# Competitive Time Marching Solution Methods for Systems with Friction-Induced Nonlinearities

^{*}

## Abstract

**:**

## 1. Introduction

- a reliable model for the force-displacement relation at the contact interface;
- an efficient yet accurate technique to solve the nonlinear equilibrium equations that obtain the nonlinear response in a timely manner, compatible with design purposes.

- Section 2 will recapitulate the dynamic governing equations and will present the test case used as a demonstrator throughout this article.
- Section 3 will recapitulate the main features of the contact model and present an alternative way to represent contact forces, as a sequence of linear states.

## 2. Governing Equations

## 3. The Contact Model and Its Piecewise Linearity

- Lift-off: There is a gap between the platform and damper contact points (i.e., ${\mathrm{n}}_{\mathrm{iP}}-{\mathrm{n}}_{\mathrm{iD}}\le 0$), and contact forces are null $\mathrm{T}=\mathrm{N}=0$.
- Stick: There is interference between the platform and damper contact points (i.e., ${\mathrm{n}}_{\mathrm{iP}}-{\mathrm{n}}_{\mathrm{iD}}>0$). Furthermore, tangential relative displacement is limited, and a proportional relation between tangential forces and displacements exists: ${\mathrm{T}}_{\mathrm{i}}={\mathrm{k}}_{\mathrm{t}}\xb7\left({\mathrm{t}}_{\mathrm{iP}}-{\mathrm{t}}_{\mathrm{iD}}-{\mathrm{s}}_{\mathrm{i}}\right)$, where ${\mathrm{s}}_{\mathrm{i}}$ is the relative displacement of the tip of the slider with respect to the contact point on the damper.
- Slip: There is interference between the platform and damper contact points (i.e., ${\mathrm{n}}_{\mathrm{iP}}-{\mathrm{n}}_{\mathrm{iD}}>0$). Furthermore, the tangential relative displacement exceeds a threshold value, and a proportional relation between tangential and normal forces exists: ${T}_{i}=\mu \xb7{N}_{i}$.

**nonlinearity is obtained through a time sequence of linear systems**. The nonlinear system is actually a piecewise-linear system.

## 4. Direct Time Integration Tailored to Systems with Friction Nonlinearities

#### 4.1. Selection of the Direct Time Integration Method

- it is reasonably accurate, especially for small $\Delta \tau $, and does not introduce spurious damping and period elongation;
- it is suited for 2nd order ODEs, and no further processing is required;
- the method is implicit, which has a positive impact on the required time step size $\Delta \tau $ and on stability.

#### 4.2. Adaptive Time Step Selection for Direct Time Integration Methods

**Remark**

**1.**

#### 4.3. A Novel “Piecewise-Linear” Implementation for Direct Time Integration Methods

- the contact state changes only a few times per period of vibration (4–7 in cases encountered in this work);
- if the contact state does not change from time step ${\tau}_{n}$ to ${\tau}_{\mathrm{n}+1}$, the system can be considered perfectly linear (see Section 3)

## 5. A Time-Marching Method for Systems with Negligible Inertial Effects

#### 5.1. The Influence of Damper Dynamics on the Overall Damper Equilibrium

- the magnitude of the damper inertia forces;
- the damper “inner resonance” (i.e., the natural frequencies of the damper oscillating on the contact springs).

**Remark**

**2.**

**Remark**

**3.**

#### 5.2. Quasi-Static Solution Method

- the spurious numerical oscillations will disappear (see Figure 7), and
- inertial forces will cease to participate in the equilibrium, with negligible effects up to $\omega =2\xb7\pi \xb7$30 kHz.

## 6. A Critical Comparison of the Proposed Methods

- a
**standard**Newmark-$\beta $ algorithm, as described in Appendix A.3, to be used as a reference case, which applies standard techniques and no particular adaptation to friction damped systems; - a
**piecewise-linear**implementation of the Newmark-$\beta $ algorithm, as described in Section 4.3 and Appendix A.4; - a
**quasi-static**solution method, as described in Section 5 and Appendix B with a user-defined 300 time steps per period.

## 7. Conclusions

- the development of an adaptive time step selection for DTI algorithms resulting in a 20% decrease in the computational time;
- the development of a novel “piecewise-linear” formulation for the Newmark-$\beta $ scheme, adaptable to other DTI algorithms, and resulting in a 50% decrease in the computational time;
- the development of a novel quasi-static method to solve equilibrium equations where inertial effects are negligible, capable of reducing computational times by more than one order of magnitude.

