ME scope Application Note 16

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Bartholomew Sharp
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ME scoe Alication Note 16 Integration & Differentiation of FFs and Mode Shaes NOTE: The stes used in this Alication Note can be dulicated using any Package that includes the VES-36 Advanced Signal

INTODUCTION Deending on the units of the sensors used to acquire real-world vibration data, both integration & differentiation can be used to convert the data to different units.

1 ME scoe Alication Note 16 Integration & Differentiation of FFs and Mode Shaes NOTE: The stes used in this Alication Note can be dulicated using any Package that includes the VES-36 Advanced Signal Processing otion. Click here to download the ME scoe Demo Project file for this Alication Note. INTODUCTION Deending on the units of the sensors used to acquire real-world vibration data, both integration & differentiation can be used to convert the data to different units. Normally, integration & differentiation are only erformed on time domain waveforms, but it will be shown in this A Note how integration & differentiation can also be erformed on FFs and mode shaes, secifically esidue mode shaes. Integration & differentiation can be erformed on time or frequency domain waveforms in any Data Block in ME scoe. In addition, integration & differentiation can be erformed on esidue mode shaes in any Shae Table in ME scoe. In this A Note, the formulas for FFs in terms of modal arameters are first develoed. These formulas are develoed for the SDOF mass-sring-damer in the figure below. The formulas for an FF in terms of modal arameters are also used for FF-based modal arameter estimation (or curve fitting) in ME'scoe. The formulas for an IF in terms of modal arameters give the clearest exlanation of the role that each modal arameter lays in the vibration of a structure. SDOF Mass-Sring-Damer. The modal roerties of real-world structures are analyzed using multi-degree-of-freedom (MDOF) dynamic models. In this A Note, the single degree-of-freedom (SDOF) mass-sring-damer model shown in the figure above will be used. The dynamics of real-world MDOF structures are better understood by analyzing the dynamics of this SDOF mass-sring-damer structure. BACKGOUND MATH The dynamic behavior of the SDOF mass-sring-damer in the figure above is reresented by the single differential equation, equation (1) shown below. The dynamic behavior of an MDOF structure is reresented by multile equations of the same form as equation (1), but the mass, stiffness, and daming terms in equation (1) are relaced with a mass matrix multilied by a vector of accelerations, a daming matrix multilied by a vector of velocities, and a stiffness matrix multilied by a vector of dislacements. Page 1 of 15

The time domain equation of motion for the SDOF mass-sring-damer is reresented by Newton s Second Law, (t) + C (1) where, M= mass value C = daming coefficient K = sring stiffness Lalace Transforms

of motion results, [M s C s K] X(s) F(s) () where, X(s) = Lalace transform of the dislacement F(s) = Lalace transform of the force Lalace variable (comlex frequency s j Transfer Function Equation ()

2 The time domain equation of motion for the SDOF mass-sring-damer is reresented by Newton s Second Law, (t) + C (1) where, M= mass value C = daming coefficient K = sring stiffness Lalace Transforms (t) = acceleration = velocity = dislacement = excitation force By taking Lalace transforms of the terms in equation (1) and setting initial conditions to zero, an equivalent frequency domain equation of motion results, [M s C s K] X(s) F(s) () where, X(s) = Lalace transform of the dislacement F(s) = Lalace transform of the force Lalace variable (comlex frequency s j Transfer Function Equation () can be rewritten by dividing both sides by the coefficients of the left-hand side. X 1 (s) F(s) Ms C s K (3) The new coefficient on the right hand side of equation (3) is called a Transfer Function. X(s) (s) F(s) Ms H 1 C s K (4) A Transfer Function is comlex valued, and therefore has two arts; real & imaginary or equivalently magnitude & hase. The two arts of the Transfer Function can be lotted on the comlex Lalace lane (the s-lane), as shown in the figure below. Transfer Function on the s-plane. Page of 15

