5. Derivation of the initial equations

Let us turn further to the longitudinal oscillations.

The interaction of the elastic wave is largely with the interface of the media due to the wave impedance, which is determined by the relation [34]. It follows that

$$-\frac{\sigma}{\nu} = \rho a\_1,\tag{33}$$

where a<sup>1</sup> is the velocity of propagation of longitudinal oscillations in the medium.

When considering harmonic oscillations propagating along a line, we usually study the wave resistance in operator form or mechanical impedance

$$Z\_b(j\rho) = \frac{\sigma(j\rho)}{\nu(j\rho)}.\tag{34}$$

Referring to Formula (34) and taking into account that

$$a\_1 = \Theta \sqrt{\frac{E}{\rho}},\tag{35}$$

where [34]

$$\Theta = \sqrt{\frac{1-\mu}{(1+\mu)(1-2\mu)}},\tag{36}$$

we write

$$Z\_b(\
ja) = \rho(o)\Theta\sqrt{\frac{E\_o(\
ja)}{\rho(\
ja)}} = \Theta\sqrt{E\_o(\
ja)\rho(\
ja)}.\tag{37}$$

we obtain

The solution is

Let at x = 0

ρsυðÞ¼� s

<sup>E</sup> <sup>d</sup>υð Þ<sup>s</sup>

Dynamic Stability of Open Two-Link Mechanical Structures

DOI: http://dx.doi.org/10.5772/intechopen.91045

the line, depend on the physical and geometric properties of the line.

dυ(s)/dx using Eq. (24), and applying relation (20), we obtain

∂2 σð Þs <sup>∂</sup>x<sup>2</sup> � <sup>θ</sup><sup>2</sup>

σð Þ¼ s, x σ1ð Þ s, 0 ;

�<sup>1</sup>θð Þ<sup>s</sup> <sup>E</sup>υ1ð Þ <sup>s</sup>, 0

After substituting these dependencies, the solution will be

When we introduce hyperbolic functions, then we get

σð Þ¼ s, x σ1ð Þ s, 0 ch½ �� θð Þs x θð Þs s

∂2 υð Þs <sup>∂</sup>x<sup>2</sup> � <sup>θ</sup><sup>2</sup>

Then, taking into account (49), we get

� s

<sup>C</sup><sup>1</sup> <sup>¼</sup> <sup>σ</sup>1ð Þ� <sup>s</sup>, 0 <sup>s</sup>

manner described, we obtain

13

σð Þ¼ s, x

Differentiating Eq. (23) with respect to x, then eliminating the derivative

This equation is a second-order differential equation with constant coefficients.

The integration constants C<sup>1</sup> and C<sup>2</sup> are determined by the boundary conditions.

<sup>∂</sup><sup>x</sup> ¼ � <sup>E</sup>

<sup>∂</sup>σð Þ <sup>s</sup>, <sup>x</sup>

The last condition from (49) is obtained from (45) by replacing ρs = θ<sup>2</sup>

<sup>2</sup> ; <sup>С</sup><sup>2</sup> <sup>¼</sup> <sup>σ</sup>1ð Þþ <sup>s</sup>, 0 <sup>s</sup>

σ1ð Þ s, 0 f g exp ½ �þ θð Þs x exp ½ � �θð Þs x 2

Having solved the system of Eqs. (47) and (48) with respect to υ(s, x) in the

�<sup>1</sup>θð Þ<sup>s</sup> <sup>E</sup>σ1ð Þ <sup>s</sup>, 0 f g exp ½ �� <sup>θ</sup>ð Þ<sup>s</sup> <sup>x</sup> exp ½ � �θð Þ<sup>s</sup> <sup>x</sup>

�1

σðÞ¼ s C<sup>1</sup> exp ½ �þ θð Þs x C<sup>2</sup> exp ½ � �θð Þs : (48)

s θ2

dσð Þs

The solution of the system of Eqs. (45) and (46) allows to find for the selected section the instantaneous deviations from the steady-state values of stress and speed of movement sections of rod. Each of these quantities will be the sum of the quantities of the same name, determined in the front of the perturbation propagating in the forward and reverse directions. The instantaneous deviations of the marked variables, as well as the peculiarities of the disturbance propagation along

dx ; (45)

dx ¼ �sσð Þ<sup>s</sup> : (46)

ð Þs σðÞ¼ s 0: (47)

ð Þs υ1ð Þ s, 0 : (49)

<sup>2</sup> : (50)

�<sup>1</sup>θð Þ<sup>s</sup> <sup>E</sup>υ1ð Þ <sup>s</sup>, 0

<sup>2</sup> : (51)

Eυ1ð Þ s, 0 sh½ � θð Þs x : (52)

ð Þs υðÞ¼ s 0: (53)

(s)Е/s.

Note that the ratio Eυ/Eu = tg ξ characterizes the magnitude of internal friction and ξ determines the phase on which the change of stress is ahead of the change in deformation.

Then Eq. (37) becomes

$$Z\_b(\ j a) = \frac{\sigma(\ j a)}{\nu(\ j a)} = \Theta \sqrt{E\_u(o)\rho(o)} \sqrt{\mathbf{1} + j \frac{E\_v(o)}{E\_u(o)}}.\tag{38}$$

If ρ = const and Θ = 1, then the last equation may be written in the form

$$Z\_b(j\omega) = \frac{j\alpha \mathcal{E}\_u(\alpha)\sqrt{\mathfrak{p}/\mathcal{E}\_u(\alpha)}\sqrt{\mathfrak{1} + j\mathcal{E}\_v(\alpha)/\mathcal{E}\_u(\alpha)}}{j\alpha} = \frac{\mathcal{E}\_u(\alpha)\mathfrak{d}(j\omega)}{j\alpha}.\tag{39}$$

Here

$$\Theta(j\rho) = \pm \sqrt{j \frac{\alpha}{E\_u(\rho)} [\rho j \rho + \wp(\rho)]}; \ \wp(\rho) = -\frac{\rho \alpha E\_v(\rho)}{E\_u(\rho)}. \tag{40}$$

Internal friction in solids ψ can play a significant role. For example, it is known that magnesium alloys and a number of other materials have very good vibrationinsulating properties, largely due to internal friction. At the same time, for steels this value is small and it is often neglected.

The dynamic features of lines with parameters distributed over length (in principle, parameters are distributed in any line) are characterized by the operator coefficient of wave propagation, which, in Laplace images, can be written in the form [29, 31]

$$\Theta(s) = \pm \sqrt{\frac{s}{E\_u(o)} [\rho s + \varphi(o)]}.\tag{41}$$

Then, when E<sup>υ</sup> = 0, ρ = const, and Eu = E, the wave resistance will be

$$Z\_b(\mathfrak{s}) = \frac{\sigma(\mathfrak{s})}{\nu(\mathfrak{s})} = \frac{\theta(\mathfrak{s})E}{\mathfrak{s}}.\tag{42}$$

From where it also follows

$$
\theta(\mathfrak{s}) = \frac{sZ\_b(\mathfrak{s})}{E}.\tag{43}
$$

Conducting a one-dimensional Laplace transform [30] of Eqs. (13) and (14) for longitudinal oscillations with zero initial conditions and taking into account that with the accepted assumptions

$$
\Theta(\mathfrak{s}) = \pm \mathfrak{s} \sqrt{\frac{\mathfrak{p}}{E}} \tag{44}
$$

Dynamic Stability of Open Two-Link Mechanical Structures DOI: http://dx.doi.org/10.5772/intechopen.91045

we obtain

we write

deformation.