## Author Contributions

## Conflicts of Interest

## Appendix A. Direct Time Integration

#### Appendix A.1. Classification of DTI Methods

Optimized for 1st Order ODEs ${}^{1}$ | for 2nd Order ODEs | ||
---|---|---|---|

General Multi-Steps | Runge-Kutta | Newmark’s Family | |

Methods | Methods | of Methods | |

Explicit | Forward Euler (I) | Classic (I-VIII) | Central Difference (I) |

Adams-Bashfort (I-V) | |||

Implicit | Trapezoidal Rule (II) | Constant average acc. (II) | |

Backward Euler (II) | Linear acceleration (I) | ||

Adams-Moulton (I–V) | Gauss-Legendre (II–X) | Wilson-$\theta $ (<II) | |

Houbolt (<II) |

#### Appendix A.1.1. Methods for 1st Order vs. 2nd Order ODEs

#### Appendix A.1.2. Explicit vs. Implicit

#### Appendix A.1.3. Stability, Accuracy and Convergence

#### Appendix A.2. The Newmark-β Method

#### Appendix A.3. Standard Implementation

- The values of ${}^{\mathrm{n}}{\mathbf{q}}_{\mathbf{D}}$, ${}^{\mathrm{n}}{\dot{\mathbf{q}}}_{\mathbf{D}}$, ${}^{\mathrm{n}}{\ddot{\mathbf{q}}}_{\mathbf{D}}$, ${\mathbf{f}}_{\mathbf{ED}}\left({\tau}_{\mathrm{n}+1}\right)$, and ${\mathbf{q}}_{\mathbf{iP}}\left({\tau}_{\mathrm{n}+1}\right)$ $\forall i$ are known either from initial conditions or the previous integration step.
- PREDICTOR: An assumption on one of the unknown vectors at ${\tau}_{\mathrm{n}+1}$ must be performed, e.g., ${}^{\mathrm{n}+1}\tilde{{\mathbf{q}}_{\mathbf{D}}}={}^{\mathrm{n}}{\mathbf{q}}_{\mathbf{D}}$ is postulated.
- The values of ${}^{\mathrm{n}+1}\tilde{{\dot{\mathbf{q}}}_{\mathbf{D}}}$ and ${}^{\mathrm{n}+1}\tilde{{\ddot{\mathbf{q}}}_{\mathbf{D}}}$ are computed using Newmark’s relations in Equation (A7) and the assumption at Step 2.
- Vector ${}^{\mathrm{n}+1}\tilde{{\mathbf{q}}_{\mathbf{D}}}$ is transformed into local displacements at the contact ${}^{\mathrm{n}+1}\tilde{{\mathbf{q}}_{\mathbf{iD}}^{*}}$ using Equation (4). The same is done with the displacements of the platform contact points, thus obtaining ${\mathbf{q}}_{\mathbf{iP}}^{*}({\tau}_{n}+\Delta \tau )$ $\forall i$.
- The vectors of local displacements at the contacts are fed into the contact model. Thus, the local vectors of contact forces ${}^{\mathrm{n}+1}\tilde{{\mathbf{f}}_{\mathbf{i}}^{*}}$ $\forall i$ are obtained and transformed into vector ${}^{\mathrm{n}+1}{\mathbf{f}}_{\mathbf{CD}}$ using the transformation matrices from Equation (4).
- The residual for Equation (2), also known as unbalanced force, is computed:$$\mathbf{R}={\mathbf{M}}_{\mathbf{D}}\xb7{}^{\mathrm{n}+1}{\tilde{\ddot{\mathbf{q}}}}_{\mathbf{D}}-\left({\mathbf{f}}_{\mathbf{ED}}\left({\tau}_{\mathrm{n}+1}\right){+}^{\mathrm{n}+1}{\tilde{\mathbf{f}}}_{\mathbf{C}}\right).$$
- The incremental displacement is computed:$$\Delta \mathbf{q}={\left(\frac{1}{\beta \Delta {\tau}^{2}}{\mathbf{M}}_{\mathbf{D}}\right)}^{-1}\mathbf{R}.$$
- CORRECTOR: If the incremental displacement is below a given tolerance $|\Delta \mathbf{q}|<\mathrm{tol}$, then the equilibrium is satisfied and the integration goes to ${\tau}_{\mathrm{n}+2}$. Otherwise, the vectors of unknowns are corrected as follows (corrections factors derived from Newmark’s relations):$$\begin{array}{cc}\hfill {}^{\mathrm{n}+1}{\widehat{\mathbf{q}}}_{\mathbf{D}}& ={}^{\mathrm{n}+1}{\tilde{\mathbf{q}}}_{\mathbf{D}}+\Delta \mathbf{q}\hfill \\ \hfill {}^{\mathrm{n}+1}{\widehat{\dot{\mathbf{q}}}}_{\mathbf{D}}& ={}^{\mathrm{n}+1}{\tilde{\dot{\mathbf{q}}}}_{\mathbf{D}}+\frac{\gamma}{\beta \Delta \tau}\Delta \mathbf{q}\hfill \\ \hfill {}^{\mathrm{n}+1}{\widehat{\ddot{\mathbf{q}}}}_{\mathbf{D}}& ={}^{\mathrm{n}+1}{\tilde{\ddot{\mathbf{q}}}}_{\mathbf{D}}+\frac{1}{\beta \Delta {\tau}^{2}}\Delta \mathbf{q}\hfill \end{array}.$$

