How internal vibro-impact nonlinearities yield enhanced vibration mitigation

Abstract

We study a fully passive and robust mechanism to mitigate the transient response of resonantly base-excited single- and multi-DOF structures by drastically reducing the portion of input energy that enters into the structures from base excitations. Unlike traditional approaches which are essentially resonance-driven and rely on adding either linear or nonlinear, typically dissipative, secondary attachment(s) to a main (primary) structure, the presented approach is based on incorporating intentional strong non-smooth nonlinearities due to Hertzian interactions. We show that these nonlinear interactions effectively “stiffen” the system and drive the structural response out of resonance by breaking the monotonic increase of input energy for the resonantly excited structure. To elucidate this mechanism, first we consider a resonantly base-excited single-floor frame structure incorporating a single-sided clearance. Subsequently the study is extended to a three-floor frame structure with an internal flexible core, featuring double-sided asymmetrical clearances, subjected to resonant excitation by ground acceleration. Three distinct structural configurations are considered, namely, a linear model serving as a reference for the unprotected structure, and two nonlinear configurations incorporating purely elastic and inelastic Hertzian contacts. Moreover, the influence of the type of contact stiffness magnitude is explored across a wide range, spanning from soft to stiff Hertzian contacts. For both the single- and three-floor structures, the results reveal an unexpectedly favorable regime of contact stiffness, characterized by multiple beneficial attributes, wherein soft, and even purely elastic, contacts exhibit a significant reduction in both input energy and acceleration levels along with an enhancement in the dissipation rate compared to the linear case. It was observed that the occurrence of contacts facilitates energy transfer to and from the primary structure, acting as a mechanism to drive the system out of resonance, and disrupting the monotonic increase in input energy when the base excitation is applied. As the contact stiffness increases, the amount of input energy further decreases and the dissipation rate is enhanced; however, this mitigation performance is accompanied by a modest rise in floor acceleration levels attributed to the stiff impacts. Surprisingly, it was found that for a fixed contact stiffness, the amount of input energy increases when inelasticity in the Hertzian contacts is added. The present study underscores the importance of intentional strong nonlinearities in structural vibration mitigation. Further applications of this concept are discussed.

1 Introduction

Mitigating the undesired transient structural response of engineered structures subjected to base excitation, such as ground motion (e.g., earthquakes), has been a longstanding concern for engineers and scientists in the field of seismic or base isolation. The primary provisions are to minimize hazards to life, avoid possible damage due to excessive and repeated motion, and increase the expected performance of a building [1,2,3,4,5,6]. When a structure is subjected to ground motion, energy is directly imparted into the structure within a certain duration. Throughout this interval, a portion of the instantaneous input energy is temporarily stored within the structure in the form of mechanical energy (i.e., the sum of kinetic and potential energies), while the remaining energy is dissipated, either through the inherent viscous damping of the structural modes, or through inelastic deformations. Eventually, the entirety of the introduced energy to the structure should dissipate. Well designed and constructed structures are expected to rapidly absorb and dissipate the imparted energy during ground motion, particularly under severe excitations which are typically characterized by monotonic increase of input energy, without resulting in any loss of life and minimizing potential damage.

To mitigate the effects of undesired transient responses, passive, semi-active, and fully active innovative structural protection devices have been proposed, developed, and implemented. The primary aim of such devices is to absorb or reflect a portion of the input energy, not by the structure itself, but by an auxiliary protective system. This concept is based on the following instantaneous energy balance relation [7, 8],

$$ E_{m} + E_{h} + E_{d} = E_{I} $$

(1)

where ${E}_{I}$ is the total input energy, ${E}_{m}$ the mechanical energy (i.e., the summation of the kinetic and the recoverable potential energies), ${E}_{h}$ the nonrecoverable energy dissipated by the structure through material hysteresis or other intrinsic dissipative effects, and ${E}_{d}$ the energy dissipated by the protective system. Among these strategies, fully passive protective systems are commonly favored due to their cost-effectiveness, reliability, durability, maintainability, and lower power requirements compared to systems involving arrays of sensors and actuators that dynamically adjust the system’s properties in real-time. This preference limits the widespread use of semi- and fully active protective devices. However, fully passive devices typically are not robust when faced with significant variability in loading characteristics.

Passive protective devices are employed to absorb and locally dissipate energy to reduce the transient structural response. Such devices can be classified according to the nature of their operation, namely, linear or nonlinear isolators. Among many linear passive devices, base isolation of low-rise and mid-rise buildings [5, 6, 9,10,11], and tuned mass dampers in tall buildings [12,13,14,15] are used as vibration absorbers. Despite the simplicity of these linear passive devices, their effectiveness depends on the proximity of the absorber’s natural frequency to the excitation frequency, meaning that they are most effective near resonances when the structure is subjected to persistent narrowband excitations. To overcome the narrowband limitation of the linear passive devices, a category of relatively lightweight, dissipative absorbers known as nonlinear energy sinks (NESs), exhibiting essential (i.e., non-linearizable) nonlinearity in their stiffness characteristic, has been suggested [16,17,18,19]. These NESs have been successfully applied in diverse linear and nonlinear structures, providing fully passive, broadband mitigation that competes with the effectiveness of active and semi-active protective systems through targeted energy transfer (TET), i.e., one-way (almost irreversible) directed transfer of energy from the primary structure to the nonlinear dissipative attachment (i.e., the NES) [20].

Distinguished from traditional linear absorbers, these devices employ essentially nonlinear spring components that are intentionally introduced at the design stage. These strong nonlinearities are an important prerequisite for the formation of isolated or cascades of nonlinear transient resonance captures, enabling the NES to passively absorb and dissipate vibration energy from the primary structure in a broadband fashion. Interesting peculiarities of NESs, as passive broadband nonlinear absorbers, offer great potential for a passive strategy aimed at response mitigation, with many documented applications and experimental realizations on multiple platforms across scales [16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32]. Recently, the prospective research venues concerning TET in dynamical and acoustical systems were briefly outlined and discussed in [33].

However, conventional passive protective strategies, whether linear or nonlinear, are predominantly resonance-driven approaches, facilitated by gradual modulations of modal amplitudes, and operate on a relatively slow time scale (since “resonance takes time”). This limits their efficiency for mitigating undesired vibrations in applications involving extreme loads where the mitigation must be rapid, and effective even at the early response cycles. This is particularly significant in the initial stage of the response when the system’s energy is at its peak, and the risk of structural damage is most pronounced. In addition, these strategies necessitate the addition of a secondary attachment, typically dissipative, to the primary linear system for response mitigation. This introduces additional degrees of freedom to the primary structure, expanding the complexity of the nonlinear dynamics. As a result, dealing with intricate and unpredictable transient dynamics, such as chaotic transients, becomes common, often resulting in challenges related to control and predictive accuracy.

The authors recently demonstrated that achieving the TET is not solely reliant on nonlinear resonance; rather, it can also be realized through a non-resonant mechanism that incorporates fast-scale and intense non-smooth effects, such as vibro-impacts [34]. This non-resonant mechanism is based on rapid nonlinear scattering of the applied energy from low- to high-frequency structural modes, which is referred to as intermodal targeted energy transfer (IMTET). It is worth noting that in traditional TET-based NES designs, energy is transferred (redistributed) within the physical space between the primary structure and the NES(s), while IMTET relies on transferring (redistributing) energy within the modal space of the structure itself. The IMTET mechanism was first demonstrated for rapid mitigation of the transient response of a blast-excited two-DOF linear system with a single-sided clearance in [34]. Thereafter, this concept was extended to passively mitigate the transient response of a blast-excited nine-floor steel structure modified by installing an internal rigid core structure with specified clearance distribution along its floors [35]. In addition, the IMTET was experimentally demonstrated in an impulse-excited cantilever beam system with vibro-impacts [36]. Recently, the IMTET concept was implemented for passive seismic protection of a benchmark large-scale model of a twenty-floor steel building subjected to strong historical earthquakes [37]. An optimization study based on a multi-objective genetic algorithm (MOGA) was performed to improve the IMTET performance [35, 37]. The implementation of IMTET in the fields of blast [35] and seismic [37] mitigation brings forth substantial benefits, including a radical enhancement in the energy dissipation rate (i.e., the dissipative capacity) of the structure itself. This improvement is attributed to the active engagement of a broader spectrum of high-frequency vibration modes in the system’s response, coupled with a significant reduction in the overall level of system vibration. Noteworthy is the IMTET mechanism’s ability to passively redistribute energy within the modal space without introducing extra degrees of freedom, additional mass, or damping to the primary structure. This achievement is realized through the intentional incorporation of strong non-smooth nonlinearity. In contrast to the traditional passive protective strategies, either linear (e.g., tuned mass dampers—TMDs) or nonlinear (e.g., NES), that essentially rely on adding either linear or nonlinear, typically dissipative, secondary attachments to the primary structure. However, as reported in [35, 37], the remarkable mitigation performance of IMTET was accompanied by moderate increases in the floor accelerations and local stresses. Elevated acceleration levels pose risks of structural damage and fatigue failure, the latter being particularly concerning for structures subjected to cyclic loading. Hence, there is a critical need to reduce acceleration levels. Moreover, as the IMTET concept necessitates a minimum of two structural modes, its application is considered impractical for a single degree of freedom (SDOF) system. The SDOF system serves as a simplified but widely used model in engineering and physics to effectively represent the dynamical behavior of certain systems. For example, it is possible to estimate with sufficient accuracy the input energy to a multi-floor building from the SDOF input energy spectra using the fundamental period of the multi-floor structure [8].

In [37], it was reported that intentionally incorporating strong vibro-impacts led to a substantial decrease in the seismic input energy imparted into a twenty-floor building incorporating a flexible internal core compared to the linear case. Inspired by this observation and aiming to address the limitations related to the implementation of the IMTET concept, the present study explores in detail the implementation of non-smooth nonlinearity, in the form of Hertzian contacts, on mitigating the transient response of single- and multi-floor frame structures resonantly excited at the base. Emphasizing our objective to gain a deeper understanding of reducing input energy through the internal vibro-impacts, we opt for simpler structures. This is achieved by passively and effectively limiting the amount of input energy imparted into the structure during the base excitation, while simultaneously maintaining low acceleration levels, and enhanced dissipative rate compared to the linear unprotected structure. To this end, this work is structured as follows. Section 2 is devoted to the exploration of the single-floor frame structure with a single-sided clearance. Then, the study is extended to a multi-floor frame structure with an internal flexible core, featuring double-sided asymmetrical clearances, and demonstrated for a three-floor model in Sect. 3. Finally, concluding remarks and some avenues for future research are discussed in Sect. 4.

