Frequency Response of Axial Vibration of Continuous Beam as Five Lumped Masses

Beam

structures are widely used in many engineering applications; such as

airplane

wings, buildings, bridges, micromechanical

systems,

overhead transmission lines, as well as many others in the aerospace, mechanical, and civil industries. Numerous authors have studied the transverse vibrations of

beams

carrying masses or spring-mass-damper

system.

Most of the studies reported in the literature are based on linear vibration models. ^1–9 1. M. Gürgöze, Journal of Sound and Vibration 217, 585 (1998). https://doi.org/10.1006/jsvi.1998.1796 2. H.-Y. Lin and Y.-C. Tsai, Journal of Sound and Vibration 302, 442 (2007). https://doi.org/10.1016/j.jsv.2006.06.080 3. P. Hassanpour, W. Cleghorn, J. Mills, and E. Esmailzadeh, Journal of Vibration and Control 13, 1723 (2007). https://doi.org/10.1177/1077546307076285 4. J.-S. Wu and C.-G. Huang, Communications in Numerical Methods in Engineering 11, 743 (1995). https://doi.org/10.1002/cnm.1640110905 5. S. Naguleswaran, International Journal of Mechanical Sciences 43, 2737 (2001). https://doi.org/10.1016/S0020-7403(01)00072-8 6. C. Kirk and S. Wiedemann, Journal of Sound and Vibration 254, 939 (2002). https://doi.org/10.1006/jsvi.2001.4138 7. S. O. Maiz, D. Bambill, C. Rossit, and P. Laura, Journal of Sound and Vibration 303, 895 (2007). https://doi.org/10.1016/j.jsv.2006.12.028 8. C. Chang, Journal of Sound and Vibration 237, 733 (2000). https://doi.org/10.1006/jsvi.2000.2990 9. J.-S. Wu and T.-L. Lin, Journal of Sound and Vibration 136, 201 (1990). https://doi.org/10.1016/0022-460X(90)90851-P A few studies can be found on lateral vibration of

beams

under axial loads. ^10–12 10. A. Bokaian, Journal of Sound and Vibration 142, 481 (1990). https://doi.org/10.1016/0022-460X(90)90663-K 11. O. Barry, D. Oguamanam, and J. Zu, Shock and Vibration 2014 (2014). https://doi.org/10.1155/2014/485630 12. S. Woinowsky-Krieger, J. Appl. Mech 17, 35 (1950). It is important to note that the lateral vibrations of

beams

under tensile axial load are also of practical interest in many engineering applications. In the design of large flexible solar arrays, the boom that supports the array is under pre-tensile stresses due to the tension that must be maintained in the

solar cell

substrate. Bokaian et al. examined a free

vibration analysis

for an axially loaded

beam

with different combinations of

boundary conditions.

¹⁰ 10. A. Bokaian, Journal of Sound and Vibration 142, 481 (1990). https://doi.org/10.1016/0022-460X(90)90663-K The authors demonstrated that the

beam

behaves like a string if the dimensionless tension parameter was greater than. ¹² 12. S. Woinowsky-Krieger, J. Appl. Mech 17, 35 (1950). Barry et al. studied both free and forced vibration of an axially loaded

beam

carrying multiple spring-mass-damper

system.

¹¹ 11. O. Barry, D. Oguamanam, and J. Zu, Shock and Vibration 2014 (2014). https://doi.org/10.1155/2014/485630 They presented a generalized orthogonality conditions and showed that using the classical orthogonality condition for the

vibration analysis

of a loaded

beam

can lead to erroneous results. All the reported studies so far are based on linear vibration models, which are usually sufficient for predicting the dynamic characteristics of the

system

when dealing with small deformations. However, when dealing with higher deformation, nonlinearity should be included for accurate modeling. For

beam

problems under immovable

boundary conditions,

the most common nonlinearity is attributed to mid-plane stretching. A thorough review of the subject was examined by Nayfeh et al. ^13,14 13. A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations (John Wiley & Sons, 2008). 14. A. H. Nayfeh, Introduction to Perturbation Techniques (John Wiley & Sons, 2011). Several authors have also investigated

