Impact analytical models for earthquake-induced pounding simulation

Abstract

Structural pounding under earthquake has been recently extensively investigated using various impact analytical models. In this paper, a brief review on the commonly used impact analytical models is conducted. Based on this review, the formula used to determine the damping constant related to the impact spring stiffness, coefficient of restitution, and relative approaching velocity in the Hertz model with nonlinear damping is found to be incorrect. To correct this error, a more accurate approximating formula for the damping constant is theoretically derived and numerically verified. At the same time, a modified Kelvin impact model, which can reasonably account for the physical nature of pounding and conveniently implemented in the earthquake-induced pounding simulation of structural engineering is proposed.

Keywords

structural pounding / Hertz model / Kelvin model / nonlinear damping / coefficient of restitution

Introduction

Due to the structural damage resulted from the earthquake-induced pounding between closely spaced fixed-supporting structures, a great deal of relevant research have been conducted by numerous researchers [1-6]. Based on those valuable research results, modifications in seismic-code and various countermeasures to mitigate the hazards from the poundings of adjacent fixed-supported structures have been proposed. Structural pounding is a complex phenomenon which makes the numerical analysis of this type of problem difficult. To study the global structural response because of pounding under severe earthquakes, various impact analytical models or methodologies have been used or developed by some researchers. The most widely used impact analytical models include the restitution-based stereomechanical model [7], the contact force-based linear elastic model, nonlinear elastic model (Hertz model [2]), linear viscoelastic model (Kelvin model [8]) and corresponding variation accounting for plastic deformations of the colliding structures suggested by Komodromos et al. [9], nonlinear viscoelastic model proposed by Jankowski [10], and Hertz model with nonlinear damping implemented by Muthukumar et al. [11], which is also used in other areas such as robotics and multibody systems [12].

In this paper, review on the commonly used impact analytical models is conducted. Based on this review, the formula used to determine the damping constant related to the impact spring stiffness, coefficient of restitution, and relative approaching velocity in the Hertz model with nonlinear damping is found to be incorrect. To correct this error, a more accurate approximating formula for the damping constant is theoretically derived. At the same time, a modified Kelvin impact model, which can reasonably account for the physical nature of pounding and conveniently be implemented in the earthquake-induced pounding simulation of structural engineering, is proposed.

Review on analytical impact models

Seismic pounding is essentially a problem of dynamic impact. The forces produced during collision act for a short period, where energy is dissipated as heat because of random molecular vibrations and the internal friction of the colliding bodies. Usually, contact is modeled using either a continuous force model or via a stereomechanical (coefficient of restitution) approach, as mentioned earlier. The analytical formulations of the various impact models are outlined below.

Stereomechanical model

This approach uses momentum conservation principle and the coefficient of restitution to model impact. The duration of impact is neglected. The coefficient of restitution (e) is defined as the ratio of separation velocities of the bodies after impact to their approaching velocities before impact [7]. Because this is not a force-based approach, the effect of impact is accounted by adjusting the velocities of the colliding bodies, as shown in the following equations:

(1)\begin{array}{ll}{v}_{\mathrm{1}}^{\prime }=v_{\mathrm{1}}-(\mathrm{1}+e){\displaystyle \frac{m_{\mathrm{2}}(v_{\mathrm{1}}-v_{\mathrm{2}})}{m_{\mathrm{1}}+m_{\mathrm{2}}}},& v_{\mathrm{2}}^{\prime }=v_{\mathrm{2}}+(\mathrm{1}+e){\displaystyle \frac{m_{\mathrm{1}}(v_{\mathrm{1}}-v_{\mathrm{2}})}{m_{\mathrm{1}}+m_{\mathrm{2}}}},\end{array}

where v_{\mathrm{1}}^{\prime } and v_{\mathrm{2}}^{\prime } are the velocities of the colliding masses (m_{\mathrm{1}}, m_{\mathrm{2}}) after impact, v_{\mathrm{1}} and v_{\mathrm{2}} are the velocities before impact, and e is the coefficient of restitution, which also can be obtained from the following equation:

(2)e={\displaystyle \frac{v_{\mathrm{2}}^{\prime }-v_{\mathrm{1}}^{\prime }}{v_{\mathrm{1}}-v_{\mathrm{2}}}}.