#### Appendix A.4. Piecewise Linear Implementation

- The values of ${}^{\mathrm{n}}{\mathbf{q}}_{\mathbf{D}}$, ${}^{\mathrm{n}}{\dot{\mathbf{q}}}_{\mathbf{D}}$, ${}^{\mathrm{n}}{\ddot{\mathbf{q}}}_{\mathbf{D}}$, ${\mathbf{f}}_{\mathbf{ED}}\left({\tau}_{\mathrm{n}+1}\right)$, ${\mathbf{q}}_{\mathbf{iP}}\left({\tau}_{\mathrm{n}+1}\right)$ $\forall i$, and the contact states’ effective stiffness matrices ${}^{\mathrm{n}}{\mathbf{K}}_{\mathbf{i}}$ and ${}^{\mathrm{n}}{\mathbf{K}}_{\mathbf{Si}}$ are known either from initial conditions or the previous integration step.
- PREDICTOR: It is assumed that the contact state at ${\tau}_{n+1}$ is the same as that of the previous time step, i.e.,$$\begin{array}{cc}\hfill {}^{\mathrm{n}+1}{\tilde{\mathbf{K}}}_{\mathbf{i}}& {=}^{\mathrm{n}}{\mathbf{K}}_{\mathbf{i}}\hfill \\ \hfill {}^{\mathrm{n}+1}{\tilde{\mathbf{K}}}_{\mathbf{Si}}& {=}^{\mathrm{n}}{\mathbf{K}}_{\mathbf{Si}}.\hfill \end{array}$$It is therefore possible to express a tentative version of the equilibrium equations at time ${\tau}_{n+1}$ as (see Equation (7)):$$\begin{array}{c}\hfill {\mathbf{M}}_{\mathbf{D}}\xb7{}^{\mathrm{n}+1}{\tilde{\ddot{\mathbf{q}}}}_{\mathbf{D}}+\left(\sum _{\mathrm{i}=\mathrm{L}1}^{\mathrm{R}}\left({\Gamma}_{\mathbf{iD}}^{\prime}{\xb7}^{\mathrm{n}+1}{\tilde{\mathbf{K}}}_{\mathbf{i}}\xb7{\Gamma}_{\mathbf{iD}}\right)\right){\xb7}^{\mathrm{n}+1}{\mathbf{q}}_{\mathbf{D}}={\mathbf{f}}_{\mathbf{ED}}\left({\tau}_{\mathrm{n}+1}\right)+\\ \hfill +\sum _{\mathrm{i}=\mathrm{L}1}^{\mathrm{R}}\left({\Gamma}_{\mathbf{iD}}^{\prime}{\xb7}^{\mathrm{n}+1}{\tilde{\mathbf{K}}}_{\mathbf{i}}\xb7{\Gamma}_{\mathbf{iP}}\xb7{\mathbf{q}}_{\mathbf{iP}}\left({\tau}_{\mathrm{n}+1}\right)\right)+\sum _{\mathrm{i}=\mathrm{L}1}^{\mathrm{R}}\left({\Gamma}_{\mathbf{iD}}^{\prime}{\xb7}^{\mathrm{n}+1}{\tilde{\mathbf{K}}}_{\mathbf{Si}}{\xb7}^{\mathrm{n}}{\mathrm{s}}_{\mathrm{i}}\right)\end{array}.$$Let us rename the following quantities for the sake of brevity:$$\begin{array}{cc}\hfill {}^{\mathrm{n}+1}\tilde{\mathbf{g}}& ={\mathbf{f}}_{\mathbf{ED}}\left({\tau}_{\mathrm{n}+1}\right)+\sum _{\mathrm{i}=\mathrm{L}1}^{\mathrm{R}}\left({\Gamma}_{\mathbf{iD}}^{\prime}{\xb7}^{\mathrm{n}+1}{\tilde{\mathbf{K}}}_{\mathbf{i}}\xb7{\Gamma}_{\mathbf{iP}}\xb7{\mathbf{q}}_{\mathbf{iP}}\left({\tau}_{\mathrm{n}+1}\right)\right)+\sum _{\mathrm{i}=\mathrm{L}1}^{\mathrm{R}}\left({\Gamma}_{\mathbf{iD}}^{\prime}{\xb7}^{\mathrm{n}+1}{\tilde{\mathbf{K}}}_{\mathbf{Si}}{\xb7}^{\mathrm{n}}{\mathrm{s}}_{\mathrm{i}}\right)\hfill \\ \hfill {}^{\mathrm{n}+1}{\tilde{\mathbf{K}}}_{\mathbf{C}}& =\sum _{\mathrm{i}=\mathrm{L}1}^{\mathrm{R}}\left({\Gamma}_{\mathbf{iD}}^{\prime}{\xb7}^{\mathrm{n}+1}{\tilde{\mathbf{K}}}_{\mathbf{i}}\xb7{\Gamma}_{\mathbf{iD}}\right)\hfill \end{array}.$$