nonlinear vibrations

due to mid-plane stretching. ^15–22 15. D. Burgreen, "Free vibrations of a pin-ended column with constant distance between pin ends." Tech. Rep. (DTIC Document, 1950). 16. R. Mestrom, R. Fey, K. Phan, and H. Nijmeijer, Sensors and Actuators A: Physical 162, 225 (2010). https://doi.org/10.1016/j.sna.2010.04.020 17. K. Low, Journal of Sound and Vibration 207, 284 (1997). https://doi.org/10.1006/jsvi.1997.1135 18. E. Özkaya, M. Pakdemirli, and H. Öz, Journal of Sound and Vibration 199, 679 (1997). https://doi.org/10.1006/jsvi.1996.0643 19. E. Özkaya and M. Pakdemirli, Journal of Sound and Vibration 221, 491 (1999). 20. E. Özkaya, Journal of Sound and Vibration 257, 413 (2002). 21. M. Pakdemirli and A. Nayfeh, Journal of Vibration and Acoustics 116, 433 (1994). https://doi.org/10.1115/1.2930446 22. O. Barry, D. Oguamanam, and J. Zu, Nonlinear Dynamics 77, 1597 (2014). https://doi.org/10.1007/s11071-014-1402-5 Burgreen studied the free

vibration analysis

of pin-ended column. ¹⁵ 15. D. Burgreen, "Free vibrations of a pin-ended column with constant distance between pin ends." Tech. Rep. (DTIC Document, 1950). Ozkaya et al. studied the

nonlinear vibration

of

beam

with clamped-clamped

boundary conditions

and carrying one intermediate point mass. ¹⁸ 18. E. Özkaya, M. Pakdemirli, and H. Öz, Journal of Sound and Vibration 199, 679 (1997). https://doi.org/10.1006/jsvi.1996.0643 They extended their work by investigating the same problem but with various

boundary condition

¹⁸ 18. E. Özkaya, M. Pakdemirli, and H. Öz, Journal of Sound and Vibration 199, 679 (1997). https://doi.org/10.1006/jsvi.1996.0643 and with multiple intermediate point masses. ²⁰ 20. E. Özkaya, Journal of Sound and Vibration 257, 413 (2002). All their works demonstrated a hardening type nonlinearity. The

nonlinear vibration

of a

beam

carrying one intermediate spring-mass

system

was examined by Pakdemirli et al. ²¹ 21. M. Pakdemirli and A. Nayfeh, Journal of Vibration and Acoustics 116, 433 (1994). https://doi.org/10.1115/1.2930446 They postulated that the mid-plane stretching and the spring-mass

system

had a great effect on the frequency-response curves. Barry et al. extended the work of Pakdemirli et al. by including multiple intermediate mass-spring-damper support, and various

boundary conditions.

²² 22. O. Barry, D. Oguamanam, and J. Zu, Nonlinear Dynamics 77, 1597 (2014). https://doi.org/10.1007/s11071-014-1402-5 However, they treated the intermediate masses as point masses therefore neglecting the mass rotational inertia. At this point it is worth mentioning that in all the aforementioned

nonlinear vibration

references, the authors treated the mass as particles instead of rigid bodies. In the present work, we analytically examined for the first time the

nonlinear vibration

of an axially loaded

beam

carrying multiple rigid masses. This work is an extension of our previous work ²² 22. O. Barry, D. Oguamanam, and J. Zu, Nonlinear Dynamics 77, 1597 (2014). https://doi.org/10.1007/s11071-014-1402-5 and the work of Ozkaya. ²⁰ 20. E. Özkaya, Journal of Sound and Vibration 257, 413 (2002). We presented explicit expressions for the frequency

equation,

mode shapes, nonlinear frequency, and the modulation

equations

for the phase and amplitude. The validity of these analytical expressions were demonstrated through

finite element analysis

and via comparison with results in the literature. We conducted parametric studies to predict the effect of the mass moment of inertia and tension on the nonlinear frequency and response of the

system.

