Forced Response ........................................................................................................................... 14
Step, Impulse, Ramp and Sine Test Inputs .................................................................................................................................... 14
Interpreting Step Reponses .......................................................................................................................................................... 15
Sketching the response of a 1st order system to a step input, with correct final value and time constant ................................ 16
Week 7: 2nd Order Step Response..................................................................................................................................................... 17
Week 8: Transfer Functions .............................................................................................................................................................. 18
Part A ............................................................................................................................................................................................ 18
Part B............................................................................................................................................................................................. 20
Week 9 .............................................................................................................................................................................................. 22
Part A ............................................................................................................................................................................................ 22
Why frequency response matters ................................................................................................................................................. 22
1
, Effect of a 1st order system upon a sine wave input (gain and phase) ........................................................................................ 23
Gain and Phase ............................................................................................................................................................................. 25
Lecture 10: ........................................................................................................................................................................................ 28
Frequency response derived from the Transfer Function ............................................................................................................ 28
Interpreting Bode Plots ................................................................................................................................................................. 29
How damping factor and undamped natural frequency affects the frequency response of a 2 nd order system ......................... 29
Sketching/Obtaining a graph for frequency response using data book or MATLAB..................................................................... 30
Week 2
Lecture 0
Lecture 1: Translational Mechanical Systems
Two basic variables that are used to describe the dynamic behaviour of translational mechanical systems:
• Force: f(t) (measured in newtons, N)
• Displacement: x(t) (measures in metres, m)
• Velocity: v(t) (measured in metres/second, ms-1)
𝑥 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡
𝑑𝑥
𝑥̇ = 𝑣 = 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦
𝑑𝑡
𝑑2 𝑥
𝑥̈ = 𝑣̇ = 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛
𝑑𝑡2
Three key elements for translational mechanical systems:
Element Laws
Mass
• Stores kinetic energy and potential energy
• Reversible
Note: The elemental
equation is basically F=ma
(Newton’s 2nd Law)
Spring
• Stores potential energy (throughout stretching/compression)
• Reversible
• e.g. coiled spring, diving board, leaf spring used in suspension
2
, Hooke’s Law i.e. F=kx, where the force is linearly proprtional to the extension. In reality, springs are not linear. The diagram
above shows the theoretical line if Hooke’s Law was true. The lines are consistent in the midregion.
Damper
• Dissipates energy as heat (removes energy from the system)
• Non-Reversible
• Critical for stability (takes energy out of the system)
• e.g. shock absorbers
• Used to reduce oscillations (take the kinetic energy out of the system) or increase the quality of feel of a product
Note: Force produced by the damper is not directly
related to its displacement at the ends of the
damper, rather the velocity
Friction (Viscous Friction Element)
• Also causes a damping effect
• One mass on top of another mass/surface separated by a fluid with laminar oil flow, then a damping force proportional
to the velocity is produced
• Either the velocity of the mass on the surface or the difference in velocity between the two masses
• The sliding friction between a mass and its surface is proportional to the contact force between then (F=μR)
Assumptions
• Mass: The object does not bend (velocity on one side is the same on the other side)
• Spring: The spring does not have a mass, no change of force with a change in temperature, and a linear relationship
between the force it produces and its extension
3
Contents
Week 2 ................................................................................................................................................................................................ 2
Lecture 0 ......................................................................................................................................................................................... 2
Lecture 1: Translational Mechanical Systems ................................................................................................................................. 2
Element Laws .............................................................................................................................................................................. 2
Differential Equations (UNDERSTAND THIS BIT LATER) .............................................................................................................. 4
Lecture 2: Rotational Mechanical Systems ..................................................................................................................................... 4
Definitions and Classifications for Dynamics Systems ................................................................................................................ 4
Element Laws .............................................................................................................................................................................. 5
Interconnection Laws ...................................................................................................................................................................... 7
Differential Equations (FINISH OFF LATER) ................................................................................................................................. 7
Week 3: Electrical Systems ................................................................................................................................................................. 7
Applications for Electrical Systems ................................................................................................................................................. 7
Element Laws (Resistor, Capacitor, Inductor) ................................................................................................................................. 7