- The system of equations composed of Equation (A13) and the constitutive Newmark relations Equation (A7) is solved. Below is the expression for ${}^{\mathrm{n}+1}{\tilde{\ddot{\mathbf{q}}}}_{\mathbf{D}}$:$$\begin{array}{cc}\hfill {}^{\mathrm{n}+1}{\tilde{\ddot{\mathbf{q}}}}_{\mathbf{D}}& ={\left({\mathbf{M}}_{\mathbf{D}}+\beta \Delta {\tau}^{2}{\xb7}^{\mathrm{n}+1}{\tilde{\mathbf{K}}}_{\mathbf{C}}\right)}^{-1}({}^{\mathrm{n}+1}{\tilde{\mathbf{K}}}_{\mathbf{C}}\xb7({}^{\mathrm{n}}{\mathbf{q}}_{\mathbf{D}}+\Delta \tau \xb7{}^{\mathrm{n}}{\dot{\mathbf{q}}}_{\mathbf{D}}+\hfill \\ & +\left(\frac{1}{2}-\beta \right)\Delta {\tau}^{2}\xb7{}^{\mathrm{n}}{\ddot{\mathbf{q}}}_{\mathbf{D}}\left){+}^{\mathrm{n}+1}\tilde{\mathbf{g}}\right)\hfill \end{array}.$$The expressions for ${}^{\mathrm{n}+1}{\tilde{\mathbf{q}}}_{\mathbf{D}}$ and ${}^{\mathrm{n}+1}{\tilde{\dot{\mathbf{q}}}}_{\mathbf{D}}$ are computed using Newmark’s equations (Equation (A7)).
- Vector ${}^{\mathrm{n}+1}{\tilde{\mathbf{q}}}_{\mathbf{D}}$ is transformed into local displacements at the contact ${}^{\mathrm{n}+1}{\tilde{{\mathbf{q}}^{*}}}_{\mathbf{iD}}$ using Equation (4). The same is done with the displacements of the platform contact points, thus obtaining ${\mathbf{q}}_{\mathbf{iP}}^{*}({\tau}_{n}+\Delta \tau )$ $\forall i$.
- The vectors of local displacements at the contacts (of both the platforms and the dampers), together with the slider variable ${\mathrm{s}}_{\mathrm{i}}$ are fed into a function (similar to the contact element function) that checks whether their values are compatible with the postulated contact condition. In other words, for each i, whether the structure of ${}^{\mathrm{n}+1}{\tilde{\mathbf{K}}}_{\mathbf{i}}$ and ${}^{\mathrm{n}+1}{\tilde{\mathbf{K}}}_{\mathbf{Si}}$ is compatible with the values of local platforms and damper displacements is checked (see Table 1).
- CORRECTOR: If the compatibility is verified, then the equilibrium postulated in Equation (A13) is indeed satisfied, and the integration goes to ${\tau}_{\mathrm{n}+2}$. In this case, all slider variables are updated according to the verified contact condition (${}^{\mathrm{n}+1}{\mathrm{s}}_{\mathrm{i}}$, to be used at time step ${\tau}_{\mathrm{n}+2}$).Otherwise, the structure of all effective contact stiffness matrices is updated according to the indications of the function mentioned at Step 5—see also Table 1:$$\begin{array}{cc}\hfill {}^{\mathrm{n}+1}{\widehat{\mathbf{K}}}_{\mathbf{i}}& =\mathcal{F}\left({}^{\mathrm{n}+1}{\tilde{\mathbf{q}}}_{\mathbf{iD}}^{*},{}^{\mathrm{n}+1}{\mathbf{q}}_{\mathbf{iP}}^{*},{}^{\mathrm{n}}{\mathrm{s}}_{\mathrm{i}}\right)\hfill \\ \hfill {}^{\mathrm{n}+1}{\widehat{\mathbf{K}}}_{\mathbf{Si}}& ={\mathcal{F}}_{S}\left({}^{\mathrm{n}+1}{\tilde{\mathbf{q}}}_{\mathbf{iD}}^{*},{}^{\mathrm{n}+1}{\mathbf{q}}_{\mathbf{iP}}^{*},{}^{\mathrm{n}}{\mathrm{s}}_{\mathrm{i}}\right)\hfill \end{array}.$$The algorithm is then repeated starting from Step 3.

