On the importance and application of the extended Galerkin's method to distributed systems

Abstract

The method known as Galerkin's method, developed in the early twentieth Century for problems in elasticity is still used today, especially for the approximate solution of boundary value problems. In order to apply this method, which is more general and powerful than the Rayleigh-Ritz method, the trial functions used must satisfy all geometric and dynamic boundary conditions accompanying the differential equation. This could be considered a disadvantage of the method. This led with the contributions of H. Leipholz, to the development of the “extended Galerkin's method”. This modification generalized the method to problems where the trial functions do not satisfy some of the conditions, especially dynamic boundary conditions. While the classical Galerkin's method is widely described in the textbooks, the extended Galerkin's method is unfortunately found less frequently. Moreover, with the exception of two reference works, the book of Leipholz and partly an another book of Wunderlich and Pilkey, only the expression is given in all of them. The aim of this paper is two-fold. Firstly, to draw the reader's attention to the extended Galerkin's method, an important and very useful method, in the opinion of the authors of this paper. Secondly, they wish to discuss some important issues, including a critique of the literature. The derivation of the method, its application and related general information will be described with reference to the example of a longitudinally vibrating elastic rod carrying a mass at its free end.

Introduction

The classical Galerkin's method (CGM), developed by the Soviet scientist Boris Grigoryevich Galerkin (1871–1945), is a method included in the vast majority of structural dynamics and vibration dynamics textbooks.^1–6

The method is an efficient discretization method, mainly used for the approximate solution of boundary value problems and belongs to the class of “weighted residuals” techniques. The method was actually first applied to problems of the Theory of Elasticity by the Soviet mathematician I.G. Bubnov (1873–1919) in 1913 and then generalized by Galerkin in 1915. Thus, the method is also called the Bubnov-Galerkin method in some sources.^7,8

The essence of the method is as follows. The first step for the approximate solution of a boundary value problem is to substitute a linear combination of trial functions with initially unknown coefficients to the left of the differential equation of the boundary value problem. The important point here is that the trial functions must satisfy all the boundary conditions in question, that is, they must be of the “comparison function” class.

It is clear that this linear combination does not exactly satisfy the differential equation and when it is substituted, an error, termed the “residual” is introduced. This is what is done in the second step: The integral of the error multiplied by each trial function, i.e., in a sense, by the weighting function, in the corresponding solution domain is set equal to zero to derive as many linear and homogeneous algebraic equations as there are trial functions.

Since we are searching for an approximate solution in a finite dimensional space, we can force the residual to have a zero projection on the chosen basis, that is, the trial functions of this space. ⁸

By solving this set of equations, the coefficient in front of each trial function in the linear combination is obtained and thus the approximate solution of the problem is established.

The Galerkin's method is more general and powerful than the Rayleigh-Ritz method, which is often used to solve similar problems. It is able to approximately solve non-self adjoint problems, such as non-linear partial differential equations.

The Rayleigh-Ritz procedure and the Galerkin's method are equally effective if the system is self-adjoint, i.e., conservative. For non self-adjoint systems, (that is, non-conservative systems), only the Galerkin's method can be used. ⁵ The Galerkin's method is more general and is applicable to both conservative and non-conservative systems. ⁶

The requirement that the trial functions used in the Galerkin's method must be of the “comparison functions” class, that is, they must satisfy all the geometric and dynamic boundary conditions accompanying the differential equation of the boundary value problem, is quite restrictive for the applicability of the method. In the 1960s, modifications to the CGM by Leipholz in particular, ⁹ have led to what is known in the literature as the “extended Galerkin's method” (EGM) and allows these constraints to be relaxed. For the sake of brevity, we will henceforth refer to this modification as EGM.

In the EGM the dynamic boundary conditions are included in the calculations, just like the differential equation, so that the trial functions are only of the class of “admissible functions”, i.e., they only satisfy the geometric boundary conditions. This is clearly a great advantage. Unfortunately, this method, which has not received the recognition it deserves, in the view of the authors of this paper, is mentioned in only a few textbooks, as far as we can determine.^10–17 Although Mura and Koya ¹¹ and Raamachandran ¹² call it “Galerkin's method”, it is actually the EGM. Since the number of books that include the Galerkin's method is very large, only a few of them are given in the introduction. It should be noted that here we have listed all the books that we were able to identify.

