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# Strength Analysis of Flat Spring of the Resonant Vibro-Impact Module

Volodymyr Gursky, Igor Kuzio

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Volodymyr Gursky, Igor Kuzio. Strength Analysis of Flat Spring of the Resonant Vibro-Impact Module. Mechanics, Materials Science & Engineering Journal, 2016, 10.13140/RG.2.1.2504.0240. hal-01325251

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# Strength Analysis of Flat Spring of the Resonant Vibro-Impact Module

Volodymyr Gursky, Igor Kuzio

1 – Lviv Politechnic National University, Ukraine

<figure><img src="image_3.png" /></figure>

DOI 10.13140/RG.2.1.2504.0240

**Keywords:** flat spring, vibro-impact, resonance, finite element method (FEM), eigenfrequencies, contact stress.

**ABSTRACT.** The rod model of the resonant vibro-impact module with an electromagnetic drive is considered. Construction's design implemented an asymmetrical elastic characteristic by one flat spring with two absolutely rigid intermediate supports. Eigenfrequency is defined for corresponding location intermediate supports based on the finite element method. Stress-strain state of the elastic element is graphically represented at the expense of static displacement of local mass. Contact task is considered and contact force between the flat spring and cylindrical support is calculated. Also, contact stiffness is determinate. The parameters of volumetric stress state of the contact, calculated analytically and by modeling in SolidWorks Simulation are shown. The dynamics of the vibro-impact rod system with kinematic's disturbance is modeled. Contact and equivalent stresses during operation of the vibro-impact rod system are determined.

**1. Introduction.** Vibro-impact systems are the basis of technological machines and processes of environments with challenging physical and mechanical characteristics. This is due the presence challenging modes as imposing the main and sub-harmonic oscillations, acceleration polyfrequency range by asymmetry of displacement of the operating mass. The resulting polyfrequency system generates wave processes, resonance phenomena and power conversion in environments more effective than harmonic movement law of worker mass. So, the use of frequency vibro-impact processes and systems is almost justified and promising. However, this requires challenging practical solutions for the implementation of nonlinear systems and special theoretical analysis to evaluation the relevant modes.

**2. Analysis of recent research and literature.** The application of structural nonlinearities is most appropriate among to the known practical solutions for the implementation of vibro-impact systems [1, 2]. A few practical solutions are in patents [3–6]. The feature of such systems functioning is the emergence of sub- and superresonance modes with different multiplicities orders [7, 8]. Implementation of such is possible by matching of the system parameters and power options [3].

**3. Aim of the article.** The main objective for developing the new vibration module is providing regulated resonant harmonic and vibro-impact modes. Proposed to implement it by resetting the stiffness properties of a flat spring relative to its base design. The task of reducing the complexity of operations for the implementation of the various modes as a practical matter is pursued. This is not enough in presented solutions [3-6].

Adjustable asymmetrical resilient characteristic (fig. 1) will be implemented by the flat spring with presented functional condition:

$$F_{\text{stiffness}}[y(t)] = \begin{cases} c_{I}y(t), & y(t) + \Delta \ge 0 \\ c_{II}y(t) + (c_{II} - c_{I})\Delta, & y(t) + \Delta < 0 \end{cases} \quad (1)$$
<figure><img src="image_4.png" /><figcaption>Fig. 1. Renewable characteristic of stiffness element.</figcaption></figure>

Implementation of the regulated modes is carried out by a constructive influence on the stiffness characteristic. It is supposed that the following conditions: $c_I = \text{const}$, a $c_{II} = \text{var}$, $\Delta = \text{var}$, $\frac{c_{II}}{c_I} = n_c = 1; 2; 3$.

**4. Problem statement.** Implementation of vibro-impact systems by movement limiters is accompanied by contact phenomena. That is confirmed in subsequent works [9-12]. It should be noted that the elastic elements are an important hub of the resonant vibration machines. Strength characteristics along with the task of rigidity must be complying. Therefore, the lack of integrated techniques for analyzing stress-strain states in the implementation of the vibro-impact modes is a problematic.