2 Single floor system incorporating a single-sided vibro-impact

2.1 Model description

We consider a single-floor system with mass $m$, stiffness $k$, and viscous damping $c$. The floor is considered infinitely rigid, i.e., a shear-type frame, and the structure is subjected to base excitation (ground acceleration), denoted by ${d}^{2}{u}_{g}\left(t\right)/d{t}^{2}$, along the horizontal direction. As a result, the structure vibrates translationally along the loading direction only, with relative displacement of $u\left(t\right)$ with respect to the ground. The symmetry of the system is intentionally perturbed by introducing a non-smooth nonlinearity in the form of a single-sided barrier with clearance gap denoted by $\Delta$. The floor interacts with the barrier through a flexible impact bumper (attached to the floor) and a rigid core (attached to the ground). A schematic representation of the considered single-floor structure with a single sided clearance is shown in Fig. 1(a).

Fig. 1

Base-excited single-floor frame system with a single sided clearance (a), and the normalized (non-dimensional) ground acceleration time series with parameters $A=1, \overline{\Omega }=1, {\omega }_{n}=1, N=20$ (b)

It is assumed that the contact force between the structure and the rigid core is governed by the Hertz law together with a nonlinear viscous-elastic element, specifically the nonlinear Hunt and Crossley contact model [38] that represents energy dissipation due to inelastic contact. The equation of motion for the single-floor frame structure undergoing Hertzian contacts with the internal rigid core is expressed as follows,

$$ m\frac{{d^{2} u }}{{dt^{2} }} + c\frac{du}{{dt}} + ku + f_{NL}^{1 - story} \left( {u,\frac{du}{{dt}}} \right) = - m\frac{{d^{2} u_{g} }}{{dt^{2} }} $$

(2)

where ${f}_{NL}^{1-story}\left(u,\frac{du}{dt}\right)$ is the nonlinear contact force describing the dissipative Hertzian contact interaction, and is expressed in the form:

$$ f_{NL}^{1 - story} \left( {u,\frac{du}{{dt}}} \right) = k_{c} \left[ {u - {\Delta }} \right]_{ + }^{3/2 } \left( {1 + \frac{{3\left( {1 - r} \right)}}{{2\left. {\frac{du}{{dt}}} \right|_{{t_{c}^{ - } }} }}\frac{du}{{dt}}} \right) $$

(3)

In the expression above ${k}_{c}$ is a stiffness coefficient, $r$ a coefficient of restitution, and ${\left.\frac{du}{dt}\right|}_{{t}_{c}^{-}}$ the contact velocity of the structure just before the impact instant ${t}_{c}$. The subscript (+) indicates that only non-negative values of the arguments in the brackets should be taken into account, with zero values being assigned otherwise; this law models the possible contacts and separations between the structure (through the flexible bumper) and the rigid core and is a source of strong stiffness nonlinearity in the structural dynamics (2). To this end, it is assumed that the shape of the contact surface is a semi-sphere of radius $R$ attached to the impact bumper, whereas a plane surface represents the contact point of the rigid core. Let $E$ and $\nu$ denote the Young’s modulus and Poisson’s ratio, respectively, of the contacting semi-sphere attached to the impact bumper. Hence, the generalized stiffness coefficient becomes ${k}_{c}=4E\sqrt{R}/3\left(1-{\nu }^{2}\right)$. Without loss of generality, the ground acceleration (i.e., the base excitation) is modeled as a harmonic excitation with a finite number of cycles followed by relaxation as follows,

$$ \frac{{d^{2} u_{g} }}{{dt^{2} }} = Acos\left( {{\Omega }t} \right)\left( {H\left( t \right) - H\left( {t - NT} \right)} \right) $$

(4)

where $A$ is the excitation amplitude, $\Omega$ the excitation frequency, $H\left(t\right)$ the Heaviside function, $N$ the number of excitation cycles, and $T$ the excitation period, $T=2\pi /\Omega$.

Let ${\omega }_{n}$ denote the natural frequency of the underlying linear dynamics of Eq. (2), i.e., for the case ${f}_{NL}^{1-story}\left(u,du/dt\right)=0$. By substituting Eqs. (3) and (4) into (2), defining the non-dimensional relative displacement of the structure as $v\equiv u/\Delta$, and introducing the non-dimensional time $\tau ={\omega }_{n}t$ as a new independent variable, the equation of motion (2) can be rewritten in the following non-dimensional form:

$$ \ddot{v} + 2\zeta \dot{v} + v + {\overline{\Omega }}_{H}^{2} \left[ {v - 1} \right]_{ + }^{3/2 } \left( {1 + \frac{{3\left( {1 - r} \right)}}{{2\dot{v}^{ - } }}\dot{v}} \right) = - \ddot{v}_{g} = - Acos\left( {{\overline{\Omega }}\tau } \right)\left( {H\left( {\frac{\tau }{{\omega_{n} }}} \right) - H\left( {\frac{\tau }{{\omega_{n} }} - NT} \right)} \right) $$

(5)

where overdots denote differentiations with respect to the non-dimensional time $\tau$, ${\ddot{v}}_{g}$ represents the normalized non-dimensional ground acceleration, and the following normalizations apply:

$$ \omega_{n} = \sqrt{\frac{k}{m}} , \zeta = \frac{c}{{2\sqrt {km} }} , {\Omega }_{H} = \sqrt {\frac{{{\Delta }^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}} k_{c} }}{m}} , {\overline{\Omega }}_{H} = \frac{{{\Omega }_{H} }}{{\omega_{n} }} , {\overline{\Omega }} = \frac{{\Omega }}{{\omega_{n} }} , v_{g} = \frac{{u_{g} }}{{\Delta }} $$

(6)

The normalized Hertzian stiffness coefficient, ${\overline{\Omega } }_{H}$, characterizes the contact type of the non-smooth nonlinearity in (5), so varying this important parameter several distinct scenarios can be identified, including: (i) linear dynamics (i.e., case of no barrier) for the case ${\overline{\Omega } }_{H}\to 0$, (ii) elastic or inelastic Hertzian (flexible) contacts for the case ${0\ll \overline{\Omega } }_{H}\ll \infty$, and (iii) elastic or inelastic rigid contacts, i.e., Newtonian impacts, for the case ${\overline{\Omega } }_{H}\to \infty$. Therefore, the normalized Hertzian stiffness coefficient will serve a crucial role for understanding and analyzing the underlying dynamics governing the flow of input energy into the structure. Moreover, the effect of the coefficient of restitution $r$ on the input energy will be addressed as well. An illustration of the considered ground motion is presented in Fig. 1(b), showing a severe base excitation, that is, with excitation frequency equaling the natural frequency of the underlying linear structure.

2.2 Deriving the instantaneous and total input energies

To investigate the effects of non-smooth nonlinearity and the coefficient of restitution on the flow of input energy into the system from base excitation, we initially consider the case of single-sided Hertzian contacts. This analysis incorporates the effects on energy dissipation by inelastic vibro-impacts, besides the inherent viscous damping of the structure. Subsequently, we derive two special cases, namely, a linear model, i.e., with no barrier, which serves as a reference model for an unprotected structure, and the case of single-sided elastic Hertzian contacts, emphasizing the role of non-smooth nonlinearity. The flow of input energy will be formulated for each of these structural configurations.

Integrating Eq. (5) with respect to the non-dimensional relative displacement $v$ and using the relationship $dv=\dot{v}d\tau$, provides:

$$ \mathop \smallint \limits_{0}^{\tau } \ddot{v}\dot{v}d\tau + 2\zeta \mathop \smallint \limits_{0}^{\tau } \dot{v}^{2} d\tau + \mathop \smallint \limits_{0}^{\tau } v\dot{v}d\tau + \mathop \smallint \limits_{0}^{\tau } {\overline{\Omega }}_{H}^{2} \left[ {v - 1} \right]_{ + }^{3/2 } \left( {1 + \frac{{3\left( {1 - r} \right)}}{{2\dot{v}^{ - } }}\dot{v}} \right)\dot{v}d\tau = - \mathop \smallint \limits_{0}^{\tau } \ddot{v}_{g} \dot{v}d\tau $$

(7)

Equation (7) represents the instantaneous energy balance of the nonlinear single-floor model with single-sided inelastic Hertzian contacts based on the relative motion of the store, which can be further expressed as,

$$ \begin{aligned} E_{k} \left( \tau \right) + & E_{d} \left( \tau \right) + E_{p} \left( \tau \right) + W_{{Hertz}}^{{NC}} \left( \tau \right) \\ & = W_{{ext}} \left( \tau \right) + W_{{Hertz}}^{C} \left( \tau \right) \\ \end{aligned} $$

(8)

where ${E}_{k}\left(\tau \right)$ is the instantaneous kinetic energy, ${E}_{d}\left(\tau \right)$ the cumulative dissipative energy (i.e., energy dissipated by viscous damping), ${E}_{p}\left(\tau \right)$ the instantaneous potential energy, ${W}_{Hertz}^{C}\left(\tau \right)$ the potential energy associated with Hertzian contacts, i.e., the work performed by the elastic component of the Hertzian contact force, ${W}_{Hertz}^{NC}\left(\tau \right)$ the work performed by the nonlinear viscoelastic element of the Hertzian contact force, and ${W}_{ext}\left(\tau \right)$ the work done by the external base excitation (i.e., the ground acceleration). Assuming that the structure is initially at rest, i.e., $\dot{v}\left(\tau =0\right)=v\left(\tau =0\right)=0$, the instantaneous energy components in Eq. (8) can be expressed as follows:

$$ \begin{aligned} E_{k} \left( \tau \right) & = \mathop \smallint \limits_{0}^{\tau } \ddot{v}\dot{v}d\tau = \frac{1}{2}\dot{v}^{2} \left( \tau \right),E_{p} \left( \tau \right) = \mathop \smallint \limits_{0}^{\tau } v\dot{v}d\tau = \frac{1}{2}v^{2} \left( \tau \right), \\ E_{d} \left( \tau \right) & = 2\zeta \mathop \smallint \limits_{0}^{\tau } \dot{v}^{2} \left( \tau \right)d\tau ,W_{ext} \left( \tau \right) = - \mathop \smallint \limits_{0}^{\tau } \ddot{v}_{g} \dot{v}d\tau , \\ W_{Hertz}^{C} \left( \tau \right) & = - \overline{\Omega }_{H}^{2} \mathop \smallint \limits_{0}^{\tau } \left[ {v - 1} \right]_{ + }^{\frac{3}{2}} \dot{v}d\tau = - \frac{2}{5}\overline{\Omega }_{H}^{2} \left[ {v\left( \tau \right) - 1} \right]_{ + }^{\frac{5}{2}} , \\ W_{Hertz}^{NC} \left( \tau \right) & = \overline{\Omega }_{H}^{2} \mathop \smallint \limits_{0}^{\tau } \frac{{3\left( {1 - r} \right)}}{{2\dot{v}^{ - } }}\left[ {v - 1} \right]_{ + }^{\frac{3}{2}} \dot{v}^{2} d\tau \\ \end{aligned} $$

(9)

The contact force, as described in Eq. (3), entails contributions from conservative and nonconservative dissipative forces. Therefore, the superscript $C$ in ${W}_{Hertz}^{C}\left(\tau \right)$ indicates that the work done by the elastic component of the Hertzian interaction originates from a conservative force, while the superscript $NC$ in ${W}_{Hertz}^{NC}\left(\tau \right)$ is associated with the nonconservative dissipative component of the contact force which accounts for the energy dissipation during the contact event.

