# Vibro-Impact Response Analysis of Collision with Clearance: A Tutorial

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- Step 1:
- Establish the dynamic equation of the nonlinear structure with local clearance.
- Step 2:
- Select the description method of clearance and the model of nonlinear impact force.
- Step 3:
- Select the solving method of the nonlinear dynamic equations.

## 2. Vibro-Impact Response Analysis

#### 2.1. Nonlinear Dynamics Equations of Structures with Clearances

**x**is the displacement matrix, $\dot{\mathit{x}}$ is the velocity matrix, $\ddot{\mathit{x}}$ is the acceleration matrix,

**M**is the mass matrix of the structure,

**C**is the damping matrix,

**K**is the stiffness matrix and

**F**is the external force matrix.

**F**

_{n}is the contact force matrix. In the dynamic equation of structure with clearance, the contact force vector

**F**

_{n}determines the dynamic features in the process of collision. Different clearance description methods produce different impact force models. It is very important to compare different clearance description methods to select the appropriate impact force model for accurately describing the dynamic characteristics of structures with clearance.

#### 2.2. Model of Nonlinear Impact Force

- (1)
- Hertz contact force model

- (2)
- Hunt–Crossley (H-C) contact force model

- (3)
- Lankarani–Nikravesh (L–N) contact force model [35]

- (4)
- Gonthier contact force model

_{e}

^{2}.

- (5)
- Flores contact force model [38]

_{max}is the maximum elastic deformation; then the internal damping coefficient χ in the collision process can be obtained

_{e}= 0.5, ball mass m = 0.04 kg, radius R

_{1}= 10 mm for sphere 1 and R

_{2}= 9.9 mm for sphere 2. Each of these values is substituted into the formula in Table 2. The relation between the elastic deformation of the ball and the impact force is shown in Figure 2.

#### 2.3. Solving Method of Nonlinear Dynamic Equations

- (1)
- Newmark-β method combined with Newton-Raphson method [41]

_{n}time, ${\dot{\mathit{x}}}_{n}$ is the velocity at t

_{n}time, ${\ddot{\mathit{x}}}_{n}$ is the acceleration at t

_{n}time, Δt = t

_{n}

_{+1}− t

_{n}, two parameters γ and β are introduced, and are usually required

_{n}

_{+1}can be obtained

_{n}

_{+1}can be obtained from Equation (15)

_{n}

_{+1}

_{n+1}. Then, the speed at t

_{n+1}could be calculated by using Equation (14). At this point, the motion state at t

_{n+1}has been calculated, and they can be used as the starting values to calculate the motion state at the next moment, and so on.

_{n+1}. When there is a nonlinear term in the structure, Equation (21) can be changed to

**f**

_{nl}is the nonlinear term and

**x**

_{n}

_{+1}is taken as the variable. Equation (20) can be written as follows

**Ψ**

_{n}

_{+1}first-order is continuously differentiable in Equation (23), let the initial approximation obtained by the Newmark-β method be ${\mathit{x}}_{n+1}^{0}$, and the k-th iteration approximation obtained by the Newton–Raphson method be ${\mathit{x}}_{n+1}^{k}$, and

**Ψ**

_{n}

_{+1}is the Taylor expansion and the higher-order term is dropped

**x**

_{n}

_{+1}in Equation (24)

**K**

_{T}is the tangent stiffness matrix, and the expression is

- (2)
- Generalized α method

- (3)
- Precise Adams Multi-step method (precise integration method)

_{k}, t

_{k}

_{+1}] can be derived from the above formula

_{k}, t

_{k}

_{+1}], the nonlinear term f(

**x**, t) is approximated by an m-degree polynomial

_{0,k}, f

_{1,k}, … ] denotes the coefficient of the approximate polynomial in section k, and Equation (38) can be expressed as

_{m}(η) can be called the Duhamel integration matrix of external force.

_{k}, t

_{k}

_{+1}], we can also use the information of the previous section [t

_{k}

_{−2}, t

_{k}

_{+1}], [t

_{k}

_{−1}, t

_{k}] for the polynomial approximation of f(

**x**,t); the most commonly used is the Adams linear multi-step method.

_{k}

_{−2}, t

_{k}

_{−1}and t

_{k}as interpolation points, approximate Lagrange polynomials of f(

**x**,t) are obtained

_{0}(η), …, Φ

_{−m}(η) is determined by the Duhamel integration matrix.

