An effective reduction method with Caughey damping for spurious oscillations in dynamic problems

Open Access
12-08-2025
Research

Published in:

Authors: Dániel Serfözö; Balázs Pere
Published in: Meccanica | Issue 9/2025

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Patentsearch

Off

Abstract

This article delves into the persistent issue of spurious oscillations in dynamic problems, particularly in contact and impact scenarios. It introduces a novel approach that leverages the Central Difference Method (CDM) coupled with Caughey damping to mitigate these oscillations effectively. The article meticulously examines various damping characteristics, including degressive, linear, and progressive types, and compares their efficacy in reducing spurious oscillations. Through rigorous testing on both one-dimensional (1D) and two-dimensional (2D) models, the proposed method demonstrates superior performance in maintaining accuracy while significantly reducing oscillations. The study also explores the impact of different parameters on the damping characteristics and provides insights into the optimal settings for practical applications. Furthermore, the article offers a detailed comparison with widely used methods like the HHT- and Bulk Viscosity (BV) methods, showcasing the advantages of the proposed approach. The results indicate that the proposed method can be effectively applied to more complex models, making it a valuable tool for engineers and researchers in the field of mechanical engineering.

AI Generated

This summary of the content was generated with the help of AI.

Notes

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

1 Introduction

Spurious oscillations have a numerical origin and are problematic in many fields of mechanical engineering. Numerical solutions of dynamic problems are especially prone to include undesired oscillations emerging from varying sources. These oscillations have adverse effects on the solution, distorting its accuracy or even causing divergence. The primary origin of spurious oscillations is the spatial discretization which is mainly realized by the Finite Element Method (FEM) [1]. As the continuum can only be considered approximately in the Finite Element (FE) model, the highest eigenmodes are not included in the solution [2]. This approximation may induce severe oscillations in the solution that must be eliminated or significantly reduced.

Many types of dynamic problems are concerned with the adverse effect of spurious oscillations. One of the main fields where these oscillations usually cause serious difficulties are contact problems [3]. Similar concerns are involved in impact problems [4, 5] which signify a special case of contact problems. Undesired oscillations are present in the time evolution of velocity and contact pressure in both frictional [6] and non-frictional [7] models. The numerical modeling of wave propagation problems [8, 9] are also affected by spurious oscillations. In fluid dynamics, the numerical solution of equations of gas dynamics contains numerical distortions too. The pressure and the temperature distribution of shock waves are both susceptible to include undesired oscillations [10]. Besides, the appropriate numerical treatment of supersonic flows are especially problematic due to the wave’s interaction with the boundaries of the model [11].

Spurious oscillations may be treated according to many different approaches. Most attempts aim to reduce the oscillations with a damping effect included in the numerical method. This damping can be exerted in many ways, but there are two main approaches. One branch of numerical methods operates with numerical damping that is involved in the scheme. This kind of methods can be subdivided into forward [12, 13] and backward increment [14, 15], single-stage [16, 17] and multistage [18] schemes. They also may be distinguished whether the damping effect may be altered [19, 20] or not [21]. Besides, the shape of the damping characteristics is also important, as it can be both progressive [18], degressive [20] or mixed [22, 23]. The well-known Newmark method [24] possesses a backward increment scheme and a controllable numerical damping with one parameter. As it is unconditionally stable and simply formulated, the Newmark method is the standard time stepping method in many FE software like Abaqus or Ansys. The improved version of the Newmark scheme, the HHT-$\alpha$ method [20] is also widely applied. It has a degressive damping character that can be set with an additional $\alpha$ parameter. Kim’s method [18] represents a novel, two-stage formulation with progressive damping characteristics. On one hand, its forward increment scheme requires less computational capacity, but the amount damping which may be exerted is very limited. The method developed by Kolay and Ricles [22] (KR-$\alpha$) as well as the scheme of Noh and Bathe ($\rho _{\infty }$-Bathe) [23] possess a mixed damping character whose initial stage is progressive and changes to degressive at the higher eigenfrequencies.The KR-$\alpha$ and the $\rho _{\infty }$-Bathe schemes both have second order accuracy and unconditional stability. The KR-$\alpha$ method has a forward increment formulation, while the $\rho _{\infty }$-Bathe method’s formulation is implicit.