Furthermore, it should also be noted that there are published papers with EGM in the title.^18,19 However, in them, there is an approximate solution of a non-linear differential equation and what is done is on the additional integration over one period of harmonic vibrations, on the averaged and weighted equation of motion, which can yield excellent results for free and forced vibration analysis with typical nonlinear vibration problems.

For the sake of completeness, in the following, we will first try to give a brief summary of the CGM, with our notations, on the example of a fixed-free elastic rod vibrating longitudinally, compiled from the book by Benaroya and Nagurka, pp. 812–813. ⁵

Classical Galerkin's method

The longitudinal vibration system in Figure 1 has an equation of motion and boundary conditions at both ends.

$$EA{\displaystyle \frac{{\mathrm{\partial }}^{2}u(x,t)}{\mathrm{\partial }{x}^2}-m{\displaystyle \frac{{\mathrm{\partial }}^{2}u(x,t)}{\mathrm{\partial }{t}^2}=0}}$$

(1)

$$u(0,t)=0,{{\displaystyle \frac{\mathrm{\partial }u(x,t)}{\mathrm{\partial }x}}|}_{x=L}=0$$

(2)

Figure 1.

Axially vibrating bare rod.

An approximate solution $u_{\alpha }(x,t)$ of the boundary value problem (1) and (2) is written as

$$u_{\alpha }(x,t)=\sum _{r=1}^P{\psi }_r(x){\alpha }_r(t)$$

(3)

where the functions ${\psi }_r(x)$ are taken to satisfy the boundary conditions (2). Since the approximate solution (3) does not exactly satisfy the governing equation of motion, i.e., Eq. 1, there is an error. This error is minimized by the Galerkin's method, which will be given very briefly below.

If the approximate solution (3) is substituted into the differential equation (1), the following expression is obtained:

$$EA{\displaystyle \frac{{\mathrm{\partial }}^2{u}_{\alpha }(x,t)}{\mathrm{\partial }{x}^2}-m{\displaystyle \frac{{\mathrm{\partial }}^2{u}_{\alpha }(x,t)}{\mathrm{\partial }{t}^2}=R_{error}}}$$

(4)

where R_error, known as the residual error, exists if $u_{\alpha }(x,t)$ is not the exact solution. Otherwise, if it is the exact solution, then R_error = 0. In the Rayleigh-Ritz procedure it is known that the expression for the eigenfrequency is minimized. In the Galerkin's method an expression that includes the residual error is minimized. However, since the error depends on the quality of the functions ${\psi }_r(x)$ chosen above, the minimization equations are taken with respect to these functions.

The essence of the Galerkin's method is expressed in the following equations:

$$\underset{0}{\overset{L}{\int }}\left[EA\frac{{\mathrm{\partial }}^2{u}_{\alpha }\left(x,t\right)}{\mathrm{\partial }{x}^2}-m\frac{{\mathrm{\partial }}^2{u}_{\alpha }\left(x,t\right)}{\mathrm{\partial }{t}^2}\right]{\psi }_r\left(x\right)dx=\underset{0}{\overset{L}{\int }}{R}_{error}{\psi }_r\left(x\right)dx=0\left(r=1,2,...,P\right)$$

(5)

To minimize the error in $u_{\alpha }(x,t)$ , the integral of the weighted residual error must be minimized. The criterion for selecting ${\psi }_r(x)$ is that the integral on the right-hand side be as close to zero as possible. The weighting functions ${\psi }_r(x)$ are selected such that the residual error is minimum over the domain 0 ÷ L.

The evaluation of the integrals in (5) leads to a set of P second order differential equations for the unknown functions ${\alpha }_r(t)$ .

In the next step, assuming an exponential solution as:

$${\alpha }_r(t)={\overline{\alpha }}_r{e}^{\lambda t},(r=1,2,\dots ,P)$$

(6)

we arrive at a matrix eigenvalue problem, the solution of which leads first to the ${\overline{\alpha }}_r$ 's and then to the approximate solution $u_{\alpha }(x,t)$ in (3), and then, via $\lambda$ to the ${\omega }_r$ 's, the eigenfrequencies of the system.

Extraction and implementation of EGM on the example system

To extract and apply the EGM, let us consider the vibration system in Figure 2. We know that the exact solution for the motion of this simple system is available in the literature. ²⁰ This example is chosen to evaluate the precision of the results of the EGM.