**5. Research materials.** Resonant vibration module designed by the two-mass oscillation system with active 1 and reactive 2 masses, which are connected by one flat spring 3. Active mass 1 is designed as a frame with an horizontal grooves for hard fixing brackets 4. Brackets 4 have vertical slots for adjustable cylindrical rods 5. Two core 6 are placed on the active mass 1. The reactive mass 2 contains anchors 7. The resonance module can be the basis for creating of the vibratory technological machine. Module also can be used as a stand for dynamic testing.

<figure><img src="image_5.png" /><figcaption>Fig. 2. General view of the vibro-impact module.</figcaption></figure>
This structural implementation allows to use one a flat spring 3 to implement the basic harmonic mode in the lower frequency region by stiffness coefficient $c_I$. Cylindrical shafts 5 are mounted in vertical grooves of brackets 4 and are used as adjustable intermediate stops for flat springs 3. Using the horizontal grooves changes the location of the intermediate support along the flat springs. Accordingly is changes stiffness coefficient $c_{II}$. The appropriate gap $\Delta$ between flat spring and intermediate stops set by using the vertical grooves. So, adjustable asymmetric piecewise linear characteristics of a flat spring and adjustable oscillations modes with different amplitude and frequency spectrum will be realized.

With appropriate settlement schemes, which describe asymmetric transverse stiffness change according to (1), also taking $\Delta= 0$, the stiffness coefficient $c_{II}$ is a necessary to define with the stress-strain state analysis.

For the analysis of rod systems appropriate to use finite element method [13]. Finite-element scheme of the vibration system in vibro-impact mode based on flat spring implemented on fig. 2.

The standard matrix of nodal reactions (transverse forces and bending moments) of one finite element rod has the form [13]:

$$[\bar{\mathbf{Z}}_{i,i+1}] = \begin{Bmatrix} R_i \\ M_i \\ R_{i+1} \\ M_{i+1} \end{Bmatrix} = EJ \begin{bmatrix} 12/l_i^3 & -6/l_i^2 & -12/l_i^3 & -6/l_i^2 \\ -6/l_i^2 & 4/l_i & 6/l_i^2 & 2/l_i \\ -12/l_i^3 & 6/l_i^2 & 12/l_i^3 & 6/l_i^2 \\ -6/l_i^2 & 2/l_i & 6/l_i^2 & 4/l_i \end{bmatrix} \quad (2)$$

where EJ – is a bending stiffness of the rod.

<figure><img src="image_6.png" /><figcaption>Fig. 3. Operation scheme of the vibro-impact system.</figcaption></figure>

Bending rigidity EJ of a flat spring is calculated to realization of the necessary natural frequency $\omega_{0I} = \sqrt{c_I/m}$ of the harmonic oscillation mode for scheme I:
$$EJ = mL^3 \omega_0 l^2 / 192, \qquad (3)$$

where $m = \frac{m_1 m_2}{(m_1+m_2)}$ – is reduced mass of vibrating system.

Finite-element scheme during movement in a negative direction is based on these boundary conditions:

$$\begin{pmatrix} y_0 = 0 \\ \varphi_0 = 0 \end{pmatrix}, \begin{pmatrix} y_1 = 0 \\ \varphi_1 = \text{var} \end{pmatrix}, \begin{pmatrix} y_2 = \text{var} \\ \varphi_2 = \text{var} \end{pmatrix}, \begin{pmatrix} y_3 = 0 \\ \varphi_3 = \text{var} \end{pmatrix}, \begin{pmatrix} y_4 = 0 \\ \varphi_4 = 0 \end{pmatrix}. \qquad (4)$$