Here

form [29, 31]

Then Eq. (37) becomes

Zbð Þ¼ jω ρð Þ ω Θ

Truss and Frames - Recent Advances and New Perspectives

Zbð Þ¼ <sup>j</sup><sup>ω</sup> <sup>σ</sup>ð Þ <sup>j</sup><sup>ω</sup>

Zbð Þ¼ <sup>j</sup><sup>ω</sup> <sup>j</sup>ωEuð Þ <sup>ω</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

θð Þ¼� jω

this value is small and it is often neglected.

From where it also follows

with the accepted assumptions

12

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Eωð Þ jω ρð Þ jω

Note that the ratio Eυ/Eu = tg ξ characterizes the magnitude of internal friction and ξ determines the phase on which the change of stress is ahead of the change in

<sup>υ</sup>ð Þ <sup>j</sup><sup>ω</sup> <sup>¼</sup> <sup>Θ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

If ρ = const and Θ = 1, then the last equation may be written in the form

<sup>ρ</sup>=Euð Þ <sup>ω</sup> <sup>p</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Euð Þ <sup>ω</sup> ½ � <sup>ρ</sup>j<sup>ω</sup> <sup>þ</sup> ψ ωð Þ

Internal friction in solids ψ can play a significant role. For example, it is known that magnesium alloys and a number of other materials have very good vibrationinsulating properties, largely due to internal friction. At the same time, for steels

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s Euð Þ <sup>ω</sup> ½ � <sup>ρ</sup><sup>s</sup> <sup>þ</sup> ψ ωð Þ

The dynamic features of lines with parameters distributed over length (in principle, parameters are distributed in any line) are characterized by the operator coefficient of wave propagation, which, in Laplace images, can be written in the

r

Then, when E<sup>υ</sup> = 0, ρ = const, and Eu = E, the wave resistance will be

θðÞ¼ s

θðÞ¼� s s

σð Þs <sup>υ</sup>ð Þ<sup>s</sup> <sup>¼</sup> <sup>θ</sup>ð Þ<sup>s</sup> <sup>E</sup>

sZbð Þs

Conducting a one-dimensional Laplace transform [30] of Eqs. (13) and (14) for longitudinal oscillations with zero initial conditions and taking into account that

> ffiffiffi ρ E r

ZbðÞ¼ s

<sup>j</sup> <sup>ω</sup>

θðÞ¼� s

r

<sup>1</sup> <sup>þ</sup> jEυð Þ <sup>ω</sup> <sup>=</sup>Euð Þ <sup>ω</sup> <sup>p</sup>

¼ Θ

q

Euð Þ <sup>ω</sup> <sup>ρ</sup>ð Þ <sup>ω</sup> <sup>p</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ j

<sup>j</sup><sup>ω</sup> <sup>¼</sup> Euð Þ <sup>ω</sup> <sup>θ</sup>ð Þ <sup>j</sup><sup>ω</sup>

; ψ ωð Þ¼� ρωEυð Þ <sup>ω</sup>

s

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Eωð Þ jω ρð Þ jω

> Eυð Þ ω Euð Þ ω

: (37)

: (38)

<sup>j</sup><sup>ω</sup> : (39)

Euð Þ <sup>ω</sup> : (40)

: (41)

<sup>s</sup> : (42)

<sup>E</sup> : (43)

, (44)

s

$$
\rho \mathfrak{so}(\mathfrak{s}) = -\frac{d\sigma(\mathfrak{s})}{d\mathfrak{x}};\tag{45}
$$

$$E\frac{d\nu(s)}{d\kappa} = -s\sigma(s). \tag{46}$$

The solution of the system of Eqs. (45) and (46) allows to find for the selected section the instantaneous deviations from the steady-state values of stress and speed of movement sections of rod. Each of these quantities will be the sum of the quantities of the same name, determined in the front of the perturbation propagating in the forward and reverse directions. The instantaneous deviations of the marked variables, as well as the peculiarities of the disturbance propagation along the line, depend on the physical and geometric properties of the line.

Differentiating Eq. (23) with respect to x, then eliminating the derivative dυ(s)/dx using Eq. (24), and applying relation (20), we obtain

$$\frac{\partial^2 \sigma(s)}{\partial \mathbf{x}^2} - \theta^2(s)\sigma(s) = \mathbf{0}.\tag{47}$$

This equation is a second-order differential equation with constant coefficients. The solution is

$$
\sigma(\mathfrak{s}) = \mathbf{C}\_1 \exp\left[\theta(\mathfrak{s})\mathfrak{x}\right] + \mathbf{C}\_2 \exp\left[-\theta(\mathfrak{s})\right].\tag{48}
$$

The integration constants C<sup>1</sup> and C<sup>2</sup> are determined by the boundary conditions. Let at x = 0

$$
\sigma(\mathfrak{s}, \mathfrak{x}) = \sigma\_1(\mathfrak{s}, \mathbf{0}); \frac{\partial \sigma(\mathfrak{s}, \mathfrak{x})}{\partial \mathfrak{x}} = -\frac{E}{\mathfrak{s}} \Theta^2(\mathfrak{s}) \nu\_1(\mathfrak{s}, \mathbf{0}).\tag{49}
$$

The last condition from (49) is obtained from (45) by replacing ρs = θ<sup>2</sup> (s)Е/s. Then, taking into account (49), we get

$$C\_1 = \frac{\sigma\_1(\mathfrak{s}, \mathfrak{0}) - \mathfrak{s}^{-1}\theta(\mathfrak{s})E\nu\_1(\mathfrak{s}, \mathfrak{0})}{2};\\C\_2 = \frac{\sigma\_1(\mathfrak{s}, \mathfrak{0}) + \mathfrak{s}^{-1}\theta(\mathfrak{s})E\nu\_1(\mathfrak{s}, \mathfrak{0})}{2}.\tag{50}$$

After substituting these dependencies, the solution will be

$$\sigma(s,\mathbf{x}) = \frac{\sigma\_1(s,\mathbf{0})\{\exp\left[\theta(s)\mathbf{x}\right] + \exp\left[-\theta(s)\mathbf{x}\right]\}}{-\frac{s^{-1}\theta(s)E\sigma\_1(s,\mathbf{0})\{\exp\left[\theta(s)\mathbf{x}\right] - \exp\left[-\theta(s)\mathbf{x}\right]\}}{2}}.\tag{51}$$

