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Advanced Structural Analysis 5 > Dynamics > Flashcards

Dynamics Flashcards

1
Q

What is net force on an element?

A

The sum of internal and external forces

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2
Q

How is the speed of p-wave propagation, cp, found?

A

cp = sqrt(E/rhotilde)

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3
Q

What are the functions F & G which sum to the displacement u?

A

Functions of forward and backward wave movement respectively

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4
Q

How is the speed of shear wave propagation, cs, found?

A

cs = sqrt(G/rhotilde)

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5
Q

How do p-waves and shear waves differ in their propagation?

A

In p-waves, particles move in parallel with the direction of wave movement, whereas in shear waves the particles move perpendicularly to wave propagation

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6
Q

How do the speeds of p-waves and shear waves compare in the same medium?

A

P-waves are faster

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7
Q

In statics, d/dx(AE du/dx) + b = 0. What is the equivalent equation for dynamics?

A

d/dx(AE du/dx) + b = (A rhotilde) u’’

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8
Q

What is different about the displacement function u in dynamics when compared to statics?

A

It is a function of x and t, not just x

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9
Q

How is rhotilde different from rho?

A

Rhotilde is mass/volume, rho is mass/length

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10
Q

What is the standard mass matrix for a single element?

A

M = (rho le) [2 1]
/6 [1 2]

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11
Q

What is the standard mass matrix for a two element system?

A

M = (rho le) [4 2 -1]
/30 [2 16 2]
[-1 2 4]

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12
Q

How do you find the undamped force matrix F?

A

F = Md’’ + Kd, where M as the mass matrix, K is the stiffness matrix and d is the displacement vector

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13
Q

How can you integrate over a time domain?

A

You have to discretise the increments of time

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14
Q

What is the order of accuracy?

A

The reduction in the time interval required to increase the accuracy of results by a specified amount

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15
Q

If we rename d’ to v, what is v’?

A

v’ = 1/M(f-(v-Kd))

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16
Q

What are the most important values of correction factors β & γ?

A

β = 0 means the solution is explicit
γ = 1/2 means the solution has 2nd order accuracy

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17
Q

When is mass lumping used?

A

When the consistent mass matrix is not diagonal

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18
Q

How is mass lumping performed?

A

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

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19
Q

When is stability unconditional?

A

2β >= γ >= 1/2

20
Q

When is stability conditional?

A

γ >= 1/2 and β < γ/2

21
Q

What is the stability condition in terms of natural frequency?

A

ωmaxΔt <= Ωcrit

22
Q

What is the equation for an eigenvalue?

A

λ = ω^2

23
Q

How is the eigenfunction, Utilde(x), related to the displacement function?

A

u(x,t) = Utilde(x) sin ω(t - t0)

24
Q

What is the number of solutions of a finite element model, neq, equal to?

A

neq is equal to the number of degrees of freedom in the finite element model, DOF per node * No. of nodes

25
Q

What is dtilde, the common prefactor for the Ke and Me matrices with 2DoF?

A

dtilde = 6Ee/(rhotilde * le^2)

26
Q

What are the minimum and maximum values of the eigenvalue, λe, when the determinant of [Ke - λeMe] = 0?

A

λ = 0 or λ = 2dtilde

27
Q

From the eigenvalue equations, what is the time taken for an elastic wave to traverse an element?

A

le/c

28
Q

What does λ = ω = 0 signify?

A

A rigid body motion is occuring

29
Q

How do the number of zero eigenvalues relate to the number of degrees of freedom?

A

The number of zero eigenvalues is half the number of degrees of freedom

30
Q

How does the critical time step differ based on the consistency of the mass matrix?

A

Critical time steps for lumped mass matrices are typically larger

31
Q

What is the equation for maximum natural frequency for a lumped mass matrix?

A

ωmax = 2c/le

32
Q

What is the general aim of damping?

A

To remove high frequency modes

33
Q

How does changing the correction factors affect damping?

A

Making γ > 1/2 increases damping at the cost of accuracy

34
Q

What is the equation for Rayleigh Damping?

A

C = aM + bK

35
Q

How do you begin the weak form derivation for Timoshenko beam theory?

A

Test (multiply and integrate) the force balance with δw and the moment balance with δβ

36
Q

What are acceptable values of Poisson’s Ratio?

A

0 < ν < 0.5

37
Q

Why are entries in the mass matrix sometimes negative when an object can’t have negative mass?

A

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

38
Q

Why will different entries in a beam mass matrix have different units?

A

There are two degrees of freedom with different units combined in one matrix

39
Q

When can a discretised damped system be solved without the need for matrix inversion?

A

What the mass and damping matrices are both diagonal

40
Q

How do you solve for the eigenvalue of a matrix problem?

A

det[ K - λM ] = 0 or det[ MinvK - λI] = 0

41
Q

Which eigenvalue is used for calculating critical time step?

A

The highest value

42
Q

What is Ωcrit for a central difference scheme?

A

Ωcrit = 2

43
Q

What is the damping ratio for Rayleigh damping?

A

ξ = (a + bωn^2)/2ωn

44
Q

What is the value of the damping ratio at critical damping?

A

ξ = 1

45
Q

At what type of boundary will a wave invert as it is reflected?

A

Free boundary

46
Q

What is the difference between viscous and algorithmic damping?

A

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

Advanced Structural Analysis 5 (2 decks)

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