Figure 2.

Axially vibrating rod with a tip mass.

While the equation of motion (1) and the condition (2a) at the left boundary, i.e., $u(0,t)=0$ , remain valid, the following dynamic boundary condition will emerge:

$$M\ddot{u}(L,t)+EA{u}^{\mathrm{\prime }}(L,t)=0$$

(7)

this time expressing the force balance at the right end.

The superimposed dots and lines indicate the partial derivatives with respect to time and position.

Let us start with the following equation ¹⁰ :

$$-{\int }_V\mu \ddot{w}\delta \mathbf{w}dV-\delta ({\mathrm{\Pi }}_i+{\mathrm{\Pi }}_a)+{\int }_O{\mathbf{k}}_{\alpha }\delta \mathbf{w}dO+{\sum }_i{\mathbf{K}}_{\alpha \mathbf{i}}\delta {\mathbf{w}}_{\mathbf{i}}=0$$

(8)

The equation is termed the “principle of virtual work”, which is actually the “Lagrangian form of D'Alembert's principle”.

Where $\mu$ is the mass density, $\delta \mathbf{w}$ is the virtual displacement vector, ${\mathrm{\Pi }}_i$ and ${\mathrm{\Pi }}_a$ represent the potential of internal and external conservative forces, and ${\mathbf{k}}_{\alpha }$ and ${\mathbf{K}}_{\alpha \mathbf{i}}$ represent the non-conservative surface and singular external forces.

Since ${\mathrm{\Pi }}_a$ and the last two terms are not present in our system shown in Figure 2, we can write

$$-{\int }_0^{L}m\ddot{u}\delta u(x,t)dx-M\ddot{u}(L,t)\delta u(L,t)-\delta \left({\displaystyle \frac{1}{2}{\int }_0^{L}EA{u^{\mathrm{\prime }}}^2(x,t)dx}\right)=0.$$

(9)

After applying the necessary variation operations, and considering $\delta u(0,t)=0$ , we arrive at:

$${\int }_0^L\left(m\ddot{u}(x,t)-EA{u}^{\mathrm{″}}(x,t)\right)\delta u(x,t)dx+\left[EA{u}^{\mathrm{\prime }}(L,t)+M\ddot{u}(L,t)\right]\delta u(L,t)=0$$

(10)

From now on, for brevity, suppose we assume the solution in (3) without using the subindex “ $\alpha$ ” :

$$u(x,t)=\sum _{r=1}^P{\psi }_r(x){\alpha }_r(t)$$

(11)

Here, we assume that ${\psi }_r(x)$ s represents chosen functions that satisfy the geometric boundary condition (2a), ensuring following expression:

$${\psi }_r(0)=0,(r=1,2,\dots ,P)$$

(12)

In contrast, ${\alpha }_r(t)$ s are unknown functions defined as:

$${\alpha }_r(t)={\overline{\alpha }}_r{e}^{\lambda t}$$

(13)

The following solution assumption is substituted in (9)

$$u(x,t)=e^{\lambda t}\sum _{r=1}^P{\overline{\alpha }}_r{\psi }_r(x)$$

(14)

and the necessary variation operations are performed. Of note, the quantities to be varied are now only the ${\overline{\alpha }}_r$ -s ! The following expression is obtained:

$$\begin{array}{rl}& \{\underset{0}{\overset{L}{\int }}\left[m{\lambda }^2\sum _{r=1}^P{\psi }_r\left(x\right){\overline{\alpha }}_r-EA\sum _{r=1}^P{{\psi }^{\mathrm{″}}}_r\left(x\right){\overline{\alpha }}_r\right]{\psi }_1\left(x\right)dx\\ & +\left[\left[M{\lambda }^2\sum _{r=1}^P{\psi }_r\left(L\right){\overline{\alpha }}_r+EA\sum _{r=1}^P{{\psi }^{\mathrm{\prime }}}_r\left(L\right){\overline{\alpha }}_r\right]\right]{\psi }_1\left(L\right)\}\delta {\overline{\alpha }}_1\\ & +\left\{\underset{0}{\overset{L}{\int }}\left[\right]{\psi }_2\left(x\right)dx+\left[\left[\right]\right]{\psi }_2\left(L\right)\right\}\delta {\overline{\alpha }}_2+...\end{array}$$

