TRC3200 - Dynamical Systems Summary Notes
TRC3200 - Dynamical Systems Summary Notes Contents Cylindrical Coordinates ..................................................................................................... 3 Projectile Motion.................................................................................................................... 3 Angular position and velocity................................................................................................. 3 Degrees of Freedom ......................................................................................................... 4 Constraints - Jacobian ....................................................................................................... 4 Constraints Moving in Time ................................................................................................... 5 Examples ................................................................................................................................ 5 Kinematics ....................................................................................................................... 6 Finite Motions ........................................................................................................................ 6 Transformation Matrices ....................................................................................................... 6 Forward Kinematics................................................................................................................ 8 Reverse Kinematics ................................................................................................................ 8 D’Alemberts Priciple ......................................................................................................... 9 Constraints ............................................................................................................................. 9 D’Alembert Example ............................................................................................................ 10 Virtual Power ........................................................................................................................ 10 Lagrange’s Equations ...................................................................................................... 12 Standard Routine ................................................................................................................. 12 Mutual Inductance ............................................................................................................... 13 First Order Equations ........................................................................................................... 14 Hamilton Formalism ....................................................................................................... 14 Bond Graphs .................................................................................................................. 16 Single Port Elements ............................................................................................................ 16 Two Port Elements ............................................................................................................... 17 Junctions............................................................................................................................... 19 Power Direction.................................................................................................................... 20 Causality ............................................................................................................................... 21 Elements .......................................................................................................................... 21 Junctions .......................................................................................................................... 22 Differential Causality........................................................................................................ 22 Finn Andersen, 2013
1
Obtaining Differential Equations.......................................................................................... 23 Mechanical Example ........................................................................................................ 24 Summary of Methods ..................................................................................................... 25
Finn Andersen, 2013
2
Cylindrical Coordinates A 3-dimensional coordinate system which specifies point positions using: • • •
Distance from a chosen reference axis (p) Direction from the axis relative to a chosen reference direction (θ) Distance from chosen reference plane perpendicular to the axis (z)
Similar to polar coordinate system but adds a third dimension. . does not change with time (‘upwards’ z direction is constant). However, direction of can change with in the direction of :
So for change in over time: , -
are variable with time so product rule must be applied where they are involved. Direction of changes with in direction of :
,
Projectile Motion Consider projectile motion with initial velocity defined as: cos sin Horizontal component of velocity remains constant without air resistance. Displacement equations are therefore:
Combining by eliminating t gives:
Angular position and velocity || || !(
Finn Andersen, 2013
∆
! " # $ % & % $ $' ∆
( %" ) $ %* $ $' * '%$ + $%) 3
Degrees of Freedom The number of degrees of freedom is determined by the number of independent coordinates required to completely specify the position of each particle of the system (number of coordinates which are not constant (degrees of constraint)). Made up of both translation and rotation. Examples: One degree of freedom • •
Particle constrained to move along straight line (single dimension) Simple pendulum motion with fixed rotation axis.
Two degrees of freedom • •
A particle free to move in contact with plane (one dimension constant, other two variable) Dumbbell free to slide along an axis and rotate about it
The number of degrees of freedom of a system can be influenced by the number of particles in the system p, with maximum of 3p degrees of freedom. The constraints of a system are represented by equations of constraint. For a bead constrained to a straight wire in XY plane:
If wire is parabolic:
Constraints - Jacobian For particle constrained to move along curve length defined by s(t) (generalised variable), the components of its position vector 01, +, 12 are functions of s. Velocity is given by: 34 5 36
34 5 35
35 36
Expression in brackets above can be expressed in a Jacobian matrix which contains information about the geometric constraints.
Finn Andersen, 2013
4
Transformer Mechanical Example Mechanically, due to pivot arm length ratio: 7 ( 8 Since *( ;*8 , ;
< =
Also, required force ratio due to arm length ratio: 9( ):'' 98 7
Gyrator Electrical Example
The electric motor (gyrator) converts electrical current (flow) into mechanical torque (effort). > ?@A 8 Since ( ;*8 , ; ?@
Example of gyrator (ideal DC motor) with internal resistance/moment of inertia:
Finn Andersen, 2013
18
Junctions Elements are connected to other elements with bonds via junctions. • • • •
Elements cannot be connected directly together, only to junctions Junctions define whether connected elements are in series (share flow) or parallel (share effort) At ‘1’ junction, all elements share the same flow *8 *( *B *C … At ‘0’ junction, all elements share the same effort 8 ( B C …
Electrical
Mechanical
Finn Andersen, 2013
19
1
Obtaining Differential Equations.......................................................................................... 23 Mechanical Example ........................................................................................................ 24 Summary of Methods ..................................................................................................... 25
Finn Andersen, 2013
2
Cylindrical Coordinates A 3-dimensional coordinate system which specifies point positions using: • • •
Distance from a chosen reference axis (p) Direction from the axis relative to a chosen reference direction (θ) Distance from chosen reference plane perpendicular to the axis (z)
Similar to polar coordinate system but adds a third dimension. . does not change with time (‘upwards’ z direction is constant). However, direction of can change with in the direction of :
So for change in over time: , -
are variable with time so product rule must be applied where they are involved. Direction of changes with in direction of :
,
Projectile Motion Consider projectile motion with initial velocity defined as: cos sin Horizontal component of velocity remains constant without air resistance. Displacement equations are therefore:
Combining by eliminating t gives:
Angular position and velocity || || !(
Finn Andersen, 2013
∆
! " # $ % & % $ $' ∆
( %" ) $ %* $ $' * '%$ + $%) 3
Degrees of Freedom The number of degrees of freedom is determined by the number of independent coordinates required to completely specify the position of each particle of the system (number of coordinates which are not constant (degrees of constraint)). Made up of both translation and rotation. Examples: One degree of freedom • •
Particle constrained to move along straight line (single dimension) Simple pendulum motion with fixed rotation axis.
Two degrees of freedom • •
A particle free to move in contact with plane (one dimension constant, other two variable) Dumbbell free to slide along an axis and rotate about it
The number of degrees of freedom of a system can be influenced by the number of particles in the system p, with maximum of 3p degrees of freedom. The constraints of a system are represented by equations of constraint. For a bead constrained to a straight wire in XY plane:
If wire is parabolic:
Constraints - Jacobian For particle constrained to move along curve length defined by s(t) (generalised variable), the components of its position vector 01, +, 12 are functions of s. Velocity is given by: 34 5 36
34 5 35
35 36
Expression in brackets above can be expressed in a Jacobian matrix which contains information about the geometric constraints.
Finn Andersen, 2013
4
Transformer Mechanical Example Mechanically, due to pivot arm length ratio: 7 ( 8 Since *( ;*8 , ;
< =
Also, required force ratio due to arm length ratio: 9( ):'' 98 7
Gyrator Electrical Example
The electric motor (gyrator) converts electrical current (flow) into mechanical torque (effort). > ?@A 8 Since ( ;*8 , ; ?@
Example of gyrator (ideal DC motor) with internal resistance/moment of inertia:
Finn Andersen, 2013
18
Junctions Elements are connected to other elements with bonds via junctions. • • • •
Elements cannot be connected directly together, only to junctions Junctions define whether connected elements are in series (share flow) or parallel (share effort) At ‘1’ junction, all elements share the same flow *8 *( *B *C … At ‘0’ junction, all elements share the same effort 8 ( B C …
Electrical
Mechanical
Finn Andersen, 2013
19
