2. Benchmark solution#

2.1. Calculation method used for the reference solution#

2.1.1. Case 1: free end#

The equation for the beam is written as:

(2.1)#\[ \ frac {{d} ^ {4} v} {{\ mathit {xx}}} ^ {4}} =\ frac {-\ rho S} {\ mathit {EI}}}\ frac {{d}} ^ {d} ^ {d} ^ {2} v} {{2} v} {{\ mathit {dt}}} ^ {2}}\]

We are looking for a solution in the form of:

(2.2)#\[ v (x) =\ mathrm {sin} (\ omega t)\ left [A\ mathrm {cos} (\ mathit {kx}) +B\ mathrm {sin} (\ mathit {kx}) +C\ mathrm {kx}) +C\ mathrm {kx}) +C\ mathrm {kx}) +C\ mathrm {kx}) +C\ mathrm {kx}) +C\ mathrm {kx}) +C\ mathrm {kx})\ right]\]

with \(\omega =\sqrt{\frac{\mathit{EI}}{\rho S}}{k}^{2}\)

The boundary conditions are as follows:

In \(x=0\), \(v=0\) and \(\frac{{d}^{2}v}{{\mathit{dx}}^{2}}=0\)

In \(x=L\), \(\frac{{d}^{2}v}{{\mathit{dx}}^{2}}=0\) and \(\frac{{d}^{3}v}{{\mathit{dx}}^{3}}=0\)

Replacing \(v\) with the expression (), we get:

In \(x=0\), \(A=C=0\);
In \(x=L\), \(\mathrm{sin}(\mathit{kL})\mathrm{cosh}(\mathit{kL})=\mathrm{cos}(\mathit{kL})\mathrm{sinh}(\mathit{kL})\) (3)

The first six roots of () are given in the following table:

\(\mathit{kL}\)

\(0\)

\(3.9266\)

\(7.06858\)

\(10.2102\)

\(13.3518\)

\(16.4934\)

From this we deduce the values of \({\omega }_{i}\), \(i=\mathrm{1,}\mathrm{...},6\).

2.1.2. Case 2: elastic end#

The eigenvalue equation is as follows:

\[\]

: label: eq-4

mathrm {sin} (mathit {kL})left ({k}} ^ {3} {L} ^ {3}frac {mathit {EI}} {{L} ^ {3}}}mathrm {kL}}}mathrm {kL})mathrm {kL})right)mathrm {3}}}mathrm {3}}}mathrm {3}}}mathrm {cosh} (mathit {kL})right) -mathrm {cosh} (mathit {kL}) sinh} (mathit {kL})left ({k}} ^ {3} {3} {L} ^ {3}frac {mathit {EI}} {{L} ^ {3}}mathrm {cos}} (mathrm {cos}} (mathit {cos}} (mathit {cos}}) (mathit {cos}} (mathit {cos}} (mathit {kL}}) (mathit {kL})right) (mathit {kL})right) =0

The own pulsations are given by:

\[\]

: label: eq-5

omega =frac {1} {{L} ^ {2}}}sqrt {frac {mathit {EI}} {rho S}} {k}} {k} ^ {2} ^ {2} {2} {L} ^ {2}

The first six roots of () are given in the following table:

\(\mathit{kL}\)

\(2.7880886\)

\(4.5619625\)

\(7.1895724\)

\(10.249031\)

\(13.368916\)

\(16.502406\)

We then deduce the values of \({\omega }_{i}\), \(i=\mathrm{1,}\mathrm{...},6\).

2.2. Benchmark results#

2.2.1. Case 1: free end#

The structure has a rigid mode at zero frequency. The following table shows the non-zero frequencies.

Mode	Frequency (\(\mathit{Hz}\))
1	\(85.5\)
2	\(277\)
3	\(577.9\)
4	\(988.2\)
5	\(1507.9\)

2.2.2. Case 2: elastic end#

Mode	Frequency (\(\mathit{Hz}\))
1	\(43.1\)
2	\(115.4\)
3	\(286.5\)
4	\(582.3\)
5	\(990.7\)
6	\(1509.6\)

2.3. Uncertainty about the solution#

Analytical solution.