## Appendix B. Quasi-Static Method Algorithm

- The values of ${}^{\mathrm{n}}{\mathbf{q}}_{\mathbf{D}}$, ${}^{\mathrm{n}}{\dot{\mathbf{q}}}_{\mathbf{D}}$, ${}^{\mathrm{n}}{\ddot{\mathbf{q}}}_{\mathbf{D}}$, ${\mathbf{f}}_{\mathbf{ED}}\left({\tau}_{\mathrm{n}+1}\right)$, ${\mathbf{q}}_{\mathbf{iP}}\left({\tau}_{\mathrm{n}+1}\right)$ $\forall i$, and the contact states’ effective stiffness matrices ${}^{\mathrm{n}}{\mathbf{K}}_{\mathbf{i}}$ and ${}^{\mathrm{n}}{\mathbf{K}}_{\mathbf{Si}}$ are known either from initial conditions or the previous integration step.
- PREDICTOR: It is assumed that the contact state at ${\tau}_{n+1}$ is the same as that of the previous time step, i.e.,$$\begin{array}{cc}\hfill {}^{\mathrm{n}+1}{\tilde{\mathbf{K}}}_{\mathbf{i}}& {=}^{\mathrm{n}}{\mathbf{K}}_{\mathbf{i}}\hfill \\ \hfill {}^{\mathrm{n}+1}{\tilde{\mathbf{K}}}_{\mathbf{Si}}& {=}^{\mathrm{n}}{\mathbf{K}}_{\mathbf{Si}}.\hfill \end{array}$$
- It is therefore possible to compute the expression of ${}^{\mathrm{n}+1}{\tilde{\mathbf{q}}}_{\mathbf{D}}$ using the tentative version of the equilibrium equations at time ${\tau}_{n+1}$ (see Equation (10)):$$\begin{array}{c}\hfill {}^{\mathrm{n}+1}{\tilde{\mathbf{q}}}_{\mathbf{D}}{=}^{\mathrm{n}+1}{\tilde{\mathbf{K}}}_{\mathbf{C}}^{-1}\left({}^{\mathrm{n}+1}\tilde{\mathbf{g}}\right)\end{array}$$$$\begin{array}{cc}\hfill {}^{\mathrm{n}+1}\tilde{\mathbf{g}}& ={\mathbf{f}}_{\mathbf{ED}}\left({\tau}_{\mathrm{n}+1}\right)+\sum _{\mathrm{i}=\mathrm{L}1}^{\mathrm{R}}\left({\Gamma}_{\mathbf{iD}}^{\prime}{\xb7}^{\mathrm{n}+1}{\tilde{\mathbf{K}}}_{\mathbf{i}}\xb7{\Gamma}_{\mathbf{iP}}\xb7{\mathbf{q}}_{\mathbf{iP}}\left({\tau}_{\mathrm{n}+1}\right)\right)+\sum _{\mathrm{i}=\mathrm{L}1}^{\mathrm{R}}\left({\Gamma}_{\mathbf{iD}}^{\prime}{\xb7}^{\mathrm{n}+1}{\tilde{\mathbf{K}}}_{\mathbf{Si}}{\xb7}^{\mathrm{n}}{\mathrm{s}}_{\mathrm{i}}\right)\hfill \\ \hfill {}^{\mathrm{n}+1}{\tilde{\mathbf{K}}}_{\mathbf{C}}& =\sum _{\mathrm{i}=\mathrm{L}1}^{\mathrm{R}}\left({\Gamma}_{\mathbf{iD}}^{\prime}{\xb7}^{\mathrm{n}+1}{\tilde{\mathbf{K}}}_{\mathbf{i}}\xb7{\Gamma}_{\mathbf{iD}}\right)\hfill \end{array}.$$
