• •Multi-1)0F Sysit/Page 18 Example: Compare the Stiffness and Flexibility Matricies formed by using...

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• •Multi-1)0F Sysit/Page 18 Example: Compare the Stiffness and Flexibility Matricies formed by using the unit-displacement and unit-load techniques for the following system. First, find Stiffness Matrix using unit-displacement technique:'"1 IZ x3(0) (I XXXXXXXXXXt) 'I x :1;1 FF;2 k, *31= k „(a)01 ! XXXXXXXXXX)T 12 X XXXXXXXXXXPl * 1 2 FtrF P3 *32k22 (b)(I XXXXXXXXXX) "' '3 s3 I:I 14 17 Pa *13 P2 *23 FNitrFlc)P3 - 433From (a), (unit displacement of ml)kl 1k21= F1 + F XXXXXXXXXX= - • k31= 0 1' 2 31From (b) (unit displacement of m2)k12k221 1 -F • k = - F-1 ' 2 " 13Fl +12F 13• • From (c) (unit displacement of m3)Then the Stiffness Matrix is:[k] =F F XXXXXXXXXX-F0(F12k13 = 0 ; k23Multi-DOF SysilisiPage 19 = —F1 13k33 = Fl + F 1 — 13 140F-F XXXXXXXXXX-F13F  • • The Flexibility Matrix found by applying unit loads at each station:x = a l' SO a= (Do force balance to find each ad (Let = I, + + 1, + 1,)[a] -.«.0s,11111.9.„' XXXXXXXXXX.° XXXXXXXXXXmfLoading at Station 3 is similar to Station I1, XXXXXXXXXX, XXXXXXXXXX, XXXXXXXXXX, 12)(13 ' XXXXXXXXXX XXXXXXXXXX11 + 12 " 13)• Now, Prove [ay' = [k]•First, find [ adj a (Using MATLAB, MATHCAD or DERIVE): Then find del a : Divide [ adj a] by der a to find [a]': Which is the same expression as [k] ?¦¦•• 1.4¦2.0CF 21

Robert · Accepted Answer

Microsoft Word - New Microsoft Word Document _3_.doc
1) The meaning of co-efficient of stiffness matrix can be expressed as 
 
jnodeatntdisplacemeunitproducetoinodeatrequiredForceKij =  
 
Now to get the stiffness matrix of the given problem we will use unit displacement 
method. 
 
In this technique we will apply unit displacement at one node and zero displacement 
at the other nodes and then we will write the force balance equation. Now since the 
given problem is of 3 dof so first we will apply the unit displacement on mass m1 and 
in the subsequent steps on mass m2 and m3. 
 
Step 1: 
 
Let us assume mass m1 displaces by unit and rest all the masses have zero 
displacement.  
 
Now using this we will derive  312111, KandKK  
 
Now the mass m1 is attached by two strings. So after deflection let these strings make 
angle θ1 and θ2 with the horizontal. 
 
Let the forces acting on mass m1, m2 and m3 in the vertical direction be P1, P2 and 
P3 simultaneously.  
 
The force along the string 1 and string 2 after the displacement be F. 
 
Now using the force equilibrium  
 
0=∑ yF  
 
Now for mass m1 using the equilibrium we have 
 
111
2sin1sin1
KPagainand
FFP
=
+= θθ
 
 
Now we can say that  
 
2
1
2
12
)11(
1
1
1
11
ll
xSin
Similarly
xBecause
ll
xSin
==
===
θ
θ
 
So,

• • Multi-1)0F Sysit/Page 18 Example: Compare the Stiffness and Flexibility Matricies formed by using the unit-displacement and unit-load techniques for the following system. First, find Stiffness...

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