For the general case of inelastic contacts, an amount of energy is imparted into the structure during the ground motion. Part of this input energy is stored temporarily in the structure in the form of kinetic, ${E}_{k}\left(\tau \right)$, and potential, ${E}_{p}\left(\tau \right)$, energies, and the rest is dissipated either by the inherent viscous damping of the structure ${E}_{d}\left(\tau \right)$ or by the inelastic contacts themselves, ${W}_{Hertz}^{NC}\left(\tau \right)$. Ultimately, the entire energy imparted to the system should be dissipated. From (8) and (9), we obtain the following instantaneous amount of energy imparted into the structure—referred to as input energy—for the nonlinear, inelastic contact case, denoted as ${E}_{I}^{Inelastic}\left(\tau \right)$:

$$ \begin{aligned} E_{I}^{Inelastic} \left( \tau \right) & = W_{ext} \left( \tau \right) + W_{Hertz}^{C} \left( \tau \right) \\ & = - \mathop \smallint \limits_{0}^{\tau } \ddot{v}_{g} \dot{v}d\tau - \frac{2}{5}{\overline{\Omega }}_{H}^{2} \left[ {v\left( \tau \right) - 1} \right]_{ + }^{\frac{5}{2}} \\ & = E_{k} \left( \tau \right) + E_{d} \left( \tau \right) + E_{p} \left( \tau \right) + W_{Hertz}^{NC} \left( \tau \right) \\ \end{aligned} $$

(10)

In this nonlinear model, the instantaneous amount of energy, ${E}_{I}^{Inelastic}\left(\tau \right)$, imparted into the system during the ground motion is directly affected by the work performed by the base excitation and indirectly influenced by the Hertzian potential (i.e., the work performed by the elastic component of the Hertzian contact force). Let ${\tau }_{f}$ denote the duration of the ground acceleration, which for $N$ excitation cycles equals ${\tau }_{f}=2\pi N/\overline{\Omega }$. Then, the total normalized input energy ${E}_{I, tot}^{Inelastic}$ can be expressed from Eq. (10) as:

$$ \begin{aligned} E_{I,tot}^{Inelastic} & = E_{I}^{Inelastic} \left( {\tau = \tau_{f} } \right) = W_{ext} \left( {\tau_{f} } \right) + W_{Hertz}^{C} \left( {\tau_{f} } \right) \\ & = - \mathop \smallint \limits_{0}^{{\tau_{f} }} \ddot{v}_{g} \dot{v}d\tau - \frac{2}{5}{\overline{\Omega }}_{H}^{2} \left[ {v\left( {\tau_{f} } \right) - 1} \right]_{ + }^{\frac{5}{2}} \\ & = E_{k} \left( {\tau_{f} } \right) + E_{d} \left( {\tau_{f} } \right) + E_{p} \left( {\tau_{f} } \right) + W_{Hertz}^{NC} \left( {\tau_{f} } \right) \\ \end{aligned} $$

(11)

Moreover, bintegrating by parts we may express ${W}_{ext}\left({\tau }_{f}\right)$ as,

$$ W_{ext} \left( {\tau_{f} } \right) = - \mathop \smallint \limits_{0}^{{\tau_{f} }} \ddot{v}_{g} \dot{v}d\tau = \dot{v}\left( 0 \right)\dot{v}_{g} \left( 0 \right) - \dot{v}\left( {\tau_{f} } \right)\dot{v}_{g} \left( {\tau_{f} } \right) + \mathop \smallint \limits_{0}^{{\tau_{f} }} \ddot{v}\dot{v}_{g} d\tau $$

(12)

which, by introducing (5) into (12) and considering that ${\dot{v}}_{g}\left(0\right)={\dot{v}}_{g}\left({\tau }_{f}\right)=0$, yields:

$$ W_{ext} \left( {\tau_{f} } \right) = - \mathop \smallint \limits_{0}^{{\tau_{f} }} \left( {2\zeta \dot{v} + v + {\overline{\Omega }}_{H}^{2} \left[ {v - 1} \right]_{ + }^{3/2 } \left( {1 + \frac{{3\left( {1 - r} \right)}}{{2\dot{v}^{ - } }}\dot{v}} \right)} \right)\dot{v}_{g} d\tau $$

(13)

Finally, by introducing (13) into (11), the total input energy for the case of inelastic Hertzian contacts is expressed as:

$$ E_{I,tot}^{Inelastic} = - \mathop \smallint \limits_{0}^{{\tau_{f} }} \left( {2\zeta \dot{v} + v + {\overline{\Omega }}_{H}^{2} \left[ {v - 1} \right]_{ + }^{3/2 } \left( {1 + \frac{{3\left( {1 - r} \right)}}{{2\dot{v}^{ - } }}\dot{v}} \right)} \right)\dot{v}_{g} d\tau - \frac{2}{5}{\overline{\Omega }}_{H}^{2} \left[ {v\left( {\tau_{f} } \right) - 1} \right]_{ + }^{\frac{5}{2}} $$

(14)

In the second model, we study the effect of the non-smooth nonlinearity on the flow of the input energy by considering the case of purely elastic Hertzian contacts, i.e., by setting $r=1$, so that no energy dissipation is incurred during the vibro-impacts. Then, from (10), the corresponding instantaneous input energy, ${E}_{I}^{Elastic}\left(\tau \right)$, reduces to the following form:

$$ E_{I}^{Elastic} \left( \tau \right) = W_{ext} \left( \tau \right) + W_{Hertz}^{C} \left( \tau \right) = - \mathop \smallint \limits_{0}^{\tau } \ddot{v}_{g} \dot{v}d\tau - \frac{2}{5}{\overline{\Omega }}_{H}^{2} \left[ {v\left( \tau \right) - 1} \right]_{ + }^{\frac{5}{2}} = E_{k} \left( \tau \right) + E_{d} \left( \tau \right) + E_{p} \left( \tau \right) $$

(15)

We note that besides the direct effect of the base excitation, the instantaneous amount of energy ${E}_{I}^{Elastic}\left(\tau \right)$ imparted into the structure during the ground motion is indirectly influenced by the Hertzian potential. The total normalized input energy ${E}_{I, tot}^{Elastic}$ can be expressed from Eq. (14) as:

$$ E_{I,tot}^{Elastic} = - \mathop \smallint \limits_{0}^{{\tau_{f} }} \left( {2\zeta \dot{v} + v + {\overline{\Omega }}_{H}^{2} \left[ {v - 1} \right]_{ + }^{3/2 } } \right)\dot{v}_{g} d\tau - \frac{2}{5}{\overline{\Omega }}_{H}^{2} \left[ {v\left( {\tau_{f} } \right) - 1} \right]_{ + }^{\frac{5}{2}} $$

(16)

Lastly, for the limiting linear case (with no vibro-impacts—which serves as a reference model) we neglect the geometric Hertzian coefficient, i.e., ${\overline{\Omega } }_{H}=0$, so that the instantaneous input energy denoted by ${E}_{I}^{Linear}\left(\tau \right)$, simply reduces to the following form:

$$ E_{I}^{Linear} \left( \tau \right) = W_{ext} \left( \tau \right) = - \mathop \smallint \limits_{0}^{\tau } \ddot{v}_{g} \dot{v}d\tau = E_{k} \left( \tau \right) + E_{d} \left( \tau \right) + E_{p} \left( \tau \right) = \frac{1}{2}\dot{v}^{2} \left( \tau \right) + 2\zeta \mathop \smallint \limits_{0}^{\tau } \dot{v}^{2} \left( \tau \right)d\tau + \frac{1}{2}v^{2} \left( \tau \right) $$

(17)

For both the nonlinear model with pure elastic contacts and the linear model, an amount of energy, respectively denoted by ${E}_{I}^{Elastic}\left(\tau \right)$ and ${E}_{I}^{Linear}\left(\tau \right)$, is imparted into the structure during the ground motion. Part of this input energy is stored temporarily in the structure in the form of mechanical energy (i.e., the summation of kinetic ${E}_{k}\left(\tau \right)$ and potential ${E}_{p}\left(\tau \right)$ energies), and the rest is dissipated by the inherent viscous damping of the structure ${E}_{d}\left(\tau \right)$. Ultimately, the total energy imparted to a structure should be dissipated solely by the inherent viscous damping of the system.

Equation (17) shows that ${E}_{I}^{Linear}\left(\tau \right)$ is independent of the system mass and can be regarded as a characteristic of a linear single-floor model. In addition, from the linearized version of the equation of motion (5), the relative displacement, velocity and acceleration of the structure are directly proportional to the ground acceleration. In other words, if the ground acceleration is scaled by a factor $\alpha$, then the response quantities $v$, $\dot{v}$, and $\ddot{v}$ will be multiplied by the same factor $\alpha$. The input energy, however, is proportional to the square of the relative displacement and velocity, and so it will be multiplied by ${\alpha }^{2}$. The corresponding total normalized input energy ${E}_{I, tot}^{Linear}$ can be expressed from Eq. (16) as:

$$ E_{I,tot}^{Elastic} = - \mathop \smallint \limits_{0}^{{\tau_{f} }} \left( {2\zeta \dot{v} + v} \right)\dot{v}_{g} d\tau $$

(18)

2.3 Results and discussion

Here we illustrate the previous analytical derivations, and, based on these, investigate the potential for vibration mitigation of the internal vibro-impacts. Throughout this section, and unless stated otherwise, we consider the following system parameters:

$$ A = 1, {\overline{\Omega }} = 1, \omega_{n} = 1 \left[ {rad/sec} \right], N = 20, \zeta = 1 $$

(19)

Additional system parameters will be provided during the following exposition. Specifically, we aim to assess the efficacy of passively mitigating the transient response of this system through the drastic reduction of the input energy into the system caused by the intentional vibro-impacts, under the most severe condition possible. Thus, the excitation frequency was chosen to be identical to the natural frequency of the underlying linear (unprotected) system, i.e., $\overline{\Omega }=1$. The rationale here is to ensure that a significant portion of the input energy is directly imparted into the linear structure, thus realizing a worst-case scenario.

To quantify the effectiveness of the nonlinear structural configurations (with both elastic and inelastic Hertzian contacts) compared to the linear reference case, five normalized evaluation criteria are introduced, as listed in Table 1. The superscripts $"L"$ and $"NL"$ in the notations used in Table 1 denote the linear and nonlinear (either with elastic or inelastic contacts) configurations, respectively. Specifically, the evaluation criteria are defined in terms of peak floor displacement ${v}_{max}=\underset{\tau }{\text{max}}\left|v\left(\tau \right)\right| ({J}_{1})$, peak floor velocity ${\dot{v}}_{max}=\underset{\tau }{\text{max}}\left|\dot{v}\left(\tau \right)\right| ({J}_{2})$, peak floor acceleration ${\ddot{v}}_{max}=\underset{\tau }{\text{max}}\left|\ddot{v}\left(\tau \right)\right| ({J}_{3})$, total input energy ${E}_{I,tot}={W}_{ext}\left({\tau }_{f}\right)+{W}_{Hertz}^{C}\left({\tau }_{f}\right) ({J}_{4})$, and total energy dissipated, ${E}_{d,tot}$ $({J}_{5})$, by both modal viscous damping and inelastic contacts (in the nonlinear inelastic case only). Using these five evaluation criteria we assess both the overall dissipative capacity of the system, its maximum response features, and also the energy input provided by the base excitation.