## 3. Nonlinear Dynamic Response Analysis of Structures with Clearance

#### 3.1. Single Degree of Freedom Model

_{0}= 100 kg and the initial installation clearance is 5 mm.

#### 3.2. Multiple Degrees of Freedom Model

^{3}, the elastic modulus was 207 GPa and the Poisson ratio was 0.3. There are 42 nodes in the hollow shaft section and 62 nodes in the axial direction, for a total of 2064 nodes.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Description methods of clearance: (

**a**) massless bar method; (

**b**) spring-damping method; (

**c**) force description method.

**Figure 2.**Schematic diagram of single collision with clearance structure. (

**a**) Before the collision. (

**b**) State of collision.

**Figure 5.**Impact vibration model of structure with bilateral clearances, where k

_{h}is the nonlinear stiffness during collision, k

_{0}is the linear stiffness of the structure, c

_{h}is the nonlinear damping during collision, c

_{0}is the linear damping of the structure, d is the initial installation clearance and m

_{0}is the mass block of the concentrated mass.

**Figure 6.**Displacement time curve of the mass block: (

**a**) Newmark-β method combined with Newton–Raphson method; (

**b**) generalized α method; (

**c**) precise integration method.

**Figure 7.**Time curve of the mass block: (

**a**) Newmark-β method combined with Newton–Raphson method; (

**b**) generalized α method; (

**c**) precise integration method.

**Figure 16.**Finite element model: (

**a**) cutaway view of finite element model; (

**b**) slider distribution position of finite element model.

**Figure 17.**Dynamic response analysis results of simplified model: (

**a**) displacement in the reference model; (

**b**) acceleration in the reference model; (

**c**) displacement in the numerical model; (

**d**) acceleration in the numerical model.

Parameters | Definition |
---|---|

F_{n} | Nonlinear impact force |

K | Coefficient of contact stiffness |

δ | Elastic deformation |

$\dot{\delta}$ | Derivative of elastic deformation |

${\dot{\delta}}^{(-)}$ | Relative collision velocity |

Model | Expression of Impact Force |
---|---|

Hertz contact force model | ${F}_{n}=K{\delta}^{n}$ |

Hunt–Crossley contact force model | ${F}_{n}=K{\delta}^{n}+b{\delta}^{n}\dot{\delta}$ |

Lankarani–Nikravesh contact force model | ${F}_{n}=K{\delta}^{n}+D{\dot{\delta}}^{(-)}$ |

Gonthier contact force model | ${F}_{n}=K{\delta}^{n}\left[1+\frac{1-{c}_{e}^{2}}{{c}_{e}^{2}}\frac{\dot{\delta}}{{\dot{\delta}}^{(-)}}\right]$ |

Flores contact force model | ${F}_{n}=K{\delta}^{n}\left[1+\frac{8\left(1-{c}_{e}\right)}{5{c}_{e}}\frac{\dot{\delta}}{{\dot{\delta}}^{(-)}}\right]$ |

**Table 3.**The computational efficiency of different numerical integration algorithms at 0.02 s simulation time.

Numerical Integration Algorithms | Time of Calculation/s |
---|---|

Newmark-β method combined with Newton–Raphson method | 1.3 |

Generalized α method | 2.1 |

Precise integration method | 25 |

Model | Reference Model/Hz | Numerical Model/Hz | |
---|---|---|---|

Mode | |||

First bending mode | 477 | 477 | |

Second bending mode | 1035 | 1035 | |

Third bending mode | 1614 | 1637 |

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## Share and Cite

**MDPI and ACS Style**

Xu, Y.; Tian, Y.; Li, Q.; Li, Y.; Zhang, D.; Jiang, D.
Vibro-Impact Response Analysis of Collision with Clearance: A Tutorial. *Machines* **2022**, *10*, 814.
https://doi.org/10.3390/machines10090814

**AMA Style**

Xu Y, Tian Y, Li Q, Li Y, Zhang D, Jiang D.
Vibro-Impact Response Analysis of Collision with Clearance: A Tutorial. *Machines*. 2022; 10(9):814.
https://doi.org/10.3390/machines10090814

**Chicago/Turabian Style**

Xu, Yongjie, Yu Tian, Qiyu Li, Yanbin Li, Dahai Zhang, and Dong Jiang.
2022. "Vibro-Impact Response Analysis of Collision with Clearance: A Tutorial" *Machines* 10, no. 9: 814.
https://doi.org/10.3390/machines10090814