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Effects of passive vibration absorbers on the mechanisms having clearance joints

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Abstract

In the current study, the dynamic behavior of two planar mechanisms with revolute joints, in the presence of clearances is investigated. Subsequently, a control scheme with the aim of restraining the clearance and maintaining a more stable behavior is proposed. This approach is based on using tuned mass damper (TMD) in order to reduce the effects of clearances in mechanisms for passive control purpose. The applied absorber’s mass is insignificant and they are little in size. By attaching two perpendicular absorbers in order to control the clearance, the oscillations are notably reduced. The proposed methodology is applied to planar multibody mechanical systems with revolute clearance joints with a view to demonstrating its features. The absorber performance is evaluated for changes around the designed state and the results show that the designed absorbers have a good robustness with variations to changes in different parameters.

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Abbreviations

\(r_{i}\) :

Clearance radius

\(R_{B}\), \(R_{j}\) :

Bearing and journal radius

\(K\) :

Stiffness parameter

\(\nu\) :

Poisson’s ratio

\(E\) :

Young’s modulus

\(\omega\) :

Angular velocity

\(\delta\) :

Relative penetration depth

\(\dot{\delta }^{(-)}\) :

Initial impact velocity

\(L_{i}\) :

Length of link \(i\)

\(m_{i}\) :

Mass of link \(i\)

\(I_{i}\) :

Moment of inertia for link \(i\)

\(\theta _{i}\) :

Angle of link \(i\)

\(\nu _{T}\) :

Relative tangential velocity

\(C_{e}\) :

Restitution coefficient

\(\nu _{0}\), \(\nu _{1}\) :

Tolerances for the tangential velocity

\(x\) :

Slider horizontal position

\(C_{f}\) :

Coefficient friction

\(C_{d}\) :

Dynamic correction coefficient

\(F_{N}\) :

Normal contact force

\(F_{T}\) :

Tangential contact force

\(M\) :

Input moment

\(V\) :

Potential energy

\(T\) :

Kinetic energy

\(q\) :

Generalized coordinate

\(X_{G_{i}}\) :

Horizontal position of the COM for link \(i\)

\(Y_{G_{i}}\) :

Perpendicular position of the COM for link \(i\)

\(c_{P}\) :

Absorber damping coefficient

\(Q_{c}\) :

Magnitude of contact force

\(K_{p}\) :

Absorber stiffness coefficient

\(D_{p}\) :

Dissipation Rayleigh term

\(\phi\) :

Friction force angle

\(D\) :

Hysteresis damping

\(\psi \) :

Orientation of total contact force

\(\alpha\) :

Contact force angle

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Authors and Affiliations

  1. Department of Mechanical Engineering, Babol Noshirvani University of Technology, Shariati Street, P.O. Box 484, 47148-71167, Babol, Mazandaran, Iran

    Aryan Amiri, Morteza Dardel & Hamidreza Mohammadi Daniali

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  1. Aryan Amiri
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  2. Morteza Dardel
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  3. Hamidreza Mohammadi Daniali
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Correspondence to Morteza Dardel.

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Appendix: Algorithm to detect contact and computation of contact force

Appendix: Algorithm to detect contact and computation of contact force

In the MATLAB software, contacts can be accurately identified using the event option of ODE solver. The overall structure of this option is as follows:

[value,isterminal,direction] = events(t,y)

value = penetration depth;

isterminal = 0;

direction = 1;

end

The value determines the penetration depth. If penetration depth is positive, the contact force is presented, otherwise it is zero. If isterminal is equal to 1, the integration is terminated, and if it is set to zero, the solution will be proceeding. The direction is set to +1 to detect the direction of increasing penetration.

With this option, multiple collisions can also be identified. It can be easily explained that if the penetration depth is positive, the amount of force is positive and considered. But if the contacts goes away, the contact force will be zero. There are several collisions, so this option can be reliably used to solve.

Also the event option in odeset of Matlab software can be used to obtain Poincare section.

There is another method to confront with impact. For example following conditional structure can be used for this purpose.

There is a second method to detect contact and calculate contact force for each clearance joint, which can be written to the following form:

if \(\delta \leq 0\)

\(Q_{c} =0\)

if \(\dot{r} >0\)

\(\dot{r}_{1} = \dot{r}\)

end

else

\(Q_{c} = \mbox{Eq.~(13)}\)

end

This simple algorithm can be implemented in the Matlab, and can be written for each of clearance joint, and accumulation of events can be simply handled.

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Amiri, A., Dardel, M. & Daniali, H.M. Effects of passive vibration absorbers on the mechanisms having clearance joints. Multibody Syst Dyn 47, 363–395 (2019). https://doi.org/10.1007/s11044-019-09684-2

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