6. multi-dof mechanical systems
TRANSCRIPT
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Department of Mechanical Engineering, NTU
National Taiwan UniversityENGINEERINGMechatronic and Robotic Systems Laboratory
System Dynamics
Yu-Hsiu Lee
6. Multi-DOF Mechanical Systems
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11/4/2021 2Mechatronic and Robotic Systems Laboratory, Department of Mechanical Engineering, NTU
Outline
• Examples
MCK systems
Double pendulum
• Natural modes
• Natural frequency and mode shapes
• Anti-resonance modes
• Case study: Huygens’ clock
• Lagrange’s equation
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Examples
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11/4/2021 4Mechatronic and Robotic Systems Laboratory, Department of Mechanical Engineering, NTU
MCK Systems
• Example: piezo-actuator in nano-positioning system
[FA09] Fleming, Andrew J. "Nanopositioning system with force feedback for high-performance tracking and vibration control." IEEE/Asme Transactions on Mechatronics 15.3 (2009): 433-447.
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11/4/2021 5Mechatronic and Robotic Systems Laboratory, Department of Mechanical Engineering, NTU
MCK Systems
• Example: cantilever models for Atomic Force Microscopy (AFM) system
[SY08] Song, Yaxin, and Bharat Bhushan. "Atomic force microscopy dynamic modes: modeling and applications." Journal of Physics: Condensed Matter 20.22 (2008): 225012.
1-D point mass model
3-D point mass model
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11/4/2021 6Mechatronic and Robotic Systems Laboratory, Department of Mechanical Engineering, NTU
MCK Systems
• Example: car suspension system
[CH05] Chen, H., Z-Y. Liu, and P-Y. Sun. "Application of constrained H∞ control to active suspension systems on Half-Car models." (2005): 345-354.
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11/4/2021 7Mechatronic and Robotic Systems Laboratory, Department of Mechanical Engineering, NTU
MCK Systems
• 1-DOF
• 2-DOF
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11/4/2021 8Mechatronic and Robotic Systems Laboratory, Department of Mechanical Engineering, NTU
MCK Systems
• 3-DOF
Matrix form
Symmetries of M, C, and K matrices are NOT a coincidence.
• n-DOF system
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11/4/2021 9Mechatronic and Robotic Systems Laboratory, Department of Mechanical Engineering, NTU
Double Pendulum
• Schematic
Want to put this into standard form
• Free body diagram (FBD) 1:
Rotational equation of motion about point O
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11/4/2021 10Mechatronic and Robotic Systems Laboratory, Department of Mechanical Engineering, NTU
Double Pendulum
• Schematic
Want to put this into standard form
• Free body diagram (FBD) 2:
Rotational equation of motion about C.M.
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11/4/2021 11Mechatronic and Robotic Systems Laboratory, Department of Mechanical Engineering, NTU
Double Pendulum
• Schematic
Want to put this into standard form
• Free body diagram (FBD) 2:
Translational equation of motion about point O
Assume small angle around
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11/4/2021 12Mechatronic and Robotic Systems Laboratory, Department of Mechanical Engineering, NTU
Double Pendulum
• Schematic
Want to put this into standard form
• Free body diagram (FBD) 2:
Translational equation of motion about point O
Assume small angle around
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11/4/2021 13Mechatronic and Robotic Systems Laboratory, Department of Mechanical Engineering, NTU
Double Pendulum
• Schematic
Want to put this into standard form
• Link 1 rotational equation of motion about point O
• Link 2 rotational equation of motion about C.M.
Substitute for reaction forces
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11/4/2021 14Mechatronic and Robotic Systems Laboratory, Department of Mechanical Engineering, NTU
Double Pendulum
• Schematic
Want to put this into standard form
• Complete system of differential equation
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Natural Modes
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11/4/2021 16Mechatronic and Robotic Systems Laboratory, Department of Mechanical Engineering, NTU
Matrices of MCK Systems
• Consider
Assume single-input-single-output (SISO)
2-DOF example:
Actuator location
Sensor location
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11/4/2021 17Mechatronic and Robotic Systems Laboratory, Department of Mechanical Engineering, NTU
Poles of MCK Systems
• Consider
Assume single-input-single-output (SISO)
Take Laplace transform
Characteristic equation
Each DOF is 2nd order
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11/4/2021 18Mechatronic and Robotic Systems Laboratory, Department of Mechanical Engineering, NTU
MCK Matrices
• Consider
Characteristic equation
1. If all masses are fully connected to the inertial frame through springs, then
2. If all masses are fully connected to the inertial frame through dampers, then
3. If
Example:
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11/4/2021 19Mechatronic and Robotic Systems Laboratory, Department of Mechanical Engineering, NTU
MCK Matrices
• Consider
Characteristic equation
4.