However, stereomechanical approach is rather not recommended when a precise pounding-involved structural response is required, especially in the case of multiple impacts. Moreover, because this model does not trace the structural response during pounding, assuming that the impact duration can be ignored, its application is usually limited to the analysis of pounding between two structures modeled as systems of single degree of freedom. In case when the structures are simulated by models of multiple degrees of freedom, the structural response during the time when contact takes place is important. This is due to the fact that, when the structural members rebound after collision, they might come into contact with other members.

Linear spring model

A linear impact spring of stiffness (k_{\mathrm{l}}) can be used to simulate impact once the gap between adjacent bodies closes. The impact force at time t is provided by

(3)F_{\mathrm{c}}(t)=k_{\mathrm{l}}\delta (t),

where \delta (t) is the interpenetration depth of the colliding bodies. This approach is relatively straightforward and can be easily implemented in commercial software. However, energy loss during impact is not taken into account.

Kelvin model

A linear impact spring of stiffness (k_{\mathrm{k}}) is used to be in conjunction with a damper element (c_{\mathrm{k}}) that accounts for energy dissipation during impact. The impact force-penetration relation can be represented as

(4)F_{\mathrm{c}}(t)=k_{\mathrm{k}}\delta (t)+c_{\mathrm{k}}\dot{\delta }(t),

where \dot{\delta }(t) is the relative velocity between the colliding bodies at time t. The damping coefficient c_{\mathrm{k}} can be related to the coefficient of restitution e, by equating the energy losses during impact [13]:

(5)\begin{array}{ll}{c}_{\mathrm{k}}=\mathrm{2}\xi \sqrt{k_{\mathrm{k}}{\displaystyle \frac{m_{\mathrm{1}}{m}_{\mathrm{2}}}{m_{\mathrm{1}}+m_{\mathrm{2}}}}},& \xi =-{\displaystyle \frac{\ln e}{\sqrt{{\mathrm{\pi }}^{\mathrm{2}}+{(\ln e)}^{\mathrm{2}}}}}\end{array}.

The disadvantage of the Kelvin model is that its viscous component is active with the same damping coefficient during the whole time of collision. This results in the uniform dissipation during the approach and restitution periods, which is not fully consistent with the reality [7]. Moreover, this model exhibits an initial jump of the impact force values on impact because of the damping term. Furthermore, the damping force causes negative impact forces that pull the colliding bodies together, during the unloading phase, instead of pushing them apart. To avoid the tensile impact forces that arise between the colliding structures at the end of the restitution period, because of the damping term, a minor modification is proposed by Komodromos et al. for the Kelvin model [9]. In particular, when the impact force is about change sign, this impact spring and damper are removed. Therefore, the equation that provides the impact force can be written as follows:

(6)F_{\mathrm{c}}(t+\mathrm{\Delta }t)=\{\begin{array}{ll}{k}_{\mathrm{k}}\delta (t)+c_{\mathrm{k}}\dot{\delta }(t),& F_{\mathrm{c}}(t)>\mathrm{0},\\ \mathrm{0},& F_{\mathrm{c}}(t)\le \mathrm{0.}\end{array}

Hertz model

Another popular model for representing pounding is the Hertz model, which uses a nonlinear impact spring of stiffness (k_{\mathrm{h}}). The impact force representation is

(7)F_{\mathrm{c}}(t)=k_{\mathrm{h}}\delta {(t)}^n.

The use of the Hertz contact law has an intuitive appeal in modeling pounding because one would expect the contact area between the colliding structures to increase as the contact force increases, leading to nonlinear stiffness described as the Hertz coefficient (n). The Hertz coefficient n is typically taken as 3/2. Similar to the linear spring model, energy loss during impact cannot be modeled in the Hertz model as well.

