Response to Harmonic Excitation

Resonance occurs when the external excitation Ωmax has the same frequency as the natural frequency ω0 of the system. It leads to large displacements and can cause a system to exceed its elastic range and fail structurally. A popular example, that many people are familiar with, is that of a singer breaking a glass by singing.

Thus, the lowest frequency of the element ω01 must be larger than the highest excitation frequency Ωmax.



What is the difference between axial stiffness and bending stiffness?

First you need to know about stiffenss which means resistance against deformation.

Axial stiffness: stiffness which resists axial force (force through axis of body)

Bending stiffness: stiffness which resists bending forces (shear forces)


What is a phasor?

A phasor is simply a shorthand way of representing a signal that is sinusoidal in time. Though it may seem difficult at first, it makes the mathematics involved in the analysis of systems with sinusoidal inputs much simpler. To start, we take a sinusoidal signal in time defined by magnitude, phase and frequency (A, θ and ω)


and represent it in phasor form as a complex number with a magnitude and phase (A, and θ). Note that frequency (ω) is not included, but is implicit in the concept of a phasor.


We frequently write the phasor in terms of its magnitude and phase.


If we multiply the phasor, F, by ejωt, we are simply rotating F by an angle ωt (i.e., the resulting vector has an angle of (ωt+θ).


We can recover the time domain function, f(t), by taking the real part of this rotating vector.


Important things to note

  • The variable “A” determines the length of F and the amplitude of f(t). Modify “A” with slider (or text box) to demonstrate.
  • The variable “θ” determines the angle of F and the phase of the time domain function. If θ=0, we get a cosine. Modify θ to see its effect on F and f(t). Note that θ is shown in degrees, but when you actually do calculations, you will want to use radians (ωt is in radians).
  • The variable “ω” doesn’t affect F at all. Increasing ω increases the frequency of f(t), and also increases the speed of rotation of Fejωt. Modify ω to see for yourself.
  • As “t” increases, Fejωt rotates in a counterclockwise direction. You can vary t with the textbox, the slider or by clicking and dragging in the time domain (rightmost) graph.
  • The phasor, F, is a constant. We multiply by ejωt to make a rotating vector, Fejωt (so we need to know the frequency, ω).
  • f(t)=Re{Fejωt}
Key Concept: A sinusoidal signal can be represented by a vector in the complex plane called a phasor

A sinusoidal signal f(t)=A·cos(ωt+θ) can be represented by a phasor F=Ae, which is a vector in the complex plane with length A, and an angle θ measured in the counterclockwise direction. If we multiply F by ejωt, we get a vector that rotates counterclockwise with a rotational velocity of ω. The real part of this vector is equal to f(t).


The phasor, F, is a constant. We need to know the frequency, ω, in order to make a rotating vector and to determine the time domain function f(t).


Basic Frame Analysis

Fixed constraint

Fixed Constraint removes all translational and rotational DOF of the selected node or beam. 

Pinned constraint

Pinned Constraint only removes all translational DOF of the selected node or beam. At roller and hinges, the rotation of the connected structure is not prevented. Therefore, the bending moment is zero as it does not resists rotation. While in fixed supports, the bending moment is nonzero as it resists rotation of the connected structure.

Floating pinned constraint

Floating Pinned Constraint restricts rotation and translation in one plane only for a selected node or beam.

Up and Running with Autodesk Inventor Simulation 2011 (2nd edition)
A step-by-step guide to engineering design solutions
2010, Pages 383-412

What is a catenary?

In physics and geometry, a catenary is the curve that an idealized hanging chain or cable assumes under its own weight when supported only at its ends. The catenary curve has a U-like shape, superficially similar in appearance to a parabolic arch, but it is not a parabola.


Symmetric matrix

In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally,

Because equal matrices have equal dimensions, only square matrices can be symmetric.

The entries of a symmetric matrix are symmetric with respect to the main diagonal. So if a_ij denotes the entry in the i-th row and j-th column then

for all indices i and j.

Every square diagonal matrix is symmetric, since all off-diagonal elements are zero.

Symmetry of a 5×5 matrix

What is modulus of elasticity dimension?

MoE is usually expressed as change in stress / change in strain.

Strain is said to be dimensionless, since it is change in length / original length. Therefore a unit for MoE is the same as a unit for stress. In SI-land, that unit is the pascal. and pascals are newtons / metre square. I.e. N m-2. 

The MoE is material dependent. For example, the MoE of steel is about 200 GPa (29,000,000 psi), and the MoE of concrete is around 30 GPa (4,350,000 psi). The MoE of aluminum is 69 GPa (10,000,000 psi).