A schematic of the

system

is depicted in Fig. 1 . Following our previous work, ²² 22. O. Barry, D. Oguamanam, and J. Zu, Nonlinear Dynamics 77, 1597 (2014). https://doi.org/10.1007/s11071-014-1402-5 the

system

governing

equations

are

$$\begin{array}{cc}\hfill m{\ddot{w}}_{\text{i}+1}+EI{w}_{\text{i}+1}^{\mathrm{\prime \prime \prime \prime }}-T{w}_{\text{i}+1}^{″}=\hfill & \hfill \frac{EA-T}{2L}\left[\sum _{r=0}^n{\int }_{x_{\text{r}}}^{x_{\text{r}+1}}{w}_{\text{r}+1}^{\mathrm{\prime }2}dx\right]w_{\text{i}+1}^{″}\hfill \\ \hfill \text{for}\mathit{}i=0,1,2,\mathrm{\dots }n\hfill & \end{array}$$

(1)

$$\begin{array}{cc}& \hfill w_{\mathrm{p}}(x_{\mathrm{p}},t)=w_{\mathrm{p}+1}(x_{\mathrm{p}},t),\hfill \\ & \hfill w_{\mathrm{p}}^{\prime }(x_{\mathrm{p}},t)=w_{\mathrm{p}+1}^{\prime }(x_{\mathrm{p}},t),\hfill \\ & \hfill EI\left(w_{\mathrm{p}}^{″}(x_{\mathrm{p}},t)-w_{\mathrm{p}+1}^{″}(x_{\mathrm{p}},t)\right)+J_{\mathrm{p}}{\ddot{w}}_{\mathrm{p}}^{\prime }(x_{\mathrm{p}},t)=0\hfill \\ & \hfill EI\left(w_{\mathrm{p}}^{‴}(x_{\mathrm{p}},t)+w_{\mathrm{p}+1}^{‴}(x_{\mathrm{p}},t)\right)-M_{\mathrm{p}}{\ddot{w}}_{\mathrm{p}}(x_{\mathrm{p}},t)=0\hfill \end{array}$$

(2)

where $w$ is the transverse displacement of the

beam,

$x$ is the axial coordinate, $m$ is the mass per unit length of the

beam,

$T$ is the tension of the

beam,

$EI$ is the flexural rigidity of the

beam,

$M_{\mathrm{p}}$ and $J_{\mathrm{p}}$ are the $p^{th}$ in-span mass and rotational inertia, respectively.

FIG. 1. Schematic of an axially loaded simply supported

beam

carrying multiple rigid masses.

• PPT
|
• High resolution

The following dimensionless parameters can be introduced

$$\begin{array}{ccc}\hfill \zeta \hfill & \hfill =\hfill & \frac{x}{L},\mathit{}{W}_{\mathrm{p}}=\frac{w_{\mathrm{p}}}{L},\mathit{}{\xi }_{\mathrm{p}}=\frac{x_{\mathrm{p}}}{L},\mathit{}\tau =\frac{t}{L^2}\sqrt{\frac{EI}{m}},\hfill \\ \hfill {\eta }_{\mathrm{p}}\hfill & \hfill =\hfill & \frac{J_{\mathrm{p}}}{m{L}^3},\mathit{}{\alpha }_{\mathrm{mp}}=\frac{M_{\mathrm{p}}}{mL},\mathit{}R=\sqrt{\frac{I}{A{L}^2}},\hfill \\ \hfill s\hfill & \hfill =\hfill & \sqrt{\frac{T{L}^2}{2EI}},\mathit{}\lambda =\frac{1}{R^2}(1-2R^2{s}^2),\hfill \\ \hfill \alpha \hfill & \hfill =\hfill & \sqrt{-s^2+\sqrt{s^4+{\omega }^2}},\mathit{}\beta =\sqrt{s^2+\sqrt{s^4+{\omega }^2}}\hfill \end{array}$$

(3)

where $\omega$ is the circular linear natural frequency. Using the above dimensionless parameters and adding damping and forcing terms, the governing

equations

becomes

$${\ddot{W}}_{\mathrm{i}+1}+W_{\mathrm{i}+1}^{\mathrm{\prime \prime \prime \prime }}-2s^2{W}_{\mathrm{i}+1}^{″}=\frac{1}{2}\lambda \left[W_{\mathrm{i}+1}^{″}\sum _{r=0}^n{\int }_{{\xi }_{\mathrm{r}}}^{{\xi }_{\mathrm{r}+1}}{W}_{\mathrm{r}+1}^{\mathrm{\prime }2}d\zeta \right]-2\overline{\mu }{\dot{W}}_{\mathrm{i}+1}+{\overline{F}}_{\mathrm{i}+1}\cos \mathrm{\Omega }\tau$$