(15)

In the last equation, assuming that the variations of $\delta {\overline{\alpha }}_1$ , $\delta {\overline{\alpha }}_2$ , …, $\delta {\overline{\alpha }}_P$ are arbitrary, we come to these set of equations:

$${\int }_0^L\left[\right]{\psi }_q\left(x\right)dx+\left[\left[\right]\right]{\psi }_q\left(L\right)=0;\left(q=1,2,...,P\right)$$

(16)

At this point, if we recall our solution assumption (11), it is evident that the above set of equations actually corresponds to the following expression:

$$\begin{array}{rl}& \underset{0}{\overset{L}{\int }}\left[m\ddot{u}(x,t)-EA{u}^{\mathrm{″}}(x,t)\right]{\psi }_q(x)dx+\left[M\ddot{u}(L,t)+EA{u}^{\mathrm{\prime }}(L,t)\right]{\psi }_q(L)=0.\\ & (q=1,2,\dots ,P)\end{array}$$

(17)

The above equations are nothing but the application of the EGM to the mechanical vibration system at hand.

From the equations in (16), we can now reach to the conclusion.

However, before we do, let's take a closer look at the equations. Taking into account (17), it appears that the procedure to be followed is:

To the integral of the left-hand side of the differential equation of motion (1) multiplied by the qth trial function in the region 0÷L, the product of the left-hand side of the dynamic boundary condition (7) multiplied by the value of the same trial function at the boundary is added and the sum is equaled to zero.

This process is repeated for P trial functions. Thus, the dynamic boundary condition is also taken into account by means of this second term in (16). Obviously, if this second term did not exist, the CGM would be in question.

We can now proceed to the evaluation of (16). As trial functions, suppose we choose the functions which are the normalized eigenfunctions of the system in Figure 1 ²⁰

$${\psi }_r(x)=\sqrt{{\displaystyle \frac{2}{mL}}}\sin \left[{\displaystyle \frac{(2r-1)}{2L}\pi x}\right];r=1,2,\dots ,P$$

(18)

with the boundary conditions:

$${\psi }_r(0)=0,{\psi }_r^{\mathrm{\prime }}(L)\stackrel{\mathrm{\Delta }}{=}{{\displaystyle \frac{d{\psi }_r(x)}{dx}}|}_{x=L}=0$$

(19)

In addition to the conditions that comply with the above boundary conditions of the fixed-free bar, the normalized eigenfunctions also satisfy the following ortho-normality conditions:

$${\int }_0^{L}m{\psi }_r(x){\psi }_q(x)dx={\delta }_{rq};(r,q=1,2,\dots ,P)$$

(20)

When equations (16) are evaluated by considering (20), the following equation is obtained:

$$\begin{array}{rl}& \left[M\sum _{r=1}^P({\psi }_r(L){\psi }_q(L)){\overline{\alpha }}_r+{\overline{\alpha }}_q\right]{\lambda }^2+EA\sum _{r=1}^P({{\psi }^{\mathrm{\prime }}}_r(L){\psi }_q(L)){\overline{\alpha }}_r+{\overline{\omega }}_q^2{\overline{\alpha }}_q=0\\ & (q=1,2,\dots ,P)\end{array}$$

(21)

Equations (21) can be rewritten as:

$${\overline{\omega }}_r={\overline{\beta }}_r{\omega }_0,{\overline{\beta }}_r={\displaystyle \frac{(2r-1)}{2}\pi ,{\omega }_0^2={\displaystyle \frac{EA}{m{L}^2}(r=1,2,\dots ,P)}}$$

(22)

Equations (21) can be organized as:

$$\left[2{\alpha }_M\sum _{r=1}^P{f}_{qr}{\overline{\alpha }}_r+{\overline{\alpha }}_q\right]{\lambda }^2+{\overline{\omega }}_q^2{\overline{\alpha }}_q=0;(q=1,2,\dots ,P)$$

(23)

For this, these abbreviations are used:

$${\alpha }_M={\displaystyle \frac{M}{mL},f_{qr}=f_{rq}={\psi }_q(L){\psi }_r(L)=(-1{)}^{q+1}\cdot (-1{)}^{r+1}}$$

(24)

