Toolbox for Vibration Analysis of One-DOF Systems

Transcription

1 Toolbox for Vibration Analysis of One-DOF Systems Description of 1-DOF Systems Equation of motion Initial conitions m = mass or inertia (kg), c = amping coefficient (kg/s) k = spring coefficient or stiffness (N/m) F(t) = external force x = initial isplacement v = initial velocity. Problems to Solve mx &&() t + cx& () t + kx() t = F(), t t x() = x, x& () = v This Toolbox provies functions for fining solutions of the following problems: (a) Free vibration (b) Force vibration (c) Frequency response () Response to perioic excitation (e) Nonlinear vibration Copyright by Bingen Yang (3/18/25) 1

2 MATLAB Functions System Setup set1of Input system parameters (m, c, k) Winow 2.1 set1of2 Input system parameters ( ωn, ξ, m) Winow 2.1 Time Response freevib Compute an isplay free response Winow 3.1 forcevib Compute an isplay force response Winow 3.2 stepchract Compute maximum overshoot, rise time an settling time of Section step response of an unerampe system compltvib Plot total response ue to both initial an external isturbances Winow 3.9 energy Compute an isplay mechanical energy Winow 3.11 stepcharact Compute characteristic parameters of step response Section plotvib Plot time response Winow 3.12 getpts Get times of simulation Winow 3.13 animatevib Animate time response Winow 3.14 Analytical Vibration Solutions forceanl Obtain analytical expressions of force response Winow 4.1 plotanl Plot force response by exact analytical solutions Winow 4.2 Frequency Response harmonic Plot frequency response to a harmonic excitation Winow 5.1 plotfr Plot frequency response from compute ata Winow 5.2 normhar Plot normalize frequency response Winow 5.3 transm Plot force or isplacement transmissibility ue to harmonic Winow 5.4 excitation basevib Plot frequency response to a harmonic base excitation Winow 5.5 unbalance Plot the frequency response of a rotating machine with an Winow 5.6 unbalance mass isolator Design the parameters of a vibration isolator Winow 5.7 Response to Perioic Excitations perioic Plot steay-state response to a perioic excitation Winow 6.1 fseries Obtain the Fourier series for a perioic forcing function Winow 6.2 Nonlinear Vibration setnl Specify a 1-DOF nonlinear vibration system Winow 7.1 NLvib Obtain numerical solution of nonlinear vibration Winow 7.2 Utilities TBemo Show how the Toolbox works an what it can o Section 1.1 TBinfo Show the information of the Toolbox Section 1.1 RunEx Run all the numerical examples containe in this chapter Section 1.1 systinfo Display system information an step response specifications Winow 2.1 Copyright by Bingen Yang (3/18/25) 2

3 Solution Proceure To compute an iplay the response of a 1-DOF vibrating system, follow the three steps below. Step 1. Input system parameters: >> set1of(m,c,k) where m is mass, c amping coefficient, an k spring coefficient Step 2. Compute system response: For free response, type >> y = freevib(x,v); where x an v are initial isplacement an velocity For force response, type >> y = forcevib(loa_spec); where Loa_Spec specifies an external loa For the mathematical expressions of force response, type >> y = forceanl(loa_spec); For frequency response to a harmonic excitation of magnitue f, type >> y = harmonic(f); For time-omain steay-state response to a perioic excitation, type >> y = perioic(ploa); where ploa specifies a perioic external loa Step 3. Plot compute response Use plotvib(y) to plot time-omain response, an use plotfr(y) to plot frequency response, where y is a matrix of the compute response that is obtaine in Step 2. Note. For nonlinear vibration analysis, use functions setnl an Nlvib; see Section 1.7. Copyright by Bingen Yang (3/18/25) 3

4 Specification of vector Loa_Spec for functions forcevib an compltvib External Force F(t) Loa_Spec (L) Impulse Force (L1) Step Force (L2) Ramp Force (L3) Exponential (L4) Sinusoial I δ () t [ I T ] I δ ( t T ) [ I T ] F [1 F ] Fut ( T ) [1 F T ] a t [2 a ] a ( t T ) u ( t T ) [2 a T ] bt Ae [3 A b] bt ( T ) Ae u( t T ) [3 A b T ] Asin( t ) ω +φ Asin( ω t+φ ) [4 A ω φ ] ω in ra/s, φ in egrees ( ) Asin ( t T ) u( t T ) [4 A φ T ] ω +φ ω in ra/s, ω φ in egrees (L5) Pulse [5 t1 q 1 t 2 q 2] Note 1. δ () t is the elta function an u() t is the unite step function Note 2. T is a non-negative parameter representing time elay or time shift in a forcing function; see Section Copyright by Bingen Yang (3/18/25) 4

5 Specification of vector ploa for function perioic Loa P Numerical ata ploa = [ T F F L F ] 1 2 N Here Fi = F( t i ) with an t = T / N ti = i t Loa P1 Loa P2 t1 < t2 T ; q1 an q2 arbitrary numbers ploa = [1 T t q t q ] t1 < t2 < t3 T ; a arbitrary number ploa = [2 T at t t] Loa P3 Loa P4 < t1 < T ; a arbitrary number ploa = [3 T a t ] 1 t1 T ; a an b arbitrary numbers ploa = [4 T a b t ] 1 Copyright by Bingen Yang (3/18/25) 5