Table 1 Normalized evaluation criteria for the single-floor system with a single-sided clearance

Figure 2 demonstrates the dependence of each normalized evaluation criterion listed in Table 1 on the normalized Hertzian stiffness coefficient, ${\overline{\Omega } }_{H}$ (which characterizes the type of non-smooth contact). A logarithmic scale was employed to encompass a wide range of contact stiffness, namely, from linear (${\overline{\Omega } }_{H}\ll 1$) to pure (Newtonian) vibro-impacts (${\overline{\Omega } }_{H}\to \infty$). It is evident that for soft contacts, i.e., ${0<\overline{\Omega } }_{H}\ll 1$, the performance of the nonlinear system resembles that of the linear model. Conversely, increasing the normalized contact stiffness ${\overline{\Omega } }_{H}$ beyond unity (i.e., stiffening the contacts) causes substantial deviations between the nonlinear and linear cases. Specifically, there are drastic reductions in the peak floor displacement (${J}_{1}$), velocity (${J}_{2}$), total input energy (${J}_{4}$), and total dissipated energy (${J}_{5}$) for both elastic and inelastic contacts. However, as expected these reductions are accompanied by a notable variation in the acceleration (${J}_{3}$) caused by the vibro-impact nonlinearities; however, it is notable (and perhaps counter-intuitive) that in the range of contact stiffness ${0.01<\overline{\Omega } }_{H}<100$ there occurs a significant reduction in the peak floor acceleration along with drastic reductions in the other evaluation criteria. This interesting performance regime is related to intermediate-stiff Hertzian contacts and paves the way for the exciting and highly promising prospect of drastically mitigating the transient response of a resonantly excited systems through passive internal vibro-impacts with simultaneous decrease of the resulting acceleration levels!

Fig. 2

The five evaluation criteria listed in Table 1 as functions of the normalized Hertzian contact stiffness, ${\overline{\Omega } }_{H}$, for the cases of purely elastic and inelastic ($r=0.6$) vibro-impacts; the soft (${\overline{\Omega } }_{H}=1$) and stiff (${\overline{\Omega } }_{H}=31.62$) cases considered in Figs. 3 and 4 are indicated by the vertical dashed lines

Based on the results of Fig. 2, we illustrate further the important role of the contacts of the non-smooth nonlinearity for mitigating the transient response, mainly by reducing the amount of input energy imparted into the system by the base excitation, two typical cases are investigated. Namely, the case of soft contact with normalized Hertzian stiffness coefficient ${\overline{\Omega } }_{H}=1$, and the case of stiff contact with ${\overline{\Omega } }_{H}=31.62$; these cases are represented by the vertical dashed-red and dashed-blue vertical lines, respectively, in Fig. 2.

The instantaneous floor displacement, velocity and acceleration time series for the soft contact case, ${\overline{\Omega } }_{H}=1$, are illustrated in Fig. 3 By introducing single-sided soft Hertzian contacts, both the nonlinear configurations (either elastic or inelastic contacts) exhibit a significant reduction (approximately by an order of magnitude) in the transient response, including the acceleration of the floor compared to the unprotected linear model. This is perhaps counterintuitive, since one would anticipate that the occurrence of the vibro-impacts would lead to an increase in peak floor acceleration. The soft character of the Hertzian contacts is evidenced by the visible deformation of the barrier, as shown in the top-right panel of Fig. 3. However, the transient dynamics of the nonlinear configurations with such soft Hertzian contacts remain strongly nonlinear. As the Hertzian contacts stiffness increases, a further reduction in floor displacement and velocity is achieved; however, this gain is accompanied by an increase in floor acceleration, as depicted in Fig. 4 corresponding to the stiff contact case ${\overline{\Omega } }_{H}=31.62$. The nature of the stiff is underlined by the fact that the floor hardly deforms through the barrier, as shown in the top-right panel of Fig. 4.

Fig. 3

Comparison of the relative displacement (1st row), velocity (2nd row), and acceleration (3rd row) time histories corresponding to ${\overline{\Omega }=\overline{\Omega } }_{H}=1$ for the linear model (i.e., with no barrier), and nonlinear models with purely elastic and inelastic Hertzian contacts; the right plots are enlargements of the vibro-impact responses

Fig. 4

Comparison of the relative displacement (1st row), velocity (2nd row), and acceleration (3rd row) time histories corresponding to ${\overline{\Omega }=1, \overline{\Omega } }_{H}=31.6228$ for the linear model (i.e., no barrier), nonlinear model with purely elastic Hertzian contacts, and nonlinear models with purely elastic and inelastic Hertzian contacts; right plots are enlargements of the vibro-impact responses

The normalized instantaneous input energy ${E}_{I}\left(\tau \right)$, mechanical energy ${E}_{m}\left(\tau \right)$, energy dissipated by viscous damping ${E}_{d}\left(\tau \right)$, and energy dissipated by the inelastic contacts ${W}_{Hertz}^{NC}\left(\tau \right)$ are illustrated in Fig. 5 for the cases of no contacts (linear model)—panel (a), elastic contacts—panels (b) and (d), and inelastic contacts—panels (c) and (e). Soft contacts are related to panels (b) and (c), while stiff contacts to panels (d) and (e). To make the comparison between the linear model and the nonlinear configurations more pronounced, in all the cases depicted in Fig. 5 the normalizations were performed with respect to the total input energy imparted into the structure for the case of the linear (unprotected) model, i.e., ${E}_{I,tot}^{Linear}$. The accuracy of the numerical integration is illustrated in Fig. 5 (for all cases) by depicting the instantaneous energy balance between the input energy and the summation of mechanical and dissipated energies (represented by blue and dashed-red, respectively).

Fig. 5

Normalized instantaneous energies: linear model (a), purely elastic contacts with ${\overline{\Omega } }_{H}=1$ (b), inelastic contacts with ${\overline{\Omega } }_{H}=1$ (c), purely elastic contacts with ${\overline{\Omega } }_{H}=31.6228$ (d), and inelastic contacts with ${\overline{\Omega } }_{H}=31.6228$ (e). The normalization here was performed with respect to the total input energy corresponding to the linear model, i.e., ${E}_{I,tot}^{L}$

Based on the results depicted in Fig. 5, we note that for the duration of the ground excitation, i.e., in the time interval $0\le \tau \le {\tau }_{f}$, the linear model is resonantly excited, resulting in monotonic increase of the instantaneous input energy imparted into the structure, as shown in Fig. 5(a). Consequently, the instantaneous mechanical energy, temporarily stored in the structure, also monotonically grows. However, by introducing single-sided soft and purely elastic Hertzian impacts, a drastic reduction (approximately 98% compared to the linear case) in the amount of input energy imparted into the structure is achieved, as shown in Fig. 5(b). It is worth noting that this reduction and the resulting enhanced vibration mitigation is solely due to the single-sided soft, purely elastic vibro-impacts, which highly restrict the input energy into the system without the need for adding any mass, damping, or additional degrees of freedom to the system. Surprisingly, for the same contact stiffness the amount of input energy increases for the case of inelastic contacts, as shown in Fig. 5(c), compared to the case when the contacts are purely elastic. It is noteworthy that, besides dissipating energy through the viscous damping, a significant portion of the input energy is dissipated by the inelastic contacts themselves. It is evident that, although the total input energy imparted into the structure for the inelastic contacts case is higher compared to purely elastic contacts, the amounts of energy dissipated by the inherent modal viscous damping of the system in both cases are comparable.

As the contact stiffness increases, the amount of input energy decreases significantly (by about three orders of magnitude) compared to the linear model, as shown in panels (d) and (e) of Fig. 5. However, as demonstrated in Figs. 2 and 4, the resulting good mitigation performance is achieved at the cost of moderately increasing the level of floor acceleration caused by the stiff vibro-impacts. The instantaneous work performed by the elastic component of the Hertzian contact force, i.e., the energy stored, even briefly, at the contact, is incorporated into the instantaneous input, mechanical and dissipated energies as illustrated in Figs. 6 and 7 for the cases of soft and stiff contacts, respectively. In each Figure, the first row is related to elastic contacts, while the second is devoted to inelastic contacts.

Fig. 6

Incorporation of the normalized instantaneous contact energy into the input, mechanical and dissipated energies for the soft contact case, ${\overline{\Omega } }_{H}=1$: purely elastic contacts (first row), and inelastic contacts (second row); The right column provides enlargements of the dashed rectangular regions in the left column

Fig. 7

Incorporation of the normalized instantaneous contact energy into the input, mechanical and dissipated energies for the stiff contact case, i.e., ${\overline{\Omega } }_{H}=31.6228$: purely elastic contacts (first row), and inelastic contacts (second row); the right column illustrates enlargements of the dashed-magenta rectangular regions in the left column

To elucidate the physical mechanism governing the enhanced mitigation performance of the single-sided vibro-impacts (which is mainly due to its capacity to significantly reduce the input energy imparted into the system from base excitation), we focus on the right columns of Figs. 6 and 7, where it is noted that the contact duration of typical vibro-impact, ${\tau }_{c}$, is inversely proportional to the contact stiffness, ${\tau }_{c}\propto 1/{\overline{\Omega } }_{H}$, regardless of the coefficient of restitution. Hence, for a given contact stiffness both the purely elastic and inelastic vibro-impacts possess the same contact duration, however, the contact depth (i.e., vibro-impact intensity) is larger for purely elastic contacts. Each contact event consists of compression and restitution phases. During the compression phase there occurs a drop of the instantaneous input energy imparted into the system (in the form of mechanical energy). This is followed by a “relaxation” or restitution phase, whereby the energy temporarily stored in the contact is released back to the system proper. For purely elastic contacts, the rate of energy transfer in the compression phase is equal to the corresponding rate in the restitution phase, with this symmetry showing clearly in the first row of Figs. 6 and 7 for both soft and stiff contacts, respectively. For inelastic vibro-impacts, however, due to the hysteresis damping factor incorporated in the Hertzian contact law, the rate of compression is different than the rate of restitution (as per the second row of Figs. 6 and 7 for soft and stiff contacts, respectively). Even though this energy transaction at the contact sites causes a slight delay in the overall energy transfer from the base excitation to the system proper, it has a drastic effect in the dynamics of the resonantly excited system as it drives it out of resonance, breaking the monotonic increase of input energy. Consequently, the total input energy that is eventually imparted into the system is drastically reduced compared to that of the (unprotected) linear model.