The second possibility to exert damping is the application of a viscous damping term in the FE model [25]. This approach was firstly described by von Neumann and Richtmeyer in [26] whose method has been named as Bulk Viscosity (BV). The damping characteristics of the BV method is linear, its detailed formulation can be found in [27]. This scheme is usually coupled with the Central Difference Method (CDM) [28] which does not contain numerical damping, so the aimed damping effect can be directly exerted through the viscous damping term. This technique has been applied in many industrial FE software, like Ansys [29] and Abaqus [30]. In contrast, the Caughey damping [31] provides a more versatile definition for the damping character. Using this kind of damping, a wide range of damping curves may be approximated with high accuracy [32]. Applying the Caughey damping is not a resource intensive procedure, as its composition only requires the addition and multiplication of small bandwidth matrices. Moreover, the Caughey damping has already been applied for solving wave propagation problems [33], so it is relevant to use it for reducing the spurious oscillations.

Solving contact and impact problems is an important field, where the reduction of spurious oscillations is a crucial aspect. The treatment of the contact may be realized using two main approaches. On one hand, the penalty method [34] can be applied whose formulation is simpler, but its accuracy lags behind the Lagrange multiplier method [35] which is the second approach for treating contact. Here, the contact pressure, an additional variable must be considered in the equation of motion making the numerical solution computationally more expensive. Nevertheless, the more accurate solution has been favoured in this article, so the latter method has been applied for handling contact.

In this paper, a novel approach is presented aiming to reduce the spurious oscillations in dynamic problems. The proposed technique is based on the CDM, while the damping effect is exerted as viscous damping. The viscous term is formulated as Caughey damping which provides an accurate fitting to certain kind of damping characters. The aimed damping curve is defined for the Algorithmic Damping Ratio (ADR) according to [36]. In order to verify the newly developed method, it is tested on a standard test example of an impact problem. The main improvement of the research is the effective reduction of spurious oscillations by keeping the accuracy level as high as possible.

2 Methodology

2.1 Discretization scheme

In order to deal with the definition of the damping characteristics, the equation of motion must be formulated. For the spatial discretization FEM is used, while the temporal discretization is realized by the CDM. The corresponding equation of motion can be written as a system of semi-discretized second-order ordinary differential equations.

$$\begin{aligned} \textbf{M}\ddot{\textbf{u}}(t)+\textbf{C}\dot{\textbf{u}}(t)+\textbf{K}\textbf{u}(t)=\textbf{f}(t) \end{aligned}$$

(1)

where $\textbf{M}$, $\textbf{C}$ and $\textbf{K}$ mean the mass, damping and stiffness matrices, respectively. On the right hand side of the formula, $\textbf{f}(t)$ denotes the load vector, $\textbf{u}(t)$ is the nodal displacement vector, $\dot{\textbf{u}}(t)$ means the nodal velocity vector, while $\ddot{\textbf{u}}(t)$ denotes the nodal acceleration vector in the function of time t. In addition, the solution of Eq. (1) requires both boundary and initial conditions. The boundary conditions depend on the considered model, while the initial conditions must be given at $t=0$ as

$$\begin{aligned} \textbf{u}(0)=\textbf{u}_{0}\qquad {\hbox {and}}\qquad \dot{\textbf{u}}(0)=\textbf{v}_{0} \end{aligned}$$

(2)

where $\textbf{u}_{0}$ is the initial displacement vector, while $\textbf{v}_{0}$ denotes the initial velocity vector at $t=0$.