3 Poles of the Transfer Function Notice that the magnitude of the Transfer Function has two eaks. These eaks are where the value of the Transfer Function goes to infinity. The real and imaginary arts also show the same two eaks. By insection of equation (4), it is clear that the Transfer Function has a value of infinity for values on the s-plane where its denominator is zero. It is also clear from equation (4) that as s aroaches infinity, the Transfer Function aroaches zero. The denominator in equation (4) is a second order olynomial in the s variable, called the characteristic olynomial. Since it is a second order, it has two roots (values of s where it will be zero). The roots of the characteristic olynomial are called the oles of the Transfer Function. Poles always occur in airs, one ole being the comlex conjugate of the other. Poles are the locations in the s-lane where the Transfer Function has a value of infinity. The oles are also called eigenvalues. j, j (5) s-plane Nomenclature The real axis in the s-plane is called the daming axis, and the imaginary axis is called the frequency axis. The locations of the oles in the s-plane have also been given some other commonly used names, as shown in the figure below. Modal Parameters s-plane Nomenclature. The coordinates of the oles in the s-plane are also modal arameters. ewriting equation (4) in terms of its ole locations, or modal arameters, 1/ M H (s) (6) s s where, C K, (7) M M modal daming coefficient undamed modal frequency damed modal frequency Page 3 of 15

4 The ercent of critical daming ( ) is written as, C (8) M K Frequency esonse Function (FF) Notice in the figure below that the Transfer Function is only lotted for half of the s-plane. That is, it has been only been lotted for negative values of daming axis (the real art of s=σ+jω). This was done so that the values of the Transfer Function along the frequency axis (the imaginary art of s=σ+jω) are clearly seen. Definition: The Frequency esonse Function (FF) is defined as the values of the Transfer Function only along the jωaxis (or the frequency axis) in the s-lane. Since the FF is only defined along the jω-axis, the FF is merely the Transfer function evaluated along the line (s=jω) in the s-lane. FF H(j) H(s) X(s) F(s) sj sj X(j) F(j) (9) FF Plotted on the jω-axis. The FF for the SDOF mass-sring-damer is written by relacing the s-variable with j in equation (6). H(j ) (j) 1/ M (1) (j) Page 4 of 15

5 Partial Fraction Exansion of an FF Furthermore, using the oles of the characteristic olynomial, a artial fraction exansion can be erformed on equation (1). 1 H j) j j j ( (11) where: 1/ M (1) is called the modal residue. It is the amlitude of the numerator of each resonance term when an FF is written in artial fraction exansion form. Comaring equation (11) with the lot of the FF in the figure above, it is clear that the FF of the SDOF mass-sringdamer is a summation of two resonance curves, each curve forming a eak at one of the two ole locations. j & j In artial fraction exansion form, the FF for the SDOF mass-sring-damer is fully reresented by a modal frequency ( j ), modal daming ), and modal residue ( ). esidue Units ( From equation (11), it is clear that, esidue units = (FF units) x (radians/second) This is because the units of the FF denominator are (radians/second), or this means that they also have unique values. Hz. Because residues have engineering units, For the SDOF mass-sring-damer, the residue is actually art of a vector with two comonents. The second comonent corresonds to the ground, to which the sring & damer are attached. The ground has no motion, so the second comonent of the residue vector has a value of zero. esidue Mode Shae: esidues have unique values. In ME'scoe, when residues are assembled into a vector, they are called a esidue Mode Shae. Mode Shae One final ste is to reresent the FF in terms of mode shae comonents instead of residues. 1 { u} H j) j j { u} j ( (13) where: 1 { u} A M, A scaling constant (14) { Notice that the mode shae u } is a vector with two comonents, and that its second comonent is zero. The second comonent corresonds to the ground, where there is no motion. Notice also that the mode shae contains a scaling constant (A). The scaling constant (A) is necessary because the comonents of a mode shae do not have unique values. The comonents of a mode shae do not have unique values. Only the "shae" (one comonent relative to another) of a mode shae is unique. For this reason, a mode shae is also called an eigenvector. Page 5 of 15