Moreover, while the contact duration ${\tau }_{c}$ is governed by the contact stiffness ${\overline{\Omega } }_{H}$, the contact intensity is determined by the coefficient of restitution, rendering the elastic vibro-impacts more intensive compared to the inelastic ones (for the same contact stiffness). Therefore, the elastic vibro-impacts are more efficient in reducing the amount of input energy imparted into the system compared to the inelastic ones, cf. Figures 6 and 7. In addition, The stiff contacts are characterized by short-duration intensive vibro-impacts (cf. Figure 7), whereas soft contacts correspond to longer-duration ones (cf. Figure 6). This explains the increased acceleration levels for the case of stiff contacts, as shown in Figs. 2 and 3. However, it is important to note that due to the very short duration of these impacts, the likelihood of damage to the system is reduced.

In the nonlinear configurations (with purely elastic or inelastic contacts), the vibro-impacts cause significant scattering of the input energy at high frequencies (beyond the natural frequency). This strong low-to-high frequency energy transfer drives the system out of resonance, resulting in rapid suppression of the system response. High-frequency responses lead to lower vibration amplitudes, and the input energy is more efficiently dissipated at higher frequencies.

3 Multi-floor system with asymmetrical clearances

3.1 Model description and equations of motion

The previous study is now extended to the more general case of a multi-floor frame system subject to base excitation. This extension will introduce the additional feature of intermodal targeted energy transfer (IMTET) to our study, corresponding to irreversible energy transfer between the structural modes caused by the vibro-impacts; this was missing in the single-floor system considered previously (as it possessed a single structural mode). Specifically, a one-dimensional N-floor lumped-mass shear-type frame model is considered here as a primary system, with the following physical mass (${\varvec{M}}$), stiffness (${\varvec{K}}$), and damping (${\varvec{C}}$) $NxN$ matrices,

$$ {\varvec{M}} = \left[ {\begin{array}{*{20}c} {m_{1} } & 0 & 0 & 0 & 0 \\ 0 & {m_{2} } & 0 & 0 & 0 \\ 0 & 0 & \ddots & 0 & 0 \\ 0 & 0 & 0 & {m_{N - 1} } & 0 \\ 0 & 0 & 0 & 0 & {m_{N} } \\ \end{array} } \right]_{NxN} , $$

$$ {\varvec{K}} = \left[ {\begin{array}{*{20}c} {2k_{1} } & { - k_{2} } & 0 & 0 & 0 \\ { - k_{1} } & {2k_{2} } & { - k_{3} } & 0 & 0 \\ 0 & \ddots & \ddots & \ddots & 0 \\ 0 & 0 & { - k_{N - 2} } & {2k_{N - 1} } & { - k_{N} } \\ 0 & 0 & 0 & { - k_{N - 1} } & {k_{N} } \\ \end{array} } \right]_{NxN} $$

$$ {\varvec{C}} = \left( {{{\varvec{\Phi}}}^{{\text{T}}} } \right)^{ - 1} \left[ {\begin{array}{*{20}c} {2\zeta_{1} \omega_{1} } & 0 & 0 & 0 & 0 \\ 0 & {2\zeta_{2} \omega_{2} } & 0 & 0 & 0 \\ 0 & 0 & \ddots & 0 & 0 \\ 0 & 0 & 0 & {2\zeta_{N - 1} \omega_{N - 1} } & 0 \\ 0 & 0 & 0 & 0 & {2\zeta_{N} \omega_{N} } \\ \end{array} } \right]{{\varvec{\Phi}}}^{ - 1} $$

(20)

where ${m}_{i}$ and ${k}_{i}$ denote the mass and the column stiffness of the ${i}^{th}$ floor, respectively. The Wilson modal damping approach [39] was assumed here to construct the physical damping matrix, ${\varvec{C}}$, where ${\zeta }_{i}$ and ${\omega }_{i}$ represent the damping ratio and natural frequency of the ${i}^{th}$ mode, and ${\varvec{\Phi}}$ is the modal matrix. For simplicity we assume that the system is made of identical columns, so the total stiffness at each floor is expressed as,

$$ k_{1} = k_{2} = \cdots = k_{N} \equiv k = n_{c} \frac{{12EI_{c} }}{{h^{3} }} $$

(21)

where, ${n}_{c}$ is the number of columns at each floor, and $E$, ${I}_{c}$, and $h$ are the Young’s modulus, area moment of inertia, and height of single column, respectively.

Similarly to the previous section, to induce intentional non-smooth nonlinearity, an additional internal flexible core structure is introduced intentionally with distributed asymmetrical clearances with respect to the floors of the primary system. A schematic representation of a three-floor ($N=3$) frame system with an internal core subjected to base excitation is illustrated in Fig. 8(a). As the system and core respond to sufficiently strong ground excitation, the clearance gaps will result in a series of double-sided vibro-impacts between the floors and the core structure. In turn, it is expected that the amount of input energy imparted into the system will be strongly affected as well.

Fig. 8

Considered three-floor system: a Schematic representation of a base-excited three-floor frame system with internal flexible core, b asymmetrical distribution of the clearances at each floor of the primary system and the core structure, and c wavelet spectrum of the ground acceleration, where the notations ${f}_{1}$, ${f}_{2}$, and ${f}_{3}$ represent the 1st, 2nd, and 3rd natural frequencies, respectively, of the underlying linear system (without the core)

The model of the flexible core structure was based on the Euler–Bernoulli beam-column theory, yielding the mass and stiffness $NxN$ matrices denoted by ${{\varvec{M}}}_{cs}$ and ${{\varvec{K}}}_{cs}$, respectively. Subsequently, the corresponding damping matrix, denoted by ${{\varvec{C}}}_{cs}$, was created using the Wilson modal damping approach [39]. The integrated system, consisting of the primary system incorporating a flexible core is governed by the following equations of motion,

$$ \begin{array}{c} {{\varvec{M}}\ddot{u} + {{\varvec C}\dot{\varvec u}} + {\varvec{Ku}} - {\varvec{f}}_{{{\varvec{NL}}}}^{{{\varvec{N}} - {\varvec{story}}}} \left( {\dot{\varvec{u}},{\varvec{u}},\dot{\varvec{v}},{\varvec{v}}, {{\varvec{\Delta}}}_{{\varvec{L}}} ,{{\varvec{\Delta}}}_{{\varvec{R}}} } \right) = -{\varvec{M\varGamma}}\ddot{u}_{g} } \\ {{\varvec{M}}_{{{\varvec{cs}}}} {\ddot{\varvec v}} + {\varvec{C}}_{{{\varvec{cs}}}} \dot{\varvec{v}} + {\varvec{K}}_{{{\varvec{cs}}}} {\varvec{v}} + {\varvec{f}}_{{{\varvec{NL}}}}^{{{\varvec{N}} - {\varvec{story}}}} \left( {\dot{\varvec{u}},{\varvec{u}},\dot{\varvec{v}},{\varvec{v}}, {{\varvec{\Delta}}}_{{\varvec{L}}} ,{{\varvec{\Delta}}}_{{\varvec{R}}} } \right) = - {\varvec{M}}_{{{\varvec{cs}}}}{\varvec{\varGamma}}\ddot{u}_{g} } \\ \end{array} $$

(22)

where ${\varvec{u}}={\left[{u}_{1},{u}_{2},\cdots ,{u}_{N}\right]}^{T}$ and ${\varvec{v}}={\left[{v}_{1},{v}_{2},\cdots ,{v}_{N}\right]}^{T}$ represent t he relative displacement vectors of the building floors and core contact points with respect to the ground, respectively, ${\ddot{u}}_{g}$ is the ground acceleration, ${\varvec{\gamma}}$ the influence vector of the base motion, ${{\varvec{f}}}_{{\varvec{N}}{\varvec{L}}}^{{\varvec{N}}-{\varvec{s}}{\varvec{t}}{\varvec{o}}{\varvec{r}}{\varvec{y}}}={\left[{f}_{1}^{N-story},{f}_{2}^{N-story},\cdots ,{f}_{N}^{N-story}\right]}^{T}$ the Hertzian contact force vector, and ${{\varvec{\Delta}}}_{{\varvec{L}}}={\left[{\Delta }_{{L}_{1}},{\Delta }_{{L}_{2}},\cdots ,{\Delta }_{{L}_{N}}\right]}^{T}$ and ${{\varvec{\Delta}}}_{{\varvec{R}}}={\left[{\Delta }_{{R}_{1}},{\Delta }_{{R}_{2}},\cdots ,{\Delta }_{{R}_{N}}\right]}^{T}$ the asymmetrical left and right side vectors of clearance gaps between the primary system and core structure, respectively.

Similar to the case of the single-floor system discussed previously, we assume that the contact force between the primary system and the flexible core is governed by the Hertz law complemented by a nonlinear Hunt and Crossley viscoelastic contact model [38]. This model captures the dissipation of energy resulting from inelastic contact. For the ${j}^{th}$ floor, the double-sided contact force is given by,

$$ f_{j}^{N - story} = k_{c} \left[ {\left[ {v_{j} - u_{j} - {\Delta }_{{L_{j} }} } \right]_{ + }^{\frac{3}{2}} - \left[ {u_{j} - v_{j} - {\Delta }_{{R_{j} }} } \right]_{ + }^{\frac{3}{2}} } \right]\left( {1 + \frac{{3\left( {1 - r} \right)}}{{2\left( {\dot{u}_{j}^{ - } - \dot{v}_{j}^{ - } } \right)}}\left( {\dot{u}_{j} - \dot{v}_{j} } \right)} \right) $$

(23)

where ${k}_{c}$ represents the contact stiffness coefficient, reflecting the magnitude of the contacts, which is dependent on the material properties and shape of the contact surfaces. In the current study, ${k}_{c}$ will be treated as a design parameter, and its impact on the input energy imparted into the system will be investigated. To cover a wide range of contact stiffness, ${k}_{c}$ is expressed as ${k}_{c}={10}^{p} [N/m]$. In (23), $r$ is a coefficient of restitution, and $\left({\dot{u}}_{j}^{-}-{\dot{v}}_{j}^{-}\right)$ is the relative contact velocity just before a vibro-impact occurs. In addition, the subscript ( +) indicates that only non-negative values of the arguments in the brackets should be taken into account, with zero values being assigned otherwise.