(4)

$$\begin{array}{cc}& \hfill W_{\mathrm{p}}({\xi }_{\mathrm{p}},\tau )=W_{\mathrm{p}+1}({\xi }_{\mathrm{p}},\tau ),\hfill \\ & \hfill W_{\mathrm{p}}^{\prime }({\xi }_{\mathrm{p}},\tau )=W_{\mathrm{p}+1}^{\prime }({\xi }_{\mathrm{p}},\tau ),\hfill \\ & \hfill W_{\mathrm{p}}^{″}({\xi }_{\mathrm{p}},\tau )-W_{\mathrm{p}+1}^{″}({\xi }_{\mathrm{p}},\tau )=-{\eta }_{\mathrm{p}}\ddot{W}_{\prime }^{\mathrm{p}}({\xi }_{\mathrm{p}},\tau ),\hfill \\ & \hfill W_{\mathrm{p}}^{‴}({\xi }_{\mathrm{p}},\tau )+W_{\mathrm{p}+1}^{‴}({\xi }_{\mathrm{p}},\tau )={\alpha }_{\mathrm{mp}}{\ddot{W}}_{\mathrm{p}}({\xi }_{\mathrm{p}},\tau )\hfill \end{array}$$

(5)

where the dots and primes denote differentiation with respect to dimensionless time $\tau$ and dimensionless coordinate ${\xi }_{\mathrm{p}}$ , respectively. $\overline{\mu }$ is the dimensionless damping coefficient of the

beam,

${\overline{F}}_{\mathrm{i}+1}$ is the dimensionless excitation amplitude and $\mathrm{\Omega }$ is the dimensionless excitation frequency.

Due to the absence of quadratic nonlinearity, the solution of Eq. 4 is assumed to be expandable in the form

$$W_{\mathrm{i}+1}(\zeta ,\tau ,ϵ)=ϵW_{(\mathrm{i}+1)1}(\zeta ,T_0,T_2)+{ϵ}^3{W}_{(\mathrm{i}+1)3}(\zeta ,T_0,T_2)+\mathrm{\dots },$$

(6)

where $ϵ$ is a small dimensionless parameter used for book-keeping. $T_0=\tau$ is a fast-time scale and $T_2={ϵ}^2\tau$ is a slow-time scale. The present study considers primary resonances only. Hence, the damping and forcing terms are ordered to counter the effect of the nonlinear terms. The damping coefficient $\overline{\mu }$ and excitation amplitude ${\overline{F}}_{\mathrm{i}+1}$ are given as

At order $ϵ$ , the problem is linear. Hence, the solution can be assumed as

$$W_{(\mathrm{p})1}=\left[A(T_2)e^{j\omega T_0}+cc\right]Y_{\mathrm{p}}(\zeta )$$

(7)

where $cc$ denotes the complex conjugate of the preceding terms and $Y_{\mathrm{p}}(\zeta )$ is the mode shape.

Note that for one intermediate mass, it is more convenient to use two reference frames (i.e., one at each end of the

beam)

to obtain a more compact representation of the frequency

equation

and mode shapes. After some algebraic manipulation, the frequency

equation

for one intermediate mass is obtained as

$$\begin{array}{c}(\alpha {\beta }^5+2{\alpha }^3{\beta }^3+{\alpha }^5\beta )\sin \alpha \text{sinh}\beta +{\alpha }_{\mathrm{mp}}{\omega }^2[({\alpha }^3+\alpha {\beta }^2)\sin \alpha \text{sinh}\beta {\xi }_1\text{sinh}\beta {\xi }_2\hfill \\ -({\alpha }^2\beta +{\beta }^3)\text{sinh}\beta \sin \alpha {\xi }_1\sin \alpha {\xi }_2]+{\alpha }_{\mathrm{mp}}\omega 4{\eta }_{\mathrm{p}}[{\alpha }^2\cos \alpha {\xi }_1\cos \alpha {\xi }_2\text{sinh}\beta {\xi }_1\sin \beta {\xi }_2\hfill \\ +{\beta }^2\cos \alpha {\xi }_1\cos \alpha {\xi }_2\text{cosh}\alpha \beta {\xi }_1\text{cosh}\beta {\xi }_2+\alpha \beta (\sin \alpha {\xi }_1\cos \alpha {\xi }_2\text{cosh}\beta {\xi }_1\text{sinh}\beta {\xi }_2\hfill \\ +\cos \alpha {\xi }_1\sin \alpha {\xi }_2\text{sinh}\beta {\xi }_1\text{cosh}\beta {\xi }_2)]+{\omega }^2{\eta }_{\mathrm{p}}[(\alpha {\beta }^4+{\alpha }^3{\beta }^2)\sin \alpha \text{cosh}\beta {\xi }_1\text{cosh}\beta {\xi }_2\hfill \\ -({\alpha }^4\beta +{\alpha }^2{\beta }^3)\text{sinh}\beta \text{cosh}\alpha {\xi }_1\cos \alpha {\xi }_2]=0\hfill \end{array}$$