Nonlinear viscoelastic model

To be able to include energy dissipation mechanism, Jankowski [10] proposed the nonlinear viscoelastic model based on Hertz’s contact law. He incorporated a nonlinear damper parallel to the nonlinear impact spring stiffness (k_{\mathrm{h}}) during the approach phase of the contact, while during the restitution phase, the energy dissipation is omitted; thus, the impact force can be expressed as

(8)F_{\mathrm{c}}(t)=\{\begin{array}{ll}{k}_{\mathrm{h}}\delta {(t)}^{\mathrm{3}/\mathrm{2}}+c_{\mathrm{h}}(t)\dot{\delta }(t),& \dot{\delta }(t)>\mathrm{0},\\ k_{\mathrm{h}}\delta {(t)}^{\mathrm{3}/\mathrm{2}},& \dot{\delta }(t)\le \mathrm{0.}\end{array}

According to Jankowski [10], the impact-damping coefficient c_{\mathrm{h}}(t) is provided by the following formula in terms of the impact-damping ratio {\xi }_{\mathrm{h}} and the interpenetration depth \mathrm{\delta :0}(t):

(9)c_{\mathrm{h}}(t)=\mathrm{2}{\xi }_{\mathrm{h}}\sqrt{k_{\mathrm{h}}\sqrt{\delta (t)}{\displaystyle \frac{m_{\mathrm{1}}{m}_{\mathrm{2}}}{m_{\mathrm{1}}+m_{\mathrm{2}}}}}.

The impact-damping ratio {\xi }_{\mathrm{h}} can be estimated using Jankowski’s [14] formula:

(10){\xi }_{\mathrm{h}}={\displaystyle \frac{\mathrm{9}\sqrt{\mathrm{5}}}{\mathrm{2}}}{\displaystyle \frac{\mathrm{1}-e^{\mathrm{2}}}{e(e(\mathrm{9}\mathrm{\pi }-\mathrm{16})+\mathrm{16})}}.

However, the impact force-time curve obtained from this impact model is not smoothly varied between the approach phase and restitution period of the collision.

Hertz model with nonlinear damper (Hertz damp model)

The Hertz model suffers from the limitation that it cannot represent energy dissipation during contact. An improved version of Hertz model, Hertz model with nonlinear damping (Hertz damp model), already used in other areas such as robotics and multibody systems [12], is first introduced by Muthukumar et al. [11] to pounding simulation in structural engineering. The contact force of Hertz damp model can be expressed as

(11)F_{\mathrm{c}}(t)=k_{\mathrm{h}}\delta {(t)}^{\mathrm{3}/\mathrm{2}}+{\widehat{c}}_{\mathrm{h}}(t)\dot{\delta }(t).

The damping coefficient {\widehat{c}}_{\mathrm{h}}(t) is taken as follows:

(12){\widehat{c}}_{\mathrm{h}}(t)={\widehat{\xi }}_{\mathrm{h}}\delta {(t)}^{\mathrm{3}/\mathrm{2}},

where {\widehat{\xi }}_{\mathrm{h}} is the damping constant. Equating the energy loss during stereomechanical impact to the energy dissipated by the damper, an expression for the damping constant {\widehat{\xi }}_{\mathrm{h}} can be found in terms of the impact spring stiffness (k_{\mathrm{h}}), the coefficient of restitution (e), and the relative approaching velocity (v_{\mathrm{1}}-v_{\mathrm{2}}), as follows:

(13){\widehat{\xi }}_{\mathrm{h}}={\displaystyle \frac{\mathrm{3}{k}_{\mathrm{h}}(\mathrm{1}-e^{\mathrm{2}})}{\mathrm{4}(v_{\mathrm{1}}-v_{\mathrm{2}})}}.