The double-sided Hertzian contact force at the ${j}^{th}$ floor, as given in Eq. (23), comprises of both conservative and nonconservative components. The conservative part is associated with the purely elastic component, ${f}_{j}^{C}$, while the nonconservative part, ${f}_{j}^{NC}$, to the inelastic one:

$$ \begin{array}{*{20}c} {f_{j}^{N - story} = f_{j}^{C} + f_{j}^{NC} } \\ {f_{j}^{C} { = }k_{c} \left[ {\left[ {v_{j} - u_{j} - \Delta_{{L_{j} }} } \right]_{ + }^{\frac{3}{2}} - \left[ {u_{j} - v_{j} - \Delta_{{R_{j} }} } \right]_{ + }^{\frac{3}{2}} } \right]} \\ {f_{j}^{NC} { = }k_{c} \left[ {\left[ {v_{j} - u_{j} - \Delta_{{L_{j} }} } \right]_{ + }^{\frac{3}{2}} - \left[ {u_{j} - v_{j} - \Delta_{{R_{j} }} } \right]_{ + }^{\frac{3}{2}} } \right]\frac{{3\left( {1 - r} \right)}}{{2\left( {\dot{u}_{j}^{ - } - \dot{v}_{j}^{ - } } \right)}}\left( {\dot{u}_{j} - \dot{v}_{j} } \right)} \\ \end{array} $$

(24)

By introducing (24) into the primary system equation of motion, i.e., the first equation in (22), one obtains,

$$ {{\varvec M}\ddot{\varvec u}} + {{\varvec C}\dot{\varvec u}} + \varvec{Ku} - \varvec{f^{{NC}}} = \varvec{- M\varGamma} \ddot{u}_{g} + \varvec{f^{C}} $$

(25)

where, ${{\varvec{f}}}^{{\varvec{C}}}{=\left[{f}_{1}^{C},{f}_{2}^{C},\cdots ,{f}_{N}^{C}\right]}^{T}$, ${{\varvec{f}}}^{{\varvec{N}}{\varvec{C}}}{=\left[{f}_{1}^{NC},{f}_{2}^{NC},\cdots ,{f}_{N}^{NC}\right]}^{T}$. From Eq. (25), three distinct structural configurations can be identified: (i) Linear dynamics (i.e., no core) with ${{\varvec{f}}}^{{\varvec{C}}}={{\varvec{f}}}^{{\varvec{N}}{\varvec{C}}}=0$, (ii) purely elastic Hertzian contacts with ${{\varvec{f}}}^{{\varvec{C}}}\ne 0, \; {{\varvec{f}}}^{{\varvec{N}}{\varvec{C}}}\equiv 0$ (or, alternatively, $r=1$), and (iii) inelastic Hertzian contacts with ${{\varvec{f}}}^{{\varvec{C}}}\ne 0, \; {{\varvec{f}}}^{{\varvec{N}}{\varvec{C}}}\ne 0$.

In the following study it will be shown that the normalized Hertzian stiffness coefficient, ${k}_{c}$, serves a crucial role for understanding and analyzing the underlying nonlinear dynamics governing the flow of input energy into the multi-floor system from the base excitation; this is similar to what was observed for the case of the single floor system in the previous section, although in the present case the effect of IMTET is accounted as well. Moreover, the effect of the coefficient of restitution $r$ on the input energy will be addressed as well.

3.2 Instantaneous energy balance and intermodal targeted energy transfer (IMTET)

We study in detail the linear model without a core, serving as a reference model, and the nonlinear model with double-sided purely elastic Hertzian contacts, highlighting the effects on the dynamics of the non-smooth nonlinearity. Integrating Eq. (25) with respect to the relative displacement ${{\varvec{u}}}^{T}$ we obtain,

$$ \mathop \smallint \limits_{0}^{{\varvec{t}}} \dot{\varvec{u}}^{T} {\varvec{M}}\ddot{\varvec{u}}dt + \mathop \smallint \limits_{0}^{{\varvec{t}}} \dot{\varvec{u}}^{T} {{\varvec C}\dot{\varvec u}}dt + \mathop \smallint \limits_{0}^{{\varvec{t}}} \dot{\varvec{u}}^{T} {\varvec{Ku}}dt - \mathop \smallint \limits_{0}^{{\varvec{t}}} \dot{\varvec{u}}^{T} {\varvec{f}}^{{{\varvec{NC}}}} dt = - \mathop \smallint \limits_{0}^{{\varvec{t}}} \dot{\varvec{u}}^{T}{\varvec{M\varGamma}}\ddot{u}_{g} dt + \mathop \smallint \limits_{0}^{{\varvec{t}}} \dot{\varvec{u}}^{T} {\varvec{f}}^{{\varvec{C}}} dt $$

(26)

which describes the instantaneous energy balance of the base-excited multi-floor model with double-sided inelastic asymmetrical Hertzian contacts based on relative motion. This can be further expressed as,

$$ E_{k} \left( t \right) + E_{d} \left( t \right) + E_{p} \left( t \right) + W_{Hertz}^{NC} \left( t \right) = W_{ext} \left( t \right) + W_{Hertz}^{C} \left( t \right) $$

(27)

where ${E}_{k}\left(t\right)$ is instantaneous kinetic energy, ${E}_{d}\left(t\right)$ cumulative dissipated energy by the structural modes, ${E}_{p}\left(\tau \right)$ instantaneous potential energy, ${W}_{Hertz}^{C}\left(t\right)$ the work performed by the elastic component of the Hertzian contact force, ${W}_{Hertz}^{NC}\left(t\right)$ the work performed by the nonlinear viscoelastic element of the Hertzian contact force, and ${W}_{ext}\left(t\right)$ the work performed by the external base excitation (i.e., the ground acceleration). Assuming that the system is initially at rest, $\dot{{\varvec{u}}}\left(0\right)={\varvec{u}}\left(0\right)=0$, the instantaneous energy components in (27) can be expressed as follows:

$$ \begin{gathered} E_{k} \left( t \right) = \frac{1}{2}\dot{\varvec{u}}^{T} \varvec{M}\dot{\varvec{u}} = \frac{1}{2}\mathop \sum \limits_{{i = 1}}^{N} m_{i} \dot{u}_{i}^{2} , \quad E_{p} \left( t \right) = \frac{1}{2}\varvec{u}^{T} \varvec{Ku}, \hfill \\ E_{d} \left( t \right) = \mathop \smallint \limits_{0}^{t} \dot{\varvec{u}}^{T} {{\varvec C}\dot{\varvec u}}dt,W_{{ext}} \left( t \right) = - \mathop \smallint \limits_{0}^{t} \dot{\varvec{u}}^{T} \varvec{M}{\varvec{\varGamma }}\ddot{u}_{g} dt = - \mathop \smallint \limits_{0}^{t} \mathop \sum \limits_{{i = 1}}^{N} m_{i} \dot{u}_{i} \ddot{u}_{g} dt, \hfill \\ W_{{Hertz}}^{C} \left( t \right) = \mathop \smallint \limits_{{\mathbf{0}}}^{\varvec{t}} \dot{\varvec{u}}^{T} \varvec{f}^{\varvec{C}} dt = k_{c} \mathop \smallint \limits_{0}^{t} \left( {\mathop \sum \limits_{{i = 1}}^{N} \dot{u}_{i} \left[ {\left[ {v_{i} - u_{i} - \Delta _{{L_{i} }} } \right]_{ + }^{{\frac{3}{2}}} - \left[ {u_{i} - v_{i} - \Delta _{{R_{i} }} } \right]_{ + }^{{\frac{3}{2}}} } \right]} \right)dt, \hfill \\ W_{{Hertz}}^{{NC}} \left( t \right) = - \mathop \smallint \limits_{{\mathbf{0}}}^{\varvec{t}} \dot{\varvec{u}}^{T} \varvec{f}^{{\varvec{NC}}} dt = - k_{c} \mathop \smallint \limits_{0}^{t} \left( {\mathop \sum \limits_{{i = 1}}^{N} \dot{u}_{i} \left[ {\left[ {v_{i} - u_{i} - \Delta _{{L_{i} }} } \right]_{ + }^{{\frac{3}{2}}} - \left[ {u_{i} - v_{i} - \Delta _{{R_{i} }} } \right]_{ + }^{{\frac{3}{2}}} } \right]\frac{{3\left( {1 - r} \right)}}{{2\left( {\dot{u}_{i}^{ - } - \dot{v}_{i}^{ - } } \right)}}\left( {\dot{u}_{i} - \dot{v}_{i} } \right)} \right)dt \hfill \\ \end{gathered} $$

(28)

Note that ${W}_{ext}\left(t\right)$ is the summation of the work performed by the inertia force,$-{m}_{i}{\ddot{u}}_{g}$, at each floor with relative velocity ${\dot{u}}_{i}$ with respect to the base. For the case of inelastic contacts, energy is imparted into the multi-floor system during ground motion. A portion of this input energy is temporarily stored in the structural modes in the form of mechanical energy that is, ${E}_{k}\left(t\right)+{E}_{p}\left(t\right)$, while the remaining energy is dissipated, either through the inherent viscous damping of structural modes, ${E}_{d}\left(t\right)$, or by the inelastic contacts themselves,${W}_{Hertz}^{NC}\left(t\right)$. Ultimately, the entire input energy introduced into the system must be dissipated. For both the nonlinear model with purely elastic contacts and the linear model, however, the entirety of input energy is dissipated solely by the inherent modal viscous damping of the system, ${E}_{d}\left(t\right)$, i.e., the structural modes.

From Eqs. (27) and (28), we obtain the following instantaneous amount of input energy imparted into the N-floor system for the nonlinear configuration with double-sided asymmetrical inelastic contacts, denoted as ${E}_{I}^{Inelastic}\left(t\right)$:

$$ E_{I}^{Inelastic} \left( t \right) = W_{ext} \left( t \right) + W_{Hertz}^{C} \left( t \right) = E_{k} \left( t \right) + E_{d} \left( t \right) + E_{p} \left( t \right) + W_{Hertz}^{NC} \left( t \right) = - \mathop \smallint \limits_{0}^{t} \mathop \sum \limits_{i = 1}^{N} m_{i} \dot{u}_{i} \ddot{u}_{g} dt + k_{c} \mathop \smallint \limits_{0}^{t} \left( {\mathop \sum \limits_{i = 1}^{N} \dot{u}_{i} \left[ {\left[ {v_{i} - u_{i} - {\Delta }_{{L_{i} }} } \right]_{ + }^{\frac{3}{2}} - \left[ {u_{i} - v_{i} - {\Delta }_{{R_{i} }} } \right]_{ + }^{\frac{3}{2}} } \right]} \right)dt $$

(29)

The influence of non-smooth nonlinearity on the input energy flow is examined by considering double-sided, asymmetric and purely elastic Hertzian contacts with $r=1$, so that no energy dissipation is incurred during the vibro-impacts. The corresponding instantaneous input energy, denoted by ${E}_{I}^{Elastic}\left(t\right)$, from Eq. (29) becomes:

$$ E_{I}^{Elastic} \left( t \right) = W_{ext} \left( t \right) + W_{Hertz}^{C} \left( t \right) = E_{k} \left( t \right) + E_{d} \left( t \right) + E_{p} \left( t \right) = - \mathop \smallint \limits_{0}^{t} \mathop \sum \limits_{i = 1}^{N} m_{i} \dot{u}_{i} \ddot{u}_{g} dt + k_{c} \mathop \smallint \limits_{0}^{t} \left( {\mathop \sum \limits_{i = 1}^{N} \dot{u}_{i} \left[ {\left[ {v_{i} - u_{i} - {\Delta }_{{L_{i} }} } \right]_{ + }^{\frac{3}{2}} - \left[ {u_{i} - v_{i} - {\Delta }_{{R_{i} }} } \right]_{ + }^{\frac{3}{2}} } \right]} \right)dt $$

(30)

Lastly, for the linear model (no core, nor vibro-impacts), which serves as a reference model for an unprotected N-floor system, we set ${k}_{c}=0$ and obtain the following corresponding instantaneous input energy denoted by ${E}_{I}^{Linear}\left(t\right)$:

$$ E_{I}^{Linear} \left( t \right) = W_{ext} \left( t \right) = E_{k} \left( t \right) + E_{d} \left( t \right) + E_{p} \left( t \right) = - \mathop \smallint \limits_{0}^{t} \mathop \sum \limits_{i = 1}^{N} m_{i} \dot{u}_{i} \ddot{u}_{g} dt $$

(31)

In contrast to the single-floor model which possessed a single structural mode, in the N-floor system model with N structural modes, non-smooth nonlinearity enables an additional energy transfer mechanism, namely nonlinear scattering of input energy from low to high-frequency structural modes. Consequently, a better utilization of the intrinsic modal dissipative capacity of the system is achieved. This mechanism of energy transfer among vibration modes is referred to as intermodal targeted energy transfer (IMTET), and its efficacy for passive nonlinear vibration and shock mitigation was recently demonstrated in the fields of blast and seismic mitigation [34,35,36,37]. This additional feature will be taken into account in the following energy balance study.