When we introduce hyperbolic functions, then we get

$$
\sigma(\boldsymbol{\varsigma}, \boldsymbol{\varsigma}) = \sigma\_1(\boldsymbol{\varsigma}, \mathbf{0}) ch[\boldsymbol{\Theta}(\boldsymbol{\varsigma})\boldsymbol{\varkappa}] - \boldsymbol{\Theta}(\boldsymbol{\varsigma}) \boldsymbol{\varsigma}^{-1} E \nu\_1(\boldsymbol{\varsigma}, \mathbf{0}) sh[\boldsymbol{\Theta}(\boldsymbol{\varsigma})\boldsymbol{\varkappa}].\tag{52}
$$

Having solved the system of Eqs. (47) and (48) with respect to υ(s, x) in the manner described, we obtain

$$\frac{\partial^2 \nu(s)}{\partial \mathbf{x}^2} - \Theta^2(s)\nu(s) = \mathbf{0}.\tag{53}$$

For boundary conditions with x = 0

$$\nu(\mathfrak{s}, \mathfrak{x}) = \nu\_1(\mathfrak{s}, \mathbf{0}); \frac{\partial \nu(\mathfrak{s}, \mathfrak{x})}{\partial \mathfrak{x}} = -\frac{\mathfrak{s}\sigma(\mathfrak{s}, \mathbf{0})}{E},\tag{54}$$

σ2ðÞ¼ s

Dynamic Stability of Open Two-Link Mechanical Structures

DOI: http://dx.doi.org/10.5772/intechopen.91045

Ω2ð Þs 1 þ hkϑkð Þs s þ Jϑkð Þs s

GrWp<sup>2</sup>

analyzing the operation of the hydraulic drive [9, 35–37].

Here

influence of F on σ:

WFð Þ¼ <sup>j</sup><sup>ω</sup> <sup>υ</sup>2ð Þ <sup>j</sup><sup>ω</sup>

WFσð Þ¼ <sup>j</sup><sup>ω</sup> <sup>σ</sup>2ð Þ <sup>j</sup><sup>ω</sup>

<sup>W</sup>υð Þ¼ <sup>j</sup><sup>ω</sup> <sup>υ</sup>2ð Þ <sup>j</sup><sup>ω</sup>

We introduce the notation α = l ω (ρ/E)

Znð Þ¼ <sup>j</sup><sup>ω</sup> th½ � <sup>θ</sup>ð Þ <sup>j</sup><sup>ω</sup> <sup>l</sup>

WFσð Þ¼ <sup>j</sup><sup>ω</sup> <sup>σ</sup>2ð Þ <sup>j</sup><sup>ω</sup>

Insofar as

then

15

<sup>ϑ</sup>kðÞ¼ <sup>s</sup> <sup>ϑ</sup>k0Zkð Þ<sup>s</sup> ; <sup>ϑ</sup>k<sup>0</sup> <sup>¼</sup> <sup>l</sup>

FcðÞþs υ1ð Þs ð Þ hn þ ms =chA

chAk

Performing such transformations in relation to torsional vibrations, we obtain

thAk Ak

Comparing (15), (18), and (19), we can see that the processes of motion transfer in solid and liquid media can be described by similar equations. This is shown when

Eqs. (47) and (48) make it possible to calculate the frequency characteristics of the system, i.e., determine the response of the model to the harmonic change in the

For F � 0, we obtain the frequency characteristics W<sup>υ</sup> (jω), illustrating the effect

<sup>2</sup> � � <sup>¼</sup> <sup>Ω</sup>1ð Þ<sup>s</sup>

; ZkðÞ¼ s

6. Lemma on the degree of distribution of force line parameters

speed of the lead link or the resistance force acting on the driven link.

For example, for υ<sup>1</sup> � 0 and s=jω, we obtain the characteristic WF (jω), illustrating the influence of F on υ2, and the characteristic WF<sup>σ</sup> (jω), illustrating the

Fcð Þ <sup>j</sup><sup>ω</sup> ¼ � <sup>ϑ</sup>nð Þ <sup>j</sup><sup>ω</sup> <sup>j</sup><sup>ω</sup>

of υ<sup>1</sup> on υ2, and WF<sup>σ</sup> (jω), illustrating the influence of υ<sup>1</sup> on σ2. So we have

Fcð Þ <sup>j</sup><sup>ω</sup> <sup>¼</sup> <sup>1</sup>

<sup>υ</sup>1ð Þ <sup>j</sup><sup>ω</sup> ¼ � <sup>с</sup>h�<sup>1</sup>

<sup>υ</sup>1ð Þ <sup>j</sup><sup>ω</sup> <sup>¼</sup> hn <sup>þ</sup> mj<sup>ω</sup>

<sup>θ</sup>ð Þ <sup>j</sup><sup>ω</sup> <sup>l</sup> <sup>¼</sup> jtg<sup>α</sup>

From (63) to (66), it can be seen that changes in the voltage and speed of movement of the output link are lagging behind changes in input impacts.

1/2.

<sup>j</sup><sup>α</sup> <sup>¼</sup> tg<sup>α</sup>

<sup>f</sup> <sup>2</sup> <sup>1</sup> <sup>þ</sup> hnϑnð Þ<sup>s</sup> <sup>s</sup> <sup>þ</sup> <sup>m</sup>ϑnð Þ<sup>s</sup> <sup>s</sup><sup>2</sup> ½ � : (60)

; Ak ¼ θkð Þs l; θkðÞ¼� s

<sup>1</sup> <sup>þ</sup> hnϑnð Þ <sup>j</sup><sup>ω</sup> <sup>j</sup><sup>ω</sup> <sup>þ</sup> <sup>m</sup>ϑnð Þ <sup>j</sup><sup>ω</sup> ð Þ <sup>j</sup><sup>ω</sup> <sup>2</sup> ; (63)

<sup>f</sup> <sup>2</sup> <sup>1</sup> <sup>þ</sup> hnϑnð Þ <sup>j</sup><sup>ω</sup> <sup>j</sup><sup>ω</sup> <sup>þ</sup> <sup>m</sup>ϑnð Þ <sup>j</sup><sup>ω</sup> ð Þ <sup>j</sup><sup>ω</sup> <sup>2</sup> h i : (64)

<sup>1</sup> <sup>þ</sup> hnϑnð Þ <sup>j</sup><sup>ω</sup> <sup>j</sup><sup>ω</sup> <sup>þ</sup> <sup>m</sup>ϑnð Þ <sup>j</sup><sup>ω</sup> ð Þ <sup>j</sup><sup>ω</sup> <sup>2</sup> , (65)

<sup>α</sup> ; ch j ð Þ¼ <sup>α</sup> cos <sup>α</sup>, (67)

½ � θð Þ jω l

<sup>f</sup> <sup>2</sup>ch½ � <sup>θ</sup>ð Þ <sup>j</sup><sup>ω</sup> <sup>l</sup> <sup>1</sup> <sup>þ</sup> hnϑnð Þ <sup>j</sup><sup>ω</sup> <sup>j</sup><sup>ω</sup> <sup>þ</sup> <sup>m</sup>ϑnð Þ <sup>j</sup><sup>ω</sup> ð Þ <sup>j</sup><sup>ω</sup> <sup>2</sup> h i : (66)

� Mrð Þs ϑnð Þs s: (61)

ffiffiffiffi ρ G r

: (62)

We finally obtain

$$\nu(\mathfrak{s}, \mathfrak{x}) = \nu\_1(\mathfrak{s}, \mathfrak{O}) ch[\mathfrak{G}(\mathfrak{s})\mathfrak{x}] - \frac{s \sigma\_1(\mathfrak{s}, \mathfrak{O}) ch[\mathfrak{G}(\mathfrak{s})\mathfrak{x}]}{\mathfrak{G}(\mathfrak{s}) E} \,. \tag{55}$$

Movement in two-link elements that occurs within elastic limits can be viewed as the movement of the driven point (link) from the movement of the leading point (link), which is affected by the previous links of the truss, for example, which perceive wind load. If, in the process of oscillation, the output link does not allow the input impulse to pass, then waves of disturbance are reflected from the end of the lines.