According to the statement of Jankowski in Ref. [14], in the case of nonlinear viscoelastic model, a value of {\xi }_{\mathrm{h}} = 0 stands for a fully elastic collision, and a value of {\xi }_{\mathrm{h}}\to\infty stands for a fully plastic one. Although in the nonlinear elastic model, the restitution period of collision is considered to be fully elastic, it is believed that similar relations may also concern the Hertz damp model. However, the formula of damping constant {\widehat{\xi }}_{\mathrm{h}} in Eq. (13) will provide the following surprising relationship:

(14)\begin{array}{lll}e=\mathrm{1}\hfill & \to \hfill & {\widehat{\xi }}_{\mathrm{h}}=\mathrm{0},\hfill \\ e=\mathrm{0}\hfill & \to \hfill & {\widehat{\xi }}_{\mathrm{h}}\ne \infty .\hfill \end{array}

If we reexamine the derivation of the formula for damping constant {\widehat{\xi }}_{\mathrm{h}} in Hertz damp model, we will find that the derivation is based on the following two assumptions: 1) the energy dissipated during impacts is small compared with the maximum absorbed elastic energy, and 2) the penetration velocities during the compression and restitution phases are approximately equal. In other words, the Hertz damp model is valid for the case of e approximating one. According to the conclusions made by Anagnostopoulos in Ref. [1], the coefficient of restitution used to simulate real collision in structural engineering varied in the range of 0.5-0.75, which indicates that the Hertz damp model using Eq. (13) to determine the damping constant {\widehat{\xi }}_{\mathrm{h}} is incorrect for pounding simulation in structural engineering.

Modified Kelvin model

In the aforementioned impact models, the linear spring model and the Kelvin model can be categorized into linear impact models because of the linearity of impact spring stiffness; correspondingly, the Hertz model, nonlinear viscoelastic model, and Hertz damp model can be considered as nonlinear impact models because of the nonlinearity of impact spring stiffness. For the nonlinear impact models, the impact spring stiffness k_{\mathrm{h}} is a function of elastic properties and geometry of the two colliding bodies. Based on the assumption that the two colliding bodies are isotropic spheres of radii R_{\mathrm{1}} and R_{\mathrm{2}}, k_{\mathrm{h}} can be expressed as follows [12]:

(15)\begin{array}{ll}{k}_{\mathrm{h}}={\displaystyle \frac{\mathrm{4}}{\mathrm{3}\mathrm{\pi }(h_{\mathrm{1}}+h_{\mathrm{2}})}}{\left[{\displaystyle \frac{R_{\mathrm{1}}{R}_{\mathrm{2}}}{R_{\mathrm{1}}+R_{\mathrm{2}}}}\right]}^{\mathrm{1}/\mathrm{2}},& h_i={\displaystyle \frac{\mathrm{1}-{\nu }_i^{\mathrm{2}}}{\mathrm{\pi }{E}_i}},i=\mathrm{1},\mathrm{2},\end{array}

where h_{\mathrm{1}} and h_{\mathrm{2}} are the material parameters and {\nu }_i and E_i are the Poisson’s ratio and modulus of elasticity, respectively, of sphere i. However, the assumption that the two colliding structures have the spherical shape cannot be satisfied in the practical structural pounding; hence, it is difficult to determine the value of impact spring stiffness k_{\mathrm{h}}. On the contrary, using the linear impact models, the impact spring stiffness k_{\mathrm{l}} or k_{\mathrm{k}} can be estimated through the axial stiffness of the colliding structures by the rule of thumb [9]. Noting that the linear elastic and Kelvin model cannot reasonably reflect the physical nature of structural pounding, a modified Kelvin model is proposed in this paper to remedy the drawbacks of Kelvin model and simultaneously preserve the convenience of determination of the linear impact spring stiffness in the Kelvin model. The mathematical formulation of modified Kelvin model is given by

(16)F_{\mathrm{c}}(t)=k_{\mathrm{k}}\delta (t)+{\widehat{c}}_{\mathrm{k}}(t)\dot{\delta }(t).

The damping coefficient {\widehat{c}}_{\mathrm{k}}(t)is supposed to have the similar form of Eq. (12):

(17){\widehat{c}}_{\mathrm{k}}(t)={\widehat{\xi }}_{\mathrm{k}}\delta (t),

where {\widehat{\xi }}_{\mathrm{k}} is the damping constant for the modified Kelvin model. In the following, the derivation of expression for damping constant {\widehat{\xi }}_{\mathrm{k}} is presented, and corresponding numerical verification is performed.