(8)

and the mode shapes are

$$Y_{\mathrm{i}}(\zeta )=c_{1\mathrm{i}}\sin \alpha {\xi }_{\mathrm{i}}+c_{2\mathrm{i}}\text{sinh}\alpha {\xi }_{\mathrm{i}}\mathit{}\mathrm{for}\mathit{}\mathrm{i}=\mathrm{1,2}$$

(9)

where constants $c_{\mathrm{ii}}$ are

$$c_{11}=1c_{21}=-\frac{\alpha }{\beta {\gamma }_{\mathrm{pp}}}\left[({\alpha }^2+{\beta }^2)\sin \alpha \text{sinh}\beta {\xi }_2+{\omega }^2{\eta }_{\mathrm{p}}\cos \alpha {\xi }_1(\alpha \cos \alpha {\xi }_2\text{sinh}\beta {\xi }_2-\beta \sin \alpha {\xi }_2\text{cosh}\beta {\xi }_2)\right]c_{12}=\frac{1}{{\gamma }_{\mathrm{pp}}}\left[({\alpha }^2+{\beta }^2)\sin \alpha {\xi }_1\text{sinh}\beta +{\omega }^2{\eta }_{\mathrm{p}}\text{cosh}\beta {\xi }_2(\alpha \cos \alpha {\xi }_1\text{sinh}\beta {\xi }_1-\beta \sin \alpha {\xi }_1\text{cosh}\beta {\xi }_1)\right]c_{22}=\frac{\alpha }{\beta {\gamma }_{\mathrm{pp}}}\left[({\alpha }^2+{\beta }^2)\sin \alpha \text{sinh}\beta {\xi }_1+{\omega }^2{\eta }_{\mathrm{p}}\cos \alpha {\xi }_2(\alpha \cos \alpha {\xi }_1\text{sinh}\beta {\xi }_1\beta \sin \alpha {\xi }_1\text{cosh}\beta {\xi }_1)\right]\text{and}{\gamma }_{\mathrm{pp}}=[({\alpha }^2+{\beta }^2)\text{sinh}\beta \sin \alpha {\xi }_2+{\omega }^2{\eta }_{\mathrm{p}}\text{cosh}\beta {\xi }_1(\alpha \cos \alpha {\xi }_2\text{sinh}\beta {\xi }_2-\beta \sin \alpha {\xi }_2\text{cosh}\beta {\xi }_2)]$$

At order ${ϵ}^3$ , the problem is nonlinear. A solution can be obtained if a solvability condition is satisfied. This condition can be obtained by expressing the solution in the form

$$W_{(\mathrm{i}+1)3}={\mathrm{\Phi }}_{\mathrm{i}+1}(\zeta ,T_2)e^{j\omega T_0}+cc+W_{\mathrm{i}+1}(\zeta ,T_0,T_2)$$

(10)

Following the procedure in our previous work, ²² 22. O. Barry, D. Oguamanam, and J. Zu, Nonlinear Dynamics 77, 1597 (2014). https://doi.org/10.1007/s11071-014-1402-5 by substituting Eq. 10 into Eq. 4, multiplying each resulting

equation

by its corresponding linear mode shape $Y_{\mathrm{i}}$ , taking the integral and adding the two resulting

equations,

and using the orthogonality condition along with the

boundary conditions

(after substituting Eq. 10 into Eq. 5), the solvability condition for the nonlinear problem can be obtained as

$$2j\omega (A^{\prime }+\mu A)b_1+\frac{3}{2}\lambda A^2\overline{A}{b}_2^2-\frac{1}{2}f{e}^{j\sigma (T_2)}+2j\omega A^{\prime }\sum _{r=1}^n{\alpha }_{\mathrm{mr}}{Y}_{\mathrm{r}}^2({\xi }_{\mathrm{r}})+2j\omega A^{\prime }\sum _{r=1}^n{\eta }_{\mathrm{r}}{Y^{\prime }}_{\mathrm{r}}^2({\xi }_{\mathrm{r}})=0$$