Consider the case of a matched load when there are no reflected waves in the system and oscillations in the system do not affect the movement of the driven link due to the attached large mass. In this case, the boundary conditions are the following relations:

$$\begin{aligned} \nu(s,l) &= \nu\_2(s); \ \nu(s,0) = \nu\_1(s); \ \sigma(s,l) = \sigma\_2(s); \ \sigma(s,0) = \sigma\_1(s);\\ \sigma\_2(s) &= \frac{F\_\varepsilon(s) + h\_n \nu\_2(s) + ms \nu\_2(s)}{f\_2}. \end{aligned} \tag{56}$$

Here f<sup>2</sup> is the sectional area of the line in front of the slave link of mass m; hn and F<sup>с</sup> are coefficient of friction loss, proportional to the speed of movement, as well as the resistance force acting on the slave link; l is the length of the line.

Together we solve (52)–(56) by performing the following transformations

$$\begin{aligned} \sigma\_1(s) &= \frac{1}{chA} \left[ \sigma\_2(s) + \frac{1}{s} E \nu\_1(s) \theta(s) hA \right]; \\ \nu\_2(s) &= \nu\_1(s) chA - \frac{s}{E\theta(s)} shA \left\{ \frac{1}{chA} \left[ \sigma\_2(s) + \frac{1}{s} E \nu\_1(s) \theta(s) shA \right] \right\}; \\ \nu\_2(s) &= \frac{\nu\_1(s)}{chA} - \frac{s}{E\theta(s)} \sigma\_2(s) thA; \\ \left[ \frac{\nu\_1(s)}{chA} - \nu\_2(s) \right] \frac{E}{s} \frac{\theta(s)}{thA} &= \frac{1}{f\_2} [F(s) + h\_n \nu\_2(s) + m \nu\_2(s)]. \end{aligned} \tag{57}$$

After bringing similar members, we obtain the equation of motion of the driven link of the mechanical system in the form

$$
\eta\_2(\boldsymbol{\varsigma}) \left[ \mathbf{1} + h\_n \theta\_n(\boldsymbol{\varsigma}) \boldsymbol{\varsigma} + m \theta\_n(\boldsymbol{\varsigma}) \boldsymbol{\varsigma}^2 \right] = \frac{\nu\_1(\boldsymbol{\varsigma})}{ch[\boldsymbol{\Theta}(\boldsymbol{\varsigma}) \boldsymbol{l}]} - F\_c(\boldsymbol{\varsigma}) \theta\_n(\boldsymbol{\varsigma}) \boldsymbol{\varsigma}. \tag{58}
$$

Here

$$\theta\_n(s) = \theta\_{n0} Z\_n(s); \theta\_{n0} = \frac{l}{Ef\_2}; Z\_n(s) = \frac{thA}{A}; A = \theta(s)l. \tag{59}$$

Substituting (58) into the last equation of system (57), we obtain an equation describing the stresses fluctuations in the force line in the vicinity of the slave link in form

Dynamic Stability of Open Two-Link Mechanical Structures DOI: http://dx.doi.org/10.5772/intechopen.91045

$$\sigma\_2(s) = \frac{F\_c(s) + \nu\_1(s)(h\_n + ms)/chA}{f\_2[1 + h\_n\theta\_n(s)s + m\theta\_n(s)s^2]}.\tag{60}$$

Performing such transformations in relation to torsional vibrations, we obtain

$$\mathfrak{Q}\_2(\mathfrak{s}) \left[ \mathbf{1} + h\_k \mathfrak{G}\_k(\mathfrak{s}) \mathfrak{s} + J \mathfrak{G}\_k(\mathfrak{s}) \mathfrak{s}^2 \right] = \frac{\mathfrak{Q}\_1(\mathfrak{s})}{chA\_k} - M\_r(\mathfrak{s}) \mathfrak{G}\_n(\mathfrak{s}) \mathfrak{s}. \tag{61}$$

Here

For boundary conditions with x = 0

Truss and Frames - Recent Advances and New Perspectives

We finally obtain

lowing relations:

σ2ðÞ¼ s

σ1ðÞ¼ s

υ1ð Þs chA � <sup>υ</sup>2ð Þ<sup>s</sup> E

Here

in form

14

<sup>υ</sup>2ðÞ¼ <sup>s</sup> <sup>υ</sup>1ð Þ<sup>s</sup>

1

<sup>υ</sup>2ðÞ¼ <sup>s</sup> <sup>υ</sup>1ð Þ<sup>s</sup> chA � <sup>s</sup>

chA � <sup>s</sup>

Eθð Þs

s θð Þs thA <sup>¼</sup> <sup>1</sup> f 2

driven link of the mechanical system in the form

υ2ð Þs 1 þ hnϑnð Þs s þ mϑnð Þs s

<sup>ϑ</sup>nðÞ¼ <sup>s</sup> <sup>ϑ</sup><sup>n</sup>0Znð Þ<sup>s</sup> ; <sup>ϑ</sup><sup>n</sup><sup>0</sup> <sup>¼</sup> <sup>l</sup>

chA <sup>σ</sup>2ðÞþ<sup>s</sup>

υð Þ¼ s, x υ1ð Þ s, 0 ;

υð Þ¼ s, x υ1ð Þ s, 0 ch½ �� θð Þs x

<sup>∂</sup>υð Þ <sup>s</sup>, <sup>x</sup>

Movement in two-link elements that occurs within elastic limits can be viewed as the movement of the driven point (link) from the movement of the leading point (link), which is affected by the previous links of the truss, for example, which perceive wind load. If, in the process of oscillation, the output link does not allow the input impulse to pass, then waves of disturbance are reflected from the end of the lines. Consider the case of a matched load when there are no reflected waves in the system and oscillations in the system do not affect the movement of the driven link due to the attached large mass. In this case, the boundary conditions are the fol-

υð Þ¼ s, l υ2ð Þs ; υð Þ¼ s, 0 υ1ð Þs ; σð Þ¼ s, l σ2ð Þs ; σð Þ¼ s, 0 σ1ð Þs ;