Derivation of expression for damping constant {\widehat{\xi }}_{\mathrm{k}}

According to stereomechanical model, the energy loss (\mathrm{\Delta }E) during the impact can be expressed in terms of the coefficient of restitution (e) and the approaching velocities (v_{\mathrm{1}}-v_{\mathrm{2}}) of the two colliding bodies, as follows:

(18)\mathrm{\Delta }E={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{\displaystyle \frac{m_{\mathrm{1}}{m}_{\mathrm{2}}}{m_{\mathrm{1}}+m_{\mathrm{2}}}}(\mathrm{1}-e^{\mathrm{2}}){(v_{\mathrm{1}}-v_{\mathrm{2}})}^{\mathrm{2}}.

The energy dissipated by the damping force can be evaluated as

(19)\mathrm{\Delta }E={\displaystyle \oint {\widehat{c}}_{\mathrm{k}}}\dot{\delta }\mathrm{d}\delta ={\displaystyle \oint {\widehat{\xi }}_{\mathrm{k}}\delta \dot{\delta }\mathrm{d}\delta }.

To evaluate the energy loss from Eq. (19), the interpenetration velocity \dot{\delta }(t) must be expressed as a function of interpenetration \delta (t) during the period of contact. The variation of \delta (t) is illustrated in Fig. 1, where t^{-}, t^{\max }, and t^{+} denote initial time of contact, time of maximum interpenetration {\delta }_{\max }, and the time of separation of two colliding bodies, respectively. At the end of the compression phase, the two bodies move with a common velocity V.

Because of the nonlinearity of the modified Kelvin model, there is no exact formula for the interpenetration velocity during impact. However, we can attempt to use approximate functions to describe the relationship between the interpenetration velocity \dot{\delta }(t) and interpenetration \delta (t). As a matter of fact, through the transformation of reference system, the impact between two colliding bodies (shown in Fig. 2(a)) can be equivalently modeled as response of single-degree-of-freedom (SDOF) system (depicted in Fig. 2(b)) with the initial interpenetration {\delta }_{\mathrm{0}}=\delta (t=\mathrm{0})=\mathrm{0} and initial interpenetration velocity {\dot{\delta }}_{\mathrm{0}}=\dot{\delta }(t=\mathrm{0})=v_{\mathrm{1}}-v_{\mathrm{2}}. The equation of motion of such SDOF system is written as

(20)m\ddot{\delta }(t)+c\dot{\delta }(t)+k\delta (t)=\mathrm{0},

where \ddot{\delta }(t) is the interpenetration acceleration, m(=m_{\mathrm{1}}{m}_{\mathrm{2}}/(m_{\mathrm{1}}+m_{\mathrm{2}})) is the mass of equivalent SDOF system, and damping coefficient c and spring stiffness k of the equivalent SDF system are assumed to be linear to conveniently look for the approximate relationship between \dot{\delta }(t) and \delta (t) by use of the knowledge of structural dynamics. Thus, the solution to the Eq. (20) is as follows:

(21)\begin{array}{ll}\delta (t)={\mathrm{e}}^{-r\omega t}{\displaystyle \frac{{\dot{\delta }}_{\mathrm{0}}}{{\omega }_{\mathrm{d}}}}\sin ({\omega }_{\mathrm{d}}t),& \dot{\delta }(t)={\mathrm{e}}^{-r\omega t}{\dot{\delta }}_{\mathrm{0}}\cos ({\omega }_{\mathrm{d}}t)-r{\mathrm{e}}^{-r\omega t}{\displaystyle \frac{\omega {\dot{\delta }}_{\mathrm{0}}}{{\omega }_{\mathrm{d}}}}\sin ({\omega }_{\mathrm{d}}t),\end{array}

where \omega(=\sqrt{k/m}) is the radial frequency, r(=c/(\mathrm{2}m\omega )) is the damping ratio, and {\omega }_{\mathrm{d}}(=\omega \sqrt{\mathrm{1}-r^{\mathrm{2}}}) is the damped radial frequency. If the damping effect is further ignored (i.e., r=\mathrm{0}), t^{\max } and {\delta }_{\max } can be easily determined as

(22)\begin{array}{ll}{t}^{\max }={\displaystyle \frac{\mathrm{\pi }}{\mathrm{2}\omega }},& {\delta }_{\max }={\displaystyle \frac{{\dot{\delta }}_{\mathrm{0}}}{\omega }}\end{array}.