5.
6.
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11/4/2021 20Mechatronic and Robotic Systems Laboratory, Department of Mechanical Engineering, NTU
2-DOF MK System
• Example: 2-DOF MK system • Fill stiffness matrix
Easy to understand the source of symmetry
This rule applies to dampers as well
Newton’s 3rd law
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11/4/2021 21Mechatronic and Robotic Systems Laboratory, Department of Mechanical Engineering, NTU
2-DOF MK System
• Example: 2-DOF MK system • Assume
Case A:
Case B:
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11/4/2021 22Mechatronic and Robotic Systems Laboratory, Department of Mechanical Engineering, NTU
2-DOF MK System
• Example: 2-DOF MK system
Case B, 1st mode:
Case B, 2nd mode:
• Assume
Case A:
Case B:
Rigid body motion
Oscillation
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Natural Frequency and Mode shapes
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11/4/2021 24Mechatronic and Robotic Systems Laboratory, Department of Mechanical Engineering, NTU
Mode Shape and Free Response
•
n-DOF system has n modes.
•
Natural frequency
Modeshape
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11/4/2021 25Mechatronic and Robotic Systems Laboratory, Department of Mechanical Engineering, NTU
Mode Shape and Free Response
•
1-DOF free response:
General n-DOF free response:
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11/4/2021 26Mechatronic and Robotic Systems Laboratory, Department of Mechanical Engineering, NTU
Verification of the Solution
• Verify the solution of the free response by defining:
The solution satisfies the ODE of the unforced system:
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11/4/2021 27Mechatronic and Robotic Systems Laboratory, Department of Mechanical Engineering, NTU
Double Pendulum
• Schematic
System matrices
Eigenvalues
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11/4/2021 28Mechatronic and Robotic Systems Laboratory, Department of Mechanical Engineering, NTU
Double Pendulum
• Schematic
System matrices
Eigenvalues
Eigenvectors (mode shapes)
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11/4/2021 29Mechatronic and Robotic Systems Laboratory, Department of Mechanical Engineering, NTU
Double Pendulum
• Mode shapes
Free response will the superposition of two modes
Eigenvalues
Eigenvectors (mode shapes)
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11/4/2021 30Mechatronic and Robotic Systems Laboratory, Department of Mechanical Engineering, NTU
Double Pendulum
• Mode shapes
Free response will the superposition of two modes
• Example: free response with I.C.
1. Express I.C. as linear combination of eigenvectors
2. Obtain solution by superposition
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Anti-resonance Modes
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11/4/2021 32Mechatronic and Robotic Systems Laboratory, Department of Mechanical Engineering, NTU
Double Pendulum
• Schematic
System matrices
Transfer function
Zeros:
Poles:
Output on the 1st joint.
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11/4/2021 33Mechatronic and Robotic Systems Laboratory, Department of Mechanical Engineering, NTU
Double Pendulum
• Bode plot Transfer function
Zeros:
Poles:
n peaks indicate an n-DOF system, and vice versa.
