6. multi-dof mechanical systems

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