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Under Over and Critical Damping 1. Response to Damping As we saw , the unfor ced damped harmonic oscillator has equation .. . 0, (1) with 0, 0 and 0. It has characteristic equation 2 0 with characteristic oots 2 (2) Ther e ar e thr ee cases depending on the sign of the expr ession under the squar e oot: i) 2 (this will be underdamping , is small elative to and ). ii) 2 (this will be overdamping , is lar ge elative to and ). iii) 2 (this will be critical damping , is just between over and under damping. Mathematically , the easiest case is over damping because the oots ar e eal. However , most people think of the oscillatory behavior of a damped oscillator . Since thi s is connected to under damping we start with that case. Case (i) Underdamping (non-r eal complex oots) If 2 then the term under the squar e oot is negative and the characteristic oots ar e not eal. In or der for 2 the damping onstant must be elatively small. First we use the oots (2) to solve equation (1). Let 2 . Then we have characteristic oots: . leading to complex exponential solutions: , . The basic eal solutions ar e cos and sin . The general eal solution is found by taking linear combinations of the two basic solutions, that is: cos sin

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Under, Over and Critical Damping OCW 18.03SC or 1 cos 2 sin cos . (3) Let’s analyze this physically . When 0 the esponse is a sinusoid. Damping is a frictional for ce, so it generates heat and dissipates ener gy . When the damping constant is small we would expect the system to still oscillate, but with decr easing amplitude as its ener gy is converted to heat. Over time it should come to est at equilibrium. This is exactly what we see in (3). The factor cos shows the oscillation. The exponential factor has a negative exp onent and the efor e gives he decaying amplitude. As , the exponential goes asympto tically to 0, so also goes asympotically to its equilibrium position 0. e call the damped angular (or circular) frequency of the syste m. This is sometimes called a pseudo-frequency of . e need to be car eful to call t a pseudo-fr equency because is not periodic and only periodic functions have a fr equency . Nonetheless, does oscillate, cr ossing 0 twice each pseudo-period. .. Example 1. Show that the system 0 is under damped, ﬁnd its damped angular fr equency and graph the solution with initial conditions . 1, 0. Solution. Characteristic equatio n: 2 3 0. Characteristic r oots: 2 11 2. Basic r eal solutions: 2 cos 11 , 2 sin 11 . General solution: 1 cos 11 2 sin 11 2 cos 11 2 . Since the r oots have nonzer o imaginary part, the system is under damped. The damped angular fr equency is 11 2. The initial conditions ar e satisﬁed when 1 1 and 2 11. So, 2 cos 11 11 sin 11 12 2 11 cos 11 2 , wher e tan 11 . 2

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Under, Over and Critical Damping OCW 18.03SC Figur e 1: The damped oscillation for example 1. Case (ii) Overdamping (distinct r eal r oots) If 2 then the term under the squar e oot is positive and the char - acteristic oots ar e eal and distinct. In or der for 2 the damping constant must be r elatively lar ge. One extr emely important thing to notice is that in this case the roots are both negative . ou can see this by looking at the formula (2). The term under the squar e oot is positive by assumption, so the oots ar e eal. Since 2 2 the squar e oot is less than and ther efor e the oot 2 0. The other r oot is clearly negative. Now we use the r oots to solve equation (1) in this case. 2 2 Characteristic r oots: 1 , 2 Exponential solutions: , . General solution: . Let’s analyze this physic ally . When the damping is lar ge the frictional for ce is so gr eat that the system can’t oscil late. It might sound odd, but an unfor ced over damped harmonic osc illator does not oscillate. Since both exponents ar e negative every solution in this case goes asymptotically to the equilibrium 0. At the top of many doors is a spring to make them shut automatically . The spring is damped to contr ol the rate at which the door closes. If the damper is str ong enough, so that the spring is over damped, then the door just settles back to the equilibrium position (i.e. the closed position) with- out oscillating –which is usually what is wanted in this case. .. Example 2. Show that the system 0 is over damped and graph the solution with initial conditions 1, 0. Which oot contr ols how fast the solution r eturns to equilibrium? 3