Here f<sup>2</sup> is the sectional area of the line in front of the slave link of mass m; hn and F<sup>с</sup> are coefficient of friction loss, proportional to the speed of movement, as well as the resistance force acting on the slave link; l is the length of the line. Together we solve (52)–(56) by performing the following transformations

;

1 s

Eυ1ð Þs θð Þs shA

;

ch½ � <sup>θ</sup>ð Þ<sup>s</sup> <sup>l</sup> � Fcð Þ<sup>s</sup> <sup>ϑ</sup>nð Þ<sup>s</sup> <sup>s</sup>: (58)

<sup>A</sup> ; <sup>A</sup> <sup>¼</sup> <sup>θ</sup>ð Þ<sup>s</sup> <sup>l</sup>: (59)

(57)

chA <sup>σ</sup>2ðÞþ<sup>s</sup>

½ � F sðÞþ hnυ2ðÞþs msυ2ð Þs :

After bringing similar members, we obtain the equation of motion of the

Ef <sup>2</sup>

Substituting (58) into the last equation of system (57), we obtain an equation describing the stresses fluctuations in the force line in the vicinity of the slave link

; ZnðÞ¼ s

thA

<sup>2</sup> <sup>¼</sup> <sup>υ</sup>1ð Þ<sup>s</sup>

Eυ1ð Þs θð Þs shA

shA <sup>1</sup>

FcðÞþs hnυ2ðÞþs msυ2ð Þs f 2

> 1 s

Eθð Þs

σ2ð Þs thA;

<sup>∂</sup><sup>x</sup> ¼ � <sup>s</sup>σð Þ <sup>s</sup>, 0

sσ1ð Þ s, 0 sh½ � θð Þs x

<sup>E</sup> , (54)

<sup>θ</sup>ð Þ<sup>s</sup> <sup>E</sup> : (55)

: (56)

$$\theta\_k(\mathfrak{s}) = \theta\_{k0} Z\_k(\mathfrak{s}); \theta\_{k0} = \frac{l}{Gr W\_{p2}}; Z\_k(\mathfrak{s}) = \frac{thA\_k}{A\_k}; A\_k = \theta\_k(\mathfrak{s}) l; \theta\_k(\mathfrak{s}) = \pm \sqrt{\frac{\rho}{G}}.\tag{62}$$

Comparing (15), (18), and (19), we can see that the processes of motion transfer in solid and liquid media can be described by similar equations. This is shown when analyzing the operation of the hydraulic drive [9, 35–37].

### 6. Lemma on the degree of distribution of force line parameters

Eqs. (47) and (48) make it possible to calculate the frequency characteristics of the system, i.e., determine the response of the model to the harmonic change in the speed of the lead link or the resistance force acting on the driven link.

For example, for υ<sup>1</sup> � 0 and s=jω, we obtain the characteristic WF (jω), illustrating the influence of F on υ2, and the characteristic WF<sup>σ</sup> (jω), illustrating the influence of F on σ:

$$W\_F(jo) = \frac{\nu\_2(jo)}{F\_c(jo)} = -\frac{\mathfrak{d}\_n(jo)jo}{1 + h\_n \mathfrak{d}\_n(jo)jo + m\mathfrak{d}\_n(jo)(jo)};\tag{63}$$

$$W\_{F\sigma}(jao) = \frac{\sigma\_2(jao)}{F\_c(jao)} = \frac{1}{f\_2\left[1 + h\_n\theta\_n(jao)jao + m\theta\_n(jao)(jao)^2\right]}.\tag{64}$$

For F � 0, we obtain the frequency characteristics W<sup>υ</sup> (jω), illustrating the effect of υ<sup>1</sup> on υ2, and WF<sup>σ</sup> (jω), illustrating the influence of υ<sup>1</sup> on σ2. So we have

$$W\_{\nu}(jo) = \frac{\nu\_2(jo)}{\nu\_1(jo)} = -\frac{ch^{-1}[\theta(jo)l]}{1 + h\_n\theta\_n(jo)jo + m\theta\_n(jo)(jo)^2},\tag{65}$$

$$W\_{F\sigma}(jao) = \frac{\sigma\_2(jao)}{\nu\_1(jao)} = \frac{h\_n + mjo}{f\_2 ch[\theta(jao)l] \left[1 + h\_n \theta\_n(jao)jo + m\theta\_n(jao)(jo)^2\right]}.\tag{66}$$

From (63) to (66), it can be seen that changes in the voltage and speed of movement of the output link are lagging behind changes in input impacts.

We introduce the notation α = l ω (ρ/E) 1/2.

Insofar as

$$Z\_n(ja) = \frac{th[\theta(ja)l]}{\theta(ja)l} = \frac{jtga}{ja} = \frac{tga}{a};\ ch(ja) = \cos a,\tag{67}$$

then

$$
\mathfrak{G}\_{\mathfrak{n}}(j a) = \mathfrak{G}\_{\mathfrak{n}}(a) = \mathfrak{G}\_{\mathfrak{n}0} Z\_{\mathfrak{n}}(a) \tag{68}
$$

and cosα are not complex functions.

The graph of the function Zп(α) is shown in Figure 4.

On the whole we conclude that at k = 1 α ! 0, Z<sup>п</sup> ! 1; at π/2 + kπ > α > π + kπ, Z<sup>п</sup> < 0. Here k = 0, 1, 2....n.

Eq. (58), given this, can be rewritten in the form

$$
\nu\_2(\mathfrak{s}) \left[ \mathbf{1} + h\_n \mathfrak{d}\_n(a) \mathfrak{s} + m \mathfrak{d}\_n(a) \mathfrak{s}^2 \right] = \nu\_1(\mathfrak{s}) \cos^{-1}(a) - F\_\varepsilon(\mathfrak{s}) \mathfrak{s} \mathfrak{d}\_n(a). \tag{69}
$$

If α ! 0, then Eq. (69) is reduced to the well-known equation describing dynamic processes in the mechanism with short lines of force:

$$
\sigma\_2(\mathfrak{s}) \left[ \mathbf{1} + h\_n \mathfrak{g}\_{n0} \mathfrak{s} + m \mathfrak{g}\_{n0} \mathfrak{s}^2 \right] = \nu\_1(\mathfrak{s}) - F\_c(\mathfrak{s}) \mathfrak{s} \mathfrak{g}\_{n0}.\tag{70}
$$

The breaks shown in Figure 4 are mathematically related to the function of tangent. For a real mechanism, this means that motion parameters are rebuilt. Apparently, in this instant, the form of oscillations changes abruptly. Below this feature is discussed in more detail.