Interconnection Laws .................................................................................................................................................................. 8
Week 4: Thermal Systems ................................................................................................................................................................... 9
Applications for Thermal Systems .................................................................................................................................................. 9
Element Laws (Thermal Resistors and Thermal Capacitors) ........................................................................................................... 9
Interconnection Laws and Differential Equations ........................................................................................................................... 9
Week 5: Data Driven Models ............................................................................................................................................................ 10
Different Modelling Approaches ................................................................................................................................................... 10
Applications of Data-Driven Models ............................................................................................................................................. 10
Fitting Polynomial Models to data using MATLAB ........................................................................................................................ 11
Regression ................................................................................................................................................................................. 11
Appreciation of Potential Problems .............................................................................................................................................. 12
Week 6: 1st Order Step Response ..................................................................................................................................................... 12
1st Order and 2nd Order Differential Equations across domains ................................................................................................... 12
Difference between Free and Forced Response ........................................................................................................................... 14
Step, Impulse, Ramp and Sine Test Inputs .................................................................................................................................... 14
Interpreting Step Reponses .......................................................................................................................................................... 15
Sketching the response of a 1st order system to a step input, with correct final value and time constant ................................ 16
Week 7: 2nd Order Step Response..................................................................................................................................................... 17
Week 8: Transfer Functions .............................................................................................................................................................. 18
Part A ............................................................................................................................................................................................ 18
Part B............................................................................................................................................................................................. 20
Week 9 .............................................................................................................................................................................................. 22
Part A ............................................................................................................................................................................................ 22
Why frequency response matters ................................................................................................................................................. 22
1
, Effect of a 1st order system upon a sine wave input (gain and phase) ........................................................................................ 23
Gain and Phase ............................................................................................................................................................................. 25
Lecture 10: ........................................................................................................................................................................................ 28
Frequency response derived from the Transfer Function ............................................................................................................ 28
Interpreting Bode Plots ................................................................................................................................................................. 29
How damping factor and undamped natural frequency affects the frequency response of a 2 nd order system ......................... 29
Sketching/Obtaining a graph for frequency response using data book or MATLAB..................................................................... 30
Week 2
Lecture 0
Lecture 1: Translational Mechanical Systems
Two basic variables that are used to describe the dynamic behaviour of translational mechanical systems:
• Force: f(t) (measured in newtons, N)
• Displacement: x(t) (measures in metres, m)
• Velocity: v(t) (measured in metres/second, ms-1)
𝑥 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡
𝑑𝑥
𝑥̇ = 𝑣 = 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦
𝑑𝑡
𝑑2 𝑥
𝑥̈ = 𝑣̇ = 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛
𝑑𝑡2
Three key elements for translational mechanical systems:
Element Laws
Mass
• Stores kinetic energy and potential energy
• Reversible
Note: The elemental
equation is basically F=ma
(Newton’s 2nd Law)
Spring
• Stores potential energy (throughout stretching/compression)
• Reversible
• e.g. coiled spring, diving board, leaf spring used in suspension
2
, Hooke’s Law i.e. F=kx, where the force is linearly proprtional to the extension. In reality, springs are not linear. The diagram
above shows the theoretical line if Hooke’s Law was true. The lines are consistent in the midregion.
Damper
• Dissipates energy as heat (removes energy from the system)
• Non-Reversible
• Critical for stability (takes energy out of the system)
• e.g. shock absorbers
• Used to reduce oscillations (take the kinetic energy out of the system) or increase the quality of feel of a product
Note: Force produced by the damper is not directly
related to its displacement at the ends of the
damper, rather the velocity
Friction (Viscous Friction Element)
• Also causes a damping effect
• One mass on top of another mass/surface separated by a fluid with laminar oil flow, then a damping force proportional
to the velocity is produced
• Either the velocity of the mass on the surface or the difference in velocity between the two masses
• The sliding friction between a mass and its surface is proportional to the contact force between then (F=μR)
Assumptions
• Mass: The object does not bend (velocity on one side is the same on the other side)
• Spring: The spring does not have a mass, no change of force with a change in temperature, and a linear relationship
between the force it produces and its extension
3
