Spring Eqn w/ Friction Consider the equation: m d 2 y dt 2 + b dy dt - PowerPoint PPT Presentation

Spring Eqn w/ Friction Consider the equation: m d 2 y dt 2 + b dy dt + ky = 0 A spring w/ stiffness k , friction b and mass m , all positive values Free Vibrations – p. 1/5

Spring Eqn w/ Friction Consider the equation: m d 2 y dt 2 + b dy dt + ky = 0 A spring w/ stiffness k , friction b and mass m , all positive values The characteristic equation is: mr 2 + br + y = 0 √ √ b 2 − 4 mk b 2 − 4 mk which has roots r = − b ± = − b 1 2 m ± 2 m 2 m Free Vibrations – p. 1/5

Spring Eqn w/ Friction Consider the equation: m d 2 y dt 2 + b dy dt + ky = 0 A spring w/ stiffness k , friction b and mass m , all positive values The characteristic equation is: mr 2 + br + y = 0 √ √ b 2 − 4 mk b 2 − 4 mk which has roots r = − b ± = − b 1 2 m ± 2 m 2 m √ √ b 2 − 4 mkt + c 2 e − b So y H ( t ) = c 1 e − b 2 m + 1 2 m − 1 b 2 − 4 mkt 2 m 2 m Free Vibrations – p. 1/5

Underdamped Vibrations If the friction coefficient, b , is small enough so that the discriminant b 2 − 4 mk < 0 , the result is underdamped vibrations - a sinusoidal vibration whose size is exponentially dying. Free Vibrations – p. 2/5

Underdamped Vibrations If the friction coefficient, b , is small enough so that the discriminant b 2 − 4 mk < 0 , the result is underdamped vibrations - a sinusoidal vibration whose size is exponentially dying. The roots of the characteristic polynomial are a complex conjugate pair: √ √ { r 1 , r 2 } = {− b i 4 mk − b 2 , − b i 2 m + 2 m − 4 mk − b 2 } 2 m 2 m Free Vibrations – p. 2/5

Underdamped Vibrations If the friction coefficient, b , is small enough so that the discriminant b 2 − 4 mk < 0 , the result is underdamped vibrations - a sinusoidal vibration whose size is exponentially dying. The roots of the characteristic polynomial are a complex conjugate pair: √ √ { r 1 , r 2 } = {− b i 4 mk − b 2 , − b i 2 m + 2 m − 4 mk − b 2 } 2 m 2 m √ √ b 2 − 4 mkt + c 2 e − b So y H ( t ) = c 1 e − b 2 m + 1 2 m − 1 b 2 − 4 mkt 2 m 2 m Free Vibrations – p. 2/5

Rewriting as sin/cos and phase Recall Euler’s Equation: e it = cos( t ) + i sin( t ) . If the roots of the characteristic polynomial are a complex conjugate pair: { r 1 , r 2 } = { a + bi, a − bi } Then c 1 e a + bi + c 2 e a − bi can be rewritten as: Free Vibrations – p. 3/5

Rewriting as sin/cos and phase Recall Euler’s Equation: e it = cos( t ) + i sin( t ) . If the roots of the characteristic polynomial are a complex conjugate pair: { r 1 , r 2 } = { a + bi, a − bi } Then c 1 e a + bi + c 2 e a − bi can be rewritten as: c 1 ( e a e bi ) + c 2 ( e a e − bi ) Free Vibrations – p. 3/5

Rewriting as sin/cos and phase Recall Euler’s Equation: e it = cos( t ) + i sin( t ) . If the roots of the characteristic polynomial are a complex conjugate pair: { r 1 , r 2 } = { a + bi, a − bi } Then c 1 e a + bi + c 2 e a − bi can be rewritten as: c 1 ( e a e bi ) + c 2 ( e a e − bi ) = e a ( c 1 e bi + c 2 e − bi ) Free Vibrations – p. 3/5

Rewriting as sin/cos and phase Recall Euler’s Equation: e it = cos( t ) + i sin( t ) . If the roots of the characteristic polynomial are a complex conjugate pair: { r 1 , r 2 } = { a + bi, a − bi } Then c 1 e a + bi + c 2 e a − bi can be rewritten as: c 1 ( e a e bi ) + c 2 ( e a e − bi ) = e a ( c 1 e bi + c 2 e − bi ) = e a ( c 1 (cos( b ) + i sin( b )) + c 2 (cos( − b ) + i sin( − b ))) Free Vibrations – p. 3/5