The appearance of resonance is described by another expression that defines the conditions for the formation of the maximum amplitude of oscillations.

$$A\_F(\boldsymbol{\alpha}) = \frac{\mathfrak{d}\_n(\boldsymbol{a})\boldsymbol{\alpha}}{\sqrt{\left[\mathbbm{1} - m\mathfrak{d}\_n(\boldsymbol{a})\boldsymbol{\alpha}^2\right]^2 + \left[h\_n\mathfrak{d}\_n(\boldsymbol{a})\boldsymbol{\alpha}\right]^2}} = \left(\sqrt{\frac{\left[\mathbbm{1} - m\mathfrak{d}\_n(\boldsymbol{a})\boldsymbol{\alpha}^2\right]^2}{\left[\mathfrak{d}\_n(\boldsymbol{a})\boldsymbol{\alpha}^2\right]} + h\_n^2}\right)^{-1}.\tag{71}$$

Thus AF achieves its maximum when the condition

$$\frac{1 - m\theta\_n(a)\alpha^2}{\theta\_n(a)\alpha} = 0.\tag{72}$$

Let the mass of the load be m and the ratio of the mass of the rod with the cross-sectional area f to the mass of the load be a = lμ/m. Here μ is the linear mass

It is believed that the longitudinal tension of the rod during oscillations is balanced by the force of inertia of the load. This leads to the following condition at

¼ �m

At the top end, which is fixed we have y(0, t) = 0. At the initial time, the rod is stretched by the force F applied to the lower end and then without the initial speed

dy xð Þ , 0

<sup>t</sup> � � <sup>þ</sup> Bi sin pi

∂2 y ∂t2 � �

x¼l

φð Þ¼ x B cosð Þþ bx D sin ð Þ bx , (75)

<sup>t</sup> � � � � <sup>φ</sup>ð Þ <sup>x</sup> (76)

βtgβ ¼ a, (77)

: (73)

dt <sup>¼</sup> <sup>0</sup>: (74)

x¼l

y xð Þ¼ :<sup>0</sup> Fx

and parameter b values in the equation of the vibration modes

i�1

From the first boundary condition (41), it follows that B = 0. After that, from the second condition is the equation of frequencies

u xð Þ¼ , <sup>t</sup> <sup>X</sup><sup>∞</sup>

in accordance with the initial conditions (74).

Ef ;

and, moreover, to the calculation of the constants Ai and Bi of the general

Ai cos pi

The solution of the problem is reduced to the calculation of the constant B, D,

Ef <sup>∂</sup><sup>y</sup> ∂x � �

Dynamic Stability of Open Two-Link Mechanical Structures

DOI: http://dx.doi.org/10.5772/intechopen.91045

of the rod.

Figure 5.

the lower end of the rod:

Cargo suspended on a rod.

is left to itself, so that

solution

17

We show that this condition is similar to the rule for determining natural frequencies. This rule was formulated by Babakov [22].

Consider the following problem. Let the rod hang vertically. At the end of the rod, a load is fixed. The load is assumed to be point (Figure 5).

#### Figure 4.

Change of function Zn of dimensionless parameter α: I is the range of admissible function values for systems with lumped parameters; II is the range of admissible values of the function for systems with distributed parameters.

Dynamic Stability of Open Two-Link Mechanical Structures DOI: http://dx.doi.org/10.5772/intechopen.91045

ϑnð Þ¼ jω ϑnð Þ¼ α ϑn0Znð Þ α (68)

<sup>2</sup> � � <sup>¼</sup> <sup>υ</sup>1ðÞ�<sup>s</sup> Fcð Þ<sup>s</sup> <sup>s</sup>ϑn0: (70)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>m</sup>ϑnð Þ <sup>α</sup> <sup>ω</sup><sup>2</sup> ½ �<sup>2</sup>

<sup>ϑ</sup>nð Þ <sup>α</sup> <sup>ω</sup> <sup>¼</sup> <sup>0</sup>: (72)

<sup>ϑ</sup>nð Þ <sup>α</sup> <sup>ω</sup><sup>2</sup> ½ � <sup>þ</sup> <sup>h</sup><sup>2</sup>

n

1 A

�1

: (71)

ð Þ� α Fcð Þs sϑnð Þ α : (69)

and cosα are not complex functions.

Truss and Frames - Recent Advances and New Perspectives

υ2ð Þs 1 þ hnϑnð Þ α s þ mϑnð Þ α s

feature is discussed in more detail.

Figure 4.

16

AFð Þ¼ <sup>ω</sup> <sup>ϑ</sup>nð Þ <sup>α</sup> <sup>ω</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>1</sup> � <sup>m</sup>ϑnð Þ <sup>α</sup> <sup>ω</sup><sup>2</sup> ½ �<sup>2</sup> <sup>þ</sup> ½ � hnϑnð Þ <sup>α</sup> <sup>ω</sup> <sup>2</sup> q ¼

Thus AF achieves its maximum when the condition

quencies. This rule was formulated by Babakov [22].

rod, a load is fixed. The load is assumed to be point (Figure 5).

Z<sup>п</sup> < 0. Here k = 0, 1, 2....n.

The graph of the function Zп(α) is shown in Figure 4.

<sup>2</sup> � � <sup>¼</sup> <sup>υ</sup>1ð Þ<sup>s</sup> cos �<sup>1</sup>

dynamic processes in the mechanism with short lines of force:

υ2ð Þs 1 þ hnϑn0s þ mϑn0s

Eq. (58), given this, can be rewritten in the form

On the whole we conclude that at k = 1 α ! 0, Z<sup>п</sup> ! 1; at π/2 + kπ > α > π + kπ,

If α ! 0, then Eq. (69) is reduced to the well-known equation describing

The breaks shown in Figure 4 are mathematically related to the function of tangent. For a real mechanism, this means that motion parameters are rebuilt. Apparently, in this instant, the form of oscillations changes abruptly. Below this

conditions for the formation of the maximum amplitude of oscillations.

<sup>1</sup> � <sup>m</sup>ϑnð Þ <sup>α</sup> <sup>ω</sup><sup>2</sup>

We show that this condition is similar to the rule for determining natural fre-

Consider the following problem. Let the rod hang vertically. At the end of the

Change of function Zn of dimensionless parameter α: I is the range of admissible function values for systems with lumped parameters; II is the range of admissible values of the function for systems with distributed parameters.

The appearance of resonance is described by another expression that defines the

0s @

Let the mass of the load be m and the ratio of the mass of the rod with the cross-sectional area f to the mass of the load be a = lμ/m. Here μ is the linear mass of the rod.