- Vector ${}^{\mathrm{n}+1}{\tilde{\mathbf{q}}}_{\mathbf{D}}$ is transformed into local displacements at the contact ${}^{\mathrm{n}+1}{\tilde{\mathbf{q}}}_{\mathbf{iD}}^{*}$ using Equation (4). The same is done with the displacements of the platform contact points, thus obtaining ${\mathbf{q}}_{\mathbf{iP}}^{*}({\tau}_{n}+\Delta \tau )$ $\forall i$.
- The vectors of local displacements at the contacts (of both the platforms and the damper), together with the slider variable ${\mathrm{s}}_{\mathrm{i}}$ are fed into a function (similar to the contact element function) that checks whether their values are compatible with the postulated contact condition. In other words, for each i, whether the structure of ${}^{\mathrm{n}+1}{\tilde{\mathbf{K}}}_{\mathbf{i}}$ and ${}^{\mathrm{n}+1}{\tilde{\mathbf{K}}}_{\mathbf{Si}}$ is compatible with the values of local platforms and damper displacements is checked (see Table 1).
- CORRECTOR: If the compatibility is verified, then the equilibrium postulated in Equation (10) is indeed satisfied, and the integration goes to ${\tau}_{\mathrm{n}+2}$. In this case, all slider variables are updated according to the verified contact condition (${}^{\mathrm{n}+1}{\mathrm{s}}_{\mathrm{i}}$, to be used at time step ${\tau}_{\mathrm{n}+2}$).Otherwise, the structure of all effective contact stiffness matrices is updated according to the indications of the function mentioned at Step 5—see also Table 1:$$\begin{array}{cc}\hfill {}^{\mathrm{n}+1}{\widehat{\mathbf{K}}}_{\mathbf{i}}& =\mathcal{F}\left({}^{\mathrm{n}+1}{\tilde{\mathbf{q}}}_{\mathbf{iD}}^{*},{}^{\mathrm{n}+1}{\mathbf{q}}_{\mathbf{iP}}^{*},{}^{\mathrm{n}}{\mathrm{s}}_{\mathrm{i}}\right)\hfill \\ \hfill {}^{\mathrm{n}+1}{\widehat{\mathbf{K}}}_{\mathbf{Si}}& ={\mathcal{F}}_{S}\left({}^{\mathrm{n}+1}{\tilde{\mathbf{q}}}_{\mathbf{iD}}^{*},{}^{\mathrm{n}+1}{\mathbf{q}}_{\mathbf{iP}}^{*},{}^{\mathrm{n}}{\mathrm{s}}_{\mathrm{i}}\right)\hfill \end{array}.$$The algorithm is then repeated starting from Step 3.

