SD Prelim Review

Question 1

Define the inertia force.

Answer 1

F_I = - m\ddot{x}

Jan 16 Lecture

Question 2

What is dynamic equilibrium?

Answer 2

Dynamic equilibrium occurs when the sum of forces and moments acting on a structure is zero, taking into account both the applied external forces and the internal forces (including the inertia forces). In a dynamically loaded system, this equilibrium is expressed as:

SumofAppliedForces=SumofInertiaForces

Question 3

Describe D’Alembert’s principle.

Answer 3

D’Alembert’s principle is a reformulation of Newton’s second law, which allows dynamic problems to be analyzed as if they were static. According to this principle, the inertia force can be treated as an “added force” to the system, allowing us to consider a dynamic problem as a static equilibrium problem by introducing this fictitious force.

D’Alembert’s force is the inertia force, but with a negative sign, making it appear on the same side as the applied forces in the equilibrium equation. This transforms the dynamic equation of motion into a form that resembles static equilibrium.
F(t) - m\ddot{x} = 0

Question 4

What is the natural frequency?

Answer 4

The natural frequency is the frequency at which a system naturally vibrates once it has been disturbed from its equilibrium position and then allowed to oscillate freely.

wn = \sqrt{k/m}

Question 5

Define a degree of freedom.

Answer 5

The degrees of freedom are the number of displacements that must be considered to represent the effects of all significant inertia forces.

Question 6

The damping ratio is the ratio of what two properties?

Answer 6

The damping ratio c and the critical damping ratio c_{cr} = 2m\omega. it is a measure of how much the systems oscillations are reduced by damping.

Question 7

Describe the damped natural frequency.

Answer 7

The damped vibrational frequency, denoted as 𝜔𝑑, is the frequency at which a damped system oscillates when it is displaced from its equilibrium position and then released. It is different from the natural frequency 𝜔𝑛 of the system because the presence of damping reduces the frequency of oscillation.

wd = wn\sqrt{1 - \zeta^2}

Question 8

Describe the dynamic magnification factor.

Answer 8

The Dynamic Magnification Factor (DMF), also known as the Amplification Factor, is a measure of how much a system’s steady-state response to a sinusoidal force is amplified compared to the static response of the system under a constant load. It is particularly important in the context of resonant vibrations, where the response of the system can be significantly larger than the applied force due to resonance effects.

Question 9

Describe the frequency response function.

Answer 9

The Frequency Response Function (FRF) is a complex function that relates the output response of a system (such as displacement, velocity, or acceleration) to an input force as a function of frequency. It is defined as the ratio of the output response to the input force in the frequency domain.

Question 10

Describe the variational principle.

Answer 10

The variational principle states that the true path or state of a system is the one for which a certain quantity, called the functional, is stationary (usually a minimum). This principle is often used to derive equations of motion or to find the equilibrium state of a system.

Question 11

Define virtual work.

Answer 11

Virtual work is the work that would be done by the forces on a system if the system were to undergo a small, hypothetical (virtual) displacement from its current equilibrium position.

The principle of virtual work states that if a mechanical system is in equilibrium, then the total virtual work done by all external forces (including applied loads and reactions) during any virtual displacement that is consistent with the system’s constraints is zero.

Question 12

In general, what are the two types of constraints? What is the difference between the two?

Answer 12

Holonomic and Non-Holonomic.

Holonomic constraints are constraints that can be expressed as an equation relating the coordinates of the system and, possibly, time.

Non-holonomic constraints are constraints that cannot be expressed purely as equations involving only the coordinates and time. Instead, they often involve inequalities or differential relationships that include velocities or higher-order derivatives.

Question 13

How does the principle of virtual work connect to dynamic equilibrium?

Answer 13

A system is in dynamic equilibrium at a given time 𝑡 if, for any permissible virtual displacement, the virtual work done by the effective forces (including both external forces and inertial forces) is zero.

Question 14

Schleronomic and Rheonomic constraints fall under what general category?

Answer 14

Holonomic

Question 15

What is the difference between Schleronomic and Rheonomic constraints?

Answer 15

Schleronomic - time is not present
Rheonomic - time is present

Question 16

True or False
A holonomic constraint reduces the DOF by 1.

Answer 16

True.

Question 17

True or False
A non-holonomic constraint will reduce the DOFs by 1.

Answer 17

False

Question 18

Define generalized coordinates.

Answer 18

Generalized coordinates are variables that describe the configuration of a system relative to some reference configuration. These coordinates are not restricted to standard Cartesian coordinates (e.g., 𝑥, 𝑦, 𝑧) but can include angles, distances, or any other parameters that uniquely define the system’s state.