2.1.1 The central difference method

The main reasons of choosing the CDM for temporal discretization are the forward increment formulation and the second order accuracy. Another important and beneficial feature for this application is the lack of numerical damping, so the damping effect can be exerted directly as viscous damping. The major drawback of the CDM is the conditional stability. However, in the case of dynamic problems, sufficiently accurate calculations usually requires time steps significantly lower than the critical value. The original CDM can be given as the formulae of the velocity and the acceleration at the current time step t in an incremental form the following way.

$$\begin{aligned} \ddot{\textbf{u}}_i=\dfrac{\textbf{u}_{i+1}-2\textbf{u}_{i}+\textbf{u}_{i-1}}{\Delta t^{2}} \end{aligned}$$

(3)

$$\begin{aligned} \dot{\textbf{u}}_{i}=\frac{\textbf{u}_{i+1}-\textbf{u}_{i-1}}{2\Delta t} \end{aligned}$$

(4)

where $\Delta t$ denotes the time step size, while $i+1$, i and $i-1$ increments mean the numerical values at time step $t+\Delta t$, t and $t-\Delta t$, respectively. By substituting Eqs. (3) and (4) into Eq. (1) and rearranging it to $\textbf{u}_{i+1}$, the following linear system of equations shall be obtained.

$$\begin{aligned} \left( \Delta t\textbf{C}+2\textbf{M}\right) \textbf{u}_{i+1}={2\Delta }t^{2}\textbf{f}_{i}+\left( 4\textbf{M}-2\Delta t^{2}\textbf{K}\right) \textbf{u}_{i}+\left( \Delta t\textbf{C}-2\textbf{M}\right) \textbf{u}_{i-1} \end{aligned}$$

(5)

However, applying the formulation above is not beneficial, as it requires the inversion of term $\Delta t\varvec{\varvec{\hbox {C}}}+2\varvec{\textrm{M}}$ which is a computationally expensive procedure. Hence, Eq. (4) is substituted with the following formula to calculate $\dot{\textbf{u}}_{i}$.

$$\begin{aligned} \dot{\textbf{u}}_{i}=\frac{\textbf{u}_{i}-\textbf{u}_{i-1}}{\Delta t} \end{aligned}$$

(6)

Thus, substituting Eqs. (3) and (6) into Eq. (1) and deducting it to $\textbf{u}_{i+1}$ the following equation can be determined.

$$\begin{aligned} \textbf{M}\textbf{u}_{i+1}=\Delta t^{2}\textbf{f}_{i}+\left( 2\textbf{M}-\Delta t\textbf{C}-\Delta t^{2}\textbf{K}\right) \textbf{u}_{i}+\left( \Delta t\textbf{C}-\textbf{M}\right) \textbf{u}_{i-1} \end{aligned}$$

(7)

This formulation still does not contain numerical damping, but it only requires the inversion of $\textbf{M}$. Furthermore, the mass matrix can be diagonalized according to [37] which gives the so-called lumped mass matrix whose inversion is computationally cheap. The appropriate definition of the damping character requires the knowledge of the amplification matrix for the CDM. Thus, it is practical to rearrange Eq. (7) as the following.

$$\begin{aligned} \left[ \begin{array}{c} \textbf{u}_{i+1}\\ \textbf{u}_{i} \end{array}\right] =\textbf{A}^{\textrm{CDM}}\left[ \begin{array}{c} \textbf{u}_{i}\\ \textbf{u}_{i-1} \end{array}\right] +\textbf{b}^{\textrm{CDM}} \end{aligned}$$

(8)

where $\textbf{A}^{\textrm{CDM}}$ means the amplification matrix and $\textbf{b}^{\textrm{CDM}}$ contains the inhomogeneous components of Eq. (7). For later examinations, only the former quantity shall be substantial. For the CDM, this can be given as follows.

$$\begin{aligned} \textbf{A}^{\textrm{CDM}}=\left[ \begin{array}{cc} 2\textbf{I}-\Delta t\textbf{M}^{-1}\textbf{C}-{\bigtriangleup }t^{2}\textbf{M}^{-1}\textbf{K} \quad & \quad \Delta t\textbf{M}^{-1}\textbf{C}-\textbf{I}\\ \textbf{I} \quad & \quad \textbf{0} \end{array}\right] . \end{aligned}$$

(9)

where $\textbf{I}$ denotes the identity matrix and $\textbf{0}$ means the zero matrix.