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**Figure 1.**(

**a**) CAD model of a turbine bladed disk with underplatform dampers (not directly visible in the figure). (

**b**) Sketch representing a close up of three blades with interposed underplatform dampers.

**Figure 2.**(

**a**) Contact model with three contact springs. (

**b**) Contact patches on the blade platforms and on the strip damper.

**Figure 3.**(

**Left**) Contact model representation and list of its calibration parameters. (

**Right**) Standard implementation of the contact model routine through a predictor-corrector scheme.

**Figure 4.**(

**a**) Scheme representing the Newmark-$\beta $ implicit DTI scheme (standard implementation) with an adaptive time step. (

**b**) Example of result of Newmark-$\beta $ DTI with adaptive time step.

**Figure 5.**Histogram representing the number of iterations per time step of the predictor-corrector scheme implemented in the DTI algorithm: (

**a**) constant displacement assumption; (

**b**) unchanged contact state assumption.

**Figure 6.**(

**a**) Low frequency hysteresis cycle at the left contact obtained through DTI without additional post-processing. (

**b**) Low frequency hysteresis cycle at the left contact obtained through DTI and an a posteriori low-pass filter.

**Figure 7.**DTI vs. quasi-static results comparison. (

**a**) Tangential displacement of the right damper contact point. (

**b**) Vertical component of the right contact force.

**Figure 8.**Time savings introduced by the novel techniques and solution methods proposed in the paper. A standard Newmark-$\beta $ computation is taken as a reference.

Contact State | Condition | ${\mathit{K}}_{\mathit{i}}$ | ${\mathit{K}}_{\mathbf{Si}}$ |
---|---|---|---|

Lift-off | ${\mathrm{n}}_{\mathrm{iP}}-{\mathrm{n}}_{\mathrm{iD}}<0$ | $\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$ | $\left[\begin{array}{c}0\\ 0\end{array}\right]$ |

Stick | $\begin{array}{c}{\mathrm{n}}_{\mathrm{iP}}-{\mathrm{n}}_{\mathrm{iD}}>0\\ {\mathrm{k}}_{\mathrm{ti}}({\mathrm{t}}_{\mathrm{iP}}-{\mathrm{t}}_{\mathrm{iD}}-{\mathrm{s}}_{\mathrm{i}})<\mu \xb7{\mathrm{k}}_{\mathrm{ni}}\xb7({\mathrm{n}}_{\mathrm{iP}}-{\mathrm{n}}_{\mathrm{iD}})\end{array}$ | $\left[\begin{array}{cc}{\mathrm{k}}_{\mathrm{ti}}& 0\\ 0& {\mathrm{k}}_{\mathrm{ni}}\end{array}\right]$ | $\left[\begin{array}{c}-{\mathrm{k}}_{\mathrm{ti}}\\ 0\end{array}\right]$ |

Slip | $\begin{array}{c}{\mathrm{n}}_{\mathrm{iP}}-{\mathrm{n}}_{\mathrm{iD}}>0\\ {\mathrm{k}}_{\mathrm{ti}}({\mathrm{t}}_{\mathrm{iP}}-{\mathrm{t}}_{\mathrm{iD}}-{\mathrm{s}}_{\mathrm{i}})\ge |\mu \xb7{\mathrm{k}}_{\mathrm{ni}}\xb7({\mathrm{n}}_{\mathrm{iP}}-{\mathrm{n}}_{\mathrm{iD}})|\end{array}$ | $\left[\begin{array}{cc}0& \pm \mu \xb7{\mathrm{k}}_{\mathrm{ni}}\\ 0& {\mathrm{k}}_{\mathrm{ni}}\end{array}\right]$ | $\left[\begin{array}{c}0\\ 0\end{array}\right]$ |

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## Share and Cite

**MDPI and ACS Style**

Gastaldi, C.; Berruti, T.M. Competitive Time Marching Solution Methods for Systems with Friction-Induced Nonlinearities. *Appl. Sci.* **2018**, *8*, 291.
https://doi.org/10.3390/app8020291

**AMA Style**

Gastaldi C, Berruti TM. Competitive Time Marching Solution Methods for Systems with Friction-Induced Nonlinearities. *Applied Sciences*. 2018; 8(2):291.
https://doi.org/10.3390/app8020291

**Chicago/Turabian Style**

Gastaldi, Chiara, and Teresa M. Berruti. 2018. "Competitive Time Marching Solution Methods for Systems with Friction-Induced Nonlinearities" *Applied Sciences* 8, no. 2: 291.
https://doi.org/10.3390/app8020291