Quiz 2 Flashcards
(41 cards)
What is the process of form finding and how does it differ from traditional design analysis?
Form finding is when you are given a problem with boundary conditions and you give the connectivity first. You prescribe how you want the internal forces to be and then you find the unknown geometry afterwards.
This is the direct inverse of design analysis.
What were some disadvantages of physical form-finding?
- They required meticulous care in both the set-up and in taking measurements so that the data can be used in real buildings
- The types of structures that could be feasibly modelled and their complexity was very restricted.
How does the Force Density Method work?
It seeks the equilibrium of a pin-jointed network with fixed boundary nodes.
If there is a 3 bar systems with 4 nodes (1 free node and 3 fixed nodes, What does m, n, ni, nf equal?
m = bars = 3
n = nodes = 4
ni = free nodes = 1
nf = fixed nodes = 3
In this equation to find xi:
Dixi + Dfxf + px = 0
What do all the values mean?
Di = CiT * Q * Ci
CiT = transpose of the sub-matrix of the connectivity matrix for the free nodes
Q = the diagonal matrix of q which is the force density of a bar which = Fi / Li
Xi = unknown internal coordinates
Df = CiT * Q * Cf
Cf = the sub-matrix of the connectivity matrix for the fixed nodes
Xf = Known fixed boundary points
Px = External load
How can you fix the problem with the FDM in hypar form-finding when the geometry pulls in too much?
When all the force densities are constant the form pulls itself in too much. To fix this increase the size of (tighten) the force densities at the edges.
How can we improve FDM even more in compression-tension-only structures?
We can be more intelligent in how we distribute the force densities. For example, structural weight-optimised force density or even better member length-optimised solutions which use the minimum amount of material or have more uniform member lengths.
How can we used the data produced in FDM to determine the material required?
The FDM provides us with the length and forces within the elements. We can use this to find the best section and we can decide what material we use. For an elastic material the area of the material is:
Ai = Fi / fy where fy is the yield stress for the material which should be factored in practice.
The lengths provided by FDM are the stressed lengths. To find the unstressed lengths:
L0,i = Li / ( Fi/(E*Ai) + 1 )
What is Dynamic Relaxation (DR) and what is it used for?
DR is a pseudo-dynamic analysis where we find the rest state after oscillations have damped. This is used for form-finding and analysis of membrane structures, Ties/nets.
Using the DR method how is the next displacement (x(t+Δt) ) found?
x(t+Δt) = xt + Δt * vx(t+Δt)
x = displacement
v = velocity
vx(t+Δt) = vt + Δt * rx/m
rx = residule force = px - k*x
px = external force
K = stiffness
m = nodal mass
How does kinetic damping for in DR?
It is effective to dampen a dynamic system at peak kinetic energy. To do this the system’s kinetic energy is tracked and nodal velocities are set to 0 at a local kinetic energy peak. Kinetic energy declines with time as energy are extracted form the system. It approaches the final rest state with no kinetic energy and no residual forces.
How to calculate residual forces for all nodes after a system has finished oscillating
rx = px - CT * Q * C*x
Px = external force
CT = transpose of the connectivity matrix
Q = Diagonal matrix of element force densities
C = Connectivity matrix
x = coordinates of nodes
Why are mass-spring models used in engineering?
To simplify member stiffness is static/dynamic analysis
For particle springs what does velocity and acceleration equal?
V = dx/dt
a = dv/dt = 1 / m * (p - cv - Σki(Li - Li,0))
m = partical mass
p = applied load
c = drag coefficient
v = velocity
ki = stiffness
Li = current length
Li,0 = Rest length
How is Euclidean norm calculated in 3D?
L = II b-a II = √( (xb - xa)^2 + (yb - ya)^2 + (zb - za)^2)
How is the dot product calculated in 3D and how is this used to find then angle between two vectors?
ab = xaxb + yayb + zazb
ab = IIaII * IIbII * cosθ
So,
θ = arccos( (ab) / (IIaII * IIbII) )
How is the cross-product found in 3D?
a x b = I i,j,k ; xa,ya,za ; xb,yb,zb I
Which equals
(yazb - zayb)i - (xazb - zaxb)j + (xayb - yaxb)k
What are the uses of vector length, Dot Product and Cross-product?
Vector length find the length of a vector
The dot product finds the angle between to vectors
Cross-product Returns a vector that is perpendicular to the original vector. It can be used as a IIaxbII to find the area of a parallelogram.
How does translation work?
You add a vector to a point/vertex and this will translate the point by the translation vector.
How does scaling work and what are the different ways you can do it?
Scaling is when you multiply a point by a scaler
(c). There is uniform scaling about the origin or a defined point such as an object’s centroid. You can also non uniformly scale a point or objects by using cx, cy and cz
To scale point (x,y,z) by scale factors of cx, cy and cz what would the matrix equation look like?
[x’ ; y’ ; z’ ; 1] = [ cx,0,0,0 ; 0,cy,0,0 ; 0,0,cz,0 ; 0,0,0,1] [x ; y ; z ; 1]
How would you rotate point (x,y) through an angle θ to get (x’,y’)
[ x’ ; y’ ; 1] = [ cosθ,-sinθ,0 ; sinθ,cosθ,0 ; 0,0,1] * [ x ; y ;1]
How do you rotate in 3D and what are Rx, Ry, Rz?
[x’ ; y’ ; z’] = Ri * [x ; y ; z]
Rx = [ 1,0,0 ; 0,cosα,-sinα ; 0,sinα,cosα]
Ry = [ cosβ,0,sinβ ; 0,1,0 ; -sinβ,0,cosβ]
Rz = [ cosγ,-sinγ,0 ; sinγ,cosγ,0 ; 0,0,1]
What are the types of arrays and how do they work?
1) Rectangular arrays repeat geometry in the x and y direction by some factor and count
2) Box arrays repeat geometry in the x, y and z direction by some factor and count
3) Polar arrays provide a geometry, a plane for rotation, and the count and angle to rotate through.