2.1.2 Treatment of contact

The appropriate definition of boundary conditions between contacting bodies is an important aspect to be considered in a contact problem. For this purpose, the Lagrange multiplier method has been applied in this article, as it had already been mentioned in Sect. 1. According to this approach, the penetration between the contacting bodies is completely prohibited. Thus, the contact condition for the displacements is formulated in the following way.

$$\begin{aligned} \mathbf {G (u+X)=0} \end{aligned}$$

(10)

where $\textbf{X}$ means the vector of initial configuration, $\textbf{u}$ denotes the displacement vector, while $\textbf{G}$ represents the contact constraint matrix. Besides, regarding the contact pressure between contacting bodies, the equation of motion (Eq. (1)) must be modified and rewritten in the following form.

$$\begin{aligned} \textbf{M}\ddot{\textbf{u}}_i+\textbf{C}\dot{\textbf{u}}_i+\textbf{K}\textbf{u}_i + \textbf{G}_i^{\textrm{T}} \lambda _i=\textbf{f}_i \end{aligned}$$

(11)

where $\lambda _i$ means the Lagrange multiplier at the $i^{\textrm{th}}$ time instant and it is equivalent with the surface contact pressure. A more detailed description of this formulation can be found in [4]. Considering Eqs. (10) and (11), the deduction for the CDM presented in Sect. 2.1.1 can easily be performed by replacing Eq. (1) with Eq. (11). The amplification matrix ($\textbf{A}^{\textrm{CDM}}$) given by Eq. (9) remain unchanged, as the additional Lagrange multiplier term in Eq. (11) is independent from the the displacement and its derivatives.

2.2 Damping formulation

In the proposed method, the aimed damping characteristics shall be prescribed as the function of the circular eigenfrequencies. Albeit, the composition of the damping matrix does not demand the calculation of the eigenmodes, the appropriate description of the proposed method requires the transformation of the equation of motion into the frequency domain. Thus, the displacement vector can be written as

$$\begin{aligned} \textbf{u}(t)=\mathbf {T\,q}(t) \end{aligned}$$

(12)

where $\textbf{T}$ means the transformation matrix, while $\textbf{q}$ denotes the vector of the modal coordinates. The former term is composed based on the mass-normalized eigenvectors in the following way.

$$\begin{aligned} \textbf{T}=\left[ \begin{array}{ccc} c_{1}\textbf{a}_{1}&\ldots&c_{n}\textbf{a}_{n}\end{array}\right] \qquad {\hbox {where}}\qquad c_{k}=\frac{1}{\sqrt{\textbf{a}_{k}^{\textrm{T}}\textbf{M}\textbf{a}_{k}}} \end{aligned}$$

(13)

In the formulae above, $\textbf{a}_{k}$ denotes the $k^{\textrm{th}}$ eigenvector of the considered FEM model. Substituting Eq. (13) into Eq. (1) and multiplying it by $\textbf{T}^{\textrm{T}}$ on the left side, the following equation can be obtained.

$$\begin{aligned} \textbf{T}^{\textrm{T}}\textbf{M}\textbf{T}\ddot{\textbf{q}}(t)+\textbf{T}^{\textrm{T}}\textbf{C}\textbf{T}\dot{\textbf{q}}(t)+\textbf{T}^{\textrm{T}}\textbf{K}\textbf{T}\textbf{q}(t)=\textbf{T}^{\textrm{T}}\textbf{f} \end{aligned}$$

(14)

where the coefficients of $\ddot{\textbf{q}}$, $\dot{\textbf{q}}$ and $\textbf{q}$ are the corresponding terms for $\textbf{M}$, $\textbf{C}$ and $\textbf{K}$ in the frequency domain, respectively.

$$\begin{aligned} & \textbf{T}^{\textrm{T}}\textbf{M}\textbf{T}=\textbf{I} \end{aligned}$$

