Rayleigh-Ritz method
Encyclopedia
In applied mathematics
and mechanical engineering
, the Rayleigh–Ritz method (after Walther Ritz and Lord Rayleigh) is a widely used, classical method for the calculation of the natural vibration
frequency
of a structure in the second or higher order. It is a direct variational method in which the minimum of a functional defined on a normed linear space is approximated by a linear combination of elements from that space. This method will yield solutions when an analytical form for the true solution may be intractable.
The method is also widely used in quantum chemistry
.
Typically in mechanical engineering
it is used for finding the approximate real resonant frequencies of multi degree of freedom
systems, such as spring mass systems or flywheel
s on a shaft with varying cross section
. It is an extension of Rayleigh's method. It can also be used for finding buckling loads for columns.
The following discussion uses the simplest case, where the system has two lumped springs and two lumped masses, and only two mode shapes are assumed. Hence M = [m1, m2] and K = [k1, k2].
A mode shape is assumed for the system, with two terms, one of which is weighted by a factor B, eg Y = [1, 1] + B[1, −1].
Simple harmonic motion
theory says that the velocity
at the time when deflection is zero, is the angular frequency
times the deflection (y) at time of maximum deflection. In this example the kinetic energy
(KE) for each mass is
etc, and the potential energy
(PE) for each spring
is
etc. For continuous systems the expressions are more complex.
We also know, since no damping
is assumed, that KE when y=0 equals the PE when v=0 for the whole system. As there is no damping all locations reach v=0 simultaneously.
so, since KE = PE,
Note that the overall amplitude of the mode shape cancels out from each side, always. That is, the actual size of the assumed deflection does not matter, just the mode shape.
A bit of mathematical skulduggery then reveals a solution for
, in terms of B. Then differentiate
with respect to B, and find the minimum, i.e. when 
. This gives the value of B for which 
is lowest. This is an upper bound solution for 
if 
is hoped to be the predicted fundamental frequency of the system because the mode shape is assumed, but we have found the lowest value of that upper bound, given our assumptions, because B is used to find the optimal 'mix' of the two assumed mode shape functions.
There are many tricks with this method, the most important is to try and choose realistic assumed mode shapes. For example in the case of beam deflection problems it is wise to use a deformed shape that is analytically similar to the expected solution. A quartic
may fit most of the easy problems of simply linked beams even if the order of the deformed solution may be lower. The springs and masses do not have to be discrete, they can be continuous (or a mixture), and this method can be easily used in a spreadsheet
to find the natural frequencies of quite complex distributed systems, if you can describe the distributed KE and PE terms easily, or else break the continuous elements up into discrete parts.
This method could be used iteratively, adding additional mode shapes to the previous best solution, or you can build up a long expression with many Bs and many mode shapes, and then differentiate them partially.
Applied mathematics
Applied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge...
and mechanical engineering
Mechanical engineering
Mechanical engineering is a discipline of engineering that applies the principles of physics and materials science for analysis, design, manufacturing, and maintenance of mechanical systems. It is the branch of engineering that involves the production and usage of heat and mechanical power for the...
, the Rayleigh–Ritz method (after Walther Ritz and Lord Rayleigh) is a widely used, classical method for the calculation of the natural vibration
Oscillation
Oscillation is the repetitive variation, typically in time, of some measure about a central value or between two or more different states. Familiar examples include a swinging pendulum and AC power. The term vibration is sometimes used more narrowly to mean a mechanical oscillation but sometimes...
frequency
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...
of a structure in the second or higher order. It is a direct variational method in which the minimum of a functional defined on a normed linear space is approximated by a linear combination of elements from that space. This method will yield solutions when an analytical form for the true solution may be intractable.
The method is also widely used in quantum chemistry
Quantum chemistry
Quantum chemistry is a branch of chemistry whose primary focus is the application of quantum mechanics in physical models and experiments of chemical systems...
.
