Add mass-spring system

Author: mkraska

Dynamics/Discrete/MS.inp

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+** based on example spring1.inp
+*NODE,NSET=NALL
+1,0.,0.,0.
+2,1.,0.,0.
+3,2,0,0
+4,3,0,0
+5,4,0,0
+*nset,nset=response
+2,4
+*ELEMENT,TYPE=SPRINGA,ELSET=ESpring
+1,1,2
+3,3,4
+** this element enforces the existence of node 5 in the model,
+** in the expanded model it is otherwise not existent any more
+5,4,5
+*ELEMENT,TYPE=DASHPOTA,ELSET=EDash
+6,1,2
+7,3,4
+*ELEMENT,TYPE=B31,ELSET=Emass
+2,2,3
+4,4,5
+*BOUNDARY
+1,1,3
+2,2,3
+3,2,6
+4,2,6
+5,2,6
+*equation
+2
+2,1,1,3,1,-1
+*equation
+2
+4,1,1,5,1,-1
+*SPRING,ELSET=Espring
+
+10.
+*DASHPOT,elset=EDash
+
+0.000001
+*MATERIAL,NAME=EL
+*ELASTIC
+210000.,.3
+*DENSITY
+7.8E-9
+*beam section, elset=Emass,material=EL,section=rect
+1,1
+*STEP
+*frequency,storage=yes
+2
+*cload
+5,1,0
+*NODE file
+U
+*END STEP
+*STEP
+*STEADY STATE DYNAMICS
+0,12000
+*cload
+5,1,1
+*NODE file
+PU,U
+*NODE print,nset=response
+U
+*END STEP

Dynamics/Discrete/README.md

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+# Spring/Mass Oscillator
+Tested with CGX/CCX 2.10
+
++ Frequency response analysis
++ 2-Mass oszillator
++ Mass element faked using beam element and appropriate constraints.
+
+| File                     | Contents                        |
+| :-------------           | :-------------                  |
+| [run.fbd](pre.fbd)       | Top level control script for CGX|
+| [MS.inp](MS.inp)         | Input for CCX                   |
+| [V.gnu](V.gnu)           | Gnuplot script for the response function  |
+
+The system consists of two rigid bodies connected by springs and dashpots.
+
+![](System.png)
+
+| Parameter                | Value           |
+| :-------------           | :-------------  |
+| k                        | 10 N/mm         |
+| m                        | 7.8 mg          |
+| c                        | 0.000001 N/(m/s)|
+
+Run the analysis:
+```
+> cgx -b run.fbd
+```
+
+## Preprocessing
+
+There is no cgx-based preprocessing. The model is entirely defined within the input file `MS.inp`.
+Springs and dashpots are represented by the appropriate discrete elements,
+the masses are represented by beam elements which are rigidized by an equation linking their nodes.
+
+<img src="mesh.png" width=400>
+
+## Solving
+
+First, a `*frequency` step is performed and then a `*steady state dynamics` step.
+```
+> ccx MS
+```
+## Postprocessing
+
+Mode shapes
+<img src="modes.png">
+
+
+Magnitude and phase plot of the response (displacement of the masses)
+
+<img src="V.png">

Dynamics/Discrete/System.pdf

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+set term pngcairo enhanced
+# mode shapes
+set out "modes.png"
+set key top left
+set grid
+set xlabel "Position in mm"
+set ylabel "Amplitude (unitless)"
+plot "graph_0.out" using 2:3 title "Mode 1" w lp, "graph_1.out" using 2:3 title "Mode 2" w lp
+set out "V.png"
+set multiplot layout 2,1 title "Displacement response"
+set key top left
+set grid
+#set xlabel "Frequency in Hz"
+set ylabel "Magnitude in mm"
+set yrange [0:1]
+plot "graph_response_PDISP_MAG1.out" using 1:3 title "M1" w lp, "" using 1:5 title "M2" w lp
+set xlabel "Frequency in Hz"
+set ylabel "Phase in °"
+set yrange [*:*]
+set ytics 90
+set nokey
+plot "graph_response_PDISP_PHA1.out" using 1:3 title "M1" w lp, "" using 1:5 title "M2" w lp

Dynamics/Discrete/System.png

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+sys ccx MS
+read MS.frd
+read MS.inp nom
+# plot model
+seta ! all
+hcpy png
+sys mv hcpy_1.png mesh.png
+# plot mode shapes
+seta path n 1 3 5
+ds 1 e 1
+graph path length
+ds 2 e 1
+graph path length
+
+graph response time PDISP MAG1
+graph response time PDISP PHA1
+
+sys gnuplot V.gnu

Dynamics/Discrete/V.gnu

@@ -151,3 +151,8 @@ This page uses animated gifs. In Firefox, you might want to install the plugin [
   width="200"
   title="Illustration of plane stress/strain condition">
 ](Linear/PSE/)
+[<img
+  src="Dynamics/Discrete/V.png"
+  width="200"
+  title="Freqency response of a 2 DOF spring-mass-system">
+](Dynamics/Discrete/)