The system with 10 degrees of freedom obtained in general form by using the division by 4 rods finite elements: $y_i$, $y_{i+1}$ i $\varphi_i$, $\varphi_{i+1}$ – according vertical movement and angles of rotation of the rod nodal's points ($i = 0 \cdots 4$). According to general stiffness matrix of a rod (2) are formed stiffness matrix with dimension 10×10 of finite elements rods in the global coordinate system in accordance with the boundary conditions (4).

$$\mathbf{C}_{01} = EJ \begin{pmatrix} (1 & 2 & 3 & 4) \\ \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 4/l_1 \end{bmatrix} & \begin{pmatrix} 1 \\ 2 \\ 3 \\ 4 \end{pmatrix} \end{pmatrix}, \\
\mathbf{C}_{12} = EJ \begin{pmatrix} (3 & 4 & 5 & 6) \\ \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 4/l_2 & -6/l_2^2 & 2/l_2 \\ 0 & -6/l_2^2 & 12/l_2^3 & -6/l_2^2 \\ 0 & 2/l_2 & -6/l_2^2 & 4/l_2 \end{bmatrix} & \begin{pmatrix} 3 \\ 4 \\ 5 \\ 6 \end{pmatrix} \end{pmatrix}, \\
\mathbf{C}_{23} = EJ \begin{pmatrix} (5 & 6 & 7 & 8) \\ \begin{bmatrix} 12/l_2^3 & 6/l_2^2 & 0 & 6/l_2^2 \\ 6/l_2^2 & 4/l_2 & 0 & 2/l_2 \\ 0 & 0 & 1 & 0 \\ 6/l_2^2 & 2/l_2 & 0 & 4/l_2 \end{bmatrix} & \begin{pmatrix} 5 \\ 6 \\ 7 \\ 8 \end{pmatrix} \end{pmatrix}, \\
\mathbf{C}_{34} = EJ \begin{pmatrix} (7 & 8 & 9 & 10) \\ \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 4/l_1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} & \begin{pmatrix} 7 \\ 8 \\ 9 \\ 10 \end{pmatrix} \end{pmatrix} \qquad (5)$$

$$\mathbf{C}_{23} = EJ \begin{bmatrix} 12/l_2^3 & 6/l_2^2 & 0 & 6/l_2^2 \\ 6/l_2^2 & 4/l_2 & 0 & 2/l_2 \\ 0 & 0 & 1 & 0 \\ 6/l_2^2 & 2/l_2 & 0 & 4/l_2 \end{bmatrix} \begin{pmatrix} 5 \\ 6 \\ 7 \\ 8 \end{pmatrix}, \\
\mathbf{C}_{34} = EJ \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 4/l_1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \begin{pmatrix} 7 \\ 8 \\ 9 \\ 10 \end{pmatrix},$$

The resulting stiffness matrix of finite-element rod system:
$$ = EJ \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 4/l_1 + 4/l_2 & -6/l_2^2 & 2/l_2 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -6/l_2^2 & 24/l_2^3 & 0 & 0 & 6/l_2^2 & 0 & 0 \\ 0 & 0 & 0 & 2/l_2 & 0 & 8/l_2 & 0 & 2/l_2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 6/l_2^2 & 2/l_2 & 0 & 4/l_1 + 4/l_2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{bmatrix} \quad (6) $$

Free oscillations equation of a rod system is written in matrix form:

$$ M \times \ddot{X} + C_{04} \times X = 0, \quad (7) $$

where, **X** = column(y₀, φ₀, ..., y₄, φ₄) – nodal's displacements vector of the finite element;

**M** = diag(0,0,0,0, m, 0,0,0,0) – diagonal matrix of local masses inertial parameters;

**C**₀₄ – system's stiffness matrix.

Based on the determinant for the frequency equation

$$ \det(-M \times \Omega^2 + C_{04}) = 0, \quad (8) $$

is defined natural frequency of bending vibrations:

$$ \omega_{0II} = \sqrt{\frac{6EJ(l_1+4l_2)}{m(l_1+l_2)l_2^3}}, \quad (9) $$

Reduced stiffness coefficients of rod systems according to the formulas (1) and (9) have the next forms:

$$ c_I = \frac{192EJ}{L^3}, \quad c_{II} = \frac{6EJ(l_1+4l_2)}{(l_1+l_2)l_2^3}, \quad (10) $$

Thus, obtained parametric dependence of the system's stiffness coefficient as function of l₁.

