Dynamics Flashcards
What is net force on an element?
The sum of internal and external forces
How is the speed of p-wave propagation, cp, found?
cp = sqrt(E/rhotilde)
What are the functions F & G which sum to the displacement u?
Functions of forward and backward wave movement respectively
How is the speed of shear wave propagation, cs, found?
cs = sqrt(G/rhotilde)
How do p-waves and shear waves differ in their propagation?
In p-waves, particles move in parallel with the direction of wave movement, whereas in shear waves the particles move perpendicularly to wave propagation
How do the speeds of p-waves and shear waves compare in the same medium?
P-waves are faster
In statics, d/dx(AE du/dx) + b = 0. What is the equivalent equation for dynamics?
d/dx(AE du/dx) + b = (A rhotilde) u’’
What is different about the displacement function u in dynamics when compared to statics?
It is a function of x and t, not just x
How is rhotilde different from rho?
Rhotilde is mass/volume, rho is mass/length
What is the standard mass matrix for a single element?
M = (rho le) [2 1]
/6 [1 2]
What is the standard mass matrix for a two element system?
M = (rho le) [4 2 -1]
/30 [2 16 2]
[-1 2 4]
How do you find the undamped force matrix F?
F = Md’’ + Kd, where M as the mass matrix, K is the stiffness matrix and d is the displacement vector
How can you integrate over a time domain?
You have to discretise the increments of time
What is the order of accuracy?
The reduction in the time interval required to increase the accuracy of results by a specified amount
If we rename d’ to v, what is v’?
v’ = 1/M(f-(v-Kd))
What are the most important values of correction factors β & γ?
β = 0 means the solution is explicit
γ = 1/2 means the solution has 2nd order accuracy
When is mass lumping used?
When the consistent mass matrix is not diagonal
How is mass lumping performed?
All of the values in a given row or column are summed onto the leading diagonal term, while the rest of the matrix values become zero
When is stability unconditional?
2β >= γ >= 1/2
When is stability conditional?
γ >= 1/2 and β < γ/2
What is the stability condition in terms of natural frequency?
ωmaxΔt <= Ωcrit
What is the equation for an eigenvalue?
λ = ω^2
How is the eigenfunction, Utilde(x), related to the displacement function?
u(x,t) = Utilde(x) sin ω(t - t0)
What is the number of solutions of a finite element model, neq, equal to?
neq is equal to the number of degrees of freedom in the finite element model, DOF per node * No. of nodes
What is dtilde, the common prefactor for the Ke and Me matrices with 2DoF?
dtilde = 6Ee/(rhotilde * le^2)
What are the minimum and maximum values of the eigenvalue, λe, when the determinant of [Ke - λeMe] = 0?
λ = 0 or λ = 2dtilde
From the eigenvalue equations, what is the time taken for an elastic wave to traverse an element?
le/c
What does λ = ω = 0 signify?
A rigid body motion is occuring
How do the number of zero eigenvalues relate to the number of degrees of freedom?
The number of zero eigenvalues is half the number of degrees of freedom
How does the critical time step differ based on the consistency of the mass matrix?
Critical time steps for lumped mass matrices are typically larger
What is the equation for maximum natural frequency for a lumped mass matrix?
ωmax = 2c/le
What is the general aim of damping?
To remove high frequency modes
How does changing the correction factors affect damping?
Making γ > 1/2 increases damping at the cost of accuracy
What is the equation for Rayleigh Damping?
C = aM + bK
How do you begin the weak form derivation for Timoshenko beam theory?
Test (multiply and integrate) the force balance with δw and the moment balance with δβ
What are acceptable values of Poisson’s Ratio?
0 < ν < 0.5
Why are entries in the mass matrix sometimes negative when an object can’t have negative mass?
Quadratic and higher order shape functions can have negative values, meaning that the mass is not negative but it contributes negatively to the overall deflections
Why will different entries in a beam mass matrix have different units?
There are two degrees of freedom with different units combined in one matrix
When can a discretised damped system be solved without the need for matrix inversion?
What the mass and damping matrices are both diagonal
How do you solve for the eigenvalue of a matrix problem?
det[ K - λM ] = 0 or det[ MinvK - λI] = 0
Which eigenvalue is used for calculating critical time step?
The highest value
What is Ωcrit for a central difference scheme?
Ωcrit = 2
What is the damping ratio for Rayleigh damping?
ξ = (a + bωn^2)/2ωn
What is the value of the damping ratio at critical damping?
ξ = 1
At what type of boundary will a wave invert as it is reflected?
Free boundary
What is the difference between viscous and algorithmic damping?
Viscous damping is proportional to velocity and represents physical energy losses such as friction
Algorithmic damping is a feature of the time integration scheme and not associated with a physical mechanism
