Vibration analysis

– Approximate methods for finding natural frequencies and

mode shapes

Unit 4 -continuation

The exact analysis for the vibration of systems with many degrees of freedom are generally difficult, and the computations are laborious. The expansion of the characteristic determinant and the solution of the resulting nth degree polynomial equation to obtain the natural frequencies can become tedious for large values of n.

Several analytical and numerical methods have been developed to compute the natural frequencies and mode shapes for the MDOF systems. The following are some of those methods: 1) Dunkerley’s equation 2) Rayleigh’s method 3) Rayleigh- Ritz’s method 4) Holzer method 5) Jacobi diagonalization method 6) Matrix iteration method etc.

Dunkerley’s formula and Rayleigh’s method are useful for estimating the fundamental natural frequency only. Holzer’s method is essentially a tabular method that can be used to find partial or full solution for eigenvalue problems. The matrix iteration method finds one natural frequency at a time, usually starting from the lowest value. The matrix iteration can thus be terminated after finding the required number of frequencies and mode shapes.

When all the frequencies and mode shapes are required, Jacobi’s method can be used: it finds all the frequencies and mode shapes simultaneously.

Rayleigh’s method: This is a method to find the approximate value

of the fundamental natural frequency of a MDOF system.

This uses the condition that, for a conservative system, the maximum kinetic energy is equal to

the maximum potential energy. This can be employed for both continuous and discrete

systems.

a) As a Discrete system: Kinetic and Potential energy of a ‘n’ degree of freedom system can be written as,

T = (1/2) x T[m] x and

U = (1/2) x T [k] x to find the natural frequencies, we assume harmonic motion to be, x= X cosωt, where X denotes the amplitudes (mode shapes) and ω, the natural frequencies of vibration. For a conservative system Tmax= Umax.

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Tmax= ω2 XT [m] X Umax= XT[k] X equating the two and rearranging, we get,

the RHS of the above equation is known as the Rayleigh’s quotient and is denoted as R(X) It can be shown that Rayleigh’s quotient is never lower than the first eigenvalue (Fundamental natural frequency) and never higher than the highest eigenvalue.

Example: Estimate the fundamental frequency of a 3- dof spring-mass system with spring constant k and mass m. Given the mode shape (1 2 3)T

K1=k2=k3=k, m1=m2=m3=m. Solution: The equations of motion can be written as: m1x1+k1x1+k2(x1-x2) =0 m2x2+k2(x2-X1)+k3(x2-x3)=0 m3x3+k2(x3-x2)+k3x3= 0The stiffness and mass matrix can be written in the form, 2 -1 0 1 0 0 [k] = k -1 2 -1 , [m] = m 0 1 0 0 -1 1 0 0 1

By substituting the mode shape in the expression for Rayleigh’s quotient, we get, 2 -1 0 1 (1 2 3) k -1 2 1 2 0 -1 1 3 R[X] = = 1 0 0 1 (1 2 3) m 0 1 0 2 0 0 1 3

ie., ω2 = 0.2143 (k/m)

or ωn = 0.4629 (k/m)1/2

b) continuous system: consider the case of vibration of beams. This method requires the expression for kinetic energy and potential energy. (K.E.)max = (P.E.)max the K.E. of a beam can be written as: T= = ------(1) Where ρ is the density of the beam material, and y is the deflection of the beam. The max. K.E. can be obtained by assuming a harmonic variation for the deflection, y(x,t) = Y(x) cosωt

Then max. K.E. can be written as, Tmax = --------(2)

The potential energy of the beam U is equal to the work done in deforming the beam. Neglecting the work done by the shear forces, we have, U= ---------(3) where M is the bending moment and θ, the slope of the deformed beam, given by, θ=δy/δx and dθ= δ2y/δx2; also M=EI(δ2y/δx2).

Hence, U= =

the max. value of y(x,t) is Y(x), then the max. value of U is, Umax = equating Tmax and Umax we get the Rayleigh’s quotient, which is defined as, ω2 = (P.E.)max / (K.E.)max.

The natural frequency of the beam is obtained if Y(x) is known.

2. Holzer’s method: This is a tabular method used for the determination of natural frequency of a system for free or forced vibration with or without damping. It is based on successive assumptions of the natural frequency of the system, each followed by the calculation of the configuration governed by the assumed frequency. It can be used to compute all the natural frequencies of a system, and each calculation is independent of the other. The Holzer method is particularly useful for calculating the frequencies of torsional vibration in shafts.

For an assumed value of ω, begin the process by assuming unit amplitude of vibration for the first mass. The amplitudes and inertia forces for all the remaining masses are then calculated. For the last mass of the system,if the displacement or the force is compatible with the conditions prevailing there,(ie. amplitude of vibration zero for fixed ends, total inertia force zero for free ends ), the assumed frequency is one of the frequencies of the system. The remaining values (amplitudes or inertia force) for each of the assumed frequencies are then plotted against the assumed values of natural frequency to give the true frequencies of the system.

The relationship for displacement of various systems can be written as: For both ends free systems, xi = xi -1 – (ω2/ki-1) where x,ω,m, k are the displacement, natural frequency, mass and spring constant of the system respectively. For one end fixed and the other end free, xi = xi -1 – (ω2/ki-1)For both ends fixed systems, xi = xi-1 + (1/ki -1) [kixi – ω2 ]

Example: Torsional system Consider the undamped torsional semi-defenite system shown in figure.

The equations of motion of the discs can be derived as follows: -------(1) -------(2) --------(3)

Shaft 1 Shaft 2

Since the motion is harmonic in a natural mode of vibration, we assume that , , in

equations 1,2 and 3 above and we get,

Summing the equations gives,

Equation (7) states that the sum of the inertia torques of the semidefenite system must be zero. This equation can be treated as another form of the frequency equation and the trial frequency must satisfy this requirement.

In Holzer’s method, a trial frequency,ω, is assumed and is arbitrarily chosen as unity. Next is computed using eqn (4) and then is found from eqn (5).

Thus we have, = 1 ------------ (8)

These values are then substituted in (7) to verify whether the constraint is satisfied. If eqn (7) is not satisfied, then a new value of ω is assumed and the process repeated. Equations (7), (9) and (10) can be generalized for an ‘n’ disc system as follows:

and

Where, i=1,2,3,….n.

Thus the method uses equations (11) and (12) repeatedly for different trial frequencies. If the assumed frequency is not a natural frequency of the system eqn (11) is not satisfied. The resultant torque in (11) represents a torque applied at the last disc. This torque Mt is then plotted for the chosen ω. When the calculation is repeated with other values of ω, the resulting graph appears as shown here.

Fig : Resultant torque vs frequency.

Mt = Mt3

Frequency,ω

ω1 =0 ω2

ω3

From this graph, the natural frequencies of the system can be identified as the values of ω at which Mt = 0. The amplitudes, (i=1,2,3…,n) corresponding to the natural frequencies are the mode shape of the system. Holzer’s method can also be applied to systems with fixed ends. At the fixed end, the amplitude of vibration must be zero. In this case the natural frequencies can be found by plotting the resulting amplitude against the assumed frequencies. For a system with one end free and the other end fixed, eqn(12) can be used for checking the amplitude at the fixed end.

Example: The arrangement of the compressor, turbine, generator is shown in fig. Find the natural frequencies and modeshape.

This represents a free- free torsional system.