Stiffness coefficients ratio, taking into account l₂ = L/2 - l₁ has a view:

$$ n_c = \frac{c_{II}}{c_I} = \frac{L^2(L-1,5l_1)}{(L-2l_1)^3}, \quad (11) $$
The eigenfrequencies ratio is $n_\omega = \frac{\omega_{0II}}{\omega_{0I}} = \sqrt{n_c}$.

By the condition of multiplicities of natural frequencies of the rod systems

$$n_\omega = 2, \qquad (12)$$

formulas for positioning intermediate supports are the next

$$l_1 = 0.275L, \quad l_2 = 0.225L. \qquad (13)$$

For verification must be $l_1 + l_2 = 0.5L$.

**Stress-strain state analysis.** The vector of nodal displacements for statically loaded rod system type II can be determined from the relationship:

$$\mathbf{Y} = \mathbf{C}_{04}^{-1} \times \mathbf{P}, \qquad (14)$$

where **P** – vector of external loads reduced to nodes.

Nodal's displacement vectors each finite element-rod:

$$\mathbf{Y}_{01} = \begin{bmatrix} y_0 \\ \varphi_0 \\ y_1 \\ \varphi_1 \end{bmatrix}, \quad \mathbf{Y}_{12} = \begin{bmatrix} y_1 \\ \varphi_1 \\ y_2 \\ \varphi_2 \end{bmatrix}, \quad \mathbf{Y}_{23} = \begin{bmatrix} y_2 \\ \varphi_2 \\ y_3 \\ \varphi_3 \end{bmatrix}, \quad \mathbf{Y}_{34} = \begin{bmatrix} y_3 \\ \varphi_3 \\ y_4 \\ \varphi_4 \end{bmatrix}. \qquad (15)$$

The vectors of nodal reactions each finite element defines by matrix expression (2). The hogging line of each finite element-rod is based on the known forms functions at nodes single displacement [13]:

$$y_{01}(x) = \begin{bmatrix} \frac{2x^3 - 3l_1x^2 + l_1^3}{l_1^3} \\ \frac{x^3 - 2l_1x^2 + l_1^2x}{l_1^2} \\ \frac{-2x^3 + 3l_1x^2}{l_1^3} \\ \frac{x^3 - l_1x^2}{l_1^2} \end{bmatrix} \times \mathbf{Y}_{01}, \quad y_{12}(x) = \begin{bmatrix} \frac{2(x-l_1)^3 - 3l_2(x-l_1)^2 + l_2^3}{l_2^3} \\ \frac{(x-l_1)^3 - 2l_2(x-l_1)^2 + l_2^2(x-l_1)}{l_2^2} \\ \frac{-2(x-l_1)^3 + 3l_2(x-l_1)^2}{l_2^3} \\ \frac{(x-l_1)^3 - l_2(x-l_1)^2}{l_2^2} \end{bmatrix} \times \mathbf{Y}_{12}, \qquad (16)$$

