Mechanical-vibration-of-multiple.docx

MECHANICAL VIBRATION OF MULTIPLE DEGREES OF FREEDOM SYSTEMS

GROUP 2

I.

DEFINITIONS

1.

DEGREES OF FREEDOM - The number of degrees of freedom used to analyze a system is the number of kinematically independent coordinates necessary to describe the motion of every particle in the system.

2.

NATURAL FREQUENCY - There are n natural frequencies, each associated with its own mode shape, for a system having n degrees of freedom.

3.

MODE SHAPE - The mode shapes exhibit a property known as orthogonality, which can be utilized for the solution of undamped forced-vibration problems using a procedure known as modal analysis.

II.

MODELING OF MDOF

As we have studied in mechanical vibration on SDOF, most engineering systems are continuous and have an infinite number of degrees of freedom. The vibration analysis of continuous systems requires the solution of partial differential equations, which is quite difficult. For many partial differential equations, in fact, analytical solutions do not exist. The analysis of a multiple degree of- freedom system, on the other hand, requires the solution of a set of ordinary differential equations, which is relatively simple. Hence, for simplicity of analysis, continuous systems are often approximated as multiple degree-of-freedom systems. Different methods can be used to approximate a continuous system as a multiple degree-of freedom system. A simple method involves replacing the distributed mass or inertia of the system by a finite number of lumped masses or rigid bodies. The lumped masses are assumed to be connected by massless elastic and damping members. Linear (or angular) coordinates are used to describe the motion of the lumped masses (or rigid bodies). Such models are called lumped-parameter or lumped-mass or discrete-mass systems. The minimum number of coordinates necessary to describe the motion of the lumped masses and rigid bodies defines the number of degrees of freedom of the system. Naturally, the larger the number of lumped masses used in the model, the higher the accuracy of the resulting analysis.

(a) A SDOF system and its equation of motion. The assumption is of a lumped-parameter is exhibit at 1 point in the system.

(a) A model of a building structure. (b) The corresponding MDOF system with lumped parameters.

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General MDOF equation of motion: m[x’’(t)] + c[x’(t)] + k[x(t)] = F(t)

… (equation 1)

where, the bracketed portion indicates the matrix form. III.

UNDAMPED FREE VIBRATION ON MDOF SYSTEMS.

It was seen in Section II that the equations that govern discrete multi-degree of freedom systems take the general matrix form of Equation 1. We shall here consider the fundamental class of problems corresponding to undamped systems that are free from applied (external) forces. For this situation, Equation 1 reduces to the form: m[x’’] +k[x] = 0

… equation 2

Where, for an N degree of freedom system, m and k are the N × N mass and stiffness matrices of the system, respectively, and x is the corresponding N ×1 displacement matrix. To solve Eq. 2, we parallel the approach taken for solving the corresponding scalar problem for single degree of freedom systems. We thus assume a solution of the form x = Xeiωt

… equation 3

where X is a column matrix with N, as yet, unknown constants, and ω is an, as yet, unknown constant as well. The column matrix X may be considered to be the spatial distribution of the response while the exponential function is the time dependence. Based on our experience with single degree of freedom systems, we anticipate that the time dependence may be harmonic. We therefore assume solutions of the form of Eq. 3. If we find harmonic forms that satisfy the governing equations then such forms are, by definition, solutions to those equations. Substitution of Eq. 3into Eq. 2, and factoring common terms, results in the form [ k −ω2 m ] Xeiωt = 0

… equation 4.1

Assuming that ω ≥ 0 , we can divide through by the exponential term. This results in the equation [ k −ω2 m]X = 0

… equation 4.2

which may also be stated in the equivalent form, kU =ω2 mX

… equation 4.3

Natural Frequencies One obvious solution of Eq. 4.2 is the trivial solution X = 0. This corresponds to the equilibrium configuration of the system. Though this is clearly a solution corresponding to a physically realizable configuration, it is evidently uninteresting as far as vibrations are concerned. We are thus interested in physical configurations associated with nontrivial solutions. From linear algebra we know that a

matrix equation Ax = b may be row reduced. If the rows or columns of the matrix A are linearly independent (that is, no row can be expressed as a linear combination of the other rows) then the corresponding matrix equation can be reduced to diagonal form, and the solution for x can be read directly. DAMPED FREE VIBRATION ON MDOF SYSTEMS 1.

Derive the equation motion of system shown. Consider the last spring to be nonlinear where the spring force is given by

. Consider other spring and damper behaviour to be linear.

Free body diagram of part with mass 1 =0

Free body diagram of part with mass 2 .= 0

Free body diagram of part with mass 3 =0

HARMONIC EXCITAION FOR UNDAMPED FORCED MDOF VIBRATION

Harmonic excitation refers to a sinusoidal external force of a certain frequency applied to a system. The response of a system to harmonic excitation is a very important topic because it is encountered very commonly and also covers the concept of resonance. Resonance is a phenomenon in which a vibrating system or external force drives another system to oscillate with greater amplitude at specific frequencies.

As an example, we will consider the system with two springs and masses shown in the picture. Each mass is subjected to a harmonic force, which vibrates with some frequency (the forces acting on the different masses all vibrate at the same frequency). The equations of motion are

We can write these in matrix form as

or, more generally,

To find the steady-state solution, we simply assume that the masses will all vibrate harmonically at the same frequency as the forces. where

This means that

are the (unknown) amplitudes of vibration of the two masses. In vector form we could write

where Substituting this into the equation of motion gives

FORCE DAMPED VIBRATION OF MULTIPLE DEGREES OF FREEDOM SYSTEMS

Consider a viscously damped two degrees of freedom spring‐mass system shown in the figure. The motion of the system is completely described by the coordinates x1(t) and x2(t), which define the positions of the masses m1 and m2 at any time t from the respective equilibrium positions.

PREPARED BY: LOPEZ, DIONISIO JR MAGAT, RODERICK JR CASTILLO, REYNALD ANTHONY DAGMANTE, EDDIE NITURADA, RENATO III YSIT, KYLE BRYAN