A. System Identification (Week 1)

Objective

The purpose of this experiment is to understand the existence of vibrations in a rotary flexible link and the resulting mode shapes. Any beam-like structure exhibits vibrations either due to external changing loads or due to reorientation via actuators. The experiment deals with the modeling and identification of modal frequencies and mode shapes of free vibrations for a rotary flexible link. This analysis is particularly useful to understand structural vibrations and modes and how to contain them in real-world applications like aircraft and spacecraft structures as well as robotic links/manipulators.

Equipment Required

• Flexible link plant (Quanser FLEXGAGE module)

• Power amplifier (Quanser VoltPaq-X2)

Modeling and Vibration Analysis

This experiment involves system identification and modeling of the flexible link. The objective is find the stiffness and the viscous damping coefficient of the flexible link and conduct a frequency sweep across a range to determine the structural frequencies and mode shapes of the link.

Rotary Flexible Link Model

This experiment is performed using the Quanser Rotary Flexible Link mounted on an SRV-02 servo motor. This system, shown in Fig. 1, consists of an electromechanical plant, where a flexible link is rotated using a servo motor. The base of the flexible link is mounted on the load gear of the servo motor system. The servo angle, , increases positively when it rotates counter-clockwise (CCW). The servo (and thus the link) turn in the CCW direction when the control voltage is positive, i.e., .

The main components of the setup are labeled in Figs. 2 and 3 and are listed in Table 1.

Table 1. Setup Components

The FLEXGAGE module consists of the strain gauge, the strain gauge circuitry, and a sensor connector. The flexible link is attached to this module and the strain gauge is fixed at the root of the link. The module is mounted onto the servo motor, which is the actuator for this system. The strain gauge sensor produces an analog signal proportional to the deflection of the link tip.

The link can be schematically represented as shown in Fig. 4. The flexible link has a total length of , a mass of , and its moment of inertia about the pivoted end is . The deflection angle of the link is denoted as and increases positively when rotated CCW.

The complete flexible link system can be represented by the diagram shown in Fig. 5. The control variable is the input servo motor voltage, which is proportional to the angular rate of the servo motor. This generates a torque , at the load gear of the servo that rotates the base of the link, which is given by​

where the various constants are SRV02 parameters which are mentioned in Table 2.

The viscous friction coefficient of the servo is denoted by . This is the friction that opposes the torque being applied at the servo load gear. represents the moment of inertia of the SRV02 when there is no load. The friction acting on the link is represented by the viscous damping coefficient . The flexible link is modeled as a linear spring with the stiffness and with moment of inertia .

Table 2. Setup Parameters

Equations of Motion

The servo and the flexible link can be approximated as a lumped mass system interconnected by a spring and a damper, which represent the stiffness and damping coefficient of the flexible link, respectively (see Fig.5). The equations that describe the motion of the servo and the link, i.e., the dynamics, can be obtained using free body diagram (FBD) analysis of the lumped mass moments of inertia ( and ).

The torque balance on yields Eq. (1.2) and the torque balance on yields Eq. (1.3).

On rearranging Eq. (1.3) to obtain an expression for and substituting it in Eq. (1.2), the equations of motion (EOM) for the rotary flexible link system can be obtained as

Note that the rigid rotation of the link () and its flexible deflection () are inertially coupled as seen from Eqns. (1.4) and (1.5).

Part 1: Determination of Link Stiffness and Viscous Damping from Experiment

Part 1: Theory

The stiffness and the viscous damping coefficient of the flexible link can be determined from the free oscillation of the link using a second-order model. The free-oscillatory equation of motion of this second-order system is obtained by setting the term to zero in Eq. (1.5), i.e., by holding constant, yielding:

With initial conditions and , the Laplace transform of Eqn. (1.6) yields Eq (1.7) where is the Laplace transform of .

Inverse Laplace of Eq. (1.7) yields . An example plot of is shown in Fig. 6.

The characteristic polynomial for a second-order system is:

where is the damping ratio, and is the natural frequency. Equating this to the characteristic polynomial (denominator) in Eq. (1.7) yields

where is the mass moment of inertia of the link about the pivot. This can be calculated approximately by considering the link as a rod rotating about a pivot at one edge . Equations (1.8) and (1.9) can be used to determine the stiffness and damping of the flexible link once the natural frequency and damping ratio are known.

The damping ratio of this second-order system can be found from its response (underdamped system) using the logarithmic decrement given by

where is the peak of the first oscillation and is the peak of the nth oscillation. Note that , as this is a decaying response (positive damping).

The damping ratio can be shown to be related to the logarithmic decrement as

The period of oscillation in a system response can be found using the equation

where is the time of the peak is the time of the first peak, and is the number of peaks considered.

From this, the damped natural frequency (in rad/s) is

and the natural frequency is

Part 1 Experimental Procedure:

1. Mount the flexible link onto the calibration bench.

2. Download and open FlexLink_FreeOsc_Q2_USB.slx. This is the block diagram for this part of the experiment. Change the simulation time to 5-10 seconds.

Part 1: Analysis:

1. Plot the measured angular deflection of the link vs. time for each case. Ensure that the data is centered around 0°, and if it is not, adjust it so that it is. Select peaks that are smooth (focus on the region after initial transients) as shown in the figure below. Ensure that there are at least 4-5 peaks in between the two chosen peaks. Option: Usage offindpeak MATLAB function might be helpful, but not required. If you use this function, verify the peaks identified.
2. From the peaks selected, determine the logarithmic decrement (Eq 1.10) and the time period (Eq 1.12).