$$y_{23}(x) = \begin{bmatrix} \frac{2(x-l_1-l_2)^3 - 3l_2(x-l_1-l_2)^2 + l_2^3}{l_2^3} \\ \frac{(x-l_1-l_2)^3 - 2l_2(x-l_1-l_2)^2 + l_2^2(x-l_1-l_2)}{l_2^2} \\ \frac{-2(x-l_1-l_2)^3 + 3l_2(x-l_1-l_2)^2}{l_2^3} \\ \frac{(x-l_1-l_2)^3 - l_2(x-l_1-l_2)^2}{l_2^2} \end{bmatrix} \times \mathbf{Y}_{23},$$
$$y_{34}(x) = \begin{bmatrix} \frac{2(x-l_1-2l_2)^3-3l_1(x-l_1-2l_2)^2+l_1^3}{l_1^3} \\ \frac{(x-l_1-2l_2)^3-2l_1(x-l_1-2l_2)^2+l_1^2(x-l_1-2l_2)}{l_1^2} \\ \frac{(-2x-l_1-2l_2)^3+3l_1(x-l_1-2l_2)^2}{l_1^3} \\ \frac{(x-l_1-2l_2)^3-l_1(x-l_1-2l_2)^2}{l_1^2} \end{bmatrix} \times \mathbf{Y}_{34}.$$

The resulting displacement and flex curve of a rod system II are determined by the following dependency:

$$y(x) = \begin{cases} y_{01}(x), & 0 < x \le l_1, \\ y_{12}(x), & l_1 < x \le l_1 + l_2, \\ y_{23}(x), & l_1 + l_2 < x \le l_1 + 2l_2, \\ y_{34}(x), & l_1 + 2l_2 < x \le 2l_1 + 2l_2, \end{cases} \quad (17)$$

Resilient condition of the rod evaluated by the bending moment, transverse force, normal and tangential stresses:

$$M(x) = EJ \frac{d^2}{dx^2} y(x), \quad Q(x) = EJ \frac{d^3}{dx^3} y(x), \quad \sigma(x) = \frac{M(x)}{W}, \quad \tau(x) = \frac{Q(x)S_{max}}{bJ}, \quad (18)$$

where $S_{max} = bh^2/8, J = bh^3/12, W = bh^2/6$ – geometric characteristics of the flat spring’s cross-section;

b i h – width and thickness of the flat spring.

The vector of nodal loads is presented with concentrated force with peak value F, acting at the local oscillating mass. Diagrams of deflection line, angle of rotation, transverse forces and bending moments will be obtained by solving of the equations (14)-(18). The value of the maximum bending stresses are determined by the known formulas and given for appropriate settlement schemes:

$$\sigma_{maxI} = \frac{M_{maxI}}{W}, \quad \sigma_{maxII} = \frac{M_{maxII}}{W}, \quad (19)$$

where $M_{maxI} = \frac{FL}{8}, M_{maxII} = [\bar{Z}_{12}]_4 = \frac{Fl_2(l_1+2l_2)}{2(l_1+4l_2)}$ are maximum bending moments in dangerous sections for consideration rod systems.

The maximum shear stress is acting on the central section of the rod:

$$\tau_{maxI} = \tau_{maxII} = \frac{1.5F}{bh}, \quad (20)$$

The equivalent (von Mises) stress:
$$ \sigma_{eq} = \sqrt{\sigma_{max}^2 + 4\tau_{max}^2}. \qquad (21) $$

**Research results.** Analysis of stress-strain state of the rod systems offered to implement by static displacement of local mass $y_m$. The value of renewable forces is set by effected displacement for the corresponding settlement schemes with formulas: $F_I = c_I y_m$, $F_{II} = c_{II} y_m$. System parameters for analysis: $\omega_{0I} = 314 \text{ rad/s}$, $\omega_{0II} = 628 \text{ rad/s}$, m = 20 kg, L = 0.5 m, E = $2.05 \times 10^5 \text{ MPa}$, b = 0.080 m, R = 0.006 m, $EJ = 1.284 \times 10^5 \text{ Nm}^2$, $l_1 = 0.113 \text{ m}$, $l_2 = 0.137 \text{ m}$, h = $9.79 \times 10^{-3} \text{ m}$, $c_I = 1.972 \times 10^6 \text{ N/m}$, $c_{II} = 7.888 \times 10^6 \text{ N/m}$. Taking $y_m = 0.24 \text{ mm}$ received: $\sigma_{eqI} = 23.2 \text{ MPa}$, $\sigma_{eqII} = 59.5 \text{ MPa}$. Diagrams of stress-strain state for half flat springs in scheme II ($x = 0.\frac{L}{2}$) shown on the fig. 4.

