Method for the study of dynamic characteristics in the mechanisms of motion transmission

2.1. Method of research. Theoretical aspect

With longitudinal harmonic oscillations in an isotropic body, a flat longitudinal sinusoidal wave propagates in the positive direction from the source of oscillations, the equation of which has the form [8]:

where $\phi$ is potential of the velocity of the disturbed motion of the medium; $\omega$ is cyclic frequency of oscillations; $k$ is wave number, $k=\omega$/$a_1$; $b$ is wave amplitude; ${\alpha }_0$ is initial phase of oscillations points of medium.

If a wave propagating in a certain medium reaches the interface of this medium with another medium, then reflection, refraction or absorption of this wave may occur. Features of the interaction the wave with the interface of the media is largely determined by the wave resistance. For solids in the physical encyclopedia [9], such resistance is defined as the ratio of the voltage with the opposite sign to the velocity of the medium in a wave of oscillations:

here $a_1$ is the speed of sound in the line. From the manuscripts [8, 9] it follows that $a_1=\mathrm{\Theta }{\left(\rho /E\right)}^{\frac{1}{2}}$, $\mathrm{\Theta }=\left[\right(1-\mu )/[\left(1+\mu \right){\left[\left(1-2\mu \right)\right]}^{1/2}$, where $\mu$ is Poisson’s ratio.

If we consider the harmonic oscillations propagating along the force line, then mechanical impedance in the operator form is into consideration, taking into account Eq. (7), can be represented as [7]:

We note that the ratio $E_{\upsilon }$/$E_u=\mathrm{t}\mathrm{g}\xi$ is characterizes the magnitude of internal friction, and $\xi$ determines the phase by which the change in strain is ahead of the change in stress.

When $\rho =\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$, $\mathrm{\Theta }=$1, this expression will have the form:

here:

Dynamic features of lines with parameters distributed along the length are characterized by the operator coefficient of wave propagation, which in Laplace images can be written in the form [7]:

Usually, internal friction is neglected, i.e. $\psi =$ 0.

Since the relaxation time constant is small (${\tau }_{\epsilon }\approx$ 10^-8 – 10^-9 sec), it can be seen from Eq. (4) that for a sufficiently wide frequency band, one can take $E_{\upsilon }=$ 0.

Then when $\rho =\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$, $E_u=E_2=E$, the wave impedance will be:

Conducting under zero initial conditions, one-dimensional Laplace transform [10] Eqs. (5) for longitudinal oscillations and taking into account that with the accepted assumptions:

The solution of the system of Eqs. (13) allows to find for the selected section instantaneous deviations from the steady-state values of stresses and speed of movement of the sections of the rod, depending on the physical and geometrical properties of the line [7].

We present the final solution of the system of Eqs. (13) for the mechanism transmitting translational motion, under the corresponding boundary conditions [7], Fig. 1(a):

here:

Eqs. (14), (15) allow determine various frequency characteristics. Below is one of such characteristics ($W_{\sigma }$), illustrating the effect of a change in the longitudinal velocity of the input link ${\upsilon }_1$ on the change in normal voltages in the power line near the output link ${\sigma }_2$ in the absence of a resistance force ($F\equiv$ 0) [7]:

here $f_2$ is the sectional area of the trunk at the executive body of mass $m$; $c$, $h_n$, $F$ is coefficients of friction loss and resistance force acting on the executive body; $l$ is the length of the line.

In view of (16) we have $Z_n=j\mathrm{t}\mathrm{g}\alpha$/$j\alpha =\mathrm{t}\mathrm{g}\alpha$/$\alpha$, where $\alpha =l\omega {\left(\rho /E\right)}^{1/2}$, then then functions ${\vartheta }_n$$\left(j\omega \right)={\vartheta }_n\left(\alpha \right)={\vartheta }_{n0}{Z}_n\left(\alpha \right)$ not complex functions [7]. From the analysis of the function Z_n ($j\omega$) follows: with $\alpha \to$ 0, $Z_n\to$ 1; for $\pi$/2 + $k\pi$ > $\alpha$ > $\pi$ + $k\pi$, $Z_n$ < 0. Here $k=$ 0, 1, 2, ...., $n$.

This means that at 0 ≤ $\alpha$ ≤ 1, the wave processes in the lines can be neglected.

This conclusion was first published in [7]. The papers [11-13] give solutions using Eqs. (17) for the mechanisms of rotational motion transmission, Fig. 1(b).

Using Eqs. (14)-(27), neglecting for simplify of the line parameters distribution and applying the well-known methods of the theory of functions of a complex variable, one can obtain simple expressions describing motion.

So, for example, in the case of an abrupt change in the force $F\left(t\right)=F_0*1\left(t\right)$ and at ${\upsilon }_1\equiv$ 0 as a result of the solution using the theorem of residues, the speed of movement of the executive body is described by the expression:

here $T_n$ is inertial constant $\text{(}{T}_n=m/h_n\text{);}$${\zeta }_n$ is damping coefficient, ${\zeta }_n=\mathrm{0,5}{\vartheta }_{n0}h/{\left[\left(1-c\right)m{\vartheta }_{n0}\right]}^{1/2}$ [7].

If it is impossible to neglect the distributed parameters, then it is advisable to apply the method of numerical solution to determine the quantitative characteristics [14].

For example, to study the dynamics of the mechanism, where instead of the rod (Fig. 1(a)) a spring was used, a special program was developed on the basis of Eqs. (4) that made it possible to calculate the oscillations of the speed motion of the mass. The result of the calculation is shown in Fig. 3.

Fig. 3Calculated oscillogram of the transition process

The proposed method was applied to the study of dynamic processes for two-link and three-link mechanisms in relation to the technology of manufacturing deep holes. The zones of stable and unstable operation of the mechanism were identified and the cutting conditions were optimized [15].