Answer to Question #193736 in Mechanical Engineering for Lemony

Question

Using Holzer’s method, find the natural frequencies and mode shapes of the system shown in Figure, with m1 = 100 kg, m2 = 20 kg, m3 = 200 kg, k1 = 8000 N/m, and k2 = 4000 N/m.

Answer 1

The equations of motion for free vibration can be readily written as follows:

J0+ k (6-8)=0

J₂B₂ + k (0₂-B) + k, (O₂-B₂) = 0

JB + (-) + k₂ (6-8)=0 For harmonic vibration, we assume

6,-, sin ax

-²,,+k, (-₂)-0

-a²³ J₂z+k₂ (6-₂) + (6₂ -₁ ) - 0

Summing up all the equations of motion, we get

Thus:

J, 0,²-0

This is a condition to be satisfied by the natural frequency of the freely vibrating system.

Ho zer's method consists of the following iterative steps:

Step 1: Assume a trial

frequency

Step 2: Assume the first generalized coordinate - 1 say

Step 3: Compute the other d of using the equations of motion as follows:

10.5

Step 4: Sum up and verify if Eq. (10.5.4) is satisfied to the prescribed degree of accurac

If Yes, the trial frequency is a natural frequency of the system If not, redo the steps with a frequency

in order to reduce the computations, therefore one needs to start with a good trial frequency good method of choosing the next trial frequency to converge fast.

Two trial frequencies are found by trial and error such that is a small positive number respectively than the mean of these two trial frequencies(ie bisection method) wil

estimate of for which $\Sigma x\omega²_{try} = 0$