(11)

The polar form of the complex amplitude $A$ can be expressed as

where $a$ is the real amplitude and $\theta$ denotes the phase. Substituting Eq. 12 into Eq. 11 and separating real and imaginary parts yield the following modulation

equations

for the amplitude and phase

$$\omega a{b}_4{\gamma }^{\prime }=\omega a{b}_4\sigma -\frac{3}{8}{a}^3{b}_3+\frac{1}{2}f\cos \gamma$$

(13)

where

$$b_1=\sum _{r=0}^n{\int }_{{\xi }_{\mathrm{r}}}^{{\xi }_{\mathrm{r}+1}}{Y}_{\mathrm{r}+1}^{2}d\zeta ,b_2=\sum _{r=0}^n{\int }_{{\xi }_{\mathrm{r}}}^{{\xi }_{\mathrm{r}+1}}{Y}_{\mathrm{r}+1}^{\mathrm{\prime }2}d\zeta b_3=\frac{1}{2}\lambda b_2^2,b_4=b_1+\sum _{r=1}^n\left[{\alpha }_{\mathrm{r}}{Y}_{\mathrm{r}}^2({\xi }_r)+{\eta }_{\mathrm{r}}{Y}_{\mathrm{r}}^{\mathrm{\prime }2}({\xi }_r)\right]\gamma =\sigma T_2-\theta ,f=\sum _{r=0}^n{\int }_{{\xi }_{\mathrm{r}}}^{{\xi }_{\mathrm{r}+1}}{F}_{\mathrm{r}+1}{Y}_{\mathrm{r}+1}d\zeta$$

where $\sigma$ is a detuning parameter of order. ¹ 1. M. Gürgöze, Journal of Sound and Vibration 217, 585 (1998). https://doi.org/10.1006/jsvi.1998.1796 The nonlinear undamped frequencies are obtained from Eqs. 13 and 14 by taking $\mu =f=b_5=\sigma =0$

$${\omega }_{\mathrm{NL}}=\omega +k{a}^2\mathit{}\mathrm{where}\mathit{}k=\frac{3}{8}\frac{b_3}{\omega b_4}$$

(15)

In the case of periodic excitation $a^{\prime }$ and ${\gamma }^{\prime }$ are equal to zero. Hence, the detuning parameter can be expressed as

$$\sigma =\frac{3}{8}\frac{a^2{b}_3}{\omega b_4}\pm \sqrt{\frac{f^2}{4a^2{\omega }^2{b}_4^2}-\frac{b_1^2}{b_4^2}{\mu }^2}$$

(16)

The validity of the frequency

equation

is demonstrated in Tables I and II. The results in Table I indicate excellent agreement between the present work and the previous work in the literature. Table II shows a comparison between present work and the

finite element analysis.

The results also show very good agreement with a maximum percentage of error of 0.8%. Table III shows the effect of attaching multiple rigid bodies on the natural frequencies. As expected, the results indicate that the

systems

natural frequencies decreases with increasing number of intermediate rigid masses.

TABLE I. Present vs. Ref. 19,22 19. E. Özkaya and M. Pakdemirli, Journal of Sound and Vibration 221, 491 (1999). 22. O. Barry, D. Oguamanam, and J. Zu, Nonlinear Dynamics 77, 1597 (2014). https://doi.org/10.1007/s11071-014-1402-5 ; ( $\eta =0,s=0,\alpha =1,{\xi }_1=0.5$ ).