In the next step, we obtain the following equation:

$$\left[\begin{array}{c}\\ \\ {\lambda }^2\{2{\alpha }_M{\mathbf{F}}_{\mathbf{p}\mathbf{p}}+{\mathbf{I}}_{\mathbf{p}\mathbf{p}}\}+\mathbf{d}\mathbf{i}\mathbf{a}\mathbf{g}({\overline{\omega }}_r^2)\\ \\ \end{array}\right]\left[\begin{array}{c}{\overline{\alpha }}_1\\ {\overline{\alpha }}_2\\ {\overline{\alpha }}_3\\ ⋮\\ {\overline{\alpha }}_P\end{array}\right]=\mathbf{0}$$

(25)

In the last step, the determinant of the coefficients is set equal to zero to ensure that this homogeneous set of equations yields non-trivial solutions:

$$det[{\overline{\lambda }}^2\{2{\alpha }_M{\mathbf{F}}_{\mathbf{p}\mathbf{p}}+{\mathbf{I}}_{\mathbf{p}\mathbf{p}}\}+\mathbf{d}\mathbf{i}\mathbf{a}\mathbf{g}({\overline{\beta }}_r^2)]=0.$$

(26)

where the following abbreviations are introduced:

$$\overline{\lambda }=\lambda /{\omega }_0,{\mathbf{F}}_{\mathbf{p}\mathbf{p}}=\left[\begin{array}{ccccc}+1& -1& +1& -1& \dots \\ -1& +1& -1& +1& \dots \\ \dots & \dots & +1& \dots & \dots \\ & & \dots & & \\ \dots & \dots & \dots & \dots & \dots \end{array}\right];{\mathbf{I}}_{\mathbf{p}\mathbf{p}}:\mathrm{PxP}\mathrm{dimensional}\mathrm{unit}\mathrm{matrix}$$

(27)

Note that these will be purely imaginary numbers corresponding to $i{\omega }_r/{\omega }_0$ , where ${\omega }_r$ is the rth eigenfrequency of the system. The solution of equation (26) gives us the eigenvalues ${\overline{\lambda }}_r$ (r = 1, 2, …, P).

Numerical applications

The dimensionless eigenfrequency values ${\omega }_r/{\omega }_0$ (r = 1, 2,…, 5) obtained from numerical calculations are given in Tables 1 to 8.

Table 1.

The dimensionless eigenfrequency values ${\omega }_r/{\omega }_0$ (P = 5, ${\alpha }_M=0.1$ ).

r	Exact	FEM	Rayleigh-Ritz		Extended Galerkin
			Polynomial	Eigenfunctions	Polynomial	Eigenfunctions
1	1.428870	1.432855	1.428870	1.429918	1.429113	1.429918
2	4.305801	4.415928	4.306193	4.330329	4.261344	4.330329
3	7.228110	7.749397	7.248922	7.318686	6.970728	7.318686
4	10.200263	11.575233	11.195956	10.396462	11.748822	10.396462
5	13.214186	15.438611	15.779741	13.563298	21.196841	13.563298

Table 2.

The dimensionless eigenfrequency values ${\omega }_r/{\omega }_0$ (P = 10, ${\alpha }_M=0.1$ ).

r	Exact	FEM	Rayleigh-Ritz		Extended Galerkin
			Polynomial	Eigenfunctions	Polynomial	Eigenfunctions
1	1.428870	1.429868	1.428870	1.429396	1.428870	1.429396
2	4.305801	4.333772	4.305801	4.318158	4.305805	4.318158
3	7.228110	7.364981	7.228110	7.273531	7.227607	7.273531
4	10.200263	10.596863	10.200271	10.295757	10.192990	10.295757
5	13.214186	14.093820	13.217642	13.369905	13.285455	13.369905

Table 3.

The dimensionless eigenfrequency values ${\omega }_r/{\omega }_0$ (P = 5, ${\alpha }_M=0.5$ ).

r	Exact	FEM	Rayleigh-Ritz		Extended Galerkin
			Polynomial	Eigenfunctions	Polynomial	Eigenfunctions
1	1.076874	1.077786	1.076874	1.083922	1.076907	1.083922
2	3.643597	3.706599	3.643689	3.744423	3.629818	3.744423
3	6.578334	7.007746	6.581915	6.814661	6.529087	6.814661
4	9.629560	10.970845	10.402819	10.019431	10.721506	10.019431
5	12.722299	15.172382	14.465436	13.313281	15.038811	13.313281

Table 4.