To this end, the equations of motion (25) are now transformed into modal coordinates through the coordinate transformation ${\varvec{u}}={\varvec{\Phi}}{\varvec{q}}$,

$$ {\varvec{M}}{{\varvec{\Phi}}}{\ddot{\varvec q}} + {\varvec{C}}{{\varvec{\Phi}}}\dot{\varvec{q}} + {\varvec{K}}{\varvec{\Phi} {\mathbf{q}}} - {\varvec{f}}^{{{\varvec{NC}}}} = -{\varvec{M\varGamma}}\ddot{u}_{g} + {\varvec{f}}^{{\varvec{C}}} $$

(32)

where ${\varvec{q}}={\left[{q}_{1},{q}_{2},\cdots ,{q}_{N}\right]}^{T}$ represents the modal amplitude vector of the primary system, and ${\varvec{\Phi}}$ is the corresponding ($NxN$) mass orthonormalized modal matrix satisfying, ${{\varvec{\Phi}}}^{T}{\varvec{M}}{\varvec{\Phi}}=\mathbf{I}$. Pre-multiplying both sides of Eq. (32) by ${{\varvec{\Phi}}}^{T}$ yields:

$$ {\ddot{\varvec q}} + {\hat{\varvec C}\dot{\varvec q}} + {\hat{\mathbf{K}}\mathbf{q}} - {{\varvec{\Phi}}}^{T} {\varvec{f}}^{{{\varvec{NC}}}} = - {{\varvec{\Phi}}}^{T}{\varvec{M\varGamma}}\ddot{u}_{g} + {{\varvec{\Phi}}}^{T} {\varvec{f}}^{{\varvec{C}}} $$

(33)

Assuming proportional viscous damping guarantees that the resulting transformed damping and stiffness matrices $\widehat{{\varvec{C}}}$  and $\widehat{\mathbf{K}}$, respectively, are diagonal,

$$ \hat{\varvec{C}} = {{\varvec{\Phi}}}^{T} {\varvec{C}}{{\varvec{\Phi}}} = \left[ {\begin{array}{*{20}c} {\lambda_{1} } & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & {\lambda_{N} } \\ \end{array} } \right], \hat{\varvec{K}} = {{\varvec{\Phi}}}^{T} {\mathbf{K\Phi }} = \left[ {\begin{array}{*{20}c} {\omega_{1}^{2} } & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & {\omega_{N}^{2} } \\ \end{array} } \right] $$

(34)

where ${\lambda }_{j}$ and ${\omega }_{j}$ are the modal viscous damping coefficient and the natural frequency, respectively, of the ${j}^{th}$ structural mode of the primary system. As a key performance parameter to quantify nonlinear energy transfer between the structural modes (IMTET), we define ${E}_{d,j}$ as the percentage of input energy dissipated by the ${j}^{th}$ mode of the primary system,

$$ E_{d,j} = \frac{{\mathop \smallint \nolimits_{0}^{{t_{f} }} \lambda_{j} \dot{u}_{j}^{2} dt}}{{\mathop \sum \nolimits_{i = 1}^{N} \mathop \smallint \nolimits_{0}^{{t_{f} }} \lambda_{i} \dot{u}_{i}^{2} dt}} $$

(35)

where, ${t}_{f}$ represents the total duration of the base excitation.

In the next subsection, we will employ the previous energy measures to elucidate the nonlinear mechanism for energy dissipation in a three-floor system and explain how the strong nonlinearity induced by the vibro-impacts yields drastic reduction of the flow of input energy in the system, and, thus, enhanced vibration mitigation compared to the linear case.

3.3 Results and discussion

Unless specified otherwise, we maintain the following system parameters for the three-floor system:

$$ \begin{array}{*{20}l} {m_{1} = m_{2} = m_{3} = 40000~\left[ {{\text{kg}}} \right],} \hfill \\ {k_{1} = k_{2} = k_{3} = 3.55 \cdot 10^{7} \left[ {\frac{N}{m}} \right]:~~n_{c} = 3,~E = 200\left[ {{\text{GPa}}} \right],} \hfill \\ {I_{c} = 1.33 \cdot 10^{{ - 4}} \left[ {{\text{m}}^{4} } \right],~h = 3\left[ {\text{m}} \right]} \hfill \\ {\zeta _{1} = \zeta _{2} = \zeta _{3} = 0.03} \hfill \\ {k_{c} = 10^{p} \left[ {{\text{N/m}}} \right]} \hfill \\ \end{array} $$

(36)

The asymmetric distribution of the clearances between the floors of the primary system and the core structure are illustrated in Fig. 8(b). The chosen clearances in this study are arbitrary, serving to illustrate the intentional introduction of non-smooth nonlinearity to restrict input energy flow. An optimization study for optimal clearance distribution is strongly recommended for enhanced performance. Without losing generality, the ground acceleration (base excitation) is modeled as a finite number of cycles of harmonic excitation followed by relaxation as follows:

$$ \ddot{u}_{g} \left( t \right) = cos\left( {{\Omega }t} \right)\left( {H\left( t \right) - H\left( {t - N_{c} \frac{2\pi }{{\Omega }}} \right)} \right),\quad {\Omega } = \omega_{1} ,\quad N_{c} = 20 $$

(37)

As in Sect. 2 for the single-floor system, the excitation frequency was chosen to be identical to the fundamental natural frequency of the underlying linear system, i.e., $\Omega ={\omega }_{1}$. The rationale here is to ensure that a significant portion of the input energy is directly imparted into the linear system, thus realizing a worst-case scenario (the most severe condition). The wavelet spectrum of the ground acceleration is depicted in Fig. 8(c).

Five normalized evaluation criteria are introduced in Table 2 to investigate the effectiveness of the vibration mitigation for both elastic and inelastic Hertzian contacts, compared to the linear system which is regarded as reference; as previously, the superscripts $"L"$ and $"NL"$ refer to the linear and nonlinear (either with elastic or inelastic contacts) configurations, respectively. The evaluation criteria are related to peak floor displacement ${u}_{max}=\underset{t,i\in \left\{1,2,3\right\}}{\text{max}}\left|{u}_{i}\left(t\right)\right| ({J}_{1})$, peak interfloor drift ${d}_{max}=\underset{t,i\in \left\{1,2,3\right\}}{\text{max}}\left|{u}_{i}\left(t\right)-{u}_{i-1}\left(t\right)\right| ({J}_{2})$ where ${u}_{0}\left(t\right)\equiv 0$, peak floor acceleration ${\ddot{u}}_{max}=\underset{t,i\in \left\{1,2,3\right\}}{\text{max}}\left|{\ddot{u}}_{i}\left(t\right)\right| ({J}_{3})$, total input energy ${E}_{I,tot}={W}_{ext}\left({t}_{f}\right)+{W}_{Hertz}^{C}\left({t}_{f}\right) ({J}_{4})$, total energy dissipated by modal viscous damping and at the inelastic contacts $({J}_{5})$.

Table 2 Normalized evaluation criteria for the three-story structure with double-sided asymmetrical clearances

The dependence of the normalized evaluation criteria listed in Table 2 on the power coefficient, $p$, of the Hertzian stiffness, ${k}_{c}={10}^{p}$ which characterizes the magnitude of the non-smooth nonlinearity term is demonstrated in Fig. 9. It is shown that the nonlinear configurations closely resemble the linear model in the $0\le p\le 5$ range, characterized by ultra-soft vibro-impacts. However, significant deviations occur in the range $p\ge 7$, marked by substantial reduction in peak floor displacement (${J}_{1}$), interfloor drift (${J}_{2}$), total input energy (${J}_{4}$), and total dissipated energy (${J}_{5}$) for both elastic and inelastic contacts. Surprisingly, within the range of $7 \le p \le 9.5$, these decreases are associated with a notable decrease also in the acceleration level (${J}_{3}$), attributed to intermediate-stiff Hertzian contacts. This result (which extends the similar result that was observed for the single-floor system—cf. Figure 2) renders this intermediate-stiff contact regime ideal for vibration mitigation of this resonantly excited multi-floor system, alleviating concerns of increased acceleration levels due to the Hertzian vibro-impacts. As expected, beyond the stiff contact regime, $p \ge 9.5$, there occur moderate increases in the acceleration level (criterion ${J}_{3}$), which indeed are attributed to strong Hertzian contacts. The two broad saturated plateau regimes of ultra-soft contacts ($p\le 5$) and soft to stiff contacts ($p\ge 7$) are separated by a narrow transition regime, corresponding to $5 \le p \le 7$.

Fig. 9

The effect of the Hertzian contact stiffness ${k}_{c}={10}^{p}$ on the five evaluation criteria listed in Table 2 for the cases of purely elastic and inelastic contacts; the soft ($p=8$) and stiff ($p=10$) cases considered in the following Figures are indicated by the vertical dashed lines

In the following study two representative cases are selected based on the metrics depicted in Fig. 9, corresponding to (i) $p=8$ (soft contacts) and (ii) $p=10$ (stiff contacts), and being mainly differentiated by the corresponding floor acceleration levels. These cases are represented in Fig. 9 by the dashed-red and dashed-blue vertical lines, respectively. The relative displacements between the floors of the primary system and the core structure for these two cases are depicted in Fig. 10, where the soft (stiff) nature of the contacts are signified by the corresponding large (small) barrier deformations at each floor.

