Figure 3.3.1 shows a rigid bar, fixed at end 1and with an external force F2 applied to the
outside of end 2. For convenience, the system boundary has been shrunk to fit around the
physical shape of the bar but this isnt necessary. F2 is a contac
Damping is an influence within or upon an oscillatory system that has the effect of reducing,
restricting or preventing its oscillations. In physical systems, damping is produced by processes that
dissipate the energy stored in the oscillation. Ex
THE LIMITS TO MECHANICAL ENGINEERING
Engineering systems can be designed to produce a wide variety of outputs from given
there are fundamental limitations to their capabilities. These might be kinematic, i.e
limitations on movement, or t
THE ELEMENTS OF MECHANICAL SYSTEMS
In physics, mass is a property of a physical body which determines the strength of its
mutual gravitational attraction to other bodies and its resistance to being accelerated by a force.
The SI unit of mass is
1. Mechanical Engineering, Life, the Universe and Everything
Engineering is the profession charged by society with modifying the environment and
engineers should be aware of the social and environmental consequences of their work.
The all-pervading influe
ADVANCED THERMAL ENGINEERING
Heat transfer describes the exchange of thermal energy, between physical systems depending
on the temperature and pressure, by dissipating heat. The fundamental modes of heat transfer
are conduction or diffusion,
PRINCIPLES OF CONVECTION: AN INTRODUCTION TO
HYDRODYNAMIC AND THERMAL BOUNDARY LAYERS
Convective heat transfer coefficient
The heat transfer coefficient or film coefficient, in thermodynamics and in mechanics is
the proportionality coefficient between the
PRINCIPLES OF CONVECTION:
FUNDAMENTAL EQUATIONS OF CONVECTION TRANSFER IN
Conservation of mass
1. In a steady flow, the net rate in which mass enters the control volume (inflow-outflow)
must equal zero.
2. Applying this to a differential
Principles of convection
Similarities between transport of momentum and heat
The Cauchy momentum equation is a vector partial differential equation put forth
by Cauchy that describes the non-relativistic momentum transport in any continuum:
where is th
Flows over cylinders and spheres
The characteristic length for a circular tube or sphere is the external diameter, D, and the Reynolds
number is defined:
The critical Re for the flow across spheres or tubes is 2x105 . The approaching fluid to the cyl
FLOW OVER FLAT PLATES
Flows over objects are categorized as external flows. E.g. flow over a flat plat, flow over
a turbine blade, flow over a cylinder or sphere.
Numerous engineering applications such as those in:
Gas turbine blade cooling, shell and tub
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
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.