Tutorial 1: Free Vibration of Two Degree-of-Freedom Systems
1. A two-storey shear structure consists of two slabs of masses m1=3000kg and m2=2000kg,
supported by light columns of total shear stiffness k1=40x106N/m and k2=30x106N/m.
In many engineering systems there are forces that result in energy dissipation (ie nonconservative forces). Lagranges equation allows us to model such systems by adapting the
If we start with Newtons Law:
2-DOF SYSTEM EXAMPLE FOR LAGRANGIAN EQUATION
Qu:- Find the equations of motion. The generalized coordinates are (x,).
Kinetic Energy of the system is given as:
The resultant velocity of pendulum bob, is given as,
From the diagram it is given as
EXAMPLE OF 2 DoF PRINCIPAL CO-ORDINATES
Question:- Find the principal co-ordinates for 2 DoF system shown below.
Solution:- The free body diagram is shown as below
The equations of motion are therefore, given as:
Now we are required to decouple these equa
Example of 2-DoF FORCED VIBRATION
Question:- The system consists of two rigid bodies, joined by three light springs. There is
an excitation force applied to the first rigid body.
Solution:- The free body diagram is given as
The equation of motions are the
MULTI DoF LUMPED MASS SYSTEMS
Finding the Natural Frequencies for 2 DoF Systems
The natural frequencies of the system are the frequencies of the system at which it will
oscillate having been set into motion without any applied force.
Considering an undamp
EXAMPLE: FINDING ORTHOGONALITY CONDITIONS
Question:- Find the normal modes for the system shown and demonstrate that they satisfy the
Check for orthogonality with respect to mass matrix
Check for orthogonality with res
1 DoF with Forced Harmonic Excitation
The equation for motion of dynamic system is = .
From the free body diagram equation of motion can be written as
The steady state solution can be calculated by setting particular integral as
By, substituting this solu
Tutorial 2: Forced Harmonic Vibration of 2 DoF Systems (including DVAs)
For the electronic instrument described in Tutorial 1 Question 3, a harmonically
varying force of amplitude 0.2N at a frequency of 30Hz is applied to the power unit.
Normal Mode Analysis
In a two storey shear structure, each slab has a mass of 40000kg and the
stiffness of each set of columns joining the slabs is 30x106N/m. If the ratios of
the amplitude of the upper slab to the lower slab ar
For the system shown, derive the equations of motion by Lagrange's method and write
down the mass, stiffness and damping matrices.
(Ans: [ M ] = 0
k1 + k2
, K = k
Vibration 4 - Chapter One Quiz
1. Youre about to complete an experimental modal analysis, to find a transfer function which will
The relationship between the excitation and response of a structure
The technique used to extract freque