Fig. 10

The relative displacement between the floors of the primary system and the core structure: Soft Hertzian contacts with $p=8$ (a), and stiff Hertzian contacts with $p=10$ (b)

The maximum floor displacement, interfloor drift, and floor acceleration are illustrated in Fig. 11 for the soft (first row) and stiff (second row) Hertzian contacts. We deduce that the asymmetrical double-sided soft Hertzian contacts result in drastic response reduction (on the order of 70–80% compared to the linear case) for both elastic and inelastic vibro-impacts, which highlights the efficacy of the nonlinear vibration mitigation with the added benefit of significantly reduced acceleration levels. Note that, despite the soft nature of the contacts in this case, the underlying dynamics remain strongly nonlinear due to the non-smooth nature of the vibro-impacts. As the stiffness of the Hertzian contacts increases (cf. the second row in Fig. 11), a further reduction in maximum floor displacements and interfloor drifts is achieved, however, at the expense of increased maximum floor accelerations.

Fig. 11

Comparison of floor responses for soft contacts (1st row) and stiff contacts (2nd row): Peak floor displacements—${J}_{1}$ (a) and (d), peak floor interfloor drifts—${J}_{2}$ (b) and (e), and peak floor accelerations—${J}_{3}$ (c) and (f)

In Fig. 12, the normalized instantaneous input energy $- \,{E}_{I}(t)$, mechanical energy $- \,{E}_{m}(t)$, energy dissipated by the inherent (modal) damping of the structural modes ${- \,E}_{d}(t)$, and energy dissipated by inelastic contacts $- \,{W}_{Hertz}^{NC}(t)$, are presented for the linear and nonlinear system configurations, namely, the linear case, panel (a), the case of elastic vibro-impacts, panels (b) and (d), and lastly the case of inelastic vibro-impacts, panels (c) and (e). Soft contact results are depicted in panels (b) and (c), while stiff contacts are shown in panels (d) and (e). To emphasize the effectiveness for vibration mitigation of the nonlinear cases compared to the linear one, all normalizations were performed with respect to the total input energy ${E}_{I,tot}^{Linear}$ that is imparted into the reference linear three-floor system by the base excitation. The accuracy of the numerical integration is apparent from the depicted instantaneous energy balance in each case. Like in the linear single-floor system, when the linear three-floor system is resonantly excited during the ground acceleration, a monotonically increasing instantaneous input energy into the structure is observed, as depicted in Fig. 12(a); consequently, the instantaneous mechanical energy that is temporarily stored in the structure also exhibits a monotonically increasing pattern.

Fig. 12

Normalized (with respect to ${E}_{I,tot}^{Linear}$) instantaneous energies: Linear system (a); system with soft vibro-impacts with ${k}_{c}={10}^{8}[N/m]$—purely elastic (b), and inelastic (c); and system with stiff vibro-impacts with ${k}_{c}={10}^{10}[N/m]$—purely elastic (d), and inelastic (e)

It is interesting to note that the asymmetrical double-sided soft and purely elastic Hertzian vibro-impacts result in an extraordinary 96% reduction in the amount of input energy compared to the linear case, cf. Figure 12(b). Notably, this impressive performance is achieved despite having the same dissipation mechanism as the linear model (that is, solely due to the inherent structural modal damping) and without the need of any additional external attachments (e.g., NESs) or any added mass, stiffness or damping, in the primary structure. Further reduction of the input energy can be achieved by stiffening further the contacts, as shown in Fig. 12(d, e). Moreover, this notable improvement in passive vibration mitigation is accompanied by a modest increase in floor accelerations, which can be attributed to the vibro-impacts of increased stiffness. Similar to the single-floor model, the total input energy is greater for inelastic vibro-impacts compared to purely elastic ones.

Figures 13 and 14 illustrate the incorporation of the work performed by the elastic component of the Hertzian contact force to the instantaneous input, mechanical and dissipated energies for the cases of soft and stiff vibro-impacts, respectively. The case of purely elastic vibro-impacts is depicted in the first row of each figure, while the case of inelastic vibro-impacts in the second row. Besides the direct source of the input energy induced by the ground excitation, here it is crucial to consider also the work performed by the elastic components of the contact forces, which represents a secondary source of input energy imparted into the system. Indeed, omitting this secondary source of input energy would violate the fact that the total amount of energy imparted into the structure must be eventually balanced by the energy dissipated through the inherent (modal) damping of the structural modes in the case of purely elastic contacts. In fact, in contrast to the single-floor system presented in Sect. 2, the work perform by the elastic components of the Hertzian contact forces changes cumulatively over time, which leads to non-zero net contribution to the input energy.

Fig. 13

Incorporation of the normalized instantaneous contact energy into the input, mechanical and dissipated energies for the case of soft vibro-impacts, ${k}_{c}={10}^{8}[N/m]$: Purely elastic contacts (1st row), and inelastic contacts (2nd row); The right column illustrates enlargements of the dashed-magenta rectangular regions in the left column

Fig. 14

Incorporation of the normalized instantaneous contact energy to the input, mechanical and dissipated energies for the case of stiff vibro-impacts, ${k}_{c}={10}^{10}[N/m]$: Purely elastic contacts (1st row), and inelastic contacts (2nd row); the right column shows enlargements of the dashed-magenta rectangular regions in the left column

The results presented in Figs. 13 and 14 reveal that a sequence of vibro-impacts leads to a substantial decrease in instantaneous input energy compared to the underlying linear system (with no internal core, nor vibro-impacts). This effect, which is highly beneficial from the vibration mitigation point of view, is because the resonantly excited multi-floor system is, in essence, shifted out of resonance, which, in turn, interrupts the monotonic increase of input energy transfer into the system from the base excitation. Specifically, the occurrence of the vibro-impacts serves as a mechanism of driving the system out of resonance, even in the presence of soft Hertzian contacts. However, unlike the simpler case of the single-floor system with a single-sided contact, the pattern of vibro-impacts here is rather more complicated due to the occurrence of multiple near-simultaneous contacts at different floors, making the analysis of the compression and restitution phases of each vibro-impact challenging and not amenable to the simple analysis of Sect. 2.

From another perspective, soft or stiff Hertzian contacts affect differently the resulting IMTET in the nonlinear system, i.e., of low-to-high frequency nonlinear energy scattering in the underlying linear modal space. This is evidenced by the results of Fig. 15, where the percentages of input energy that are eventually dissipated by the inherent damping of each structural mode of the underlying linear system are depicted for soft (left panel) and stiff (right panel) vibro-impacts. Similar to the linear case without a core where energy dissipation occurs almost exclusively by the fundamental structural mode, in the system with soft vibro-impacts there is minimal energy transfer to the second and third (higher frequency) structural modes. In contrast, in the system with stiff vibro-impacts about 30% of the input energy is dissipated by the second and third modes, so IMTET is realized. In turn, this causes more rapid dissipation of the input energy, as evidenced in Fig. 16. Indeed, for soft contacts almost 45% of the energy is dissipated by the end of the ground acceleration, compared to only 30% for the reference linear case. The rate of energy dissipation is drastically amplified for stiff, purely elastic contacts, where 73% of the input energy is dissipated by the end of ground acceleration.

Fig. 15

Percentage of input energy dissipated by the inherent modal damping of the three-floor linear system: Case of soft contacts, ${k}_{c}={10}^{8}[N/m]$ (left panel), and stiff contacts, ${k}_{c}={10}^{10}[N/m]$ (right panel); to obtain these results the nonlinear responses are projected onto the 3D modal space of the underlying (no core) system

Fig. 16

Comparison of the normalized cumulative energy dissipated by the inherent (modal) damping of the three structural modes of the underlying linear system for soft or stiff purely elastic contacts, and no contacts; the normalizations here were performed with respect to the corresponding total input energy in each case

4 Concluding remarks

In this study, we investigated a fully passive nonlinear mechanism for mitigating the transient response of a base-excited system. This was based on the drastic reduction of the input energy imparted into the system by the base excitation through the introduction of intentional strong non-smooth nonlinearity in the form of Hertzian vibro-impacts, which, in turn, drive the system out of resonance by breaking the monotonic increase of input energy for the resonantly excited structure.

This concept was initially studied by considering the simplest case of a single-floor frame system incorporating a single-sided clearance, and then extended to a multi-floor frame system with an internal flexible core which introduced double-sided asymmetrical vibro-impacts between the floors and the core. In both cases, the systems were resonantly excited in their fundamental modes by a finite-cycle ground acceleration, representing the most severe excitation scenario. Moreover, three distinct system configurations were studied: (i) The linear system with no core, nor internal vibro-impacts (serving as reference), (ii) the nonlinear system incorporating purely elastic Hertzian vibro-impacts, and (iii) the nonlinear system with inelastic Hertzian vibro-impacts. Additionally, the type of contact stiffness was investigated, ranging from soft to stiff Hertzian contacts, highlighting its crucial role in vibration mitigation.

To assess the efficacy of the intentional induced strong nonlinearity for effective response mitigation five evaluation criteria were introduced based on peak floor displacements, peak floor velocities (for the single-floor system) or interstory drifts (for the three-floor system), peak floor accelerations, but also total input energy and total dissipated energy. These criteria were investigated over a wide range of contact stiffness, ranging from linear and ultra-soft to very stiff contacts. The results revealed that the dependence of each evaluation criterion on the contact stiffness can be divided into three different regimes: (i) A regime resembling the linear dynamics for ultra-soft contacts, (ii) a relatively narrow transition regime for soft contacts, and (iii) a strongly nonlinear regime corresponding to intermediate soft and stiff contacts. This last regime for intermediate soft contacts corresponded to significant simultaneous reductions in all criteria, including in peak floor accelerations, paving the way for enhanced passive vibration mitigation of resonantly excited system by means of internal vibro-impact nonlinearities without any harmful “side effects” such as increased accelerations due to the occurrence of vibro-impacts.

In addition, our results showed that the introduction of either a single-sided soft contact for the single-floor system, or of asymmetrical double-sided soft contact for the three-floor system result in drastic reduction of the input energy that is imparted into the system by the base excitation, compared to the linear case. Notably, this impressive finding is achieved without the need for any additional mass, damping, stiffness or degrees of freedom in the system. This notable reduction in the input energy compared to the underlying linear system, indicates that the internal vibro-impacts effectively shift the resonantly excited system out of resonance, thus drastically reducing its capacity to absorb energy from the base excitation. This paves the way for a new design paradigm, whereby internal intentional nonlinearities can be employed to drastically reduce the energy intake in a dynamical or acoustical system from external excitations.

Moreover, the phenomenon of intermodal targeted energy transfer (IMTET), involving nonlinear modal energy scattering between structural modes, was examined in the context of the resonantly excited three-floor system. This phenomenon was demonstrated for both soft and stiff Hertzian contacts. While in the system with soft contacts there occurred minimal IMTET, in the system with stiff contacts approximately 30% of the input energy was dissipated by the second and third modes, thereby enhancing the rate of internal energy dissipation of the system compared to the linear case.

The suggested concept of introducing non-smooth nonlinearity in the form of Hertzian contacts to drastically reduce the amount of input energy imparted into a resonantly base-excited single- or multi-DOF engineering structure suggests new possibilities for mitigating its transient response. Specifically, a well-designed, optimized approach that strikes a balance between reducing input energy and simultaneously controlling the resulting acceleration levels by tuning the type (stiff or soft) of vibro-impact contacts may well be elegantly achieved in this manner.