## Abstract

A common format is developed for a mass and an inerter-based resonant vibration absorber device, operating on the absolute motion and the relative motion at the location of the device, respectively. When using a resonant absorber a specific mode is targeted, but in the calibration of the device it may be important to include the effect of other non-resonant modes. The classic concept of a quasi-static correction term is here generalized to a quasi-dynamic correction with a background inertia term as well as a flexibility term. An explicit design procedure is developed, in which the background effects are included via a flexibility and an inertia coefficient, accounting for the effect of the non-resonant modes. The design procedure starts from a selected level of dynamic amplification and then determines the device parameters for an equivalent dynamic system, in which the background flexibility and inertia effects are introduced subsequently. The inclusion of background effect of the non-resonant modes leads to larger mass, stiffness and damping parameter of the device. Examples illustrate the relation between resonant absorbers based on a tuned mass or a tuned inerter element, and demonstrate the ability to attain balanced calibration of resonant absorbers also for higher modes.

## 1. Introduction

Considerations concerning resonant vibration absorbers and their ability to mitigate the vibrational response of structures date back at least to the patent of Frahm [1] from the early twentieth century, where an auxiliary absorber mass is appended to the structure via a spring. This model has since been augmented by a viscous damper to form the well-known tuned mass damper or absorber [2–6], which has been installed or proposed for damping of excessive resonant vibrations in a variety of engineering problems, such as chatter instabilities [7], flutter of bridge decks [8], optical performance of segmented mirror telescopes [9] or misaligned wind–wave loading on offshore wind turbines [10]. More recently, an internal inertia element with two element poles has been introduced as the so-called inerter [11,12], which generates an inertia force proportional to the relative acceleration between the poles. The most common realization of the inerter element relies on flywheel(s) activated by a suitable gearing mechanism using pinions or a ball-screw. The early applications of inerter devices concerned race car suspensions [12], while recently the application of inerter-based vibration absorbers for vibration control and damping of structures has been proposed, e.g. in [13,14].

Resonant vibration control relies on accurate frequency tuning with respect to the targeted vibration mode of the flexible structure. The specific tuning of vibration absorbers is therefore of particular interest, and various design procedures and explicit calibration formulae have been proposed for tuned mass absorbers. The classic fixed-point approach, dating back to Den Hartog [2,4], is based on frequency tuning to attain equal dynamic amplification at two fixed-points at which the amplification is independent of the device damping. As demonstrated by Krenk [5], this particular frequency calibration is equivalent to having equal damping of the two modes associated with the targeted vibration form of the structure. The damping parameter of the tuned mass absorber is subsequently determined to obtain a level plateau in the dynamic amplification around resonance by a condition related to the slope at the neutral frequencies [3,4] or by requiring identical dynamic amplification at the neutral frequencies and an intermediate frequency [5]. Most classic calibration formulae for the tuned mass absorber have recently been summarized by Zilletti *et al.* [15], who also demonstrate equivalence between minimizing the kinetic energy of the structure and maximizing the power outtake by the absorber. The absorber parameters can be calibrated based on the maximum decay rate associated with transient response [16], for a 2 degrees of freedom (d.f.) absorber with four neutral frequencies [17] or for tuned mass absorbers with alternative damper configurations [18]. Closed-form calibration formulae have also been derived for different types of inerter-based absorbers in Hu *et al.* [19], using the classic fixed-point calibration of Den Hartog. In the present paper, it is demonstrated that there is a unified format applying to both a device based on an inerter in series with a combined spring/damper element as well as the classic tuned mass absorber. This permits a common calibration procedure as well as a common procedure for the analysis of the resulting system of structure and absorber device.

The classic calibration procedures assume that the response of the structure can be represented by a single targeted vibration mode. This is often a valid assumption for a suitably located tuned mass absorber targeting the first vibration mode. However, for higher vibration modes or for the inerter-based absorbers, which act on relative structural motion, the influence from non-resonant vibration modes is often sufficiently large to significantly deteriorate the absorber performance. In Krenk & Høgsberg [20], the influence of non-resonant modes has been taken into account via a single background flexibility coefficient, representing the quasi-static contribution from the non-resonant modes. While this approach is typically quite accurate for the first vibration mode, it deteriorates for higher modes due to the mixed background contributions from higher and lower non-resonant modes. In the present paper, a quasi-dynamic representation of the non-resonant modes in terms of a background flexibility as well as a background inertia coefficient is derived. This representation is then developed into an explicit two-step design procedure, in which a set of equivalent absorber parameters is first obtained, and then subsequently modified for background flexibility and inertia. For a given design value of the dynamic amplification, the effect of the background flexibility and inertia is an increase of the device stiffness, mass and damping. The accuracy of the unified calibration strategy is illustrated by numerical examples, demonstrating that the desired level of damping is indeed attained and that the dynamic amplification curves exhibit a near-level plateau for both the tuned mass and inerter absorbers.