Substituting Eq. (22) into Eq. (21), the following expression can be obtained:

(23){\left({\displaystyle \frac{\delta (t)}{{\delta }_{\max }}}\right)}^{\mathrm{2}}+{\left({\displaystyle \frac{\dot{\delta }(t)}{{\dot{\delta }}_{\mathrm{0}}}}\right)}^{\mathrm{2}}=\mathrm{1.}

From Eq. (23), the relationship between \dot{\delta }(t) and \delta (t) can be approximately considered to be elliptic. Hence, at the approaching period, the interpenetration velocity \dot{\delta }(t) can be related to \delta (t) as follows:

(24)\dot{\delta }(t)={\dot{\delta }}_{\mathrm{0}}\sqrt{\mathrm{1}-{\left({\displaystyle \frac{\delta (t)}{{\delta }_{\max }}}\right)}^{\mathrm{2}}}.

Similarly, at the restitution period, the relation between \dot{\delta }(t) and \delta (t) can be defined as:

(25)\dot{\delta }(t)={\dot{\delta }}_{\mathrm{f}}\sqrt{\mathrm{1}-{\left({\displaystyle \frac{\delta (t)}{{\delta }_{\max }}}\right)}^{\mathrm{2}}},

where {\dot{\delta }}_{\mathrm{f}}(=v_{\mathrm{1}}^{\prime }-v_{\mathrm{2}}^{\prime }) is the postimpact (final) relative velocity between two colliding bodies. Substituting Eqs. (24) and (25) into Eq. (19), we get

(26)\mathrm{\Delta }E=\mathrm{\Delta }{E}_{\mathrm{1}}+\mathrm{\Delta }{E}_{\mathrm{2}}={\displaystyle {\int }_{\mathrm{0}}^{{\delta }_{\max }}{\widehat{\xi }}_{\mathrm{k}}\delta }{\dot{\delta }}_{\mathrm{0}}\sqrt{\mathrm{1}-{(\delta /{\delta }_{\max })}^{\mathrm{2}}}\mathrm{d}\delta +{\displaystyle {\int }_{\mathrm{0}}^{{\delta }_{\max }}{\widehat{\xi }}_{\mathrm{k}}\delta }\left|{\dot{\delta }}_{\mathrm{f}}\right|\sqrt{\mathrm{1}-{(\delta /{\delta }_{\max })}^{\mathrm{2}}}\mathrm{d}\delta ,

where \mathrm{\Delta }{E}_{\mathrm{1}} is equal to the dissipated energy because of the damping force during compression period, and \mathrm{\Delta }{E}_{\mathrm{2}} stands for the dissipated energy at the restitution period. After simple integral calculation,

(27)\mathrm{\Delta }E=\mathrm{\Delta }{E}_{\mathrm{1}}+\mathrm{\Delta }{E}_{\mathrm{2}}={\displaystyle \frac{\mathrm{1}}{\mathrm{3}}}{\widehat{\xi }}_{\mathrm{k}}{\dot{\delta }}_{\mathrm{0}}{\delta }_{\max }^{\mathrm{2}}+{\displaystyle \frac{\mathrm{1}}{\mathrm{3}}}{\widehat{\xi }}_{\mathrm{k}}\left|{\dot{\delta }}_{\mathrm{f}}\right|{\delta }_{\max }^{\mathrm{2}}.