Rewriting as sin/cos and phase Recall Euler’s Equation: e it = cos( t ) + i sin( t ) . If the roots of the characteristic polynomial are a complex conjugate pair: { r 1 , r 2 } = { a + bi, a − bi } Then c 1 e a + bi + c 2 e a − bi can be rewritten as: c 1 ( e a e bi ) + c 2 ( e a e − bi ) = e a ( c 1 e bi + c 2 e − bi ) = e a ( c 1 (cos( b ) + i sin( b )) + c 2 (cos( − b ) + i sin( − b ))) = e a ( c 1 (cos( b ) + i sin( b )) + c 2 (cos( b ) − i sin( b ))) Free Vibrations – p. 3/5

Rewriting as sin/cos and phase Recall Euler’s Equation: e it = cos( t ) + i sin( t ) . If the roots of the characteristic polynomial are a complex conjugate pair: { r 1 , r 2 } = { a + bi, a − bi } Then c 1 e a + bi + c 2 e a − bi can be rewritten as: c 1 ( e a e bi ) + c 2 ( e a e − bi ) = e a ( c 1 e bi + c 2 e − bi ) = e a ( c 1 (cos( b ) + i sin( b )) + c 2 (cos( − b ) + i sin( − b ))) = e a ( c 1 (cos( b ) + i sin( b )) + c 2 (cos( b ) − i sin( b ))) = e a (( c 1 + c 2 ) cos( b ) + i ( c 1 − c 2 ) sin( b )) = e a ( d 1 cos( b ) + d 2 sin( b )) (where d 1 = c 1 + c 2 and d 2 = i ( c 1 − c 2 ) ) Free Vibrations – p. 3/5

Rewriting as sin/cos and phase Recall Euler’s Equation: e it = cos( t ) + i sin( t ) . If the roots of the characteristic polynomial are a complex conjugate pair: { r 1 , r 2 } = { a + bi, a − bi } Then c 1 e a + bi + c 2 e a − bi can be rewritten as: c 1 ( e a e bi ) + c 2 ( e a e − bi ) = e a ( c 1 e bi + c 2 e − bi ) = e a ( c 1 (cos( b ) + i sin( b )) + c 2 (cos( − b ) + i sin( − b ))) = e a ( c 1 (cos( b ) + i sin( b )) + c 2 (cos( b ) − i sin( b ))) = e a (( c 1 + c 2 ) cos( b ) + i ( c 1 − c 2 ) sin( b )) = e a ( d 1 cos( b ) + d 2 sin( b )) (where d 1 = c 1 + c 2 and d 2 = i ( c 1 − c 2 ) ) Also, = Ae a cos( b + φ ) � 2 and φ = arctan( d 1 d 2 1 + d 2 (where A = d 2 ) ) Free Vibrations – p. 3/5

Underdamped (cont) If the friction coefficient, b , is small enough so that the discriminant b 2 − 4 mk < 0 , √ 4 mk − b 2 ) t + c 2 e ( − √ b i b i y H ( t ) = c 1 e ( − 2 m + 4 mk − b 2 ) t 2 m − 2 m 2 m Free Vibrations – p. 4/5

Underdamped (cont) If the friction coefficient, b , is small enough so that the discriminant b 2 − 4 mk < 0 , √ 4 mk − b 2 ) t + c 2 e ( − √ b i b i y H ( t ) = c 1 e ( − 2 m + 4 mk − b 2 ) t 2 m − 2 m 2 m 4 mk − b 2 + c 2 e − it √ √ b it 2 m t ( c 1 e 4 mk − b 2 ) = e − 2 m 2 m Free Vibrations – p. 4/5

Underdamped (cont) If the friction coefficient, b , is small enough so that the discriminant b 2 − 4 mk < 0 , √ 4 mk − b 2 ) t + c 2 e ( − √ b i b i y H ( t ) = c 1 e ( − 2 m + 4 mk − b 2 ) t 2 m − 2 m 2 m 4 mk − b 2 + c 2 e − it √ √ b it 2 m t ( c 1 e 4 mk − b 2 ) = e − 2 m 2 m √ √ b 2 m t (( c 1 + c 2 ) cos( t 4 mk − b 2 ) + i ( c 1 − c 2 ) sin( t 4 mk − b 2 )) = e − 2 m 2 m √ √ b 2 m t ( d 1 cos( t 4 mk − b 2 ) + d 2 sin( t 4 mk − b 2 )) = e − 2 m 2 m Free Vibrations – p. 4/5

