Dynamics Primer Lectures
Dermot O’Dwyer
Objectives
• Need some theory to inderstand general
dynamics
• Need more theory understand the
implementation of some of the analytical
approaches
• Theory can answer some fundamental
questions
• Try to make the presentation very applied
Topics
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Simple Harmonic Motion
Multi-degree of freedom systems
Free vibration and eigen values and vectors
Modal Analysis
Time domain analyses
Frequency domain
Simple Harmonic Motion - I
• Typical of many
structural systems
• Force is linear
function of
displacement
Therefore,
Simple Harmonic Motion - II
Which has the form
Where
Simple Harmonic Motion - III
Simple Harmonic Motion - IV
The basic homogeneous undamped differential equation
Has the following closed form solution
Where
is the initial displacement and
Is the initial velocity
Multi-Degree of Freedom Systems
Topic II
Multi-Degree of Freedom Systems - I
• Most dynamic systems require more than a single
degree of freedom to describe them
• The general equation that describes the motion of
a multi-degreee of freedom system is
Multi-Degree of Freedom Systems - II
Consider the term within the fundamental
differential equation
- Mass Matrix
- Forcing function
- Stiffness Matrix
- Displacment vector
- Damping Matrix
Multi-Degree of Freedom Systems - III
Homogeneous equation – free response
•Calculate free response to an impulse
•Calculate natural frequencies
•Calculate eigen vectors
Natural Response
Topic III
Free Vibration I
• The response of a dynamic system in the absence
of external forces is very important because the
free response of a system gives very useful
information about the system’s characteristics
– Natural Frequencies & Resonance
– Eigen Vectors – modal analysis
Free Vibration II
• Finding the Natural Frequencies of a dynamic
system
• Solve the undamped equation
This equates to finding the values for omega
for which the determinate of the above equation is zero
Resonance
Free Vibration III
• Once the natural frequencies of the system
have been calculated, the eigen values can
be found by substituting the known values
of omega back into the equation
Eigen Vectors - 3 DOF
Modal Analysis
Topic IV
Orthogonality of Eigen Vectors
• The Eigen vectors of a system are orthogonal with
respect to the stiffness and mass matrices
• Where
Modal Analysis - I
• The displaced shape of a system can be described in terms
of its Eigen vectors (natrual modes)
Thus,
where
Modal Analysis - II
• In a similar manner the dynamic response of a multi degree of
freedom system can be represented in terms of the mode shapes
Advantages
- Significant problem reduction
- Identify potential resonance
- Calculate peak effects
Disadvantages
- None unless modes are removed
Time Stepping
Topic V
Calculating the forced response
of a dynamic system
• There are a number of techniques
– In limited cases a closed form solution is
available
• Fryba’s solution
• Response of a single degree of freedom system to a
harmonic forcing function
– Numerical Integration schemes
• Time-stepping algorithms
• Du Hammel’s Integral
Time Stepping - I
• If the position, velocity and acceleration of
a dynamic system is known at t, then the
state of the system an instant later, at
,
can be calculated
Time Stepping - II
• In the general multi-degree-of-freedom case
the fundamental equation can be rearranged
into the following form
Time Stepping - III
• Need figure of acceleration versus
• Note potential problems i.e. the difficulty with
long time intervals
• Advantages include ability to deal with non-linear
systems
• There are numerous time-stepping algorithms
– Runge Kutta
– Wilson Theta
– Newmark Beta
• Stable non-stable
Runge-Kutta - I
• The Runge-Kutta algorithm proceeds by
introducing a dummy variable
• If the value of xi and yi are known then a
Taylor’s expansion can be used to calculate their
values a short time later
Runge-Kutta - II
In practice the Taylor series is truncated after the first term
And an aveage value for the first term is used, thus
The average values are calculate using Simpson’s rule
Runge-Kutta – III
Integration table
Potential Time-Stepping
Problems
• Duration of the time-step
– Should be less than one tenth of period
• Enormous quantities of Data