## 1 Introduction  
Recently, two categories of passive devices, negative stiffness mechanisms and inerters, have received great attention and have been demonstrated to be novel, effective tools in vibration suppression [1, 2]. Both have already used in passive isolators and vibration control strategies for many mechanical and structural systems. The force-displacement relationships of two devices all exhibit an apparent negative slope. The construction of a parallel connection of negative stiffness elements and positive stiffness elements can implement a relatively low dynamic stiffness but a high static stiffness for an isolator, preventing the system from large static deflection under the static load. This layout offers a so-called quasi-zero stiffness (QZS) isolator and obtains a good performance of low-frequency vibration isolation. Many researches have taken various forms of negative stiffness mechanisms in combination with positive stiffness elements to construct QZS isolators. Existing means of negative stiffness include the oblique spring mechanism [3, 4, 5], post-buckled beams, plates, shells [6, 7], scissor-like structures [8], disc springs as negative stiffness mechanisms [9], the cam-roller-spring mechanism [10, 11] and magnetic springs [12, 13]. It is worthwhile pointing out that a novel passive linear isolator was proposed in [14], and its isolation and damping properties essentially result from the stiffness elements of the system, including negative stiffness elements. Compared to the traditional tuned mass damper and QZS isolators, this novel isolator can provide comparative advantages. An inerter is an ideal mechanical two-terminal element, whoseforces applied on the two terminals are proportional to the relative accelerations of the two ends. The constant of proportionality is called inertance and is measured with a kilogram unit [15]. With the introduction of inerters, a spring-damper-inerter mechanical network can be made in complete analogy with an inductor-resistor-capacitor electrical network. Until now, the physical realizations of an inerter have mainly included the rack and pinion mechanism [16], the ball screw mechanism [17; 18], electromagnetic devices [19], and hydraulic devices [20; 21]. It should be noted that fluid inerters have several advantages over mechanical inerters: for example, they can decrease ratcheting, backlash and friction phenomena and incorporate fluid damping in response to a pressure drop. Additionally, an inerter is usually connected in series with a stiffness and/or a damper in an isolator. These isolators include, for example, tuned inerter dampers [22], and inerter-based resonant vibration absorbers and isolators [2; 23; 24]. Although the vibration isolation characteristics of QZS-based or inerter-based isolators have been studied extensively and separately, there are few studies addressing the vibration isolation performance of an isolator incorporating a QZS and an inerter simultaneously [25], to the best of the authors' knowledge. In [25], which was a short conference article, we briefly discussed the displacement transmissibility under harmonic base excitation and the effect of the isolator parameters on the natural frequencies for two linear isolators that are investigated further in this paper. These two linear isolators present two configurations of the parallel connection of quasi-zero stiffness and inerter dampers, which are modelled as a single degree of freedom (SDOF) and a two-degree of freedom system under base excitation, respectively. In this paper, the main contents that are not involved in [25] include the following: (1) The analytic expressions of the invariant point, i.e., a particular frequency, and the anti-resonant frequency arising in the displacement transmissibility curves are derived, and the damping ratio, below which the minimum of the displacement transmissibility occurs, is given for the isolator with the first configuration.