Phasor Analysis of Linear Mechanical Systemsand Linear Differential EquationsME104, Prof. B. PadenIn this set of notes, we aim to imitate for linear mechanical systems and linear differentialequations, the phasor analysis we learned for electric circuits.Recall how we derived the complex impedance for an inductor. Starting with thedifferential equation for the V-I characteristic for an inductorIdtdLV (1)we substitute complex sinusoidstjeVVˆ(2)tjeIIˆ(3)So that equation (1) becomes tjtjeIdtdLeVˆˆ(4)Differentiating and solving yieldsILjVˆˆ(5)and the impedance of the inductor is defined byˆˆVZ j LIw� =(6)where “�” denotes “defined equal to”. Having done this calculation once, we see thatwe can jump directly from (1) to (5) by making the substitutionjdtd(7)Phasor Analysis of Linear Mechanical SystemsConsider a mechanical damper (a.k.a. shock absorber) which produces a velocity-dependent force according to the linear differential equationxdtdbf (8)Making the substitutionjdtd, we getxbjfˆˆ(9)And defining the mechanical impedance to be the ratio of force to displacement, we haveˆˆfZ j bxw� =(Newtons/meter) (10)Note that the damper has a low stiffness at low frequencies, and a high stiffness at highfrequencies. The units of impedance are Newtons/meter in mechanical systems andvolts/amp = Ohms in electrical systems.For a mass, m, we have22d d df m x m xdt dtdt� �� �= =� �� �� �� �(11)Substituting jdtd yields( )22ˆˆfj mxw w= =-(12)The impedance of a mass increases very rapidly with frequency. This explains why anvilsand machine tools are massive. In summary, we have the following impedance propertiesfor these basic mechanical componentsComponentImpedancexfˆˆ (Newtons/meter)Mass, mm2Damper, bbjSpring, kkPhasor analysis of interconnected linear mechanical systemsIn analogy to the phasor analysis of electric circuits, we analyze mechanical systems withthe following procedureStep 1. Represent sinusoidal forces and displacements by phasors (complex numberswith the same amplitude and phase) denoted with hats as shown in Figure 2.. Step 2. Make the substitution jdtd in component equations. In the case of mass,springs, dampers, forces, and displacements, this means we can substitute the compleximpedances from table above in Figure 2. (Velocities, v, and accelerations, a, are relatedto displacements by differentiation, so xjvˆˆ and xaˆˆ2. Once expressed asdisplacements, mechanical impedances can still be used.)Step 3. Analyze using the laws of mechanics (treat the mechanical impedances as if theywere springs with complex stiffnesses). Step 4. Convert phasors to sinusoidal functions of time. (Since the phasor quantities havethe amplitude and phase of the corresponding sinusoid we sometimes skip this step).Example: For the mass-spring-damper system below with  )cos( ttf, solve for x(t).Figure 1. Mass-spring-damper system.Following our procedure…Step 1. Convert to phasor representation by putting hats on the time-dependent functionsmaking them phasors (i.e. complex numbers).Step 2. Make the substitutionjdtd in the differential equations. For mechanicalsystems this means we replace the mechanical parameters by the corresponding compleximpedances. We verify that there are only phasor forces and displacements asimpedances relate forces to displacements only.Figure 2. Phasor analysis of mass-spring-damper system.Step 3. Apply the laws of mechanics to the complex quantities as though they are realstiffnesses (impedances). Since the components in the figure are in parallel, the forcesproduced by the spring, mass, and damper add to equal the applied force: xmbjkxmxbjxkfZimpedanceˆˆˆˆˆ22  (13)Since the applied force, )cos( ttf, we have that1ˆf(14)Solving for xˆ yieldsmbjkx21ˆ(15)Step 4. Convert phasors to sinusoidal functions of time:ˆ( ) Re[ ]j tx t xew=(16) ˆ ˆcos( )x t xw= +�(17) ( )( )2222 21cos( arctan ( / ( )( )t b k mk m bw w ww w= - -- +(18)Phasor Analysis of Linear Differential EquationsAny sinusoidally-forced linear differential equation with constant coefficients can beanalyzed with phasors to find a sinusoidal steady state solution. Linear electric circuitsand linear mechanical systems are special cases of systems described by such differentialequations.We describe this method by example:Consider the sinusoidally-forced linear differential equation with constant coefficients:)sin(3)();(2 ttutuxxx(19)Steps 1 and 2. Applying step 1 and step 2 of the procedure we have2ˆ ˆ ˆ ˆ ˆ2 ; 3x j x x u u jw w- + + = =-(20)Step 3 becomes “apply the laws of algebra” rather than the laws of circuits or mechanics:2ˆ1ˆ; 3ˆ2 1xu jujw w= =-- + +(21)The ratio on the left is called a “transfer function.” It tells us how u affects x. In the caseof circuits and mechanical systems, the transfer function is often an impedance (e.g.IVZˆˆ.) so that impedances are special cases of transfer functions. Solving forxˆyields.23ˆ2 1jxjw w-=- + +(22)Step 4. Converting hats to sinusoids… or not. The amplitude and phase of xˆ is the sameas the corresponding sinusoid so we can plot that directly. Or we can compute:ˆ( ) Re[ ]j tx t xew=(23) ˆ ˆcos( )x t xw= +�(24) ( )22 2 23sin ( / 2 arctan(2 /(1 ))(1 ) 4tw p w ww w= - - -- +(25)Note on stability: The underlying linear differential equation my be stable (having allcharacteristic roots with negative real part) or unstable. Correspondingly, the sinusoidalsteady state computed using phasors will be stable or unstable. RLC circuits and mass-spring-damper systems are always stable since they conserve or dissipate energy.ProblemsProblem 1. Follow the derivation in equations (1) through (5) in these notes to derive the impedance of a mass, m.Problem 2. Describe a mass-spring-damper system such that the phasor displacement of the mass is given by (22).Problem 3. For equation (22), plot the amplitude and phase of the phasorxˆ.Problem 4. For the system of Figure 3(a) solve for the impedance 1ˆˆfx. (Hint: use a differential equation model rather than impedances. It is possible to use an impedance approach however)What is the limiting impedance