Vibration Analysis of an Offset Frame

[aeroguy77]

1.0 Introduction

As the thread title suggests, this is indeed referring to a lab experiment. I am not posting here to gain easy solutions to the lab manual questions, rather I'm posting here to gain knowledge on a subject I know very little about. Let me explain the lab experiment in detail so that you can subsequently provide me detailed answers to my questions.

The apparatus (pictures provided) consists of a three mass offset frame with a "shaker" (supplied by a power amp and a wave generator) and accelerometers measuring the plate acceleration via a charge amplifier and oscillator. The plates are considered to be point masses while the stiffness is to be provided by the six supporting columns. The experimental stiffness of the columns is determined by placing a weight and pulley system at the end of the apparatus in order to measure deflections by increasing load.

The following assumptions are made:
- The mass plates are rigid
- All significant vibrating mass is in these plates
- The columns are rigidly welded to these plates and the base

The analysis is to be performed for "in-plane" motion only (i.e no twisting or "out-of-plane" motion is to be considered).

At the beginning of the lab, we set the excitation level to 10 rms and inital amplitude to 880 mV. For each mass (plate) we recorded the corresponding voltages recorded by the accelerometer. We then increased the frequency and then took down additional values. There are two accelerometers. One is always placed on mass1 (plate1) (it measures input acceleration) and the other moves between mass2 and mass3.

2.0 Theory

The deflection is to be defined by a global co-ordinate system. Thus in-plane motion is said to have 3 degrees of freedom (DOF).

[x]^T = [x1 x2 x3]

For N DOF, the free vibration of a system with no damping has the following equation of motion:

[M][X]'' + [K][X] = 0

where,

[X]^T = [X1 X2 . . . Xn]

[M] = Mass matrix, N x N

[K] = Stiffness matrix, N x N

A system with N DOF has N natural frequencies and mode shapes. If the system is in simple harmonic motion with frequency w and deflection (mode) shape [X] then,

[X(t)] = [X]cos(wt) and
{[K]-w^2[M]}[X]

solving for the corresponding natural frequencies (assuming [X] =/ 0):

DET{[K]-w^2[M]} = 0 to obtain

w1^2 w2^2 w3^2 ... wN^2
and
[X]1 [X]2 [X]3 ... [X]N

The mode vectors also have orthogonality properties for (i) different modes and (ii) the same modes:

(i) [X]r^T [M] [X]s = 0
[X]r^T [K] [X]s = 0

(ii) [X]r^T [M] [X]r = Mrr
[X]r^T [K] [X]r = Krr

where Mrr and Krr are know as the modal mass and stiffness of the rth mode. The equation of motion for this mode is:

Mrrqr'' + Krrqr = 0

with natural frequency:

wr^2 = Krr/Mrr

To find the mass matrix and stiffness matrix, we employed the kinetic and strain energy functions. The lab manual conveniently derived the corresponding theoretical matrices.

3.0 My Questions

(1) What is the "excitation" level, what are its units, and how does this effect the overall outcome of the experiment?

(2) What is a mode shape? What is a natural frequency? How are the two related?

(3) Why is w1 and [X]1 the lowest (fundamental) mode frequency and mode shape?

(4) How can the properties of orthogonality be used to determine the modes on complex structures?

(5) Why does an accelerometer give readings in voltages and not accelerations and/or deflections? How can we determine the corresponding mode shape and frequency from this information?

(6) Why does the voltages measured by the accelerometer decrease as the shaker frequency increases?

Thank you very much for your time, I know that as engineers you don't have a lot of it. Also, any insight on the topic will be greatly appreciated. Again, thank you for your time.

 

[Achy47]

Thank you for providing such a detailed explanation of your lab experiment. I am happy to help answer your questions and provide insight on the topic.

1. The "excitation" level refers to the level of energy or force being applied to the system. In this case, it is the level of vibration generated by the shaker. The units for excitation level can vary depending on the type of excitation being used. For example, if the shaker is generating vibrations through an electromagnetic field, the units could be milliamps (mA). If it is generating vibrations through mechanical means, the units could be in newtons (N). The excitation level can affect the overall outcome of the experiment because it determines the intensity of the vibration applied to the system, which can impact the natural frequencies and mode shapes of the system.

2. A mode shape is a pattern of vibration that a system exhibits at a specific natural frequency. It represents the relative motion of the different components of the system at that particular frequency. A natural frequency is the frequency at which a system will naturally vibrate without any external excitation. The two are related in that the mode shape is dependent on the natural frequency of the system. As the natural frequency changes, so does the mode shape.

3. W1 and [X]1 represent the lowest (fundamental) mode frequency and mode shape because they correspond to the lowest natural frequency of the system. This is typically the mode with the largest amplitude and therefore the most dominant mode of vibration.

4. The properties of orthogonality can be used to determine the modes on complex structures by allowing us to isolate and analyze individual modes of vibration. By using the orthogonality properties, we can determine the modal mass and stiffness for each mode, which can then be used to calculate the corresponding natural frequency and mode shape.

5. An accelerometer measures acceleration by converting it into an electrical signal, which is then displayed as a voltage. This is because it is easier to measure and record voltage signals than it is to directly measure acceleration. To determine the corresponding mode shape and frequency from the accelerometer readings, we can use the equations you provided in your post, as well as the properties of orthogonality to isolate and analyze individual modes.

6. The voltages measured by the accelerometer decrease as the shaker frequency increases because as the frequency increases, the amplitude of the vibration decreases. This means that the acceleration and therefore