MEMS 431 (FL10) Lab 6

Modal Analysis of a Cantilever Beam

Objective

To estimate the natural frequencies and mode shapes of a continuous system using impact excitation.

Background

A modally-tuned impact hammer is instrumented with a piezoelectric force transducer and connected to a PC-based data acquisition system. The Fast Fourier Transform (FFT) of the impact force describes the force delivered as a function of frequency. The FFT of the impact force applied at location $k$ will be denoted $F_k(\omega)$ . A transducer measures the response (displacement or acceleration) of the beam at a fixed location. The FFT of the response signal, $X(\omega)$ , will contain peaks at the natural frequencies $\omega_n,\ n=1,2,3,\hdots,N$ .

To normalize the response with respect to the force input, the transform of the response is divided by the transform of the impact force. This normalized response is the frequency response function (FRF), also sometimes called the transfer function. The FRF between a force (input) at location k and the response (output) at another location is just the output divided by the input.

(1) $T_k(\omega)=X(\omega)/F_k(\omega)$

For a fixed response location the elements of the nth mode shape $\mathbf{u}_n$ can be estimated from

(2) $u_{nk} \approx C_1\ \mathrm{Im}[T_k(\omega_n)]$

where $u_{nk}$ is the kth element of $\mathbf{u}_n$ , $\omega _n$ is the nth natural frequency, and $C_1$ is a constant. $\mathrm{Im[\ ]}$ denotes the imaginary part. The complete mode shape is obtained by applying the impact force at many locations, $k=1,2,3,\hdots,N$ , and constructing $\mathbf{u}_n$ from the resulting estimates of $u_{nk}$ .

Theory

How does this method work? Consider the ideal discrete mechanical system

(3) $M\ddot{\mathbf{x}}+C \dot{\mathbf{x}}+K \mathbf{x}=\mathbf{F} (t)$

This system can be uncoupled using normal modes (see B.H. Tongue, Principles of Vibration, 4.7-4.9) and the equation for each modal coordinate $r_n$ can be written:

(4) $\ddot{r}_n+2\zeta _n \omega _n \dot{r}_n+{\omega _n}^2 r_n=C_2\ \mathbf{u}_n \cdot \mathbf{F}(t)$

where $\mathbf{u}_n$ is the nth mode shape and $C_2$ is a constant. Now consider also that the force is modeled by an impulse function in time, which is zero except at the kth location: $F(t)=F_0\ \delta(t)\ [0,\hdots,0,1,0,\hdots,0]^T$ . Then the uncoupled equation is

(5) $\ddot{r}_n+2\zeta _n \omega _n \dot{r}_n+{\omega _n}^2r_n=C_2\ u_{nk}\ F_0\ \delta (t)$

The excitation (right hand side), and thus the response of the nth modal coordinate, normalized with respect to the impact force $F_0$ , is simply proportional to $u_{nk}$ . This modal response is estimated from the imaginary part of the nth peak of the transfer function.

Mode shape

Why use the imaginary part of the FRF at $\omega = \omega _n$ ? Laplace transforming Equation (5), and letting $s=i \omega$ , we get the FRF between the modal response and the input force:

(6) $\frac{R(\omega)}{F_k(\omega)}=\frac{C_2\ u_{nk}}{{\omega _n}^2-\omega ^2+i2\zeta_n \omega_n \omega}$

Now let $\omega=\omega _n$ , for two reasons. (1) The response at a given site will be dominated by the nth mode when $\omega=\omega _n$ , and (2) the FRF becomes simply a constant multiplied by the corresponding element of the mode shape.

(7) $T_k (\omega _n)=\frac{X(\omega _n)}{F_k(\omega _n)}\approx \frac{C_2}{i2\zeta _n {\omega _n}^2}\ u_{nk}$

Finally, Equation (7) can be rewritten to obtain

(8) $u_{nk}=\left \{ \frac{2\zeta_n {\omega _n}^2}{C_2} \right \} \mathrm{Im}[T_k(\omega _n)]$

which is equivalent to Equation (2) (the stuff in the curly brackets = $C_1$ ).

Reading

Review sections in the text concerning modal analysis.

Procedure

Step 1: Initialize the equipment

Turn on equipment and start the data acquisition program.

Step 2: Find the natural frequencies

Gently strike the beam with the impact hammer and observe the response of the force sensor and the response transducer. In particular note the characteristics of the response FFT and the FRF. What are the first three or four natural frequencies of the beam.

Step 3: Find the mode shapes

Apply impacts at equally spaced locations along the beam. After each impulse, record 1) the value of the FRF's imaginary part at each of the first three natural frequencies, and 2) the location of the impact. Repeat the hit twice at each location.

Analysis

The un-normalized mode shape is simply the list of the peaks in the imaginary part of the FRF at each frequency. Plot each mode as a function of the location of impact. Keep track of which peaks should be plotted in the negative direction. You will end up with three plots, each of which is an estimated mode shape. Normalize your mode shapes so each mode has amplitude 1 at the tip (divide every element of the vector by the element at the tip).

Theory

Compare the experimental natural frequencies to theoretical frequencies for the first three modes of an "equivalent" uniform cantilever beam (formula provided, see Table 1). Compare the experimentally determined mode shapes to the theoretical mode shapes of a uniform cantilever beam. (You should normalize the theoretical mode shapes to have amplitude = 1 at the tip.). Use the following material parameters:

$E=205\times 10^9\ \mathrm{N/m^2}$ $I=b h^3 /12$ $\rho=7.85 \times 10^3\ \mathrm{kg/m^3}$

Use average values for height and area. Note that the mode shape is multiplied by an arbitrary factor $A_n$ , which can be used to normalize the mode shape, and that $\beta_n x=\beta_n L(x/L)$ . A good way to do this is in MATLAB (as shown below for mode 1. You will need to enter $L,\ E,\ I,\ A,\ \rho$ . You can make similar plots for modes 2 and 3):


		>> xL  =  0.0:0.01:1.0; 

		>> BL1 = 1.875;  s1  = 0.7341; 

		>> phi1 = cosh(BL1*xL) - cos(BL1*xL) - s1*(sinh(BL1*xL) - sin(BL1*xL)); 

		>> phi1 = phi1/phi1(end); 

		>> w1  = (BL1/L)^2*sqrt(E*I/(rho*A)); 

		>> subplot(3,1,1), plot(xL,phi1);

Table 1. Theoretical natural frequencies and mode shapes of any uniform cantilever (fixed-free) beam. To find the value of $\beta_n$ for a specific beam, take the value of $\beta _n L$ and divide by the length $L$ of the beam.

Report

No specific format is required. Simply turn in a clear document with: (1) values for the first three natural frequencies (theory and experiment) and (2) plots of the first three mode shapes (theory and experiment).