<figure><img src="image_7.png" /><figcaption>Fig. 4. Theoretical dependence of stress-strain state of elastic half-length rod.</figcaption></figure>

**Determination of contact stress.** Determination of contact stress is exercise on Hertz formulas by value of reactions at intermediate supports (transverse force) [11, 14]. The cylindrical support reaction is the sum of transverse forces in the rod's nodes:

$$ Q_1 = |[\bar{Z}_{01}]_3 + [\bar{Z}_{12}]_1| = \frac{F_{II}(l_1^2+4l_1l_2+3l_2^2)}{2l_1(l_1+4l_2)}. \qquad (22) $$

The pressure in contact zone is calculated by the formula [14]:
$$p_{max} = \frac{2Q_1}{\pi bt}, \qquad (23)$$

where, $t = K_t \sqrt{Q_1}$, $K_t = \sqrt{\frac{2}{\pi b} \frac{\frac{1-\mu_1^2}{E_1} + \frac{1-\mu_2^2}{E_2}}{\frac{1}{2R}}}$, $b$ – length of contact (width of flat spring);

$R$ – radius of intermediate support;

$\mu_1$ – $\mu_2$ – Poisson's ratio of materials in contact.

Vectors of volumetric stress state determined as [14]:

$$ \begin{aligned} \sigma_x &= -p_{max} \left( \frac{1+2\zeta_b(y)^2}{\sqrt{1+\zeta_b(y)^2}} - 2|\zeta_b(y)| \right), \\ \sigma_y &= -p_{max} \frac{1}{\sqrt{1+\zeta_b(y)^2}}, \\ \sigma_z &= -2\mu_1 p_{max} (\sqrt{1+\zeta_b(y)^2} - |\zeta_b(y)|). \end{aligned} \qquad (24) $$

where, $\zeta_b(y) = y/t$.

The main stresses are identical accordingly to axial components:

$$ \sigma_1 \equiv \sigma_x, \sigma_2 \equiv \sigma_z, \sigma_3 \equiv \sigma_y. \qquad (25) $$

The maximum shear and equivalent von Misses stress presented graphically (fig. 5) by the formulas for volumetric stress state [15]. Tangential and von Misses stresses:

$$ \tau_{max} = \frac{\sigma_1 - \sigma_3}{2}, \qquad (26) $$

$$ \sigma_{von \ Mises} = \sqrt{\frac{1}{2}[(\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_3 - \sigma_1)^2]}. \qquad (27) $$

The maximum value of these stresses with static displacement $y_m = 0.24 \text{ mm}$ at $\zeta_b(y) = 0.786: \tau_{max} = 104.3 \text{ MPa}$, $\sigma_{von \ Mises} = 194.7 \text{ MPa}$. Also received $Q_1 = 1.66 \text{ kN}$.
<figure><img src="image_8.png" /><figcaption>Fig. 5. The ratio of stress to the contact pressure (a); value of von Mises and shear stresses relative to the distance from the contact (b).</figcaption></figure>

For comparison the results of the stress-strain state analysis of half flat spring in SolidWorks [8] with symmetrical scheme are conducted (fig. 6). A condition as no penetration between flat spring and cylindrical support is applied in the setting of contact task. The results for the curve of deflection and von Mises stress are enough to near with the theoretical values (fig. 6, b and c). Also the contact force is set as 1.7 kN (fig. 6, a).

<figure><img src="image_9.png" /></figure>
<figure><img src="image_10.png" /><figcaption>Fig. 6. Design model of stress-strain state (a), deflection curve (b) and von Mises stress (c) parametric changes regarding of the half long of the flat spring.</figcaption></figure>

The procedure of nonlinear analysis is done with the setting as bouded contact for the flat stress. The maximum stress in the contact zone is 195.5 MPa. This value is almost identical to the theoretical result.