It is believed that the longitudinal tension of the rod during oscillations is balanced by the force of inertia of the load. This leads to the following condition at the lower end of the rod:

$$E\xi \left(\frac{\partial \mathbf{y}}{\partial \mathbf{x}}\right)\_{\mathbf{x}=\mathbf{l}} = -m \left(\frac{\partial^2 \mathbf{y}}{\partial \mathbf{t}^2}\right)\_{\mathbf{x}=\mathbf{l}}.\tag{73}$$

At the top end, which is fixed we have y(0, t) = 0. At the initial time, the rod is stretched by the force F applied to the lower end and then without the initial speed is left to itself, so that

$$\text{hyp}(\mathbf{x}, \mathbf{0}) = \frac{F\mathbf{x}}{E\mathbf{f}} ; \frac{d\mathbf{y}(\mathbf{x}, \mathbf{0})}{dt} = \mathbf{0}. \tag{74}$$

The solution of the problem is reduced to the calculation of the constant B, D, and parameter b values in the equation of the vibration modes

$$\mathfrak{sp}(\mathfrak{x}) = B\cos\left(b\mathfrak{x}\right) + D\sin\left(b\mathfrak{x}\right),\tag{75}$$

and, moreover, to the calculation of the constants Ai and Bi of the general solution

$$u(\mathbf{x},t) = \sum\_{i=1}^{\infty} \left[ A\_i \cos \left( p\_i t \right) + B\_i \sin \left( p\_i t \right) \right] \boldsymbol{\uprho}(\mathbf{x}) \tag{76}$$

in accordance with the initial conditions (74). From the first boundary condition (41), it follows that B = 0. After that, from the second condition is the equation of frequencies

$$
\beta \text{tg}\beta = a,\tag{77}
$$

where β = al. Thus, the equation of the own modes of oscillations of the rod has the form

$$\mathfrak{q}\_k(\mathfrak{x}) = D\_k \sin \left( \frac{\beta\_k \mathfrak{x}}{l} \right)\_{, (k=1,2,3,\dots)} \tag{78}$$

7. Dynamic stability of rod systems

DOI: http://dx.doi.org/10.5772/intechopen.91045

Dynamic Stability of Open Two-Link Mechanical Structures

in [9, 29, 35–37].

functions

L�<sup>1</sup>

[29, 35].

19

[9, 29, 31, 35–37].

We have then

j j <sup>F</sup>1ð Þ<sup>s</sup> <sup>¼</sup> <sup>1</sup>

2πj

ðcþj∞

c�j∞

problem of ensuring sustainable functioning inevitably arises.

We algebraically decompose Eq. (58) into two. We have

υ1

ϑnð Þ α f <sup>2</sup>σð Þs e

L�<sup>1</sup>

Then the originals (31) can be written in the form

Here α corresponds to the expression (84).

Wave phenomena take place in hydraulic, mechanical, electrical environments [4]. As shown above, wave processes take place in systems with lines of any length, but depending on the conditions, they can either be neglected or not ignored. If at the same time there is a transfer of energy to perform any effective work, then the

cos <sup>α</sup> <sup>¼</sup> <sup>υ</sup>2ðÞþ<sup>s</sup> <sup>ϑ</sup>nð Þ <sup>α</sup> <sup>f</sup> <sup>2</sup>sσð Þ<sup>s</sup> ; (85)

σð Þs f <sup>2</sup> ¼ FcðÞþs hnυ2ðÞþs msυ2ð Þs : (86)

F1ðÞ¼ s ϑnð Þ α f <sup>2</sup>sσð Þs and F2ðÞ¼ s υ1ð Þs = cos α: (87)

1 2πj ðcþj∞

sσð Þs e

stds <sup>¼</sup> <sup>ϑ</sup>nð Þ <sup>α</sup> <sup>f</sup> <sup>2</sup>

dt ; (90)

dt : (91)

dσ dt :

(88)

c�j∞

j j F2ð Þs ¼ υ1ð Þt = cos α: (89)

dσ

dυ<sup>2</sup>

Publications devoted to the study of the stability of open-loop systems with distributed parameters in the presence of significant nonlinearities, where sufficiently precise criteria are stated, could not be found. Therefore, here outlines the main points of this issue, which were developed in more detail

The accuracy of the decomposition is easily verified by the inverse solution. Moving on to the originals of Eq. (58), we do the inverse transform Laplace of

stds <sup>¼</sup> <sup>ϑ</sup>nð Þ <sup>α</sup> <sup>f</sup> <sup>2</sup>

υ1ð Þt = cos α ¼ υ2ðÞþt ϑnð Þ α f <sup>2</sup>

σð Þt f <sup>2</sup> ¼ FcðÞþt hnυ2ðÞþt m

The system of Eqs. (90, 91) may contain various nonlinearities (yield zone in the stress diagram, nonlinear friction, etc.) and is solved by the updated Runge-Kutta method [9, 38]. The admissibility of such a technique was checked by comparing the frequency characteristics constructed by formulas (63) to (66) and using the above method when introducing harmonic oscillations with different frequencies

In addition, given that the fluctuations of the speeds of movement and stresses (pressure) in mechanical and hydraulic systems can be described by similar equations, we checked the adequacy of the proposed method by means of full-scale and numerical experiments in an electric drive. The results had good convergence

In the process of modeling, it turned out that when Z < 0 (Figure 4), the solution becomes unstable. Figure 6 shows an example of the process of loss of

Here β<sup>k</sup> is the roots of Eq. (77).

The solution of Eq. (78) can be carried out graphically [21].

The lowest natural frequencies corresponding to these values are calculated by the formula

$$
\alpha = p\_1 = \frac{\beta\_1}{l} \sqrt{\frac{E\overline{f}}{\mu}}.\tag{79}
$$

Note that the linear mass of the rod μ = ρf. Expanding the coefficients, we obtain the original equation

$$
\beta \text{tg}\beta = \frac{\text{lf}\,\rho}{m}.\tag{80}
$$

We now turn to Eq. (79) and divide the numerator of the right side by the denominator

$$\text{Ef } \frac{\sqrt{\rho/E}}{\text{tg}a} = ma. \tag{81}$$

We will still carry out a number of transformations. We have

$$\text{Efl}\frac{\sqrt{\rho/E}}{m} = \text{log}\mathbf{a}; \frac{f\mathbf{l}}{m} = \frac{\text{log}\mathbf{a}}{\sqrt{E\rho}}.\tag{82}$$

After multiplying both sides of the last equation by ρ and after performing the corresponding transformations, we get

$$\frac{\rho \text{fl}}{m} = a \text{tg}a.\tag{83}$$

Since here the parameters α and β are equivalent, it can be argued that the roots of Eqs. (72) and (80) are the same. This is confirmed by the results of calculations.

The results obtained make it possible to formulate a lemma on the degree of distribution of the parameters of power lines. But for this, we introduce the notation

$$a = \operatorname{lco} \sqrt{\mathfrak{p}/\chi},\tag{84}$$

where χ is the elastic modulus of the material of the force line.

We state the lemma as follows. If the parameters of the mechanical two-link system, characterized by the oscillation frequency ω, length l, density ρ, and the elastic modulus of the material of the force line χ connecting these links, are such that magnitude of the dimensionless coefficient α lies in the interval 0 ≤ α ≤ 1, then the wave processes in the lines of force can be neglected.

This lemma is also valid for systems where torsional vibrations of rods and vibrations in hydraulic systems take place. The criterion on the degree of distribution of the system parameters was first described in the book [29].

## 7. Dynamic stability of rod systems

where β = al.

the formula

the original equation

denominator

notation

18

lines of force can be neglected.

Here β<sup>k</sup> is the roots of Eq. (77).