6 Equation (13) states that the FF of the SDOF mass-sring-damer is fully reresented by a air of eigenvalues (oles) and a air of eigenvectors (mode shaes). In artial fraction exansion form, the FF for the SDOF mass-sring-damer is fully reresented by a modal frequency j ), modal daming ), and a mode shae { u }. ( ( Imulse esonse Function (IF) An Imulse esonse Function is the Inverse FFT of an FF. An IF can also be written in terms of modal arameters, and it rovides the best hysical meaning of the modal arameters. Alying the Inverse FFT to equation (11) gives, 1 1 h (t) FFT (15) j j j or h(t) or T j t t e e (16) t h(t) T e sin( t ) where: hase angle of Equation (17) clearly shows how the IF is exressed in terms of each modal arameter. multilies the time variable (t) in the sinusoidal function sin( t ). This term defines the sinusoidal oscillation of the SDOF, hence is called modal frequency. is the coefficient in the exonential term (e t ) the modal daming coefficient or modal daming decay constant. (17) that defines the enveloe of decay of the IF. Therefore, is called In a real-world structure, the decay of each mode is caused by a combination of daming mechanisms within or about the structure. For the SDOF mass-sring-damer, the modal daming is caused by the viscous damer attached between the mass and ground. IF Units t From equation (17), it is clear that the IF is the roduct of two functions, an exonential function (T e ) and a sinusoidal function sin ( t ). The exonential decay (e t ) and sinusoidal function are dimensionless and only T and have units. Therefore, or, Imulse esonse units = (Seconds) x (esidue units) Imulse esonse units = FF units Page 6 of 15

7 Differentiating an IF Differentiation changes vibration data for dislacement units to velocity units and from velocity units to acceleration units. Differentiating the IF with (dislacement/force) units in equation (16) with resect to time yields, dh(t) dt d dt T j T j T j t t e e t t de d e dt e t dt t e Differentiating an IF is equivalent to multilying its modal residues by their resective oles. In other words, (18) Velocity Velocity Dislacement Dislacement (19) For the SDOF mass-sring-damer, both oles have real residues ( ). For MDOF structures, each comlex conjugate air of oles has a comlex conjugate air of residues. An IF with (velocity/force) units can be differentiated to yield an IF with (acceleration/force) units. It should be clear from equation (18) that each residue with (velocity/force-sec) units would be multilied by its ole again. In summary, Acceleration Velocity Dislacement () The same equation alies for conjugate residues. A conjugate residue is multilied by its corresonding conjugate ole. Three equations relating residues with (dislacement/force-sec), (velocity/force-sec), and (acceleration/force-sec) units can be written, Acceleration Velocity Dislacement Velocity Dislacement Velocity Dislacement Acceleration Acceleration (1) Conclusion: Integration or differentiation of an IF is equivalent to dividing or multilying each of its modal residues by its associated ole. Page 7 of 15

8 Differentiation and FFs Differentiation of an IF can also be written in terms of its corresonding Transfer Function using the following derivative formula for Lalace Transforms, L h (t) sh(s) h( j j j s s s s s s j s ) s j e e The derivative of an IF can be obtained by evaluating the right-hand side of equation () along the frequency axis ( s j ). Assuming the =, equation () says that differentiation of an IF is the same as multilying its corresonding FF by j. In fact, equation () is valid for any time waveform and its corresonding frequency domain function. Any time waveform can be differentiated by multilying its corresonding frequency domain function by j. Multilying the FF for the SDOF mass-sring-damer in equation (1) by j H Velocity (j ) (j) j/ M (3) j Performing a artial fraction exansion of this FF gives the same result as equation (), (4) Velocity Dislacement From the above analysis, it can be concluded, Conclusion: Differentiation or integration of an IF is the same as multilying or dividing its corresonding FF by j, and is also the same as multilying or dividing its residues by their resective oles. In ME scoe any time waveform is differentiated or integrated by multilying or dividing its corresonding frequency domain function by j. Double Differentiating an IF Now that the relationshi between differentiation & integration of an IF and multilying or dividing its FF by j has been established, what about double integration & differentiation of an IF? () Page 8 of 15

9 Writing the formula for double differentiation of the IF in terms of its Lalace Transform gives, L h (t) s j j H(s) sh( s s s s s s 1 j s ) h ( ) s j s s s Again, the second derivative of the IF can be obtained by evaluating equation (4) for s=jω or along the frequency axis. The following conclusion can be drawn from equation (4), Conclusion: Double differentiation or double integration of an IF is the same as multilying or dividing its residues by the square of their oles. But is multilying or dividing an FF by j the same as double differentiation or double integration of its IF? Let's check by multilying the FF for the SDOF mass-sring-damer by j. Multilying equation (1) by j, H Acceleration (j ) (j) j / M (5) j Performing a artial fraction exansion of this FF gives a different result than equation (4), 1 1 Accel (j) M j j j Accel H (6) where, Accelerati on Dislacement (7) (4) The velocity term is not zero. in equation (4) is not zero. The velocity of a structure at a time that is a little greater than zero The artial fraction exansion in equation (6) has an extra term in it, (1/M). What haened to this extra term in equation (4)? It was subtracted out. Evaluating gives, h ( ) j j 1 M e 1 jm j j e j (8) Page 9 of 15