## 2. Resonant vibration absorbers on structures

Figure 1 illustrates a structure with a mounted resonant vibration absorber (*a*) showing an absorber based on the absolute motion of the point at which the absorber is mounted and (*b*) showing an absorber based on the relative displacement of the two points at which the absorber is mounted. The first type is the classic tuned mass absorber, in which a mass is suspended by a spring and a damper coupled in parallel as shown in figure 1*a*, while the second type consists of an inerter component coupled in series with a spring and a damper coupled in parallel as shown in figure 1*b*. As demonstrated in the following the two devices are governed by the same force–displacement equations, the only formal difference being that the former operates on the absolute motion of a single point of fixture, while the second operates on the relative motion of its two support points. Thus, in spite of the similarity of the governing equations, the optimal location of the two devices will typically be different.

### (a) Tuned absorber on a structure

The structure is represented by a discretized model in terms of the displacement vector **u** and the corresponding stiffness matrix **K** and mass matrix **M**. The problem is considered in the frequency domain with time dependence represented implicitly by the factor **f**_{e} represents the external load, while *f*_{d} is the magnitude of the force on the device. The connectivity vector **w** identifies the fixture points of the device. For the tuned mass device, the connectivity vector **w**=[0,…,1,…,0] has only a single unit entry, while for the inerter device it has two entries identifying a degree of freedom at the two points of fixture, corresponding to the typical form **w**=[0,…,1,−1,…,0]. The connectivity vector **w** also defines the displacement of the structure *u* at the location of the device as
*u* is the motion of the single point of fixture for the tuned mass device, and the relative motion in the case of the inerter-based device.

The device is assumed to be linear, and its frequency characteristics are contained in the relation
*H*_{d}(*ω*). When this is substituted into the equation of motion (2.1), it takes the form
*H*_{d}(*ω*)**w****w**^{T} leads to coupling between the modes. These terms limit the validity of calibration procedures based on only the resonant mode **u**_{r}. A more general, but still explicit, procedure is developed in the following based on including the effect of the non-resonant terms via quasi-dynamic flexibility and inertia terms.

### (b) Resonant absorbers with mass or inerter element

The resonant absorbers are illustrated in figure 2 showing the tuned mass absorber with a single support in figure 2*a* and the resonant absorber with an inerter element and two support points in figure 2*b*. The resonant absorber device is characterized by three components: a stiffness *k*_{d} and a viscous damper with parameter *c*_{d}, coupled in parallel and a mass element represented by an actual mass *m*_{d} in the case of the tuned mass absorber and an equivalent mass *m*_{d} created via rotation in the inerter as indicated in figure 2*b*. When introducing the equivalent mass of the inerter the two devices have identical relations between the device force *f*_{d} and the structural displacement component *u* defined in (2.2), representing a force and a displacement in the case of the tuned mass absorber, and a set of opposing forces and a relative displacement in the case of the inerter. The simplest realization of a mechanical inerter is a flywheel with moment of inertia *I*=*m*_{f}*R*^{2} activated via the relative displacement on opposing sides of an axle with radius *r* [11]. When neglecting the mass of other moving elements, energy equivalence for the rotational motion gives the equivalent device mass as
*R*/*r* provides an amplification factor on the actual rotating mass.

The tuned mass absorber shown in figure 2*a* is a single pole device in which the relevant structural displacement is defined as the absolute structural displacement *u*=**w**^{T}**u**=*u*_{j} associated with node *j* and pointing to the right in the figure. The extension of the spring/damper is defined as *u*_{d}, and the absolute displacement of the mass is then given by
*f*_{d} can be determined either as the combined forces in the spring and damper, or as the force needed to accelerate the mass,
*ω*, represented via the exponential factor *u*_{d}, leading to the force–displacement relation

