where qiðtÞ are the modes of vibrations of the flexible slider– crank mechanism。 To derive the model for the flexible mechanism   the   Euler–Lagrange   equations   are   used。  Let

L ¼ T — U, where T and U are the kinetic and potential energies of the system, respectively。 The equations of motion can be obtained using the following  equation:

2。Modeling of the mechanism

Equation of motion of a flexible slider–crank mechanism  is

where  Fi  are  the  nonconservative  forces,  τi   is  the  applied

derived using the Euler–Lagrange approach [13–17]。 The mechanism is assumed to move in the horizontal plane and

torque on the system, and !ξ

is the deflection vector。

the longitudinal defections are negligible。 Schematic of the slider–crank mechanism with a flexible connecting rod is depicted in Fig。 1。 The mechanism parameters are defined    as

½ξ1; ξ2; :::; ξn þ 1]¼ ½θ; q1ðtÞ; q2ðtÞ; :::; qnðtÞ] ð6Þ

The kinetic energy of the system is then   calculated:

follows: r is the crank length; L is the connecting rod length; θ

is the crank angle; ψis the connecting rod angle with respect to

the ground; x and w are the x- and y-coordinates, respectively,

where ms  is the mass of the slider, !X

B  is the velocity of  the

of any point on the connecting rod in the !e `1 — !e 2  coordinate

system。

The location of any point on the flexible connecting rod (Fig。 1) is given  by

connecting rod end point, Ic is the moment of inertia of the crank, and ρ; A are the density and cross section of the connecting rod, respectively。

!R  ¼ !r  þ!x  þ!w

equal to

!R  ¼ ðr  cos  θ þw  cos  ψ þx  cos  ψ Þ !i

þðr  sin  θ þw  sin  ψ  — x  sin  ψ Þ!j

The y-component of the displacement of the end point of the

connecting rod at x ¼ l, which can be obtained by taking    the

The  dependent   coordinate  ψ   is  then  omitted   using  the

scalar product of the displacement vector !Rzero。 Therefore

holonomic constraint of the slider–crank mechanism (Eq。  (3))。

The potential energy of the mechanism is given by

Fig。 1。  Slider–crank mechanism。

For a single mode  model

Table 1

Mechanism’s parameters。

Variable Definition Value

R Crank length 10 cm

L Connecting rod length 30 cm

Ms Slider mass 0。5 kg

Mc Crank mass 2(ρ)(π)hr

EI Flexibility 0。2

ρ Material density 7850

H Radius of the rod