<figure><img src="image_11.png" /><figcaption>Fig. 7. Diagrams of contact stresses (a – flat section, b – the value of along the parametric length of the spring).</figcaption></figure>

The pliability of intermediate supports is caused by relative displacement Δc of flat spring and support at the contact zone. Determined by the following expression [15]:

$$ \Delta_c = 0.83 \sqrt{\frac{Q_1^2}{R} \left( \frac{1-\mu_1^2}{E_1} + \frac{1-\mu_2^2}{E_2} \right)^2}, \quad (28) $$

The stiffness coefficient in the contact point c_y = Q_1/Δc. The resulted formula has a simplified view for contact with materials steel and steel:
$$ \Delta_c = 1.23 \sqrt[3]{\left(\frac{Q_1}{E}\right)^2 \frac{1}{R}}, \qquad (29) $$

$$ c_y = 0.81 \sqrt[3]{Q_1 R E^2}. \qquad (30) $$

The last formula allows to consider the contact stiffness in intermediate support on implementation of vibro-impact modes. In this case, parametric contact stiffness is depending from the contact force through the instantaneous movement of local mass $y_m$. For $y_m = 0.24 \text{ mm}$ contact stiffness is $c_y = 6.1 \times 10^7 \text{ N/m}$.

**Dynamics of stress-strain state.** The resulting static characteristics can be used for dynamic stress state analysis of the vibro-impact system with kinematic's disturbance of local mass (fig. 8).

The characteristics of local mass movement takes in the form $y(t) = y_m \sin(314t)$. Contact force depends of the instantaneous movement of local mass according to conditions of the system:

$$ Q_c(t) = \begin{cases} 0, & y(t) \ge 0, \\ \frac{c_{II}y(t)(l_1^2 + 4l_1l_2 + 3l_2^2)}{2l_1(l_1 + 4l_2)}, & y(t) < 0, \end{cases} \qquad (31) $$

Given a parametric dependence $c_y$ contact stiffness from the contact force $Q_1$, confirmed their nonlinearity in the fig. 9.

<figure><img src="image_12.png" /><figcaption>Fig. 8. Kinematic's disturbance scheme of the vibro-impact rod system.</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>Fig. 9. Dependence of the stiffness coefficient of intermediate support of the magnitude of the contact force (by the displacement y(t) of local mass).</figcaption></figure>

The equivalent bending stress is determined by the conditions of the rod system operation:

$$ \sigma_{eq}(t) = \begin{cases} \sigma_{eqI}(t), & y(t) \ge 0, \\ \sigma_{eqII}(t), & y(t) < 0, \end{cases} \qquad (32) $$

Depending in time the equivalent stresses for bending and contact presented at the fig. 10. They have asymmetrical pulse character, caused by implementation of the vibro-impact mode scheme.

<figure><img src="image_14.png" /><figcaption>Fig. 10. The time dependence of bending (a) and contact stresses (b) in vibro-impact rod system.</figcaption></figure>

Parametric dependents of calculated stress values from the instantaneous displacement of local mass y(t) are shown in fig. 11. Contact stress is nonlinear from displacement in time.
<figure><img src="image_15.png" /><figcaption>Fig. 11. Parametric dependence of von Mises and contact stresses in the vibro-impact rod system.</figcaption></figure>

**Summary.** The construction of resonance module capable to operate in a harmonious and vibro-impact modes is made. Formula of the bending rigidity of flat spring to implement the given the asymmetric elastic characteristic is obtained by finite element method. A value and character of stress-strain state of the flat spring analytically is determined. The result is approved by modeling in SolidWorks Simulation. Contact stiffness in intermediate cylindrical support in contact with a flat spring is determined. Dynamics for example of the kinematic disturbance is considered and built a parametric characteristic of equivalent stresses.

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