Underdamped (cont) If the friction coefficient, b , is small enough so that the discriminant b 2 − 4 mk < 0 , √ 4 mk − b 2 ) t + c 2 e ( − √ b i b i y H ( t ) = c 1 e ( − 2 m + 4 mk − b 2 ) t 2 m − 2 m 2 m 4 mk − b 2 + c 2 e − it √ √ b it 2 m t ( c 1 e 4 mk − b 2 ) = e − 2 m 2 m √ √ b 2 m t (( c 1 + c 2 ) cos( t 4 mk − b 2 ) + i ( c 1 − c 2 ) sin( t 4 mk − b 2 )) = e − 2 m 2 m √ √ b 2 m t ( d 1 cos( t 4 mk − b 2 ) + d 2 sin( t 4 mk − b 2 )) = e − 2 m 2 m √ 2 m t sin( t 4 mk − b 2 + φ ) b = Ae − 2 m Free Vibrations – p. 4/5

Underdamped (cont) If the friction coefficient, b , is small enough so that the discriminant b 2 − 4 mk < 0 , √ 4 mk − b 2 ) t + c 2 e ( − √ b i b i y H ( t ) = c 1 e ( − 2 m + 4 mk − b 2 ) t 2 m − 2 m 2 m 4 mk − b 2 + c 2 e − it √ √ b it 2 m t ( c 1 e 4 mk − b 2 ) = e − 2 m 2 m √ √ b 2 m t (( c 1 + c 2 ) cos( t 4 mk − b 2 ) + i ( c 1 − c 2 ) sin( t 4 mk − b 2 )) = e − 2 m 2 m √ √ b 2 m t ( d 1 cos( t 4 mk − b 2 ) + d 2 sin( t 4 mk − b 2 )) = e − 2 m 2 m √ 2 m t sin( t 4 mk − b 2 + φ ) b = Ae − 2 m 2 1 -10 -5 5 10 -1 -2 Free Vibrations – p. 4/5

Overdamped Case If the friction coefficient, b , is large enough so that the discriminant b 2 − 4 mk > 0 , the result is overdamped - an exponentially dying curve crossing the axis at most once. Free Vibrations – p. 5/5

Overdamped Case If the friction coefficient, b , is large enough so that the discriminant b 2 − 4 mk > 0 , the result is overdamped - an exponentially dying curve crossing the axis at most once. The roots of the characteristic polynomial are two real numbers, both negative: √ √ { r 1 , r 2 } = {− b i 4 mk − b 2 , − b i 4 mk − b 2 } 2 m + 2 m − 2 m 2 m Free Vibrations – p. 5/5

Overdamped Case If the friction coefficient, b , is large enough so that the discriminant b 2 − 4 mk > 0 , the result is overdamped - an exponentially dying curve crossing the axis at most once. The roots of the characteristic polynomial are two real numbers, both negative: √ √ { r 1 , r 2 } = {− b i 4 mk − b 2 , − b i 4 mk − b 2 } 2 m + 2 m − 2 m 2 m √ b 2 − 4 mkt + c 2 e − √ b 1 b 1 2 m + b 2 − 4 mkt So y H ( t ) = c 1 e − 2 m − 2 m 2 m = c 1 e r 1 t + c 2 e r 2 t Free Vibrations – p. 5/5

Overdamped Case If the friction coefficient, b , is large enough so that the discriminant b 2 − 4 mk > 0 , the result is overdamped - an exponentially dying curve crossing the axis at most once. The roots of the characteristic polynomial are two real numbers, both negative: √ √ { r 1 , r 2 } = {− b i 4 mk − b 2 , − b i 4 mk − b 2 } 2 m + 2 m − 2 m 2 m √ b 2 − 4 mkt + c 2 e − √ b 1 b 1 2 m + b 2 − 4 mkt So y H ( t ) = c 1 e − 2 m − 2 m 2 m = c 1 e r 1 t + c 2 e r 2 t 0.4 0.3 0.2 0.1 2 4 6 8 -0.1 -0.2 Free Vibrations – p. 5/5