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Under, Over and Critical Damping OCW 18.03SC Solution. Characteristic equation: 2 3 0. Characteristic roots: (this factors) 1, 3. Exponential solutions: , . General solution: )= . Because the roots are real and different, the system is overdamped. The intial conditions are satisﬁed when 1 3/2, 2 1/2. So, )= /2 /2. Figure 2: The overdamped graph for example 2. Because goes to 0 more slowly than /2 it controls the rate at which goes to 0. (Remember, it is the term that goes to zero slowest term that controls the rate.) Case (iii) Critical Damping (repeated real roots) If 2 mk then the term under the square root is 0 and the characteristic polynomial has repeated roots, /2 , /2 . Now we use the roots to solve equation (1) in this case. We have only one exponential solution, so we need to multiply it by to get the second solution. Basic solutions: bt /2 , te bt /2 . General solution: x t bt /2 ()=( 1 . As in the overdamped case, this does not oscillate. It is worth noting that for a ﬁxed and , choosing to be the critical damping value gives the fastest return of the system to its equilibrium position. In engineering design this is often a desirable property. This can be seen by considering the roots, but we will not go through the algebra that shows this. (See ﬁgure (4).) 4

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Under, Over and Critical Damping OCW 18.03SC .. . Example 3. Show that the system 0 is critically damped and graph the solution with initial conditions )= 1, )= 0. Solution. Characteristic equation: 2 4 0. Characteristic roots: (this factors) 2, 2. Exponential solutions: (only one) . General solution: )= 1 . Because the roots are repeated, the system is critically damped. The intial conditions are satisﬁed when 1 1, 2 2. So, x t ()=( 1 . Figure 3: The critically damped graph for example 3. Notice that qualitatively the graphs for the overdamped and critically damped cases are similar. .. . The following ﬁgure shows plots for solutions to bx 0 with initial conditions )= 1, )= 0. The three plots are 1 under- damped; 2 critically damped (dashed line); 3 overdamped. Notice that the critically damped curve has the fastest decay. .. Figure 4: Plots of solutions to bx 0. 5

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Response to Damping As we saw the unfor ced damped harmonic oscillator has equation 0 1 with 0 0 and 0 It has characteristic equation 2 0 with characteristic oots 2 2 Ther e ar e thr ee cases depending on the sign of the expr ession u ID: 23378

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Page 1

Under Over and Critical Damping 1. Response to Damping As we saw , the unfor ced damped harmonic oscillator has equation .. . 0, (1) with 0, 0 and 0. It has characteristic equation 2 0 with characteristic oots 2 (2) Ther e ar e thr ee cases depending on the sign of the expr ession under the squar e oot: i) 2 (this will be underdamping , is small elative to and ). ii) 2 (this will be overdamping , is lar ge elative to and ). iii) 2 (this will be critical damping , is just between over and under damping. Mathematically , the easiest case is over damping because the oots ar e eal. However , most people think of the oscillatory behavior of a damped oscillator . Since thi s is connected to under damping we start with that case. Case (i) Underdamping (non-r eal complex oots) If 2 then the term under the squar e oot is negative and the characteristic oots ar e not eal. In or der for 2 the damping onstant must be elatively small. First we use the oots (2) to solve equation (1). Let 2 . Then we have characteristic oots: . leading to complex exponential solutions: , . The basic eal solutions ar e cos and sin . The general eal solution is found by taking linear combinations of the two basic solutions, that is: cos sin

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Under, Over and Critical Damping OCW 18.03SC or 1 cos 2 sin cos . (3) Let’s analyze this physically . When 0 the esponse is a sinusoid. Damping is a frictional for ce, so it generates heat and dissipates ener gy . When the damping constant is small we would expect the system to still oscillate, but with decr easing amplitude as its ener gy is converted to heat. Over time it should come to est at equilibrium. This is exactly what we see in (3). The factor cos shows the oscillation. The exponential factor has a negative exp onent and the efor e gives he decaying amplitude. As , the exponential goes asympto tically to 0, so also goes asympotically to its equilibrium position 0. e call the damped angular (or circular) frequency of the syste m. This is sometimes called a pseudo-frequency of . e need to be car eful to call t a pseudo-fr equency because is not periodic and only periodic functions have a fr equency . Nonetheless, does oscillate, cr ossing 0 twice each pseudo-period. .. Example 1. Show that the system 0 is under damped, ﬁnd its damped angular fr equency and graph the solution with initial conditions . 1, 0. Solution. Characteristic equatio n: 2 3 0. Characteristic r oots: 2 11 2. Basic r eal solutions: 2 cos 11 , 2 sin 11 . General solution: 1 cos 11 2 sin 11 2 cos 11 2 . Since the r oots have nonzer o imaginary part, the system is under damped. The damped angular fr equency is 11 2. The initial conditions ar e satisﬁed when 1 1 and 2 11. So, 2 cos 11 11 sin 11 12 2 11 cos 11 2 , wher e tan 11 . 2