The number of generalized coordinates corresponds to the number of degrees of freedom of the system. The degrees of freedom represent the minimum number of independent parameters needed to specify the configuration of the system completely.

Question 19

Describe Hamilton’s principle.

Answer 19

Hamilton’s principle states that the actual path taken by a system between two points in time is the one that makes the action 𝑆 stationary.

Hamilton’s principle is closely related to the principle of virtual work, particularly in dynamic systems. While the principle of virtual work states that the virtual work of effective forces must be zero for a system in equilibrium, Hamilton’s principle extends this idea to dynamic systems by considering the entire trajectory of the system over time.

Question 20

What is a phase space diagram?

Answer 20

The phase space is a mathematical space in which all possible states of a system are represented. For a system with n degrees of freedom, the phase space is 2𝑛-dimensional, with each state of the system being a point in this space.
The coordinates of the phase space are typically the generalized coordinates.

Question 21

What principle is Lagrange’s equation derived from?

Answer 21

Hamilton’s principle.

Question 22

What are some examples of internal forces?

Answer 22

1. Linking forces
2. Elastic Forces
3. Dissiption forces (in the case of non-conservative systems)

Question 23

True or False
Linking forces do not contribute to the generalized forces.

Answer 23

True.

Question 24

Dissipative forces are _______ and __________ in direction to the velocity vector.

Answer 24

Parallel
Opposite

Question 25

Describe the Rayleigh dissipation function.

Answer 25

The Rayleigh dissipation function 𝐹 is a scalar function of the generalized velocities \dot{q}_i that represents the rate at which mechanical energy is being dissipated in the system due to non-conservative forces.

It is defined such that the generalized non-conservative forces Q_i can be derived from it as:

Q_i = - \partial F / \partial \dot{q}_i

Such that

F = \frac{1}{2}\sum {i,j} c{ij} \dot{q}_i\dot{q}_j

Question 26

True or False
The Rayleigh dissipation function represents the rate at which energy is dissipated as heat or other forms of non-recoverable energy due to resistive forces acting in the system.

Answer 26

True

Question 27

What is the Rayleigh dissipation function for a linear damped harmonic oscillator?

Answer 27

F = (1/2) c\dot{x}^2

Question 28

What is dissipation power?

Answer 28

Dissipation power is the rate at which energy is dissipated by non-conservative forces in a system. It is the power associated with the work done by these forces as they oppose the motion of the system.

Question 29

What are the two general types of forces and how are they related to a potential function?

Answer 29

1. Conservative forces (obtainable from a potential function)
2. Non-conservative forces (not obtainable from a potential function)

Question 30

Describe the normal modes of vibration approach to solving MDOF systems.

Answer 30

The normal modes approach assumes that the system’s motion can be expressed as a combination of simple harmonic motions (sinusoidal functions) at distinct frequencies. Each of these harmonic motions corresponds to a normal mode of the system. The general solution for the system’s motion is a linear combination of all the normal modes.

Question 31

Describe the decoupling process when using the normal modes of vibration to solve an MDOF system.

Answer 31

By transforming the original coordinates to the modal coordinates (generalized coordinates corresponding to the normal modes), the system of coupled differential equations can be decoupled into 𝑛 independent equations, each corresponding to a single normal mode. These equations can then be solved and recombined to form the general solution in the original coordinates.

Each normal mode represents a specific pattern of motion at a distinct natural frequency, and the system’s overall motion is a combination of these modes.

Question 32

When using the normal modes of vibration (NMV) approach to solve an MDOF system, what do the eigenvalues provide?

Answer 32

The square of the natural frequencies of the system.

Question 33

Describe a normal mode of vibration.

Answer 33

A normal mode is a characteristic vibration pattern of a system in which all parts of the system move sinusoidally with the same frequency and in a fixed relationship to each other. This frequency is called a natural frequency, and it is unique to that particular mode.

Question 34

Describe a “mode shape.”

Answer 34

The mode shape is the specific pattern of displacement or motion that occurs in a normal mode. It describes how each part of the system moves relative to the others.

Extra Notes:
Any arbitrary motion of the system can be described as a superposition (a combination) of its normal modes. This means that the system’s response to any initial disturbance can be understood as a sum of the contributions from each normal mode.

Question 35

True or False
For a linear system, the normal modes are independent.

Answer 35

True.

This means that if the system is vibrating in one normal mode, the amplitude of that mode does not affect the amplitude of any other mode.

Question 36

What is a positive definite matrix?