Mode	Present	Ref. 22 22. O. Barry, D. Oguamanam, and J. Zu, Nonlinear Dynamics 77, 1597 (2014). https://doi.org/10.1007/s11071-014-1402-5	Ref. 20 20. E. Özkaya, Journal of Sound and Vibration 257, 413 (2002).
1	5.6795	5.6795	5.6795
2	39.4784	39.4784	39.4784
3	67.8883	67.8883	67.8883
4	157.9144	157.9144	157.9144
4	206.7901	206.7901	206.7901

TABLE II. Analytical vs.

FEA

(frequency in rad/s); ( $\eta$ = 0.5, s = 1, $\alpha$ = 0.5, ${\xi }_1=0.5$ ).

Mode	Analytical	FEA	% of Error
1	4.9162	4.9559	0.8075
2	7.3091	7.3367	0.3780
3	62.3020	62.6700	0.5907
4	72.2230	72.6290	0.5621
5	200.3700	200.8300	0.2296

TABLE III. First five modes for a

beam

carrying up to four rotational masses ( $s=1$ ).

$\alpha ,\eta ,\xi$	Values	Mode	Frequency (rad/s)
${\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4$	1,1,1,1	1	4.2361
[-0.5ex] ${\eta }_1,{\eta }_2,{\eta }_3,{\eta }_4$	0.5,0.5,0.5,0.5	2	5.6644
[-0.5ex] ${\xi }_1,{\xi }_2,{\xi }_3,{\xi }_4$	0.1,0.5,0.7,0.9	3	6.0103
		4	9.1118
		5	30.3810

As for the nonlinear

analysis,

the validity is demonstrated via comparison of the results in the literature and it is depicted in Fig. 2 . The results show an excellent agreement. For validation purpose, the tension is taken to be s = 0. As observed in Fig. 2 , the curves bend to the right, which is an indication of hardening type nonlinearity. The effect of the tension on the nonlinear natural frequency is depicted in Fig. 3 . The results indicate that the stretching of the curve shifts from right to left for $s>1$ . It is also observed that the stretching to the left is more pronounced with increasing tension. Fig. 4 examines the role of the mass rotational inertia on the nonlinear frequency. The results show that the stretching decreases with increasing rotational inertia. In the forced response

analysis,

the forcing amplitude is $f=5b_4$ and the damping coefficient is $\mu$ = 0.2. The influence of the tension on the frequency response curve is depicted in Fig. 5 . As seen previously, the curve tends to bend more to the left with increasing tension. In Fig. 6 , the effect of varying the rotational inertia on the frequency response curve is examined. The results show that the frequency response curve tends to bend more to the left with decreasing rotational inertia. This observation is an indication that the tension and the rotational inertia have opposite effect on the frequency response curve. The effect of attaching multiple intermediate rigid bodies is depicted in Fig. 7 . The results indicate that the stretching of the frequency response curve tends to decrease as the number of intermediate rigid bodies is increased. This is an indication of the reduction in the softening type nonlinearity. These observations are in agreement with the literature for $s=0$ . In that, the hardening nonlinearity type is more pronounced as the number of intermediate point masses increases.

FIG. 3. Effect of tension on the nonlinear frequency for $\xi =0.5,{\alpha }_1=1,{\eta }_1=0.5$ .

• PPT
|
• High resolution

FIG. 4. Effect of rotational inertia on the nonlinear frequency for $\xi =0.5,{\alpha }_1=1,s=5$ .

• PPT
|
• High resolution

FIG. 5. Effect of tension on the frequency-response-curve for $\xi =0.5,{\alpha }_1=1,{\eta }_1=0.5$ .

• PPT
|
• High resolution

FIG. 6. Effect of rotational inertia on the frequency-response-curve for $\xi =0.5,{\alpha }_1=1,{\eta }_1=0.5$ .

• PPT
|
• High resolution

FIG. 7. Effect of multiple rigid masses on the frequency-response-curve for $s=5$ .

• PPT
|
• High resolution

In conclusion, this paper presents the

nonlinear vibration analysis

of an axially loaded simply-supported

beam

carrying multiple intermediate rigid bodies. For the first time, explicit expressions are presented for the characteristic

equation,

mode shapes, nonlinear frequency, and modulation

equations

for the steady state phase and steady state amplitude. The validity of the analytical model is demonstrated using

finite element analysis

and results in the literature. The numerical simulations indicate that the presence of the tension in the

beam

shifts the nonlinearity type from hardening to softening and that the softening type nonlinearity is more pronounced with increasing tension. However this softening nonlinearity tends to decrease with both increasing mass rotational inertia and increasing number of intermediate rigid bodies.