In the case of the inerter-based resonant absorber shown in figure 2*b*, the relative displacement between the two locations *j* and *k* is expressed as *u*=**w**^{T}**u**=*u*_{j}−*u*_{k}. The extension of the spring/damper is denoted by *u*_{d}, whereby the relative shortening over the inerter element is *u*_{m}=*u*+*u*_{d}. Thus, the present notation leads to the same set of formulae (2.7) for the device force component *f*_{d} for the inerter-based device, and the transfer function *H*_{d}(*ω*) is given by (2.8) also for this device. It can therefore be concluded that the frequency function of the inerter-based device is equivalent to that of the tuned mass absorber, when accounting for the fact that *u* is now the shortening displacement across the device, *f*_{d} is a set of opposing forces and *m*_{d} is an equivalent mass defined in relation to relative motion. Either of the two devices enters the equations of motion of the structure via the common relation (2.4). This permits the development of a common calibration procedure for both resonant absorber devices as presented in the following.

### (c) Finite-element implementation

In a finite-element analysis of the structure including the vibration absorber, it is advantageous to include an additional degree of freedom associated with the absorber. Hereby, the combined system takes the traditional format of a stiffness matrix, a mass matrix and a damping matrix, and natural frequencies can be determined by a linear eigenvalue analysis. It is convenient to select the extra degree of freedom as *u*_{m}=*u*+*u*_{d}, corresponding to the absolute displacement of the mass of the tuned mass absorber, and to the relative motion across the inerter element in the inerter-based device.

The structural displacement vector **u** and the displacement *u*_{m} associated with the device mass element are combined into the augmented displacement vector
**C**, the resulting equation of motion takes the form

## 3. Background flexibility and inertia

The resonant vibration absorber discussed in this paper is to be mounted on a flexible structure to introduce an appropriate amount of damping into the resonant mode **u**_{r}. The device is mounted at a specific location on the structure and interacts with the structure via a local force—or a pair of forces. The local nature of the force(s) implies that it generates a response of all the modes of the structure. While the resonant absorber is represented in full in the structural model described above, the calibration procedure for the resonant device is based on a simplified model system equivalent to a tuned mass absorber mounted on a spring-supported mass via a connection that contains a flexibility and an inertia element. This section presents procedures for obtaining the model parameters of this equivalent system from a flexible structure.

### (a) Modal frequency response

For the purpose of deriving an approximate formula for the response in a frequency range around a resonance frequency *ω*_{r} of a resonant mode **u**_{r}, it is convenient to write the dynamic equation (2.1) in the form
**f** includes external as well as device forces. The response **u** can be expressed by its components in terms of the mode shape vectors **u**_{j} with natural angular frequencies *ω*_{j},
*n* is the number of degrees of freedom of the structural system. When introducing a representation of the response **u** in terms of the mode shape vectors **u**_{j}, the solution to (3.1) is found in the form
*n* terms, each with individual dependence on the frequency *ω*.

The calibration of the parameters of the resonant device makes use of the dynamic properties of the combined system of structure and device as expressed in terms of the local structural displacement *u* and the corresponding force *f*, defined by the connectivity vector **w** as
*k*_{j} is the modal flexibility
**u**_{j}/(**w**^{T}**u**_{j}), normalized to unity at the device.

In the following, approximate representations are obtained for the response relation in which the contributions from the non-resonant modes are represented in an approximate form that permits calibration of the device parameters by a modification of the classic 2 d.f. procedure developed in [4,5], without need for computing the properties of the non-resonant modes.

### (b) Quasi-static flexibility correction

If the resonant mode corresponds to the lowest natural frequency the effect of the higher modes may be included in the form of a quasi-static correction. The idea is to neglect dynamic effects in the non-resonant modes, corresponding to evaluation of these terms in the summation at the frequency *ω*=0. In order to maintain the flow of the presentation, the detailed derivation is given in appendix A. The local form of the response formula with quasi-static correction follows from the approximate full response relation (A.4) by introducing the local displacement and force components from (3.4),
*k*_{r} is the modal flexibility given by (3.6), and *u* from all non-resonant modes, following from (A.3) as

### (c) Quasi-dynamic flexibility and inertia correction

In the context of a local resonant device, the calibration takes place in a frequency range around the resonant frequency *ω*_{r}, and in order to obtain an improved correction for the effect of the non-resonant modes these must be included in a more general way than the quasi-static representation presented above. It is demonstrated in appendix A that a quasi-dynamic representation can be obtained, in which the non-resonant modes are included via a flexibility and an inertia term.