(15)

$$\begin{aligned} & \textbf{T}^{\textrm{T}}\textbf{K}\textbf{T}=\left[ \begin{array}{ccc} \ddots & & 0\\ & \omega _{k}^{2}\\ 0 & & \ddots \end{array}\right] \end{aligned}$$

(16)

$$\begin{aligned} & \textbf{T}^{\textrm{T}}\textbf{C}\textbf{T}=\left[ \begin{array}{ccc} \ddots & & 0\\ & 2\xi _{k}\omega _{k}\\ 0 & & \ddots \end{array}\right] \end{aligned}$$

(17)

where $\omega _{k}$ means the $k^{\textrm{th}}$ circular eigenfrequency, while $\xi _{k}$ denotes the corresponding Physical Damping Ratio (PDR). Thus, the equation of motion (Eq. (1)) can be rewritten as a series of uncoupled Single Degree of Freedom (SDoF) equations as

$$\begin{aligned} \ddot{q}_{k}\left( t\right) +2\xi _{k}\omega _{k}\dot{q}_{k}\left( t\right) +\omega _{k}^{2}q_{k}\left( t\right) ={\varphi }_{k}\left( t\right) \end{aligned}$$

(18)

where

$$\begin{aligned} \textbf{T}^{\textrm{T}}\textbf{f}=\left[ \begin{array}{c} {{\varphi }}_{1}\\ \vdots \\ {{\varphi }}_{n} \end{array}\right] \end{aligned}$$

(19)

Using Eqs. (15)–(17), the amplification matrix in the modal domain can be written as follows.

$$\begin{aligned} \textbf{A}_{\textrm{SDOF}}^{\textrm{CDM}}=\left[ \begin{array}{cc} 2-2\Delta t\xi \omega -\Delta t^{2}\omega ^{2}\quad & \quad 2\Delta t\xi \omega -1\\ 1 \quad & \quad 0 \end{array}\right] \end{aligned}$$

(20)

2.2.1 The algorithmic damping

The accurate determination of an aimed damping characteristics requires the calculation of the Algorithmic Damping Ratio (ADR). At first, the eigenvalues of the amplification matrix (Eq. (20)) must computed based on the following formula.

$$\begin{aligned} \lambda _{1,2}=\sigma \pm \varepsilon i={\hbox {exp}}\left[ \overline{\varOmega }\left( -\overline{\xi }\pm i\sqrt{1-\overline{\xi }^{2}}\right) \right] \end{aligned}$$

(21)

where $\sigma$ and $\varepsilon$ are the real and imaginary parts of the eigenvalues, respectively. $\overline{\xi }$ denotes the ADR, while $\overline{\varOmega }=\overline{\omega }\Delta t$ where $\overline{\omega }$ means the algorithmic circular eigenfrequency. Knowing $\lambda _{1,2}$, $\overline{\varOmega }$ can be calculated in the following way.

$$\begin{aligned} \overline{\varOmega }=\frac{{\hbox {arctan}}\left( {\displaystyle \frac{\varepsilon }{\sigma }}\right) }{\sqrt{1-\overline{\xi }^{2}}} \end{aligned}$$

(22)

Using the formula above, the ADR shall be written as

$$\begin{aligned} \overline{\xi }=\frac{-{\hbox {ln}}\left( \sigma ^{2}+\varepsilon ^{2}\right) }{2\overline{\varOmega }}. \end{aligned}$$

(23)

Applying the above described calculation method of the ADR for $\textbf{A}_{\textrm{SDOF}}^{\textrm{CDM}}$ given by Eq. (20), $\overline{\xi }$ can be obtained according to the following two stage definition.

$$\begin{aligned} \hat{\xi }=\frac{{\hbox {ln}}\left( 1-2\xi \Delta t\omega \right) }{2\,{\hbox {atan}}\left( {\displaystyle \frac{\Delta t\omega \sqrt{4-\left( \Delta t\omega +2\xi \right) ^{2}}}{\left( \Delta t\omega \right) ^{2}+2\xi \Delta t\omega -2}}\right) } \end{aligned}$$