Comaring equations (4), (6) and (8) leads to the following conclusion, Conclusion: Double differentiating or integrating an IF is not the same as multilying or dividing its corresonding FF by j.

Another dialog box will oen. Enter Number of Shaes =1 and Number of M#s =1, and click on OK Enter the following frequency & daming into the uer sreadsheet in the Shae Table window as shown below, 9.

10 Comaring equations (4), (6) and (8) leads to the following conclusion, Conclusion: Double differentiating or integrating an IF is not the same as multilying or dividing its corresonding FF by j. NUMEICAL EXAMPLE Oen ANote16.VTrj, the ME scoe Demo Project file for this Alication Note Execute File New Shae Table from the ME scoe window. A dialog box will oen. Tye "SDOF Mode" and click on OK. Another dialog box will oen. Enter Number of Shaes =1 and Number of M#s =1, and click on OK Enter the following frequency & daming into the uer sreadsheet in the Shae Table window as shown below, Hz.5Hz 5% Enter the following into the lower sreadsheet in the Shae Table window, Measurement Tye = esidue mode shae DOFs = 1Z:1Z Units = in/lbf-sec Synthesizing an FF Using Modal Parameters SDOF Mode Shae Table. An FF, like a Transfer Function, defines the dynamic characteristics between two DOFs of a structure. MDOF systems have many DOF airs for which FFs can be calculated, either from measured data or from modal arameters. The mode shae of the SDOF mass-sring-damer has only one meaningful DOF. The other DOF is for the ground oint where the sring & damer are attached and where there is no motion. Page 1 of 15

Driving Point FF: Any FF between a DOF and itself is called a driving oint FF.

Synthesizing a (Dislacement/Force) FF Execute Tools Synthesize FFs in the SHP: SDOF Mode window.

Enter "Dislacement FF" into the dialog box, and click on OK Execute Dislay M# Bode in the BLK: Dislacement FF window to dislay the FF, as shown below.

11 Driving Point FF: Any FF between a DOF and itself is called a driving oint FF. A driving oint FF between DOF 1Z & DOF 1Z can be synthesized using the residue mode shae in the SHP: SDOF Mode window. Synthesizing a (Dislacement/Force) FF Execute Tools Synthesize FFs in the SHP: SDOF Mode window. The FF Synthesis dialog box will oen as shown below Enter an Ending Frequency = 1 Hz, select the oving DOF (1Z:1Z), and click on OK Synthesize FF Dialog Box. Enter "Dislacement FF" into the dialog box, and click on OK Execute Dislay M# Bode in the BLK: Dislacement FF window to dislay the FF, as shown below. Differentiating to obtain a (Velocity/Force) FF A (velocity/force) FF can be obtained in two ways, (Dislacement/Force) FF. 1. Multily the residue mode shae with (dislacement/force-sec) units by its ole, and synthesize a new FF using the residue mode shae with (velocity/force-sec) units. Multily the (dislacement/force) FF by j Page 11 of 15

Execute Tools Differentiate in the SHP: SDOF Mode window Notice that the residue mode shae in the figure below now contains velocity units in its numerator. (Velocity/Force-sec) esidue Mode Shae.