Part 2: Determination of Modal Frequencies

Part 2 Theory:

The flexible link can be considered as a thin continuous (uniform) cantilever beam anchored at one end and free at the other end. Using the Euler-Bernoulli beam theory, the equation of motion can be written as

where is the Young's modulus of the material of the beam (assumed constant), is the area moment of inertia of the beam cross-section (assumed constant), is the displacement in direction at a distance from the fixed end at time , is the circular natural frequency, is the mass per unit length (, is the material density, is the cross-section area), is the distance measured from the fixed end and is the external applied force per unit length.

Also, the angle of deflection is related to the displacement as

The general solution to Eq. (1.15) can be obtained using separation of variables, as in Eq. (1.17) below.

Substituting (1.17) in (1.15), setting and rearranging gives

Equation (1.18) can be rewritten as

Since the left side of Eq. (1.19) is only a function of and the right side of Eq. (1.19) is only a function of , they both must equal a constant. Let this constant be . Thus, Eq. (1.19) can be written as the following two equations

The solution of Eq. (1.20) gives , which is of the form

where and are unknown constants. The general solution of Eq. (1.21) is given by

where , and are unknown constants.

The constants in Eq. (1.22) are determined from four boundary conditions, while the constants in Eq. (1.23) are determined from two initial conditions.

For a clamped-free or cantilever beam, the geometric boundary conditions are

and the natural boundary conditions are

where represents the bending moment and represents the shear force.

On substitution of the geometric boundary conditions at in Eq. (1.22) and its derivative, the following relations can be obtained

Hence, Eq. (1.22) becomes

On further substitution of the natural boundary conditions at in the derivatives of Eq. (1.24) yields

or

For a non-trivial solution, the determinant of the above matrix must be 0, i.e.

which gives the following characteristic equation

The above equation simplifies to

There are infinite solutions to this characteristic equation, which are given by

Thus, the mode shapes are

Since , the modal frequencies are given by

or

The mode shapes of a uniform cantilever beam are shown in Fig. 7.

Part 2 Experimental Procedure:

In order to determine the modal frequencies and mode shapes, the link is rotated using a sinusoidal input to the link via external means. When the frequency of the input coincides with either the fundamental frequency or higher frequency modes, the corresponding modes will be excited due to resonance and their mode shapes can be visually observed. Hence, the following experiment involves a frequency sweep across a range provided as input to the flexible link via the servo motor to identify the frequencies and observe the corresponding mode shapes.

1. Mount the flexible link onto the rotary servo base.

2. Download and open FlexLink_ExciteMode.mdl

3. To build the model, click the down arrow on

Part 2 Analysis:

1. Plot the measured angular deflection of the link vs. time. Using the time domain plot, perform frequency analysis using Fast Fourier Transform (FFT) [refer to the Appendix for the code] or a similar technique to identify the number of dominant frequencies and their magnitudes present in the signal.

2. Using the system parameters provided in Table 2, determine the mass per unit length of the beam .

3. Using Eqn. 1.26 and the values of necessary system parameters, calculate the first and second modal frequencies.

Part 3 Observation of Mode Shapes

Part 3 Theory:

Analytical expressions for the mode shapes are given in Eq 1.25.

Part 3: Experimental Procedure

1. Open the FlexLink_ExciteMode.mdl Simulink file used in part 2.

2. To build the model, click the down arrow on Monitor & Tune under the Hardware tab and then click Build for monitoring . This generates the controller code.

Part 3: Analysis

1. Plot the mode shapes using Eq. (1.25) in MATLAB (plot v(x) vs x/L) and record the node locations. Plot both theoretical shape and experiment shape. For theoretical, use and from and for experimental, use two modal frequencies from FFT and to find .

2. Record the number of nodes and their locations for each mode. Determine the locations as a ratio of the link length and compare them with the values between the theoretical and experimental mode shape plots.

Results for Report

(A) From Part 1

1. Plot of the measured angular deflection of the link vs. time with the chosen peaks marked for each dataset, i.e., three plots corresponding to three perturbation locations. Plot as three subplots in one figure.

2. Equations for time period of oscillation, damped frequency, undamped frequency, damping ratio, stiffness, and viscous damping coefficient.

3. Values of logarithmic decrement and time period

(B) From Part 2

1. Plot of measured angular deflection of the link vs. time

2. Fast Fourier Transform (FFT) plot of link deflection with dominant frequencies and their magnitudes identified (either marked on the plot or mentioned in writing).

3. Calculation of mass per unit length of the link

(C) From Part 3

1. 1st and 2nd Mode shapes (theoretical and experimental) are plotted on the same graph with node locations marked. (plot of v(x) vs x/L)

2. Number of nodes and their locations for each mode (theoretical and experimental). Use Table B.2 to document the values.

3. Comparison of experimental and theoretical mode shapes/node locations and state the reason(s) for why they differ.

Table B.1 System Identification Parameters

Table B.2 Locations of Nodes

Questions for Report

1. Compare the damped and undamped frequencies of the link and report your observation. What does this signify with respect to the flexible link damping?

Appendix

The following code performs FFT analysis on the flexible link angle response