Considering the momentum and energy balance between the start and the end of the compression phase, we have

(28)\begin{array}{ll}{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{m}_{\mathrm{1}}{v}_{\mathrm{1}}^{\mathrm{2}}+{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{m}_{\mathrm{2}}{v}_{\mathrm{2}}^{\mathrm{2}}=U_{\mathrm{m}}+\mathrm{\Delta }{E}_{\mathrm{1}}+{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}(m_{\mathrm{1}}+m_{\mathrm{2}})V^{\mathrm{2}},& m_{\mathrm{1}}{v}_{\mathrm{1}}+m_{\mathrm{2}}{v}_{\mathrm{2}}=(m_{\mathrm{1}}+m_{\mathrm{2}})V,\end{array}

where U_{\mathrm{m}} and \mathrm{\Delta }{E}_{\mathrm{1}} are the stored maximum strain energy and the dissipated energy because of the damping force during compression period, respectively. Their mathematical expressions are as follows:

(29)\begin{array}{ll}{U}_{\mathrm{m}}={\displaystyle {\int }_{\mathrm{0}}^{{\delta }_{\max }}{k}_{\mathrm{k}}}\delta \mathrm{d}\delta ={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{k}_{\mathrm{k}}{\delta }_{\max }^{\mathrm{2}},& \mathrm{\Delta }{E}_{\mathrm{1}}={\displaystyle \frac{\mathrm{1}}{\mathrm{3}}}{\widehat{\xi }}_{\mathrm{k}}{\dot{\delta }}_{\mathrm{0}}{\delta }_{\max }^{\mathrm{2}}\mathrm{.}\end{array}

Substituting Eq. (29) into Eq. (28), the formulation for {\delta }_{\max }^{\mathrm{2}} can be evaluated as

(30){\delta }_{\max }^{\mathrm{2}}={\displaystyle \frac{m_{\mathrm{1}}{m}_{\mathrm{2}}}{m_{\mathrm{1}}+m_{\mathrm{2}}}}{\displaystyle \frac{\mathrm{3}}{\mathrm{2}{\widehat{\xi }}_{\mathrm{k}}{\dot{\delta }}_{\mathrm{0}}+\mathrm{3}{k}_{\mathrm{k}}}}{\dot{\delta }}_{\mathrm{0}}^{\mathrm{2}}.

Substituting {\delta }_{\max }^{\mathrm{2}} from Eq. (30) and equating energy loss Eqs. (26) and (18) together with the definition of the coefficient of restitution e(=\left|{\dot{\delta }}_{\mathrm{f}}\right|/{\delta }_{\mathrm{0}}), approximate expression for damping constant {\widehat{\xi }}_{\mathrm{k}} can be found in terms of the spring stiffness (k_{\mathrm{k}}), the coefficient of restitution (e), and the relative approaching velocity (v_{\mathrm{1}}-v_{\mathrm{2}}):

(31){\widehat{\xi }}_{\mathrm{k}}={\displaystyle \frac{\mathrm{3}{k}_{\mathrm{k}}(\mathrm{1}-e)}{\mathrm{2}e{\dot{\delta }}_{\mathrm{0}}}}={\displaystyle \frac{\mathrm{3}{k}_{\mathrm{k}}(\mathrm{1}-e)}{\mathrm{2}e(v_{\mathrm{1}}-v_{\mathrm{2}})}}.

Numerical verification of formula for damping constant {\widehat{\xi }}_{\mathrm{h}}

Having derived the formula for damping constant ({\widehat{\xi }}_{\mathrm{k}}), it is necessary to verify the correctness of the formula and reasonability of the corresponding theoretical derivation. Noting that the formula for damping constant ({\widehat{\xi }}_{\mathrm{k}}) is related to the coefficient of restitution (e), therefore, the verification can be conducted using the following procedures: 1) selecting a case of pounding simulation; 2) prespecifying a value of coefficient of restitution e_{\mathrm{p}\mathrm{r}\mathrm{e}}; 3) performing the pounding simulation; 4) calculating the coefficient of restitution e_{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{t}} from the results of the proceeding step; and 5) evaluating the difference between e_{\mathrm{p}\mathrm{r}\mathrm{e}} and e_{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{t}} by computing the relative error (=|e_{\mathrm{p}\mathrm{r}\mathrm{e}}-e_{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{t}}|/e_{\mathrm{p}\mathrm{r}\mathrm{e}}). Theoretically, e_{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{t}} should be equal to e_{\mathrm{p}\mathrm{r}\mathrm{e}}. Repeating the steps from 2) to 3), comparison in the case of various e_{\mathrm{p}\mathrm{r}\mathrm{e}} can be made.