The local response relation between *u* and *f* including the quasi-dynamic contributions from the non-resonant modes is obtained by introducing the component relations (3.4) into (A.18). This leads to the approximate one-dimensional response relation
*k*_{r}, the second term represents the background flexibility, while the last term corresponds to a background inertial effect. The modal flexibility 1/*k*_{r} corresponding to a normalized mode shape vector was defined in (3.6), while the non-resonant flexibility **K**_{r} is a modified form of the original stiffness matrix **K** combining a frequency shift with removal of the mass matrix contributions corresponding to the resonant mode. This is explained in detail in appendix A, and the modified stiffness matrix **K**_{r} is given in (A.12). The scalar response equation (3.11) is obtained by reduction of the similar matrix equation (A.18). As demonstrated in the appendix the underlying matrices are positive definite, and thus the coefficients

## 4. Equivalent absorber parameters

The local effect of the non-resonant modes is illustrated in figure 3*a* for the case of a tuned mass device. The case of the tuned inerter follows by analogy, e.g. by observing that the two systems are identical, if the right pole of the inerter-based system is locked, corresponding to *U*_{k}=0. The figure shows a representation of the local background effect in terms of a spring with stiffness

The traditional calibration procedures for resonant vibration absorbers—either based on fixed-points on the amplification curve [4] or on equal modal damping via the complex poles of the characteristic polynomial [5]—are based on properties of the quartic characteristic polynomial. The introduction of an extra spring in series with the spring/damper of the device introduces an extra degree of freedom in the simple basic system and thereby raises the degree of the characteristic polynomial from four to five. In order to stay within the simple explicit design format of resonant vibration absorbers, the effect of the background flexibility is introduced via a modification of the device parameters. This idea was used in an approximate form in [20], but in the case of inerter-based devices the background flexibility typically becomes larger and an improved equivalence is derived in this section.

The first step is to combine the inertia effects from the device mass *m*_{d} and the equivalent inerter with mass *m*′_{r}, representing background deformation of the structure. The device mass *m*_{d} is the only ‘translational mass’ in the system and, therefore, the force *f*_{d} has the same value across the spring, the inerter and the combined spring/damper. As a consequence the inerter can be moved to the right side of the spring/damper element, as shown in figure 3*b*. In this configuration, the inerter mass *m*′_{r} and the device mass *m*_{d} are coupled in series, and it is a simple matter to demonstrate that the effect of these two masses can be represented in the form of an equivalent device mass of magnitude
*m*′_{d} is shown in figure 3*c*.

Finally, the background flexibility spring is absorbed into the parallel spring/damper system as illustrated in figure 3*c*. As shown in the figure, the displacement across the spring/damper of the device is *u*_{d}, while the displacement across the background flexibility spring is

The device force *f*_{d} can be expressed both by the combination of absorber spring and damper forces, and by the background spring force. This gives the following two relations:
*u*_{d} is eliminated by use of (4.3), whereby
*f*_{d} by the second expression in (4.4), whereby
*f*_{d} in terms of equivalent device stiffness and damping parameters *k*′_{d} and *c*′_{d}.

The relation is considered in the frequency domain with the time factor *f*_{d} is then determined in terms of the equivalent spring/damper extension *u*′_{d} as
*ω*^{2} are small and can be omitted. First, the device damping coefficient *c*_{d} is represented via the device damping ratio *ζ*_{d} as
*ω*^{2}_{d}=*k*_{d}/*m*_{d} for the device natural frequency *ω*_{d} that
*ω*≃*ω*_{d} and furthermore typical device damping ratios satisfy *ω*^{2} terms in (4.7) can be omitted, when considering the relation at near-resonance frequencies, as is the case for device calibration. The device force can then be expressed approximately as

## 5. Resonant absorber calibration

The design procedure for the resonant absorber is based on the similar procedure for the idealized case of an absorber interacting with a concentrated mass. The idealized case corresponds to using the modal stiffness *k*_{r} and modal mass *m*′_{d}, *k*′_{d} and *c*′_{d}. The results for the equivalent parameters are then used to obtain expressions for the actual device parameters *m*_{d}, *k*_{d} and *c*_{d}.