(24)

$$\begin{aligned} \overline{\xi }=\frac{\hat{\xi }}{\sqrt{1+\hat{\xi }^{2}}} \end{aligned}$$

(25)

The formula above determines the damping characteristics that actually prevails in the numerical solution. The aimed damping coefficients belonging to each eigenfrequency must be defined as the ADR. As it has been proved in [36], this is the only way to compose the equivalent $\textbf{C}$ matrix accurately. Knowing the characteristics in $\overline{\xi }$, the belonging $\xi$ may only be calculated by numerical root finding methods, as it cannot be expressed analytically. For this purpose, the Krawczyk method [38] offers an accurate solution with reasonable computational demand.

As it has been deducted in [36], there is a great freedom in the determination of the damping curve. In the region of $0\le \overline{\xi }<1$ any arbitrary value can be given for every $\omega$, by any $\Delta t<\Delta t_{\textrm{max}}$ where $\Delta t_{\textrm{max}}$ is the maximal applicable time step size. For practical reasons detailed in [36], $\Delta t_{\textrm{max}}$ is given at the bifurcation point according to the following equation.

$$\begin{aligned} \Delta t_{\textrm{max}}=\Delta t_{\textrm{b}}={\hbox {min}} \left( \frac{2}{\omega } \left( 1-\xi \right) \right) \end{aligned}$$

(26)

2.2.2 Matrix composition

In order to implement an aimed damping character, the belonging damping matrix must be composed. For the effective usability, matrix $\textbf{C}$ shall be defined as Caughey damping. The Caughey damping can be given according to the formulation presented by Adhikari [32] as

$$\begin{aligned} \textbf{C}=\textbf{M}\sum _{j=0}^{N-1}\alpha _{j}\left( \textbf{M}^{-1}\textbf{K}\right) ^{j} \end{aligned}$$

(27)

where $\alpha _{j}$ denotes the coefficient belonging to $\textbf{M}^{-1}\textbf{K}$ on the $j^{\textrm{th}}$ power, while N means the number of terms considered in the summation. Transforming the equation above into the frequency domain using Eqs. (15)–(17) results a following formulation.

$$\begin{aligned} 2\xi _{k}\omega _{k}=\sum _{j=0}^{N-1}\alpha _{j}\omega _{k}^{2j} \end{aligned}$$

(28)

which can rewritten as an algebraic system of equations according to

$$\begin{aligned} \left[ \begin{array}{cccc} 1 & \omega _{1}^{2} & \omega _{1}^{4}\ldots & \omega _{1}^{2N-2}\\ 1 & \omega _{2}^{2} & \omega _{2}^{4}\ldots & \omega _{2}^{2N-2}\\ \vdots & \vdots & \vdots & \vdots \\ 1 & \omega _{n}^{2} & \omega _{n}^{4}\ldots & \omega _{n}^{2N-2} \end{array}\right] \left\{ \begin{array}{c} \alpha _{0}\\ \alpha _{1}\\ \vdots \\ \alpha _{N-1} \end{array}\right\} =\left\{ \begin{array}{c} 2\omega _{1}\xi _{1}\\ 2\omega _{2}\xi _{2}\\ \vdots \\ 2\omega _{n}\xi _{n} \end{array}\right\} \end{aligned}$$

(29)

Rearranging Eq. (28) to $\xi _{k}$ results the following polynomial formulation for the PDR.

$$\begin{aligned} & \xi _{k}=\frac{1}{2}{\mathop {j}\limits ^{[}}=0]{N-1}{\sum }\alpha _{j}\omega _{k}^{2j-1}\nonumber \\ & \quad =\frac{1}{2}\left( \frac{\alpha _{0}}{\omega _{k}}+\alpha _{1}\omega _{k}+\alpha _{2}\omega _{k}^{3}+\ldots +\alpha _{N-1}\omega _{k}^{2N-3}\right) \end{aligned}$$

(30)

