Examples • Verification Problems
Free Vibration of a Cantilever Beam (FLAC3D)
Problem Statement
Note
The project file for this example is available to be viewed/run in FLAC3D.[1] The main data file is shown at the end of this example. The remaining data files can be found in the project.
A cantilever beam is placed vertically (See Figure 1). At the beam top, the density is much bigger than elsewhere on the beam (Figure 1a). A horizontal force \(P_0 = k U_0\) is applied at the top mass centroid to ensure a static horizontal displacement of 0.001 at the beam top. The force is then released to let the beam vibrate. The model schematically shown in Figure 1a can be simplified into Figure 1b with a lumped mass \(m\) at the beam top. It can be further idealized as a single degreeoffreedom vibration system shown in Figure 1c, with a mass \(m\) and a stiffness \(k=EI/L^3\). The damping coefficient \(c\) will be assumed with various values. The FLAC3D model is based on Figure 1a using beam elements, and the FLAC3D solution will be compared with the analytical solution based on Figure 1c.
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The geometry and material parameters are summarized as follows.
Input Parameters: 

\(E\) 
2×10^{10} 
\(I\) 
1 
\(L\) 
10 
\(\rho\) 
7.60×10^{3} 
\(A\) 
1 
\(P_0\) 
6×10^{4} 
Derived Parameters: 

\(m\) 
1.52×10^{4} 
\(k\) 
6×10^{7} 
\(U_0\) 
0.001 
\(T_n\) 
0.1 
\(\omega_n\) 
20 \(\pi\) 
The governing differential equation of free vibration for a single degree system with damping is
where \(u\) is the displacement, \(m\) is the mass, \(k\) is the stiffness, and \(c\) is the damping coefficient. The natural circular frequency is \(\omega_n = \sqrt{k/m}\), and the natural period is \(T_n = 2\pi/\omega_n\).
The critical damping coefficient is defined as \(c_{cr} = 2m\omega_n = 2\sqrt{km}\), and the damping ratio is defined as \(\zeta = c/c_{cr}\).
In this example, Rayleigh damping (\(c = \beta k + \alpha m\)) is assumed. For simplicity, only massproportional Rayleigh damping (\(\beta = 0\)) is considered here, so
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In this example, the initial condition at the dynamic stage is \(u(0) = U_0\) and \(\dot{u}(0)=0\). With these initial conditions, the analytical solution (Argris 1991) can be summarized as
No Damping (\(\zeta=0\) or \(\alpha=0\)):
(3)\[u(t) = U_0 \cos{(\omega_n t)}\]Under Damping (\(0 < \zeta < 1\) or \(0 < \alpha < 2\omega_n = 4\pi/T_n\)):
(4)\[u(t) = U_0 e^{\zeta \omega_n t} \left[ \cos{(\omega_D t)} + {\zeta \over {\sqrt{1\zeta^2}}} \sin{(\omega_D t)} \right]\]where \(\omega_D = \omega_n \sqrt{1\zeta^2}\).
Critical Damping (\(\zeta = 1\) or \(\alpha = 2\omega_n = 4\pi/T_n\)):
(5)\[u(t) = U_0 (1+\omega_n t) e^{\omega_n t}\]Over Damping (\(\zeta > 1\) or \(\alpha > 2\omega_n = 4\pi/T_n\)):
(6)\[u(t) = {U_0 \over 2\sqrt{(\zeta^21)}} \left[ c_2 e^{c_1 \omega_n t}  c_1 e^{c_2 \omega_n t} \right]\]where \(c_1 = \zeta\sqrt{\zeta^21}\) and \(c_2 = \zeta+\sqrt{\zeta^21}\).
In the FLAC3D model, the beam is divided into 10 beam elements. A static horizontal force \(P_0\) is applied at the centroid of the lumped mass. After equilibrium is reached, the static displacement should be \(U_0 = P_0/k = 0.001\). The force is then released to let the cantilever beam freely vibrate at four different damping ratios, \(\zeta\) = 0%, 5%, 100%, and 500%, or \(\alpha\) = 0, \(2\pi\), \(40\pi\), and \(200\pi\) (since \(T_n\) = 0.1), to represent four cases, namely, no, under, critical, and overdamping.
The FLAC3D solutions for all four cases are plotted in Figure 2. Comparisons with the analytical solutions are plotted in Figures 3 to 6. In all cases, a good fit is observed between the FLAC3D solution and the analytical solution.
Reference
Argyris, J. & Mlejnek, H.P. Dynamics of Structures. NorthHolland, Amsterdam (1991).
Data File
FreeVibration.dat
model new
model title 'Free Vibration of a Cantilever Beam'
model largestrain off
model configure dynamic
model dynamic active off
;
[global L = 10.0]
[global E = 2.0e10]
[global I = 1.0]
[global A = 1.0]
[global k = 3.0*E*I/(L*L*L)]
[global T = 0.1]
[global m = T*T/(4*math.pi*math.pi)*k]
[global r = m/A/2.0]
[global c = 4.0*math.pi/T]
;
structure beam create byline (0,0,0) (0,0,11) segments 11 id 1 ...
elementtype eulerbernoulli
structure beam cmodel assign elastic
structure beam property young [E] poisson 0.3
structure beam property density [0.001*r]
structure beam property density [r] range positionz 9.0 11.0
structure beam property crosssectionalarea [A] moiz [I] moiy [I] ...
torsionconstant 0
structure node fix velocity range componentid 1
structure node fix rotation range componentid 1
structure node apply system global
structure node apply force [0.001*k] 0 0 range componentid 11
model solve ratiolocal 1e6
structure node apply remove force
model dynamic active on
model dynamic timetotal 0
model history dynamic timetotal
structure node history displacementx componentid 11
structure node history velocityx componentid 11
history interval 50
model save 'dynamic0'
; No Damping
model restore 'dynamic0'
model solve timetotal 0.4
history export '2' vs '1' table '1'
table '1' export 'NoDamping' truncate
model save 'NoDamping'
; Under Damping
model restore 'dynamic0'
[alpha = 0.05*c]
structure dynamic damping rayleigh 0 [alpha]
model solve timetotal 0.4
history export '2' vs '1' table '1'
table '1' export 'UnderDamping' truncate
model save 'UnderDamping'
; Critical Damping
model restore 'dynamic0'
[alpha = c]
structure dynamic damping rayleigh 0 [alpha]
model solve timetotal 0.4
history export '2' vs '1' table '1'
table '1' export 'CriticalDamping' truncate
model save 'CriticalDamping'
; Over Damping
model restore 'dynamic0'
[alpha = 5.0*c]
structure dynamic damping rayleigh 0 [alpha]
model solve timetotal 0.4
history export '2' vs '1' table '1'
table '1' export 'OverDamping' truncate
model save 'OverDamping'
Endnote
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