6. multi-dof mechanical systems
TRANSCRIPT
Department of Mechanical Engineering, NTU
National Taiwan UniversityENGINEERINGMechatronic and Robotic Systems Laboratory
System Dynamics
Yu-Hsiu Lee
6. Multi-DOF Mechanical Systems
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
Examples
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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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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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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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MCK Systems
• 1-DOF
• 2-DOF
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MCK Systems
• 3-DOF
Matrix form
Symmetries of M, C, and K matrices are NOT a coincidence.
• n-DOF system
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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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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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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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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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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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Double Pendulum
• Schematic
Want to put this into standard form
• Complete system of differential equation
Natural Modes
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Matrices of MCK Systems
• Consider
Assume single-input-single-output (SISO)
2-DOF example:
Actuator location
Sensor location
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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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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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MCK Matrices
• Consider
Characteristic equation
4.
5.
6.
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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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2-DOF MK System
• Example: 2-DOF MK system • Assume
Case A:
Case B:
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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
Natural Frequency and Mode shapes
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Mode Shape and Free Response
•
n-DOF system has n modes.
•
Natural frequency
Modeshape
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Mode Shape and Free Response
•
1-DOF free response:
General n-DOF free response:
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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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Double Pendulum
• Schematic
System matrices
Eigenvalues
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Double Pendulum
• Schematic
System matrices
Eigenvalues
Eigenvectors (mode shapes)
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Double Pendulum
• Mode shapes
Free response will the superposition of two modes
Eigenvalues
Eigenvectors (mode shapes)
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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
Anti-resonance Modes
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Double Pendulum
• Schematic
System matrices
Transfer function
Zeros:
Poles:
Output on the 1st joint.
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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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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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MCK System
• 3-DOF • 4-DOF
Case Study: Huygens’ Clock
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Equation of Motion
• Schematic
• FBD1
• FBD2
Mass location
x
y
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Equation of Motion
• Schematic
• FBD1
• FBD2
Small angle approximation
x
y
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Equation of Motion
• Schematic
• FBD1
• FBD2
• Complete system of ODE
No verticalmotion
x
y
Assumemassless link
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Equation of Motion
• Schematic
Matrix form:
x
y
(a) FBD1 (b) FBD2
Assumemassless link
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Natural Modes
• System matrices and eigenvalues
Undamped natural frequencies
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Mode Shapes
• Eigenvectors
(2) Out-of-phase (3) In-phase
(1) Translation
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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!
Lagrange’s Equation
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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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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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Lagrange’s Equation
• General form
• Procedure
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Lagrange’s Equation
• General form
• Example: 1-DOF MCK system
Procedure
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Lagrange’s Equation
• Example: cart-pulley-mass system
Procedure
No gravitationaleffect
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Lagrange’s Equation
• Example: cart-pulley-mass system
Procedure
Matrix formNo gravitationaleffect
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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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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