The dimensionless eigenfrequency values ${\omega }_r/{\omega }_0$ (P = 10, ${\alpha }_M=0.5$ ).

r	Exact	FEM	Rayleigh-Ritz		Extended Galerkin
			Polynomial	Eigenfunctions	Polynomial	Eigenfunctions
1	1.076874	1.077103	1.076874	1.080405	1.076874	1.080405
2	3.643597	3.659519	3.643597	3.694090	3.643597	3.694090
3	6.578334	6.687259	6.578334	6.694668	6.578242	6.694668
4	9.629560	9.987402	9.629561	9.813307	9.628816	9.813307
5	12.722299	13.561601	12.724385	12.975380	12.740089	12.975380

Table 5.

The dimensionless eigenfrequency values ${\omega }_r/{\omega }_0$ (P = 5, ${\alpha }_M=1$ ).

r	Exact	FEM	Rayleigh-Ritz		Extended Galerkin
			Polynomial	Eigenfunctions	Polynomial	Eigenfunctions
1	0.860334	0.860618	0.860334	0.869805	0.860340	0.869805
2	3.425618	3.482542	3.425670	3.545770	3.419077	3.545770
3	6.437298	6.859830	6.439806	6.695933	6.416815	6.695933
4	9.529334	10.875124	10.258138	9.944556	10.433212	9.944556
5	12.645287	15.133927	14.309413	13.268021	14.556958	13.268021

Table 6.

The dimensionless eigenfrequency values ${\omega }_r/{\omega }_0$ (P = 10, ${\alpha }_M=1$ ).

r	Exact	FEM	Rayleigh-Ritz		Extended Galerkin
			Polynomial	Eigenfunctions	Polynomial	Eigenfunctions
1	0.860334	0.860405	0.860334	0.865059	0.860334	0.865059
2	3.425618	3.439912	3.425618	3.485186	3.425618	3.485186
3	6.437298	6.543522	6.437298	6.562886	6.437257	6.562886
4	9.529334	9.883893	9.529335	9.722047	9.529036	9.722047
5	12.645287	13.481529	12.647193	12.907616	12.654978	12.907616

Table 7.

The dimensionless eigenfrequency values ${\omega }_r/{\omega }_0$ (P = 5, ${\alpha }_M=5$ ).

r	Exact	FEM	Rayleigh-Ritz		Extended Galerkin
			Polynomial	Eigenfunctions	Polynomial	Eigenfunctions
1	0.432841	0.432849	0.432841	0.440697	0.432841	0.440697
2	3.203935	3.256785	3.203963	3.335866	3.202791	3.335866
3	6.314846	6.732667	6.316790	6.587059	6.312916	6.587059
4	9.445948	10.795903	10.136858	9.878446	10.173786	9.878446
5	12.582265	15.102348	14.189109	13.228690	14.232557	13.228690

Table 8.

The dimensionless eigenfrequency values ${\omega }_r/{\omega }_0$ (P = 10, ${\alpha }_M=5$ ).

r	Exact	FEM	Rayleigh-Ritz		Extended Galerkin
			Polynomial	Eigenfunctions	Polynomial	Eigenfunctions
1	0.432841	0.432843	0.432841	0.436730	0.432841	0.436730
2	3.203935	3.217126	3.203935	3.268515	3.203935	3.268515
3	6.314846	6.419119	6.314846	6.445153	6.314839	6.445153
4	9.445948	9.798048	9.445949	9.643687	9.445902	9.643687
5	12.582265	13.416189	12.584047	12.850321	12.585574	12.850321

The values in column 1, labeled “Exact”, are the values obtained from the solution of the following frequency equations given indirectly in the book for the system in Figure 2 via the torsional system. ²⁰

$$\tan \beta ={\displaystyle \frac{1}{{\alpha }_M\beta };(\beta \stackrel{\mathrm{\Delta }}{=}\omega /{\omega }_0)}$$

(28)

The values in the column 2 are the values obtained using P elements in FEM. The values in columns 3 and 5 are the values obtained with the polynomial formed by ${\psi }_r(x)=(x/L{)}^r$ (r = 1, 2, …, P) functions.