Typically in mechanical engineering
Mechanical engineering
Mechanical engineering is a discipline of engineering that applies the principles of physics and materials science for analysis, design, manufacturing, and maintenance of mechanical systems. It is the branch of engineering that involves the production and usage of heat and mechanical power for the...
it is used for finding the approximate real resonant frequencies of multi degree of freedom
Degrees of freedom (physics and chemistry)
A degree of freedom is an independent physical parameter, often called a dimension, in the formal description of the state of a physical system...
systems, such as spring mass systems or flywheel
Flywheel
A flywheel is a rotating mechanical device that is used to store rotational energy. Flywheels have a significant moment of inertia, and thus resist changes in rotational speed. The amount of energy stored in a flywheel is proportional to the square of its rotational speed...
s on a shaft with varying cross section
Cross section (geometry)
In geometry, a cross-section is the intersection of a figure in 2-dimensional space with a line, or of a body in 3-dimensional space with a plane, etc...
. It is an extension of Rayleigh's method. It can also be used for finding buckling loads for columns.
The following discussion uses the simplest case, where the system has two lumped springs and two lumped masses, and only two mode shapes are assumed. Hence M = [m1, m2] and K = [k1, k2].
A mode shape is assumed for the system, with two terms, one of which is weighted by a factor B, eg Y = [1, 1] + B[1, −1].
Simple harmonic motion
Simple harmonic motion
Simple harmonic motion can serve as a mathematical model of a variety of motions, such as the oscillation of a spring. Additionally, other phenomena can be approximated by simple harmonic motion, including the motion of a simple pendulum and molecular vibration....
theory says that the velocity
Velocity
In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...
at the time when deflection is zero, is the angular frequency
Angular frequency
In physics, angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...
times the deflection (y) at time of maximum deflection. In this example the kinetic energyKinetic energy
The kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...
(KE) for each mass is
etc, and the potential energyPotential energy
In physics, potential energy is the energy stored in a body or in a system due to its position in a force field or due to its configuration. The SI unit of measure for energy and work is the Joule...
(PE) for each spring
Spring (device)
A spring is an elastic object used to store mechanical energy. Springs are usually made out of spring steel. Small springs can be wound from pre-hardened stock, while larger ones are made from annealed steel and hardened after fabrication...
is
etc. For continuous systems the expressions are more complex.We also know, since no damping
Damping
In physics, damping is any effect that tends to reduce the amplitude of oscillations in an oscillatory system, particularly the harmonic oscillator.In mechanics, friction is one such damping effect...
is assumed, that KE when y=0 equals the PE when v=0 for the whole system. As there is no damping all locations reach v=0 simultaneously.
so, since KE = PE,
Note that the overall amplitude of the mode shape cancels out from each side, always. That is, the actual size of the assumed deflection does not matter, just the mode shape.
A bit of mathematical skulduggery then reveals a solution for
, in terms of B. Then differentiateDerivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
with respect to B, and find the minimum, i.e. when . This gives the value of B for which is lowest. This is an upper bound solution for if is hoped to be the predicted fundamental frequency of the system because the mode shape is assumed, but we have found the lowest value of that upper bound, given our assumptions, because B is used to find the optimal 'mix' of the two assumed mode shape functions.There are many tricks with this method, the most important is to try and choose realistic assumed mode shapes. For example in the case of beam deflection problems it is wise to use a deformed shape that is analytically similar to the expected solution. A quartic
Quartic function
In mathematics, a quartic function, or equation of the fourth degree, is a function of the formf=ax^4+bx^3+cx^2+dx+e \,where a is nonzero; or in other words, a polynomial of degree four...
may fit most of the easy problems of simply linked beams even if the order of the deformed solution may be lower. The springs and masses do not have to be discrete, they can be continuous (or a mixture), and this method can be easily used in a spreadsheet
Spreadsheet
A spreadsheet is a computer application that simulates a paper accounting worksheet. It displays multiple cells usually in a two-dimensional matrix or grid consisting of rows and columns. Each cell contains alphanumeric text, numeric values or formulas...
to find the natural frequencies of quite complex distributed systems, if you can describe the distributed KE and PE terms easily, or else break the continuous elements up into discrete parts.
This method could be used iteratively, adding additional mode shapes to the previous best solution, or you can build up a long expression with many Bs and many mode shapes, and then differentiate them partially.
External links
- http://www.math.nps.navy.mil/~bneta/4311.pdf - Course on Calculus of Variations, has a section on Rayleigh–Ritz method.