1.8
2.2
3.4
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11/4/2021 34Mechatronic and Robotic Systems Laboratory, Department of Mechanical Engineering, NTU
Double Pendulum
• Bode plot Anti-resonance:
Mechanism:
I/O mass: actuation and sensing are collocated
1.8
2.2
3.4
Principle ofvibration absorber
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11/4/2021 35Mechatronic and Robotic Systems Laboratory, Department of Mechanical Engineering, NTU
MCK System
• 3-DOF • 4-DOF
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Case Study: Huygens’ Clock
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11/4/2021 37Mechatronic and Robotic Systems Laboratory, Department of Mechanical Engineering, NTU
Equation of Motion
• Schematic
• FBD1
• FBD2
Mass location
x
y
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11/4/2021 38Mechatronic and Robotic Systems Laboratory, Department of Mechanical Engineering, NTU
Equation of Motion
• Schematic
• FBD1
• FBD2
Small angle approximation
x
y
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11/4/2021 39Mechatronic and Robotic Systems Laboratory, Department of Mechanical Engineering, NTU
Equation of Motion
• Schematic
• FBD1
• FBD2
• Complete system of ODE
No verticalmotion
x
y
Assumemassless link
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11/4/2021 40Mechatronic and Robotic Systems Laboratory, Department of Mechanical Engineering, NTU
Equation of Motion
• Schematic
Matrix form:
x
y
(a) FBD1 (b) FBD2
Assumemassless link
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11/4/2021 41Mechatronic and Robotic Systems Laboratory, Department of Mechanical Engineering, NTU
Natural Modes
• System matrices and eigenvalues
Undamped natural frequencies
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11/4/2021 42Mechatronic and Robotic Systems Laboratory, Department of Mechanical Engineering, NTU
Mode Shapes
• Eigenvectors
(2) Out-of-phase (3) In-phase
(1) Translation
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11/4/2021 43Mechatronic and Robotic Systems Laboratory, Department of Mechanical Engineering, NTU
Mode Shapes
• If there exists viscous damping:
(2) Out-of-phase (3) In-phase(1) Translation
Bias and exp. decay
Out-of-phaseoscillation
Dampedoscillation
This requires the twopendulums to be identical!
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Lagrange’s Equation
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11/4/2021 45Mechatronic and Robotic Systems Laboratory, Department of Mechanical Engineering, NTU
Motivation
• Newton’s method requires analyzing each part with free body diagram
Example: double pendulum
The forces at interconnections may not be of interest
Equations of motion can be derived by considering energies in the system
• Lagrange’s equation
Indirect approach that can be applied for (but not limited to) mechanical systems
An energy-based method that does not compute the forces at interconnections
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11/4/2021 46Mechatronic and Robotic Systems Laboratory, Department of Mechanical Engineering, NTU
Lagrange’s Equation
• General form
• Variable definitions
For a sketch of proof, refer to the PDF document provided on the course website.
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11/4/2021 47Mechatronic and Robotic Systems Laboratory, Department of Mechanical Engineering, NTU
Lagrange’s Equation
• General form
• Procedure
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11/4/2021 48Mechatronic and Robotic Systems Laboratory, Department of Mechanical Engineering, NTU
Lagrange’s Equation
• General form
• Example: 1-DOF MCK system
Procedure
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11/4/2021 49Mechatronic and Robotic Systems Laboratory, Department of Mechanical Engineering, NTU
Lagrange’s Equation
• Example: cart-pulley-mass system
Procedure
No gravitationaleffect
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11/4/2021 50Mechatronic and Robotic Systems Laboratory, Department of Mechanical Engineering, NTU
Lagrange’s Equation
• Example: cart-pulley-mass system
Procedure
Matrix formNo gravitationaleffect
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11/4/2021 51Mechatronic and Robotic Systems Laboratory, Department of Mechanical Engineering, NTU
Lagrange’s Equation
• Equation
• Features
1. Only position and velocity are required
2. Involve scalar equations
3. No need for FBD and constraint forces
• Application: robot rigid body dynamics [KR06]
In i-th generalized coordinate (i-th joint)
Matrix form
[KR06] Kelly, Rafael, Victor Santibáñez Davila, and Julio Antonio Loría Perez. Control of robot manipulators in joint space. Springer Science & Business Media, 2006.
No. of DOFs
Function of potential and kinetic energy
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11/4/2021 52Mechatronic and Robotic Systems Laboratory, Department of Mechanical Engineering, NTU
Lagrange’s Equation
• Equation
• Robot rigid body dynamics
Assume
Apply Lagrange’s equation
Centrifugal and Coriolis effect
Gravitational effect
See an example by Prof. Lynch at Northwestern:https://www.youtube.com/watch?v=1U6y_68CjeY