### (a) Calibration of equivalent absorber system

In the classic presentation of the tuned mass absorber by Den Hartog [4], dealing specifically with an equivalent 2 d.f. system, the device mass is selected and the device stiffness and the device damping ratio are then determined as function of the mass ratio expressing the ratio of device mass to the appropriate mass of the structure. In most cases, it is convenient to modify that procedure such that it starts by selecting the value of the amplification factor of the resonant mode, translates the amplification factor into an equivalent damping ratio and then proceeds to determination of the device mass and the device stiffness [5,20]. Here, the results are first summarized in terms of the mass ratio, and then reverted to provide the design flow starting from a desired dynamic amplification factor.

In the equivalent system with a spring supported mass equipped with an equivalent tuned mass absorber as illustrated in figure 3*c*, the response of the optimally calibrated system exhibits a dynamic amplification curve as shown in figure 4 (see [5,20] for details). The mass ratio of the equivalent system is defined as the ratio of the equivalent device mass *m*′_{d} to the structural mass, here represented by the modal mass *m*_{r} of the resonant mode,

In a design situation, it is typically more convenient to start from the dynamic amplification of the equivalent system shown in figure 4. When introducing a tuned mass absorber the original resonant mode splits into two complex-valued modes with equal damping ratio *ζ*_{r1} and *ζ*_{r2}. It has been demonstrated in [5] that the frequency tuning (5.2) of an idealized 2 d.f. model of a structure with a tuned mass absorber leads to identical damping of these two modes, and furthermore that this common modal damping ratio is closely equal to the device damping ratio. In the present notation, this amounts to *ζ*_{s}, the device damping ratio can be determined from the dynamic amplification factor as
*ω*′_{d} given by (5.2).

### (b) From equivalent to actual device parameters

If there were no effect of the background flexibility of the structure, the primed device parameters would be the actual parameters and the device calibration would be completed. However, in the presence of background flexibility effects the actual device parameters must be identified from the parameters of the equivalent system by use of the relations derived in §4. For this purpose, the background flexibility and inertial effects are described by the two non-dimensional parameters

The first device parameter to be determined is the absorber mass *m*_{d}, characterized by the non-dimensional mass ratio
*m*_{d} and the absorber mass in the equivalent system was given in (4.2), conveniently written as
*μ* in terms of the equivalent mass ratio and the inertia factor as
*μ*′ is the mass ratio (5.7) corresponding to an ideal structure without flexibility effects. It is observed that when using the quasi-dynamic correction procedure the background flexibility represented via an extra inertial term always leads to a larger device mass, *μ*>*μ*′. A quasi-static calibration procedure would leave the device mass unchanged. It is also observed that the relation (5.12) imposes the upper limit

The device stiffness *k*_{d} is determined from the relation (4.11*a*) between the relative and the actual stiffness. By rearrangement of the background correction factor the following relation is obtained:
*μ*′ and the flexibility coefficient

The natural frequency of the device follows from the device stiffness (5.15) and the mass ratio (5.12) as:

It is convenient to express the device damping coefficient *c*_{d} as normalized with the geometric stiffness of the structural mode. Hereby, relation (4.11*b*) between the equivalent device damping *c*′_{d} and the actual damping coefficient *c*_{d} takes the form
*κ*_{d}) can be identified as the first factor in (5.15), the equivalent stiffness ratio is given by (5.3), and the equivalent damping ratio by (5.4). When combining these substitutions, the device damping ratio is given in the normalized form
*c*_{d}. The corresponding device damping ratio is found by similar manipulations of the previous formulae as

This completes the derivation of the formulae governing a tuned mass or inerter absorber. The actual calibration is quite simple. Start with selecting a suitable equivalent device damping ratio, e.g. from a design condition on the dynamic amplification as given in (5.6). Then calculate the mass ratio *μ*′ of an equivalent idealized system by (5.7). The structure is characterized by the modal stiffness *k*_{r} and natural frequency *ω*_{r}, and the two background flexibility and inertia coefficients *m*_{d} by (5.12), the device stiffness *k*_{d} by (5.15) and finally the device damping parameter *c*_{d} by (5.18).

## 6. Examples

The following two examples illustrate the design procedure for the tuned mass absorber and a similar tuned inerter absorber. The first example uses a simplified shear building to illustrate the role of the placement of the resonant absorber, and to compare design without correction for non-resonant modes with design including a quasi-static background correction, and design including quasi-dynamic correction with background flexibility and inertia. The second example uses a cantilever beam to demonstrate the role of the quasi-dynamic correction terms when targeting a higher mode—here illustrated by the third mode. The examples demonstrate the design procedure with quasi-dynamic correction to be robust and leading to balanced modal damping even in cases with high level of background correction.