Answer 36

A square matrix for which all the eigenvalues are positive.

Question 37

What is a rigid body mode, and how does it relate to the normal modes of vibration analysis?

Answer 37

Rigid body modes are associated with a natural frequency of zero because there is no restoring force acting to bring the system back to an equilibrium position after a rigid body motion. This is because such motions do not involve any deformation or strain energy. In a rigid body mode, since there is no deformation, there are no internal restoring forces (like those from springs or structural stiffness) trying to resist the motion.

For a system with n DOFs, if the system has free unconstrained movement in one direction, the normal modes will correspond to rigid body modes.

Question 38

True or False
The eigenvectors (mode shapes) are linearly dependent.

Answer 38

False.

Linear independence ensures that each eigenvector represents a unique mode shape of the system, with distinct patterns of motion. The set of all linearly independent eigenvectors forms a complete basis for the space of possible displacements of the system. This means any possible motion of the system can be expressed as a linear combination of the system’s normal modes (eigenvectors).

Question 39

What does linear independence of eigenvectors mean?

Answer 39

Linear independence of eigenvectors means that none of the eigenvectors (mode shapes) can be expressed as a linear combination of the others. This property ensures that each eigenvector represents a unique mode of vibration, contributing uniquely to the system’s overall motion.

Question 40

Describe the orthogonality properties of the mode shapes.

Answer 40

For a system with symmetric mass and stiffness matrices, the eigenvectors corresponding to different eigenvalues (natural frequencies) are orthogonal. This means that the inner product of any two distinct eigenvectors is zero when weighted by the mass or stiffness matrix.

X^T MX = 0
X^T KX = 0

The orthogonality of eigenvectors allows the equations of motion for an MDOF system to be decoupled. This decoupling is achieved by transforming the original coordinates (generalized displacements) into modal coordinates. In these new coordinates, each equation corresponds to a single mode of vibration and is independent of the others. This simplification is possible because, due to orthogonality, cross-terms involving different modes disappear when projecting the system’s dynamics onto the modal coordinates.

Question 41

Describe how Duhamel’s integral can be used to solve for q_n(t).

Answer 41

Duhamel’s integral is a method for determining the response of a linear, time-invariant MDOF system to arbitrary time-varying forces.

By integrating the system’s impulse response over time, it allows for the calculation of the displacement at each degree of freedom, accounting for the contributions from forces applied at all other degrees of freedom.

Extra Notes: The method leverages the principle of superposition and is particularly powerful when combined with modal analysis to simplify the computation of responses in complex systems.

Question 42

What is the Gram-Schmidt orthogonalization algorithm?

Answer 42

The Gram-Schmidt orthogonalization algorithm is a method used in linear algebra to take a set of linearly independent vectors and generate an orthogonal (or orthonormal) set of vectors that spans the same subspace.

Question 43

Describe the method of spectral expansion.

Answer 43

The method of spectral expansion in the context of Multi-Degree-of-Freedom (MDOF) systems is a technique used to solve the equations of motion by expressing the system’s response as a sum of modal contributions. It leverages the system’s natural modes of vibration (eigenvectors) and natural frequencies (eigenvalues).

Question 44

What is response truncation?

Answer 44

“Response truncation” refers to the practice of approximating the response of a Multi-Degree-of-Freedom (MDOF) system by considering only a subset of its modes, rather than the full set of modes obtained from modal analysis. In many practical cases, only a few modes (typically those with the lowest natural frequencies) contribute significantly to the system’s dynamic response, especially for low-frequency excitations. Higher-frequency modes might have minimal influence and can be neglected.

Question 45

How is response truncation related to the spatial factor?

Answer 45

The concept of the spatial factor is closely related to response truncation in the context of modal analysis for Multi-Degree-of-Freedom (MDOF) systems. The spatial factor, often referred to as the modal participation factor or modal participation coefficient, quantifies how much a particular mode contributes to the overall response of the system at a specific location or degree of freedom. When performing response truncation, modes with high spatial factors are typically retained because they contribute significantly to the system’s response at the location of interest.

Question 46

How is response truncation related to the temporal factor?

Answer 46

The temporal factor in the context of modal analysis is related to the time-dependent behavior of a system’s response, particularly how each mode evolves over time due to dynamic loading. While the spatial factor (or modal participation factor) quantifies the contribution of each mode to the response at a specific location, the temporal factor refers to how the amplitude of each mode changes over time.

Question 47

When considering damping, what two cases are considered where the nodal EOM is uncoupled?

Answer 47

1. Rayleigh Proportional Damping
2. Lightly Damped Scenarios

Question 48

Describe Rayleigh proportional damping.

