DIN 9250 Junker Dynamic Simulation: Single-DOF Transient Integration with Friction Model

1. Simulation Purpose and System Description

The Junker transverse vibration test is a standard method for evaluating the self-loosening resistance of bolted connections. By establishing a single-degree-of-freedom dynamic model to simulate the motion of clamped components under sinusoidal transverse excitation, combined with a Coulomb + Stribeck friction model to describe the tangential force at the washer interface, the real-time preload decay during vibration and the final residual preload ratio can be predicted.

Simplified Physical System:

• Clamped component mass $m$, supported by transverse spring $k_x$;
• External sinusoidal displacement $x_{base}(t) = A \sin(2\pi f t)$ applied through the spring to the mass;
• Friction force $F_f$ at the interface between the mass and the bolt-washer system prevents relative slip, with friction characteristics described by Coulomb and Stribeck effects;
• Preload $F_M$ gradually decreases during frictional slip due to tooth tip wear and plastic deformation, with the decay rate related to slip work or slip amplitude.

Frequency Sweep Range: 1–100 Hz, covering common industrial vibration frequencies; the frequency range where washer anti-loosening performance is worst can be identified through frequency sweeping.

2. Governing Equations

The equation of motion for the mass (assuming the washer-bolt system mass is negligible and preload only affects interface pressure, not transverse dynamics) is:

where:

• $m$ — effective mass of clamped component (kg)
• $c$ — viscous damping coefficient (N·s/m), representing material internal damping and minor energy losses
• $k_x$ — transverse stiffness of clamped component (N/m)
• $x_{base}(t) = A \sin(2\pi f t)$ — base excitation displacement (m)
• $x$ — absolute displacement of mass (m)
• $F_f$ — interface friction force (N), always opposite to relative velocity $\dot{x} - \dot{x}_{base}$

Let relative displacement $\xi = x - x_{base}$, then $\dot{\xi} = \dot{x} - \dot{x}_{base}$, $\ddot{\xi} = \ddot{x} - \ddot{x}_{base}$. Substituting gives:

Given $\ddot{x}_{base} = -A (2\pi f)^2 \sin(2\pi f t)$. The equation can be written as:

This is a second-order ordinary differential equation solvable by numerical integration.

3. Coulomb + Stribeck Friction Model

The interface friction force includes not only constant Coulomb friction but is also affected by relative velocity: the Stribeck effect occurs at low speeds (friction decreases with increasing speed), while at high speeds it tends to a constant value. The model uses:

• $\mu_C$ — Coulomb dynamic friction coefficient (steel-steel tooth surface equivalent, e.g., low-speed limit of $\mu_{serr}$)
• $\mu_S$ — static friction coefficient (slightly higher than $\mu_C$)
• $v_s$ — Stribeck characteristic velocity (m/s), determining the transition point of friction decrease
• $\delta_s$ — shape exponent (typically 1 or 2)
• $F_M$ — current preload (N); note that $F_M$ decays during simulation due to accumulated slip
• $\text{sign}(\dot{\xi})$ ensures friction force direction opposes relative motion

When relative velocity is zero, friction force equals static friction (dependent on external force); numerical integration must handle zero-velocity sticking. A simple approach uses linear transition near zero velocity or directly applies the Stribeck expression (no singularity when velocity is non-zero).

4. Preload Decay Model

During each macroscopic slip event, tooth tip wear or plastic deformation causes preload $F_M$ to decrease. A simple wear accumulation model is used:

or based on slip work increment:

where $\Delta W_{fric} = \int |F_f \cdot \dot{\xi}| dt$ is the slip dissipation energy; $\alpha, \beta$ are wear coefficients determined experimentally.

In simulation, cycle-based accumulation can be applied: when the relative slip amplitude exceeds the tooth characteristic size $\delta_{tooth}$, preload decreases by a certain proportion.

5. RK4 Transient Integration Method

State-space representation: let $\mathbf{y} = [\xi, \dot{\xi}]^T$, then:

The fourth-order Runge-Kutta method (RK4) is used for time stepping. The time step $\Delta t$ must satisfy sampling theorem and accuracy requirements (typically 1/40 of the highest excitation frequency). Pseudocode:

for t = 0 to T_end step Δt:
k1 = f(t, y)
k2 = f(t+Δt/2, y + Δt/2 * k1)
k3 = f(t+Δt/2, y + Δt/2 * k2)
k4 = f(t+Δt, y + Δt * k3)
y = y + (Δt/6) * (k1 + 2k2 + 2k3 + k4)
update F_M based on wear

Function f(t,y) contains the above differential equations and friction force calculation.

6. Implementation of Frequency Sweep 1–100 Hz

To evaluate anti-loosening performance at different frequencies, multiple simulations are performed, each maintaining constant amplitude $A$ while varying excitation frequency $f$, gradually increasing from 1 Hz to 100 Hz (exponential or linear steps). The residual preload ratio $R_F$ at the end of each simulation (e.g., after 10^4 cycles) is recorded for each frequency.

The frequency sweep process can use pseudo-parallel or single-thread loops. The frequency range where $R_F$ falls below the allowable value (e.g., 0.7) is identified as the dangerous frequency band for that washer.

7. Simulation Parameters and Output

Typical Input Parameters:

• $m = 1.0$ kg (equivalent mass)
• $k_x = 1\times10^6$ N/m (transverse stiffness)
• $c = 20$ N·s/m (low damping)
• Amplitude $A = 0.3$ mm (standard Junker value)
• Friction parameters: $\mu_C = 0.5, \mu_S = 0.6, v_s = 0.001$ m/s, $\delta_s = 1$
• Initial preload $F_{M0} = 20$ kN, wear coefficient $\beta$ such that dissipation energy per cycle corresponds to approximately 0.01% preload loss

Output:

• Time history: relative displacement $\xi(t)$, friction force $F_f(t)$, preload $F_M(t)$
• Steady-state residual preload ratio $R_F(f)$ vs. frequency curve

8. Correspondence with Junker Test

Although simplified, this single-DOF model can reproduce key phenomena of the Junker test:

• When amplitude exceeds the tooth locking stroke, macroscopic slip occurs and preload decreases rapidly;
• A critical frequency exists where interface slip amplitude is maximum, making anti-loosening most unfavorable;
• The Stribeck effect in the friction model can simulate nonlinear behavior where friction increases (tighter) or decreases (slipperier) at low speeds.

Simulation results can be used to optimize washer tooth geometry (adjusting $\delta_{tooth}$) or select better friction pairings to reduce vibration-induced loosening risk.

Summary:
This dynamic simulation framework solves a single-DOF equation of motion using RK4, combined with Coulomb+Stribeck friction and preload wear models. Frequency sweeping from 1–100 Hz enables comprehensive evaluation of DIN 9250 washer anti-loosening performance at different vibration frequencies, providing quantitative basis for anti-vibration design.