The pounding between a falling spherical ball and a stationary rigid surface, shown in Fig. 3,, is selected as the case of pounding simulation. The dynamic equation of motion for such a model can be written as

(32)m\ddot{y}(t)+F(t)=mg,

where m is the mass of the ball, \ddot{y}(t) is its vertical acceleration, g(=9.8 m/s²) stands for the acceleration of gravity, and F(t) is the pounding force, which is equal to zero when y(t)\le h (h is a drop height) and is defined by Eq. (16) when y(t)>h, where interpenetration \delta (t) is expressed as

(33)\delta (t)=y(t)-h.

In the numerical analysis, the following parameters have been used: m=\mathrm{1.0} kg, h=\mathrm{0.5} m, and k_{\mathrm{k}}=\mathrm{2.0}\times {\mathrm{10}}^{\mathrm{7}} N/m. The prespecifying coefficient of restitution e_{\mathrm{p}\mathrm{r}\mathrm{e}} varies from 0.1 to 1.0, with an interval of 0.1. Fourth-order Runge-Kutta with adaptive time step has been applied to solve Eq. (32) numerically.

Comparison of e_{\mathrm{p}\mathrm{r}\mathrm{e}} and e_{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{t}} is listed in Table 1. It can be seen from the table that difference between e_{\mathrm{p}\mathrm{r}\mathrm{e}} and e_{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{t}} is relatively small in the case of e_{\mathrm{p}\mathrm{r}\mathrm{e}}>\mathrm{0.3} and somewhat large when , which indicates that the assumption for relationship between interpenetration velocity and interpenetration is reasonable for the case of e_{\mathrm{p}\mathrm{r}\mathrm{e}}>\mathrm{0.3} and unsuitable for the case of . According to the conclusions in Ref. [10], the coefficient of restitution used to simulate real collision between structures varies in the range of 0.5 and 0.75. Therefore, the correctness of the formula and reasonability of the corresponding theoretical derivation is verified, and reliable results of pound simulation in structural engineering can be provided using the modified Kelvin model proposed in this paper. Although the assumption that c is equal to zero and k is supposed to be linear in the equivalent SDOF system is unreasonable, the target using the unreasonable assumption is to obtain approximating relationship between penetration and penetration velocity. Through the numerical verification, it can be found that such unreasonable assumption used in the theoretical derivation can be acceptable.

The correctness of Eq. (31) for damping constant {\widehat{\xi }}_{\mathrm{k}} in the modified Kelvin model and reasonability of the corresponding theoretical derivation have been verified by the numerical experiment. Therefore, the correction of Eq. (13) for damping constant {\widehat{\xi }}_{\mathrm{h}} in the Hertz damp model can be derived in the similar procedure, and the corrected formula for damping constant {\widehat{\xi }}_{\mathrm{h}} is as follows:

(34){\widehat{\xi }}_{\mathrm{h}}={\displaystyle \frac{\mathrm{8}}{\mathrm{5}}}{\displaystyle \frac{k_{\mathrm{h}}(\mathrm{1}-e)}{e{\dot{\delta }}_{\mathrm{0}}}}.

The correctness of Eq. (34) for damping constant {\widehat{\xi }}_{\mathrm{h}} is numerically verified as well, which are not presented here because of the similarity of numerical verification.

Conclusions

Structural pounding under earthquake has been recently extensively investigated using various impact analytical models. In this paper, a brief review on the commonly used impact analytical models is conducted. Based on this review, the formula used to determine the damping constant related to the impact spring stiffness, coefficient of restitution, and relative approaching velocity in the Hertz model with nonlinear damping is found to be incorrect. To correct this error, a more accurate approximating formula for the damping constant is theoretically derived and numerically verified. At the same time, a modified Kelvin impact model, which can reasonably account for the physical nature of pounding and conveniently implemented in the earthquake-induced pounding simulation of structural engineering is proposed.

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