Truss and Frames - Recent Advances and New Perspectives

Thus, the equation of the own modes of oscillations of the rod has the form

l � �

The lowest natural frequencies corresponding to these values are calculated by

l

Note that the linear mass of the rod μ = ρf. Expanding the coefficients, we obtain

<sup>β</sup>tg<sup>β</sup> <sup>¼</sup> lf <sup>ρ</sup>

We now turn to Eq. (79) and divide the numerator of the right side by the

ffiffiffiffiffiffiffiffi ρ=E p

<sup>m</sup> <sup>¼</sup> <sup>l</sup>ωtgα; fl

ρfl

After multiplying both sides of the last equation by ρ and after performing the

Since here the parameters α and β are equivalent, it can be argued that the roots of Eqs. (72) and (80) are the same. This is confirmed by the results of calculations. The results obtained make it possible to formulate a lemma on the degree of distribution of the parameters of power lines. But for this, we introduce the

<sup>α</sup> <sup>¼</sup> <sup>l</sup><sup>ω</sup> ffiffiffiffiffiffiffi

This lemma is also valid for systems where torsional vibrations of rods and vibrations in hydraulic systems take place. The criterion on the degree of distribu-

We state the lemma as follows. If the parameters of the mechanical two-link system, characterized by the oscillation frequency ω, length l, density ρ, and the elastic modulus of the material of the force line χ connecting these links, are such that magnitude of the dimensionless coefficient α lies in the interval 0 ≤ α ≤ 1, then the wave processes in the

where χ is the elastic modulus of the material of the force line.

tion of the system parameters was first described in the book [29].

<sup>m</sup> <sup>¼</sup> <sup>l</sup>ωtg<sup>α</sup> ffiffiffiffiffi

Ef

We will still carry out a number of transformations. We have

ffiffiffiffiffiffiffiffi ρ=E p

Efl

corresponding transformations, we get

ffiffiffiffiffi Ef μ

s

<sup>ω</sup> <sup>¼</sup> <sup>p</sup><sup>1</sup> <sup>¼</sup> <sup>β</sup><sup>1</sup>

, ð Þ k¼1,2,3, … :

: (79)

<sup>m</sup> : (80)

tg<sup>α</sup> <sup>¼</sup> <sup>m</sup>ω: (81)

<sup>m</sup> <sup>¼</sup> <sup>α</sup>tgα: (83)

ρ=χ p , (84)

<sup>E</sup><sup>ρ</sup> <sup>p</sup> : (82)

(78)

<sup>φ</sup>kð Þ¼ <sup>x</sup> Dk sin <sup>β</sup>kx

The solution of Eq. (78) can be carried out graphically [21].

Wave phenomena take place in hydraulic, mechanical, electrical environments [4]. As shown above, wave processes take place in systems with lines of any length, but depending on the conditions, they can either be neglected or not ignored. If at the same time there is a transfer of energy to perform any effective work, then the problem of ensuring sustainable functioning inevitably arises.

Publications devoted to the study of the stability of open-loop systems with distributed parameters in the presence of significant nonlinearities, where sufficiently precise criteria are stated, could not be found. Therefore, here outlines the main points of this issue, which were developed in more detail in [9, 29, 35–37].

We algebraically decompose Eq. (58) into two. We have

$$\frac{\nu\_1}{\cos a} = \nu\_2(s) + \theta\_n(a) f\_2 s \sigma(s);\tag{85}$$

$$
\sigma(\mathfrak{s})f\_2 = F\_\mathfrak{c}(\mathfrak{s}) + h\_n \nu\_2(\mathfrak{s}) + m \mathfrak{s} \nu\_2(\mathfrak{s}).\tag{86}
$$

Here α corresponds to the expression (84).

The accuracy of the decomposition is easily verified by the inverse solution. Moving on to the originals of Eq. (58), we do the inverse transform Laplace of functions

$$F\_1(\mathfrak{s}) = \theta\_n(a) f\_{\mathfrak{z}} \mathfrak{s} \sigma(\mathfrak{s}) \text{ and } F\_2(\mathfrak{s}) = \nu\_1(\mathfrak{s}) / \cos a. \tag{87}$$

We have then

$$L^{-1}|F\_1(s)| = \frac{1}{2\pi j} \int\_{c - j\circ\sigma}^{c + j\circ\sigma} \theta\_n(a) f\_2 \sigma(s) e^{\mathfrak{s}t} ds = \theta\_n(a) f\_2 \frac{1}{2\pi j} \int\_{c - j\circ\sigma}^{c + j\circ\sigma} s\sigma(s) e^{\mathfrak{s}t} ds = \theta\_n(a) f\_2 \frac{d\sigma}{dt}.\tag{88}$$

$$L^{-1}|F\_2(\mathfrak{s})| = \nu\_1(\mathfrak{t})/\cos a. \tag{89}$$

Then the originals (31) can be written in the form

$$\nu\_1(t) / \cos a = \nu\_2(t) + \theta\_n(a) f\_2 \frac{d\sigma}{dt};\tag{90}$$

$$
\sigma(t)f\_2 = F\_c(t) + h\_n v\_2(t) + m\frac{d\nu\_2}{dt}.\tag{91}
$$

The system of Eqs. (90, 91) may contain various nonlinearities (yield zone in the stress diagram, nonlinear friction, etc.) and is solved by the updated Runge-Kutta method [9, 38]. The admissibility of such a technique was checked by comparing the frequency characteristics constructed by formulas (63) to (66) and using the above method when introducing harmonic oscillations with different frequencies [29, 35].

In addition, given that the fluctuations of the speeds of movement and stresses (pressure) in mechanical and hydraulic systems can be described by similar equations, we checked the adequacy of the proposed method by means of full-scale and numerical experiments in an electric drive. The results had good convergence [9, 29, 31, 35–37].

In the process of modeling, it turned out that when Z < 0 (Figure 4), the solution becomes unstable. Figure 6 shows an example of the process of loss of

#### Figure 6.

Loss of stable operation in the mechanical element during longitudinal movement.

#### Figure 7.

Zones of steady (shaded) and unstable operation of the mechanical system of longitudinal movement.

dynamic stability in a mechanical element with longitudinal vibrations. For a truss, this means breaking one of the rods.

The areas of dynamic stability for mechanical elements with longitudinal vibrations are shown in Figure 7.

Author details

21

Leonid Kondratenko\* and Lubov Mironova

provided the original work is properly cited.

Moscow Aviation Institute (SNRU), Moscow, Russia

Dynamic Stability of Open Two-Link Mechanical Structures

DOI: http://dx.doi.org/10.5772/intechopen.91045

\*Address all correspondence to: kondrat.leonid@yandex.ru

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### 8. Conclusions

In trusses under the influence of variable loads, vibrations occur periodically in different elements, characterized by fluctuations in speed and stress. Under certain conditions, due to wave motions in the rods, the shape of the oscillations may change, i.e., lost stability of motion. At this instant, the destruction of the carrier element will occur. The proposed chapter considers the conditions for the occurrence of such an event.

Dynamic Stability of Open Two-Link Mechanical Structures DOI: http://dx.doi.org/10.5772/intechopen.91045