Thus, the damping characteristics can be defined as the linear combination of the odd powers of $\omega _{k}$. The lack of even powers results that calculating the square roots of the mass and stiffness matrices is not required. Moreover, this formulation allows a very versatile determination for the damping character. Using the polynomial above, several different damping curve shapes may be defined from degressive to linear and progressive. The greatest benefit of determining the aimed damping character this way is that it only requires the computation of $\omega _{\textrm{max}}$, so the calculation of every eigenmode is not necessary. This feature is substantial for the widespread application of the proposed method, as the solution of the frequency equation is a computationally expensive procedure. Instead, the polynomial characteristics shall be determined by a damping matrix composed using the appropriate linear combination of the mass and stiffness matrices. This approach demands only the addition and multiplication of small bandwidth matrices which are not resource-intensive operations. Moreover, the resulting damping matrix’s bandwidth remains relatively small. Thus, it can be treated as a sparse matrix which requires much less memory capacity compared to dense matrices.

2.3 Application of a specific damping character

When choosing a concrete damping characteristics, the reasonable definition through Caughey damping and the effective reduction of spurious components are major aspects to be considered. The PDR character resulted by Caughey damping only contains terms of $\omega$ of odd powers (see. Equation (30)). This feature provides a very precise approximation for functions whose Taylor series expansion only contains terms of odd powers. In addition, the precise adjustability in a wide range is an important feature too. As the hyperbolic sine function fulfills all of these criteria, it shall be very promising to apply it for reducing spurious oscillations.

$$\begin{aligned} \overline{\xi } \left( \omega \right) = \frac{c_2}{{\hbox {sh}}\left( c_1 \right) } \; {\hbox {sh}} \left( \frac{c_1}{\omega _{\textrm{max}}} \omega \right) \end{aligned}$$

(31)

For the sake of practical applicability, the damping character has been prescribed by two parameters according to Eq. (31). Thus, a very versatile definition is possible for the damping curve, as the maximal value of the damping curve at $\omega _{\textrm{max}}$ can be adjusted by $c_2$, while the curvature of the character (which also means the initial slope) shall be influenced by $c_1$. Defining the damping character in this way is unique, it is the most important feature of the proposed method. Similar definition had already been applied in the past work of authors [39], but the above described form proved to be more straightforward and practical.

2.3.1 Composition of damping matrix

As it has been written in Sect. 2.2.2, a practical way to exert an aimed damping character is the application of Caughey damping. In order to write Eq. (31) in a polynomial form which is needed in Eq. (30), it has to be expanded into a Taylor-series.

$$\begin{aligned} \frac{c_2}{{\hbox {sh}}\left( c_1 \right) } \; {\hbox {sh}} \left( \frac{c_1}{\omega _{\textrm{max}}} \omega \right) \approx \frac{c_2}{{\hbox {sh}}\left( c_1 \right) } \sum _{l=1}^{L} \frac{\left[ {\hbox {sh}} \left( \frac{c_1}{\omega _{\textrm{max}}} \omega \right) \right] ^{(l)} (\omega =0)}{l !} \omega ^l \end{aligned}$$

(32)

The Taylor-series of the damping curve given by Eq. (31) is defined as Eq. (32), where ${\hbox {sh}} \left( c_1 \omega / \omega _{\textrm{max}} \right)$ can be expanded in the following way.

$$\begin{aligned} {\hbox {sh}} \left( \frac{c_1}{\omega _{\textrm{max}}} \omega \right) \approx \frac{c_1}{\omega _\textrm{max}} \omega + \frac{1}{6} \left( \frac{c_1}{\omega _\textrm{max}} \right) ^3 \omega ^3 +\ldots + \frac{1}{L !} \left( \frac{c_1}{\omega _\textrm{max}} \right) ^L \omega ^L \end{aligned}$$

(33)

The equation above is in accordance with Eq. (30), both of them contains only members of $\omega$ of odd powers. However, as the aimed damping curve must be determined for the ADR (see [36]), the polynomial in Eq. (33) cannot be used directly. Based on the character given as Eq. (31), the corresponding $\xi$ values (PDR) must be calculated using the Krawczyk root finding method [38].