12 We have already concluded from the math that both of these methods will yield the same result. In ME'scoe, method #1 is carried out by executing Tools Differentiate followed by the Tools Synthesize FFs command in a Shae Table window. Execute Tools Differentiate in the SHP: SDOF Mode window Notice that the residue mode shae in the figure below now contains velocity units in its numerator. (Velocity/Force-sec) esidue Mode Shae. Execute Tools Synthesize FFs in the SHP: SDOF Mode window and enter the same arameters into the dialog box that were used to synthesize the (dislacement/force) FF Select New File in the next dialog box, enter "Velocity FF" into the next dialog box, and click on OK Execute Dislay M# Bode in the BLK: Velocity FF window Double click on the Line Color column header in the M#s sreadsheet and select any color but black from the dialog box that oens The resulting (velocity/force) FF is shown in the figure below. (Velocity/Force) FF. In ME'scoe, method # is carried out by executing Tools Differentiate in a Data Block window. First, the (dislacement/force) FF will be added to the BLK: Velocity FF window. Then it will be differentiated by selecting its M# and executing Tools Differentiate. Finally, the two M#s will be overlaid to comare them, Execute M#s Coy to File in the BLK: Dislacement FF window Select BLK: Velocity FF in the dialog that oens, and ress Add To Page 1 of 15

to one another. Differentiating to obtain an (Acceleration/Force) FF An (acceleration/force) FF can be obtained in two ways, 1.

13 Execute Format Overlaid in the BLK: Velocity FF window The two M#s should be dislayed as shown below. Select M# in the BLK: Velocity FF window Execute Tools Differentiate Select and un-select M# As you select and un-select M#, you will see that the two FFs in the BLK: Velocity FF window are identical to one another. Differentiating to obtain an (Acceleration/Force) FF An (acceleration/force) FF can be obtained in two ways, 1. Multily the residue mode shae with (velocity/force-sec) units by its ole and synthesize a new FF using the residue mode shae with (acceleration/force-sec) units. Multily the (dislacement/force) FF by j We have already concluded from the math that these two methods will not yield the same results. Execute Tools Differentiate in the SHP: SDOF Mode window Notice that the residue mode shae shown below contains acceleration units in its numerator. Page 13 of 15

(Acceleration/Force-sec) esidue Mode Shae.

File in the next dialog box, enter "Acceleration FF" into the next dialog box, and click on OK Execute Dislay M# Bode in the BLK: Acceleration FF window Double click on

(Acceleration /Force) FF. Method # will be carried out by executing Tools Differentiate twice.

14 (Acceleration/Force-sec) esidue Mode Shae. Execute Tools Synthesize FFs in the SHP: SDOF Mode window and enter the same arameters into the dialog box that were used to synthesis the (velocity/force) FF Select New File in the next dialog box, enter "Acceleration FF" into the next dialog box, and click on OK Execute Dislay M# Bode in the BLK: Acceleration FF window Double click on the Line Color column header in the M#s sreadsheet and select any color but black from the dialog box that oens The (acceleration/force) FF is shown in the figure below. (Acceleration /Force) FF. Method # will be carried out by executing Tools Differentiate twice. First, the (dislacement/force) FF will be added to the BLK: Acceleration FF window. Then it will be differentiated by selecting its M# and executing Tools Differentiate twice. Finally, the two M#s will be overlaid to comare them, Execute M#s Coy to File in the BLK: Dislacement FF window Select BLK: Acceleration FF in the dialog that oens, and ress Add To Page 14 of 15

$The 1/M line in M# (the extra term in its artial fraction exansion) is clearly visible. Conclusions The following conclusions can be drawn from this exercise, 1.$

15 The two M#s should be dislayed as shown below. Select M# in the BLK: Acceleration FF window Execute Tools Differentiate twice As shown in the figure below, the two M#s in the BLK: Acceleration FF window are different from one another. The 1/M line in M# (the extra term in its artial fraction exansion) is clearly visible. Conclusions The following conclusions can be drawn from this exercise, 1. Multilying & dividing an FF by j is the same as differentiating & integrating its IF. Multilying & dividing an FF by j is not the same as double differentiating & integrating its IF 3. Multilying & dividing residue mode shaes by their ole is the same as differentiating & integrating the IF corresonding to an FF that is synthesized from the residues 4. Multilying & dividing residue mode shaes by their ole squared is the same as double differentiating & integrating the IF corresonding to an FF that is synthesized from the residues 5. Any time waveform can be differentiating & integrating by multilying & dividing its corresonding frequency domain function by j Page 15 of 15

ME scope Application Note 28

ME scope Application Note 28 App Note 8 www.vibetech.com 3/7/17 ME scope Application Note 8 Mathematics of a Mass-Spring-Damper System INTRODUCTION In this note, the capabilities of ME scope will be used to build a model of the mass-spring-damper