Department of Mechanical Engineering, NTU

National Taiwan UniversityENGINEERINGMechatronic and Robotic Systems Laboratory

System Dynamics

Yu-Hsiu Lee

5. Frequency Response Functions

11/24/2021 2Mechatronic and Robotic Systems Laboratory, Department of Mechanical Engineering, NTU

Outline

• Frequency response function

Mechanical resonance

• Bode plot

Basic elements

Bode plot of basic elements

Examples

Frequency Response Function

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Frequency Response Function (FRF)

• nth order LTI system

• Response under sinusoidal input

• Particular solution when input is

LTISystem

0 when time is large.FRF does NOT address transient response.

11/24/2021 5Mechatronic and Robotic Systems Laboratory, Department of Mechanical Engineering, NTU

Frequency Response Function (FRF)

• Perspective of convolution

• FRF is the Fourier transform of the impulse response

Refer to chap. 2 fordetailed derivation.

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Frequency Response Function (FRF)

• Example: flapping wing

Substitute

• Magnitude and phase plot

Gain = 1.29

Delay = 15 [deg]

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Frequency Response Function (FRF)

• Example: flapping wing

Substitute

• Time domain response

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Mechanical Resonance

• Example: flapping wing

If

If

• Frequency domain magnitude plot

Resonance frequency

Minimizer

Bode Plot

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Bode Plot

• A special way of plotting FRF

Magnitude and phase plot

Log x-axis: display a wide range of frequencies

Log y-axis: multiplication is addition in log scale

Reason for using dB: originates from power ratio:

• Hand sketch and verify the trend in software rapidly.

MATLAB routines: bode, freqs, evalfr, …

• Example: flapping wing

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Basic Elements

• 1st order denominator • Bode plot

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Basic Elements

• 1st order numerator

Magnitude has mirror image with respect to 0 [dB]

Phase has mirror image with respect to 0 [deg]

• Bode plot

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Basic Elements

• Integrator • Bode plot

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Basic Elements

• Differentiator • Bode plot

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Basic Elements

• 2nd order denominator

Slope conversion: log-log to dB-log

• Bode plot

Multiplied by 20

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Basic Elements

• 2nd order numerator

Magnitude has mirror image with respect to 0 [dB]

Phase has mirror image with respect to 0 [deg]

• Bode plot

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Examples

• Example: band-pass filter • Bode plot

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Examples

• Example: lead filter • Bode plot

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Examples

• Example: notch filter • Bode plot