$$\begin{aligned} \xi (\omega ) \approx \sum _{j=1}^{N} \frac{a_j}{\omega _{\textrm{max}}^{2j-1}} \omega ^{2j-1}= \frac{a_1}{\omega _{\textrm{max}}} \omega + \frac{a_2}{\omega _{\textrm{max}}^3} \omega ^3 +\ldots + \frac{a_N}{\omega _{\textrm{max}}^{2N-1}} \omega ^{2 N -1} \end{aligned}$$

(34)

Considering the resulting discrete points, the curve fitting of polynomial (34) can be done using the least squares method.

$$\begin{aligned} \alpha _j = 2 \frac{a_j}{\omega _{\textrm{max}}^{2j-1}} \end{aligned}$$

(35)

Knowing $a_j$ parameters, the coefficients in Eq. (27) can be calculated as Eq. (35). For this purpose, only the calculation of the maximal circular eigenfrequency must be performed. All the other $\omega$ values are free to choose within the interval of $(0,\omega _{\textrm{max}})$.

3 Results

3.1 Analysis of damping characteristics

3.1.1 Model

In order to examine the efficacy of different damping characters in reducing the spurious oscillations emerging in a contact problem, a simple numerical example has been considered. In Fig. 1 the mechanical model of a simplified impact problem is shown. The model represents the time moment of the impact, when the linear elastic rod is colliding with the rigid wall. The right side endpoint ($\textbf{B}$) of the rod of length l is fixed while a $v_{0}$ initial velocity is defined for every other point of the rod. Point $\textbf{A}$ signifies the centerpoint of the rod which shall be important for examinations in the forthcoming sections. Because of the fixed endpoint, an elastic deformation occurs alongside the x direction which can be described as a propagating wave. Similar examples are used oftentimes to test experimental numerical methods [2, 18, 40].

Fig. 1

The mechanical model of the wave propagation problem

Method name	Character type	Increment type	Parameter value
Newmark	Degressive	Backward	\(\delta =0.5\)
HHT-\(\alpha\)	Degressive		\(\alpha =-0.33\)
\(\rho _{\infty }\)-Bathe	Mixed		\(\rho _{\infty }=0\); \(\gamma =1.74\)
BV	Linear	Forward	\(c_b=0.05\)
KR-\(\alpha\)	Mixed		\(\rho _{\infty }=0.21\)
Kim	Progressive		\(\alpha _2=0.19\)

Springer Professional

Abstract

Publisher's Note

1 Introduction

2 Methodology

2.1 Discretization scheme

2.1.1 The central difference method

2.1.2 Treatment of contact

2.2 Damping formulation

2.2.1 The algorithmic damping

2.2.2 Matrix composition

2.3 Application of a specific damping character

2.3.1 Composition of damping matrix

3 Results

3.1 Analysis of damping characteristics

3.1.1 Model

3.1.2 Numerical methods

3.1.3 Comparison of numerical results

3.2 Accuracy of approximation

3.3 Parameter identification

3.4 Comparison with 1D model

3.4.1 The effect of time step size and element number

3.5 Comparison with 2D model

3.5.1 Identifying spurious oscillations

3.5.2 Comparison by indicators

3.5.3 Qualitative comparison

4 Discussion

4.1 Insights for calculations on the 1D model

4.2 Insights for calculations on the 2D model

5 Conclusion

Acknowledgements

Declarations

Conflict of interest

Publisher's Note

Experimental validation of the analytic-centre based tension distribution algorithm for cable driven parallel robots

An analytic iterative method for tension distribution of cable-driven robots with a variable objective point

Controlling nonlinear vehicular motions by exploiting linearized feedback law under delay-tolerance: stability, gain-scheduling, and validation

A systematic method for free and forced vibration analysis of axially loaded hybrid double-beam systems

Numerical analysis of ARJ21 passenger aircraft ditching dynamics using meshless methods

Probabilistic risk assessment for heavy duty vehicles rollover under crosswind environment