The values in columns 4 and 6 are the values obtained using the eigenfunctions of the system in Figure 1. P = 5 in Tables 1, 3, 5 and 7 and P = 10 in Tables 2, 4, 6 and 8. The striking result of the comparison of the numerical values in the tables is that the Rayleigh-Ritz and EGM give exactly the same results if the solution assumption is based on the eigenfunctions of the system in Figure 1.

The polynomial results, although not exactly the same, are closer to the exact values for Rayleigh-Ritz, especially for P = 10.

As expected, as the number of trial functions, P, increases, the numerical results approach the exact values.

As the dimensionless mass parameter ${\alpha }_M$ increases, the numerical values given by all methods converge closer to the exact results. The explanation for this is that in the transcendental equation (28), ${\alpha }_M$ is in the denominator, so the results converge towards one value.

A critique of the literature

After explaining the EGM using the example of an elastic rod carrying a mass at its free end and vibrating longitudinally, we would like to mention a point that we have noticed and criticized from the literature. For this, if we go back to equation (17), we see that the left-hand side of the dynamic boundary condition (i.e., a power expression) is multiplied by the value of the trial function at the boundary and, in a sense, is indirectly taken into account as a virtual work.

This is because of the structure of the mechanical vibration system. We have noticed that in some of the references cited, the impression is given that this situation is general and the left-hand sides of all dynamic boundary conditions should be multiplied by the value of the corresponding trial function at the boundary.^13,15,17 However, depending on the nature of the dynamic boundary condition, sometimes it is not the trial function itself, but its derivative with respect to the position coordinate, that is multiplied by its value at the boundary. Let us illustrate the situation with the bending vibration system in Figure 3.

Figure 3.

Transversely vibrating beam with a tip mass.

The equivalent of equation (9) for this system is equation (29):

$$-{\int }_0^{L}m\ddot{w}(x,t)\delta w(x,t)dx-M\ddot{w}(L,t)\delta w(L,t)-\delta \left({\displaystyle \frac{1}{2}EI{\int }_0^L{w^{\mathrm{″}}}^2(x,t)dx}\right)=0$$

(29)

Similar to what we have done before, but this time with slightly longer processes and assuming an approximate solution as,

$$w(x,t)=\sum _{i=1}^N{\varphi }_i(x){\eta }_i(t)$$

(30)

the EGM leads us to the following equations:

$$\underset{0}{\overset{L}{\int }}\left[m\ddot{w}(x,t)+EI{w}^{IV}(x,t)\right]{\varphi }_j(x)dx+\left[\left[M\ddot{w}(L,t)-EI{w}^{\mathrm{″}\mathrm{\prime }}(L,t)\right]\right]{\varphi }_j(L)+[[[EI{w}^{\mathrm{″}}(L,t)]]]{\varphi }_j^{\mathrm{\prime }}(L)=0;{\varphi }_j^{\mathrm{\prime }}(L)={{\displaystyle \frac{d{\varphi }_j(x)}{dx}}|}_{x=L}(j=1,2,\dots ,N)$$

(31)

The double brackets are related to the shear force at the end and the triple brackets are related to the bending moment at the end. This means that the moment expression is not multiplied by $\varphi (L)$ but by its derivative ${\varphi }^{\mathrm{\prime }}(L)$ .

Conclusions

In the context of the approximate solution of boundary value problems, the CGM is an effective method that is included in almost all books, especially in the field of vibration dynamics and structural dynamics. On the other hand, the EGM, which was developed latterly, has found a place in relatively few books, although it is a very useful and effective method in the case of complex dynamic boundary conditions that are difficult to be satisfied by the trial functions used in approximate solution assumptions. Of these, only one of them contains information on its derivation, ¹⁰ based on the article of the same author. ⁹

In this paper, we have attempted to draw attention to the EGM, an important and useful method. The derivation of the method, its application and related general information are illustrated using the example of an elastic rod vibrating in the axial direction and carrying a mass at its free end. Numerical evaluations show that the method gives exactly the same results as the Rayleigh-Ritz method in some parameter regions. The analysis here could be viewed as a very useful and effective application of the Galerkin's Method. Furthermore, building on previous works,^18,19 we are of the opinion that the EGM reported here can also be applied to nonlinear problems.