REFERENCES

Section:

1. 1. M. Gürgöze, Journal of Sound and Vibration 217, 585 (1998). https://doi.org/10.1006/jsvi.1998.1796, Google Scholar Crossref
2. 2. H.-Y. Lin and Y.-C. Tsai, Journal of Sound and Vibration 302, 442 (2007). https://doi.org/10.1016/j.jsv.2006.06.080, Google Scholar Crossref
3. 3. P. Hassanpour, W. Cleghorn, J. Mills, and E. Esmailzadeh, Journal of Vibration and Control 13, 1723 (2007). https://doi.org/10.1177/1077546307076285, Google Scholar Crossref
4. 4. J.-S. Wu and C.-G. Huang, Communications in Numerical Methods in Engineering 11, 743 (1995). https://doi.org/10.1002/cnm.1640110905, Google Scholar Crossref
5. 5. S. Naguleswaran, International Journal of Mechanical Sciences 43, 2737 (2001). https://doi.org/10.1016/S0020-7403(01)00072-8, Google Scholar Crossref
6. 6. C. Kirk and S. Wiedemann, Journal of Sound and Vibration 254, 939 (2002). https://doi.org/10.1006/jsvi.2001.4138, Google Scholar Crossref
7. 7. S. O. Maiz, D. Bambill, C. Rossit, and P. Laura, Journal of Sound and Vibration 303, 895 (2007). https://doi.org/10.1016/j.jsv.2006.12.028, Google Scholar Crossref
8. 8. C. Chang, Journal of Sound and Vibration 237, 733 (2000). https://doi.org/10.1006/jsvi.2000.2990, Google Scholar Crossref
9. 9. J.-S. Wu and T.-L. Lin, Journal of Sound and Vibration 136, 201 (1990). https://doi.org/10.1016/0022-460X(90)90851-P, Google Scholar Crossref
10. 10. A. Bokaian, Journal of Sound and Vibration 142, 481 (1990). https://doi.org/10.1016/0022-460X(90)90663-K, Google Scholar Crossref
11. 11. O. Barry, D. Oguamanam, and J. Zu, Shock and Vibration 2014 (2014). https://doi.org/10.1155/2014/485630, Google Scholar Crossref
12. 12. S. Woinowsky-Krieger, J. Appl. Mech 17, 35 (1950). Google Scholar Crossref
13. 13. A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations (John Wiley & Sons, 2008). Google Scholar
14. 14. A. H. Nayfeh, Introduction to Perturbation Techniques (John Wiley & Sons, 2011). Google Scholar
15. 15. D. Burgreen, "Free vibrations of a pin-ended column with constant distance between pin ends." Tech. Rep. (DTIC Document, 1950). Google Scholar
16. 16. R. Mestrom, R. Fey, K. Phan, and H. Nijmeijer, Sensors and Actuators A: Physical 162, 225 (2010). https://doi.org/10.1016/j.sna.2010.04.020, Google Scholar Crossref
17. 17. K. Low, Journal of Sound and Vibration 207, 284 (1997). https://doi.org/10.1006/jsvi.1997.1135, Google Scholar Crossref
18. 18. E. Özkaya, M. Pakdemirli, and H. Öz, Journal of Sound and Vibration 199, 679 (1997). https://doi.org/10.1006/jsvi.1996.0643, Google Scholar Crossref
19. 19. E. Özkaya and M. Pakdemirli, Journal of Sound and Vibration 221, 491 (1999). Google Scholar Crossref
20. 20. E. Özkaya, Journal of Sound and Vibration 257, 413 (2002). Google Scholar Crossref
21. 21. M. Pakdemirli and A. Nayfeh, Journal of Vibration and Acoustics 116, 433 (1994). https://doi.org/10.1115/1.2930446, Google Scholar Crossref
22. 22. O. Barry, D. Oguamanam, and J. Zu, Nonlinear Dynamics 77, 1597 (2014). https://doi.org/10.1007/s11071-014-1402-5, Google Scholar Crossref

1. © 2016 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

raebuthationd.blogspot.com

Source: https://aip.scitation.org/doi/full/10.1063/1.4973334

Share :