# Presentation: 442079

## Weidong Zhu, Nengan Zheng, and Chun-Nam Wong
Department of Mechanical Engineering
University of Maryland, Baltimore County (UMBC)
Baltimore, MD 21250

- Department of Mechanical Engineering
- University of Maryland, Baltimore County (UMBC)
- Baltimore, MD 21250

**A Novel Stochastic Model for the Random Impact Series Method in Modal Testing**

## Shaker Test 

**Advantages:**
-  Persistent excitation – large energy input and high signal-to-noise ratio
-  Random excitation – can average out slight nonlinearities, such as those arising from opening and closing of cracks and loosening of bolted joints, that can exist in the structure and extract the linearized parameters
**Disadvantages:**
- Inconvenient and expensive

## Single-Impact Hammer Test

**Advantages:**
- Convenient
- Inexpensive
- Portable
**Disadvantages:**
- Low energy input
- Low signal-to-noise ratio
- No randomization of input – not good for nonlinear systems

## Development of a Random Impact Test Method and a Random Impact Device

**Combine the advantages of the two excitation methods**
- Increased energy input to the structure
- Randomized input – can average out slight nonlinearities that can exist in the structure and extract the linearized parameters
- Convenient, inexpensive, and portable
**Additional advantages**
- A random impact device can be designed to excite very large structures
- It cab be used to concentrate the input power in a desired frequency range

## Novel Stochastic Models of a RandomImpact Series 

**Random impact series (RIS) (Zhu, Zheng, and Wong, JVA, in press)**
-     Random arrival times and pulse amplitudes, with the same deterministic pulse shape
**Random impact series with a controlled spectrum (RISCS)**
-     Controlled arrival times and random pulse amplitudes, with the same deterministic pulse shape

## Previous Work on the Random Impact Test  

- 	Huo and Zhang, Int. J. of Analytical and Exp. Modal Analysis, 1988
- Modeled the pulses as a half-sine wave, which is usually not the case in practice
-  Analysis essentially deterministic in nature
- The number of pulses in a time duration was modeled as a constant
- No stochastic averages were determined
- The mean value of a sum involving products of pulse amplitudes were erroneously concluded to be zero

## N     - total number of force pulses 
y(∙)   - shape function of all force pulses 
        - arrival time of the i-th force pulse 
        - amplitude of the i-th force pulse 
        - duration of all force pulses

*y*(∙)   - shape function of all force pulses
-         - arrival time of the *i*-th force pulse
-         - amplitude of the *i-*th force pulse
-         - duration of all force pulses

**Mathematical model of an impact series**** **

## The amplitude of the force spectrum for the impact series

- Deterministic arrive times with deterministic or random amplitudes

- Random arrive times with deterministic or random amplitudes

## N(T)  - total number of force pulses that has arrived  
            within the time interval (0,T]
y(∙)  - arbitrary deterministic shape function of all force pulses 
        - random arrival time of the i-th force pulse 
        - random amplitude of the i-th force pulse 
        - duration of all force pulses

-             within the time interval (*0,T*]
*y*(∙)  - arbitrary deterministic shape function of all force pulses
-         - random arrival time of the *i*-th force pulse
-         - random amplitude of the *i-*th force pulse
-         - duration of all force pulses

**Time Function of the RIS********* **

- Challenge:  A finite time random process with stationary and non-stationary parts

*******Zhu et al., JVA, in press**

## Probability Density Function (PDF) of the Poisson Process N(T)

*n*=*N*(*T*)    - number of arrived pulses
*λ* - constant arrival rate of the pulses

## PDF of the Identically, Uniformly Distributed Arrival Times

## PDF of the Identically, Normally Distributed Pulse Amplitudes(Used in numerical simulations)

- - amplitude of the force pulses
- - mean of
- - variance of

## Mean function of x(t)

## Autocorrelation Function of x(t)

*x*(*t*) is a wide-sense stationary random  process in

- where

-  When

## Average Power Densities of x(t)

- Non-stationary at the beginning and the end of the process

- Wide sense stationary

- Average power densities

- The expectations of average power densities

- where

## Averaged, Normalized Shape Function 

## Comparison of Analytical and Numerical Results for Stochastic Averages 

- Mean function of *x*(*t*)

## Expectation of the average power density of         in

**Expectation of the average power density of         in**

- Increasing the Pulse Arrival Rate Increases the Energy Input

## A single degree of freedom system under single and random impact excitations

**A single degree of freedom system under single and random impact excitations**

- Sampling time

- Excitation time

*Ts*=16 s

*T*=16 s  (Continuous)

*T*=11.39 s  (Burst)

- Arrival times

## A single degree of freedom system under multiple impact excitations (cont.)

**A single degree of freedom system under multiple impact excitations (cont.)**

- Sampling time

- Arrival times

*Ts*=16 s

- Excitation time

*T*=11.39 s

*Where i*=1, 2,..., 66

## Random Impact Series with a Controlled Spectrum (RISCS)

- RISCS can concentrate the energy to a desired frequency range.  For example, if one wants to excite natural frequencies between 7-13 Hz.  The frequency of impacts can gradually increase from 7 Hz at t=0 s to 13 Hz at t=8 s.

## Random Impact Series with a Controlled Spectrum (RISCS) (cont.)

- RISCS Can** **Concentrate the Input Energy in a Desired Frequency Range and the Randomness of the Amplitude Can Greatly Increase the Energy Levels of the Valleys in the Spectrum

## Conclusions 

- Novel stochastic models were developed to describe a random impact series in modal testing.  They can be used to develop random impact devices, and to improve the measured frequency response functions.
- The analytical solutions were validated numerically.
- The random impact hammer test can yield more accurate test results for the damping ratios than the single impact hammer test.

## Acknowledgement

**Vibration-Based Structural Damage Detection: Theory and Applications, Award CMS-0600559 from the Dynamical Systems Program of the National Science Foundation**
**Maryland Technology Development Corporation (TEDCO) **
**Baltimore Gas and Electric Company (BGE)**
**Pratt & Whitney**