### (a) Effect of placement on shear frame

This example investigates the calibration and effect of a tuned mass or a tuned inerter absorber acting on a 10-storey shear frame structure with transverse floor displacements, as illustrated in figure 1. This generic structure is suitable for demonstrating the influence of non-resonant vibration modes because of the rather closely spaced natural frequencies, with an almost odd-integer scaling of the lowest modes: *ω*_{1}, *ω*_{2}≃3*ω*_{1}, *ω*_{3}≃5*ω*_{1}. The first three vibration modes of the shear frame are shown in figure 5. The values to the left of each vibration mode represent the modal mass *m*_{j} associated with a mass absorber attached to that particular floor, while the corresponding modal masses for the inerter-based absorber placed at the individual storeys between floor and ceiling are given to the right. A small modal mass corresponds to large absorber authority and therefore good absorber placement. It is seen that for all three modes the top floor (*n*_{d}=10) is a suitable location for the tuned mass absorber, while the bottom storey (*n*_{d}=1) is quite good although not fully optimal for the tuned inerter absorber. It follows from figure 5 that the floor 10 modal mass increases with the mode number, while modal mass for the bottom storey location decreases. This indicates that the tuned mass damper will be particularly influenced by mode shapes below the targeted vibration form, while the tuned inerter absorber will be more sensitive with respect to the higher modes.

The following three calibration procedures are investigated for both the tuned mass absorber and the tuned inerter absorber: without correction for non-resonant modes, quasi-static correction with background flexibility 1/*k*_{0}′ defined in (3.8), and quasi-dynamic correction with background flexibility 1/*k*′ and background inertia 1/*k*′′ defined in (3.10) and (3.11), respectively. The results of the parameter calibration and the damping ratios are presented in table 1 for each of the first three vibration modes (*r*=1,2,3) as the targeted mode with the tuned mass absorber results in the upper half and the tuned inerter results in the lower half. The design dynamic amplification is *u*_{dyn}/*u*_{stat}=10, and (5.7) gives the equivalent mass ratio *μ*′=0.0204.

It is seen that for the lowest vibration mode calibration without correction leads to almost equal modal damping, while a slight unbalance appears for modes 2 and 3. For the calibration with quasi-static correction, the relative magnitude of the background flexibility *k*_{r}/*k*_{0}′ is given in the table. While excellent for the lowest mode this procedure deteriorates for modes 2 and 3, because the quasi-static correction does not include the inertia-based influence from the lower mode(s). However, the quasi-dynamic calibration with both flexibility *k*_{r}/*k*_{0}′ secures balanced damping for the first vibration form and a slight detuning for the higher modes. A consistently balanced calibration with almost equal damping for all vibration forms is again obtained by the quasi-dynamic correction with a large flexibility coefficient

The frequency response properties are illustrated in figure 6 in terms of the dynamic amplification of the shear frame top floor (*a*,*b*) and the relative absorber displacement (*c*,*d*). The results in figures (*a*,*c*) represent the tuned mass absorber, while the tuned inerter absorber is shown in (*b*,*d*). The curves in each sub-figure represent the three calibration procedures without background correction (dashed line), with quasi-static flexibility (dashed-dotted line), and with quasi-dynamic flexibility and inertia (solid line). The external load excites the second vibration mode by a spatial distribution that by construction is orthogonal to the other modes. Thus, any modal interaction arises only due to the presence of the absorber and the results in figure 6 therefore verify the free vibration properties presented in table 1. For example, in the case of the tuned mass absorber (*a*,*c*) the response curves without correction and with quasi-dynamic correction are almost identical, while the quasi-static correction (dashed-dotted line) leads to a significant increase in dynamic amplification at resonance (|*u*|/*u*^{0}≃27.7) and a notable increase in the corresponding absorber amplitude. For the tuned inerter damper in figure 6*b*,*d*, it is instead the calibration without correction (dashed line) that is unable to retain the desired flat plateau around resonance. Finally, it is observed that for the tuned mass damper in (*a*,*c*) it is the low-frequency peak that is too lightly damped, while for the tuned inerter absorber it is the other (higher frequency) peak that dominates. These findings correspond with the results in table 1 for the damping ratios, where the second last column represents the damping ratio for the mode with lowest frequency, while the last column gives the damping ratio for the other mode with higher frequency.