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Under, Over and Critical Damping OCW 18.03SC Figur e 1: The damped oscillation for example 1. Case (ii) Overdamping (distinct r eal r oots) If 2 then the term under the squar e oot is positive and the char - acteristic oots ar e eal and distinct. In or der for 2 the damping constant must be r elatively lar ge. One extr emely important thing to notice is that in this case the roots are both negative . ou can see this by looking at the formula (2). The term under the squar e oot is positive by assumption, so the oots ar e eal. Since 2 2 the squar e oot is less than and ther efor e the oot 2 0. The other r oot is clearly negative. Now we use the r oots to solve equation (1) in this case. 2 2 Characteristic r oots: 1 , 2 Exponential solutions: , . General solution: . Let’s analyze this physic ally . When the damping is lar ge the frictional for ce is so gr eat that the system can’t oscil late. It might sound odd, but an unfor ced over damped harmonic osc illator does not oscillate. Since both exponents ar e negative every solution in this case goes asymptotically to the equilibrium 0. At the top of many doors is a spring to make them shut automatically . The spring is damped to contr ol the rate at which the door closes. If the damper is str ong enough, so that the spring is over damped, then the door just settles back to the equilibrium position (i.e. the closed position) with- out oscillating –which is usually what is wanted in this case. .. Example 2. Show that the system 0 is over damped and graph the solution with initial conditions 1, 0. Which oot contr ols how fast the solution r eturns to equilibrium? 3

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Under, Over and Critical Damping OCW 18.03SC Solution. Characteristic equation: 2 3 0. Characteristic roots: (this factors) 1, 3. Exponential solutions: , . General solution: )= . Because the roots are real and different, the system is overdamped. The intial conditions are satisﬁed when 1 3/2, 2 1/2. So, )= /2 /2. Figure 2: The overdamped graph for example 2. Because goes to 0 more slowly than /2 it controls the rate at which goes to 0. (Remember, it is the term that goes to zero slowest term that controls the rate.) Case (iii) Critical Damping (repeated real roots) If 2 mk then the term under the square root is 0 and the characteristic polynomial has repeated roots, /2 , /2 . Now we use the roots to solve equation (1) in this case. We have only one exponential solution, so we need to multiply it by to get the second solution. Basic solutions: bt /2 , te bt /2 . General solution: x t bt /2 ()=( 1 . As in the overdamped case, this does not oscillate. It is worth noting that for a ﬁxed and , choosing to be the critical damping value gives the fastest return of the system to its equilibrium position. In engineering design this is often a desirable property. This can be seen by considering the roots, but we will not go through the algebra that shows this. (See ﬁgure (4).) 4

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Under, Over and Critical Damping OCW 18.03SC .. . Example 3. Show that the system 0 is critically damped and graph the solution with initial conditions )= 1, )= 0. Solution. Characteristic equation: 2 4 0. Characteristic roots: (this factors) 2, 2. Exponential solutions: (only one) . General solution: )= 1 . Because the roots are repeated, the system is critically damped. The intial conditions are satisﬁed when 1 1, 2 2. So, x t ()=( 1 . Figure 3: The critically damped graph for example 3. Notice that qualitatively the graphs for the overdamped and critically damped cases are similar. .. . The following ﬁgure shows plots for solutions to bx 0 with initial conditions )= 1, )= 0. The three plots are 1 under- damped; 2 critically damped (dashed line); 3 overdamped. Notice that the critically damped curve has the fastest decay. .. Figure 4: Plots of solutions to bx 0. 5

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