Purpose Inerter-based mechanical networks have been widely used in vibration control. However, the influence of inerter on the fundamental characteristic of vibration systems—natural frequencies, has not been sufficiently investigated. This work is devoted to analyzing the influence of adding one inerter on natural frequencies of one-dimensional (1D) vibration systems. Methods In correspondence with inerter-based networks used in vibration control, the way of adding one inerter could be connecting the inerter in series or in parallel with any spring in the system, instead of only connecting two adjacent masses. To synthesize all networks comprising one inerter and more than one spring, the element extraction method is used to obtain the general form of admittance function. Results and Conclusion We find that after adding one inerter into one part of a spring network <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42417_2025_1768_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$k_i$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>k</mi> <mi>i</mi> </msub> </math></EquationSource> </InlineEquation>, the number of natural frequencies increases by one. Besides, the relation of the natural frequencies of the system after adding an inerter (the new system) and the system before adding an inerter (the original system) has and only has two different forms, which is determined by whether or not the subsystems on the left and right sides of the spring network <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42417_2025_1768_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$k_i$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>k</mi> <mi>i</mi> </msub> </math></EquationSource> </InlineEquation> have common natural frequencies. Sensitivity analysis also indicates that the natural frequencies of the new system can be reduced by increasing the inertance of the added inerter. This work provides theoretical support for the changes of natural frequencies of inerter-based vibration systems, and could be potentially important for inerter-based system analysis and natural frequency assignment.

Influence of Adding One Inerter on Natural Frequencies of 1D Vibration Systems

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发表日期：

2025年2月27日

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Hongchao Li,
Michael Z. Q. Chen,
Chanying Li

摘要

Purpose

Inerter-based mechanical networks have been widely used in vibration control. However, the influence of inerter on the fundamental characteristic of vibration systems—natural frequencies, has not been sufficiently investigated. This work is devoted to analyzing the influence of adding one inerter on natural frequencies of one-dimensional (1D) vibration systems.

Methods

In correspondence with inerter-based networks used in vibration control, the way of adding one inerter could be connecting the inerter in series or in parallel with any spring in the system, instead of only connecting two adjacent masses. To synthesize all networks comprising one inerter and more than one spring, the element extraction method is used to obtain the general form of admittance function.

Results and Conclusion

We find that after adding one inerter into one part of a spring network $k_i$ $k_{i}$ , the number of natural frequencies increases by one. Besides, the relation of the natural frequencies of the system after adding an inerter (the new system) and the system before adding an inerter (the original system) has and only has two different forms, which is determined by whether or not the subsystems on the left and right sides of the spring network $k_i$ $k_{i}$ have common natural frequencies. Sensitivity analysis also indicates that the natural frequencies of the new system can be reduced by increasing the inertance of the added inerter. This work provides theoretical support for the changes of natural frequencies of inerter-based vibration systems, and could be potentially important for inerter-based system analysis and natural frequency assignment.