Answer 48

The damping matrix, C, is approximated as
C = aK + bM where a and b are determined via experiment.

Question 49

Describe the “lightly damped” scenario.

Answer 49

When using the “lightly damped” approximation, the damping forces are assumed to be an order of magnitude less than inertial or elastic forces. n the lightly damped approximation, the equations of motion for the system can still be decoupled in modal coordinates. Each mode can be treated as a lightly damped single-degree-of-freedom (SDOF) system.

Question 50

True or False
In the lightly damped approximation, repeating natural frequencies are allowed.

Answer 50

False.

Question 51

What is the dynamic influence coefficient matrix?

Answer 51

The dynamic influence coefficient matrix is a matrix that characterizes the dynamic response of a Multi-Degree-of-Freedom (MDOF) system due to applied forces. It relates the displacements of the system’s degrees of freedom to the forces acting on the system, taking into account the system’s dynamic properties, such as mass, stiffness, and damping.

Question 52

Describe the state space approach to solving MDOF systems with damping.

Answer 52

The state-space approach involves converting the second-order differential equations into a set of first-order differential equations by introducing a state vector z(t), which includes both the displacements and velocities.

Question 53

With respect to the dynamics of continuous structures, what is a traction force?

Answer 53

Traction forces are the forces per unit area acting on a surface within a material or at its boundary. They are directly related to the stress tensor and can be decomposed into normal and tangential components, representing normal and shear stresses, respectively.

Question 54

With respect to the dynamics of continuous structures, what is the stress tensor?

Answer 54

The stress tensor is a mathematical representation of the internal forces (stresses) that develop within a material or structure in response to external loads. It is a second-order tensor that describes the state of stress at a point within the material by providing the components of stress acting on various planes passing through that point. It is typically represented as a 3x3 matrix.

Question 55

With respect to the dynamics of continuous structures, describe the strain energy density function.

Answer 55

The strain energy density function (often abbreviated as SEDF) is a scalar function that describes the amount of strain energy stored in a material per unit volume due to deformation.

Strain energy is the energy stored in a material when it is deformed elastically. When an external force is applied to a material, it deforms, and the work done by the force is stored as strain energy within the material. If the material returns to its original shape when the force is removed, the strain energy can be fully recovered, which is characteristic of elastic deformation.

Question 56

What is the purpose of Lame’s constants?

Answer 56

Lamé’s constants are two material-specific parameters, denoted by
𝜆 (Lamé’s first constant) and 𝜇 (Lamé’s second constant or the shear modulus), used in the linear theory of elasticity. These constants are fundamental in describing the elastic behavior of isotropic materials and are often used in the constitutive equations that relate stress and strain in a material.

Question 57

What are the elastodynamic equations?

Answer 57

The elastodynamic equations are derived from Newton’s second law of motion combined with the constitutive laws of elasticity. They describe how stress, strain, and displacement evolve over time in an elastic medium.

Question 58

What is the purpose of the strain displacement matrix?

Answer 58

The strain-displacement matrix often denoted as the B-matrix plays a crucial role in the finite element method (FEM) and other numerical methods used in structural analysis and continuum mechanics. Its primary purpose is to relate the displacements of a structure or material (which are the primary unknowns in a typical finite element analysis) to the resulting strains within the element.

Question 59

Describe the Rayleigh-Ritz method.

Answer 59

The Rayleigh Ritz method approximates the solution over the entire domain using a series of global, continuous trial functions (basis functions) that satisfy the boundary conditions of the problem.

The basis functions (trial functions) used in the Rayleigh-Ritz method are chosen to be globally defined over the entire domain. These functions are often selected based on physical intuition, experience, or the nature of the problem.

Question 60

Describe the Finite Element Method.

Answer 60

FEM involves discretizing the domain into smaller, simpler subdomains called elements. Within each element, the solution is approximated using shape functions (basis functions) that are typically low-order polynomials and are defined locally within each element. The overall solution is assembled from the contributions of all elements, leading to a piecewise continuous solution over the entire domain.

In FEM, the basis functions are typically low-order polynomials (e.g., linear or quadratic) and are defined locally within each element. These functions are simple and easy to compute but are only continuous within each element.

FEM typically requires the solution of a large system of algebraic equations due to the discretization into many elements. The number of equations grows with the number of elements and degrees of freedom in the mesh.

Question 61

What is the potential energy equation for a linear spring?

Answer 61

v = (1/2) kx^2

Question 62

Express the damping force in terms of the generalized coordinates.

Answer 62

F_x = - c\dot{q}