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Purpose With the development of the modern technology and aerospace industry, the noise pollution is remarkably affecting people’s daily life and has been become a serious issue. Therefore, it is the most important task to develop efficient sound attenuation barriers, especially for the low-frequency audible range. However, low-frequency sound attenuation is usually difficult to achieve for the constraints of the conventional mass-density law of sound transmission. The traditional acoustic materials are reasonably effective at high frequency range. This paper aims to discuss this issue. Design/methodology/approach Membrane-type local resonant acoustic metamaterial is an ideal low-frequency sound insulation material for its structure is simple and lightweight. In this paper, the finite element method is used to study the low-frequency sound insulation performances of the coupled-membrane type acoustic metamaterial (CMAM). It consists of two identical tensioned circular membranes with fixed boundary. The upper membrane is decorated by a rigid platelet attached to the center. The sublayer membrane is attached with two weights, a central rigid platelet and a concentric ring with inner radius e. The influences of the distribution and number of the attached mass, also asymmetric structure on the acoustic attenuation characteristics of the CMAM, are discussed. Findings In this paper, the acoustic performance of asymmetric coupled-membrane metamaterial structure is discussed. The influences of mass number, the symmetric and asymmetry structure on the sound insulation performance are analyzed. It is shown that increasing the number of mass attached on membrane, structure exhibits low-frequency and multi-frequency acoustic insulation phenomenon. Compared with the symmetrical structure, asymmetric structure shows the characteristics of lightweight and multi-frequency sound insulation, and the sound insulation performance can be tuned by adjusting the distribution mode and location of mass blocks. Originality/value Membrane-type local resonant acoustic metamaterial is an ideal low-frequency sound insulation material for its structure is simple and lightweight. How to effectively broaden the acoustic attenuation band at low frequency is still a problem. But most of researchers focus on symmetric structures. In this study, the asymmetric coupled-membrane acoustic metamaterial structure is examined. It is demonstrated that the asymmetric structure has better sound insulation performances than symmetric structure.