### (b) Resonant damping of higher mode

The influence of the absorber location is investigated by targeting the third vibration mode of a cantilever beam of length ℓ=20*a* with rigid support at the left end. The beam is discretized by 20 Bernoulli beam elements of length *a* with transverse displacement and cross-sectional rotation as the two nodal degrees of freedom. The third vibration form of the cantilever is shown in figure 7*a* with the dots indicating the individual nodes. The tuned mass absorber acts on the transverse displacement of a single node, while the tuned inerter absorber acts on the curvature of the beam represented by the increment in rotation between two adjacent nodes. The curvature of the discretized beam model is shown in figure 7*b*. It is seen that the maximum displacement occurs at the free end of the beam, while the maximum curvature is attained at the fixed support. However, because of the near-harmonic shape of the third vibration mode, two intermediate nodes at 0.3ℓ and 0.7ℓ represent extremes in both displacement and curvature, and these individual locations are therefore used to illustrate the influence of the absorber placement. The tuned inerter absorber covers a single-beam element at either element *n*_{d}=6 or 14 (thick line), while the tuned mass absorber is placed at the right node of the respective elements (square marker). The elongation *u* of the inerter absorber is represented by the change in cross-sectional rotation over the element via the connectivity vector **w**^{T}=[0…,0,−*a*,0,*a*,…0] with offset *a* relative to the neutral beam axis.

The calibration results and attainable damping ratios associated with the third vibration mode are summarized in table 2 for both absorber types and absorber locations. For the tuned mass absorber in the top half of the table, the calibration without correction gives fairly balanced damping of the two modes. However, the quasi-static correction parameter *k*_{r}/*k*′_{0} becomes so large that it leads to severe detuning and for the outer location at *n*_{d}=14, it even gives negative absorber parameters and therefore inconsistent results. For the quasi-dynamic correction with background flexibility *k*_{r}/*k*′_{0} is seen to increase the attainable modal damping. As for the tuned mass absorber the full quasi-dynamic correction provides the best results in terms of equal modal damping, with only a small difference between the two mode-three damping ratios.

The frequency response properties are investigated by assuming an external harmonic loading on the structure with spatial distribution orthogonal to the non-resonant modes of the discretized beam model. Figures 8 and 9 show the dynamic amplification of the beam tip and the frequency amplitude of the relative absorber displacement for the absorber locations *n*_{d}=6 and 14, respectively. The line styles refer to the three correction strategies previously used in figure 6 for the shear frame. For the tuned mass absorber (*a*,*c*) the quasi-static correction (dashed-dotted line) leads to significant resonant peaks, in particular for *n*_{d}=14 in figure 9, while for the tuned inerter absorber (*b*,*d*) it is the calibration without correction (dashed line) that results in a substantial increase in vibration amplitude. The quasi-dynamic calibration with both background flexibility

## 7. Conclusion

A unified format has been developed for the calibration and analysis of the effect of mounting a resonant vibration absorber on a structure. A unified format is identified for the classic tuned mass absorber and a similar type of vibration absorber based on an inerter element. While the tuned mass absorber relies on the absolute motion of a mass element suspended by a parallel spring/damper, the corresponding inerter based device is governed by the relative motion of two points on the structure. Thus, while the basic theory is the same, the optimal location of the device and the specifics of the resulting parameters may differ considerably.

An explicit calibration procedure for damping of a targeted mode has been developed within this common format for the two types of resonant vibration absorber. In this procedure, the effect of the non-resonant modes is approximated by a quasi-dynamic representation. The quasi-dynamic background effect is equivalent to introducing an extra flexibility and an inertial element in the connection between the structure and the device, and the effect of this is a device with larger mass, stiffness and damping parameter than that resulting from a classic design based on a single mode. The calibration procedure consists of two parts: a rather traditional calibration of an idealized dynamic system based on a single mode, followed by a modification of the device parameters in which the background flexibility and inertia effects are introduced via two non-dimensional parameters, characterizing the deformation characteristics of the structure at the location of the device. The representation of the effect of the non-resonant modes by a quasi-dynamic approximation in terms of a flexibility parameter as well as an inertia parameter leads to smaller values of each of the corrections, and renders this approximation more representative as well as more robust than use of the quasi-static representation in terms of a single background flexibility parameter. The examples demonstrate that the present design procedure leads to balanced dynamic amplification curves with amplification levels as initially prescribed in the design requirement, and also enables balanced damping of higher modes.

