Fractional derivatives are useful for describing the occurrence of vibrations in engineering practice. The studies involving fractional calculus and its applications to mechanical problems appear widely in different studies [1]. The advances in fractional calculus focus on modern examples in differential and integral equations, physics, signal processing, fluid mechanics, viscoelasticity, mathematical biology and electrochemistry [2].

The general solution procedure including all the problems instead of separately solving each problem is quite advantageous. Many different linear or nonlinear models addressing vibrations of continuum appear in the literature. Some of these works are as follows: Pakdemirli [3] developed a general operator technique to analyse the vibrations of a continuous system with an arbitrary number of coupled differential equations. Özhan and Pakdemirli [4–6] suggested the general solution procedure to investigate a more general class of continuous systems such as gyroscopic and viscoelastic systems. Ghayesh *et al.* [7] considered a general solution procedure for the vibrations of systems with cubic nonlinearities subjected to nonlinear and time-dependent boundary conditions. Hence, a general solution is adapted to solve the dynamic problems constituting continuum.

In recent years, there has been a growing interest in the area of fractional variational calculus and its applications [8, 9]. Fractional calculus, which is used successfully in various fields such as mathematics, science and engineering, is one of the generalisations of classical calculus. The merits of using a fractional differential operator lie in the fact that few parameters are needed to accurately describe the constitutive law of damping materials [10]. Bagley and Calico [11] modelled the mechanical properties of damping materials by fractional order time derivatives. The mechanical scientific community recognised the significance of fractional calculus for modelling viscoelastic material behaviour thanks to Bagley *et al.* [12]. Also, they studied longitudinal vibrations of rods and flexural vibrations of beams based on viscoelastic fractional derivative models [13]. For solving dynamic problems with a fractional derivative, the analysis of free damped vibrations of various mechanical systems, whose behaviour is described by linear viscoelastic models with fractional derivatives, were studied by Rossikhin and Shitikova [14]. Mainardi [15] considered the problems in continuum mechanics related to mathematical modelling of viscoelastic bodies. Cooke *et al.* [16] investigated the response of a viscoelastic beam with a fractional derivative. Skaar *et al.* [17] used a fractional standard linear solid model. French and Rogers [18] presented a small group of structural dynamics problems for which fractional calculus was adopted.

The general solution allows one to investigate the effects on a dynamic analysis of continuum whose damping term is modelled by a fractional derivative. An engineering problem which is a special application of the general model developed in this study was formerly considered in [19]. In our previous study [20], the analysis of primary and parametric resonance for the external excitation term having *ε*-order was performed. As the forced term is obtained in one-order, sum or difference type of resonance also appears in the present model. The method of multiple scales is used in the analysis. Thus, the amplitude and phase modulation equations are produced in terms of operators. In addition, the variations of the curves with respect to the dimensionless parameters are presented. Finally, the effects of fractional damping on the linear vibrations of continuum are investigated in detail.