Although not treated explicitly in this paper, an earlier study of multiple resonant dampers targeting different modes suggests that a near-balanced calibration can be obtained by a two-step procedure in which the first step calibrates all resonant dampers as if acting alone, while the second step adjusts the modal properties by including the masses and springs of the devices as estimated by the first step [21].

## Authors' contributions

S.K. conceived the basic concept and wrote the first draft. J.H. wrote first draft of the introduction with background references, and designed and wrote the examples. The final paper was reworked by S.K. and J.H.

## Competing interests

We declare we have no competing interests.

## Funding

We received no funding for this study.

## Acknowledgements

The work has been carried out as free research by the authors.

## Appendix A. Modal response with quasi-dynamic contributions

The purpose of this appendix is to give a concise derivation of two approximate representations of the frequency response to the dynamic equation
**u**_{r}.

**(a) Quasi-static flexibility correction**

If the resonant mode corresponds to the lowest natural frequency the effect of the higher modes may be included in the form of a quasi-static correction. The idea is to neglect dynamic effects in the non-resonant modes, corresponding to evaluation of these terms in the summation of the full solution (3.3) at the frequency *ω*=0. This leads to the approximate formula
*ω*_{r}. An important point in the procedure is the evaluation of the sum without computing the non-resonant mode shape vectors.

The static limit *ω*=0 gives an expansion of the inverse stiffness matrix in terms of the mode shape vectors

**(b) Quasi-dynamic flexibility and inertia correction**

In the context of a local resonant device, the calibration takes place in a frequency range around the resonant frequency *ω*_{r}, and in order to obtain an improved correction for the effect of the non-resonant modes these must be included in a more general way than the quasi-static representation presented above. First, the modal response formula (3.3) is written as the sum of the response of the resonant mode and a sum of the response of the non-resonant modes
*ω*_{r} in a form that can be evaluated without computing the non-resonant mode shapes and frequencies.

It is now demonstrated, how simultaneous introduction of a flexibility and an inertia term leads to a consistent dynamic background correction for representing the effect of the non-resonant modes. To this purpose, the frequency-dependent factors in the non-resonant terms are approximated in the form
*A*_{j} and *B*_{j} are determined by the conditions that the approximation gives the correct value and the correct derivative with respect to *ω*^{2} at the resonant frequency *ω*_{r}. The derivative condition follows from (A.6) in the form
*ω*=*ω*_{r}, this determines the parameter *B*_{j} as
*B*_{j} is always positive. With this expression for *B*_{j} direct matching of (A.6) at *ω*=*ω*_{r} gives the remaining parameter as
*A*_{j} is always positive.

Upon substitution of the approximation (A.6) with parameters *A*_{j} and *B*_{j} given in (A.9) and (A.8), respectively, into the non-resonant terms of the response formula (A.5), the following approximate response formula is obtained
**u**_{r} plus a flexibility contribution and an inertia contribution from each of the non-resonant modes.

Both the flexibility term and the inertia term from the non-resonant modes can be evaluated directly from matrices of the system, without the need to compute the non-resonant modes. The specific formulae are obtained by using a modified form of the classic modal response formula (3.3). In order to avoid the singularity at *ω*=*ω*_{r}, the mass matrix is modified by subtracting the contribution associated with the resonant mode **u**_{r},
**M**_{r} is symmetric and has the property that **M**_{r}**u**_{r}=**0**, while the similar product for any non-resonant mode is unaffected by the extra term in **M**_{r}. A modified stiffness matrix is now introduced as
**M**_{r}. The result follows from the response formula (A.5) when observing that now the mass of the resonant mode has been removed, and the frequency *ω* is replaced by *ω*_{r},

The coefficients in the desired representation (A.10) are fractions with numerator and denominator in the form of quartic expressions of the system frequencies. This suggests that the summation can be obtained by using the product form **K** has the modal representation
*j*≠*k*. This expansion determines the first term in the summation in (A.10). The second term in the summation, proportional to (*ω*_{r}/*ω*)^{2}, is determined by a simple reformulation of the frequency term in (A.15). By adding and subtracting

The approximate response relation (A.10) with quasi-dynamic flexibility and inertia correction can now be expressed in direct form by eliminating the summations by use of (A.15) and (A.17),
*u* and *f* is obtained by introducing the component relations (3.4) into (A.18). It follows from the positive definiteness of the matrices in the square brackets in (A.18) that the resulting coefficients are positive.

- Received October 16, 2015.
- Accepted December 15, 2015.

- © 2016 The Author(s)