( ) Zp THE VIBRATION ABSORBER. Preamble - A NEED arises: lbf in. sec. X p () t = Z p. cos Ω t. Z p () r. ω np. F o. cos Ω t. X p. δ s.

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1 THE VIBRATION ABSORBER Preable - A NEED arie: Lui San Andre (c) 8 MEEN Conider the periodic forced repone of a yte (Kp-Mp) defined by : lbf in : 1 3 lb (t) It natural frequency i: : ec F(t) The EOM (fro SEP) for periodic force excitation with agnitude Fo and frequency Ω i: d d t The olution of [1] i of the for Subtitution of [] into [1] give: or Z p () r : Zp ( 1 r ) + Y p co Ωt [1] Ω [3] () t Z p. co Ωt with: Let: [] r : Zp Ω 1lbf ( Ω ) a the frequency ratio Thu, the periodic force repone of the yte (Kp,Mp) i : () t δ ( 1 r ) co Ωt δ 1 r co Ωt + φ [4] with δ : and the phae angle φ i zero degree for excitation frequencie (Ω) < the natural frequency, φ -18 deg for Ω > natural frequency, and φ-9 degree for Ω natural frequency.

2 The repone Zp (aplitude and phae) a a function of the excitation frequency i: Z P ( Ω) : 1 δ Ω Frequency Repone of M-riary yte.4.1 δ Zp () ec frequency (rad/ec) Note the very large aplitude of otion (unbounded) for excitation at the yte natural frequency. In the graph above, Zp> ean in phae with the external force, Zp< ean 18 deg out of phae with external periodic force. Clearly, the yte cannot be operated at frequencie cloe (or at) the natural frequency. Since there i NO daping, the yte will jut fail b/c the aplitude of otion i jut TOO LARGE! X (t) Now, conider a econdary (K-M) yte attached to the yte (original). In thi cae, the EOM fro the SEP are: (t) F(t) M S M d dt X + + X ( ) [5] co Ω t The yte forced repone i alo periodic, i.e. with identical frequency a that of the periodic excitation force. Thu, let Z p X Z [6] co Ω t The cobined yte i -DOF. Thu, TWO natural frequencie (and natural ode) will appear.

3 ΔΩ + Ω Subtitution of [6] into [5] lead to the algebraic et of equation: The deterinant of the yte of equation [7] i Ω M ( + Ω ) Ω ( M ) Z p Z [7] [8] The olution to the algebraic yte of equation [7] i iple, - ue Craer' rule, for exaple. The repone aplitude for the and econdary ae are: Z p ( Ω) Ω M Note fro Eq. [9] that if ΔΩ Z ( Ω) ΔΩ [9] then Z p ( Ω X ) Ω M [1] at a certain frequency That i, the otion of the yte i zero (NULL)! Ω Ω X A iple and practical SOLUTION: A vibration aborber i a echanical device which perit to reduce (even eliinate) aplitude of vibration at certain excitation frequencie, in particular thoe at which the original yte howed a highly undeirable repone. For exaple, if zero aplitude vibration i deired for excitation at the natural frequency of the original yte, the deigner elect Ω X alo known a a TUNED ABSORBER Which then deterine that the tiffne and a of the econdary yte hould be uch that: ( Ω X M ) ω M np [11] i.e, the natural frequency of the econday yte MUST coincide with that of the original yte

4 The coponent of the vibration aborver (econdary yte) need NOT be the ae ize a the original yte. In practice, the agnitude of K and M are ubtantially aller. Say for then thu, for operation with excitation frequency a 1 M Z p ( Ω) Z p ( Ω) Note a : 1 : and the yte repone are Ω M ΔΩ Z ( ) : ec Z a ΔΩ δ 1 M : equal : Z ( Ω) a ( + Ω ) Ω ( M ) : Ω ΔΩ : 1 3 Z ec ec The SOFTER the econdary yte i (K << Kp), the larget the otion of the econdary yte at the deired frequency

5 The graph below how the FRF (aplitude and phae) of the vibration aborber: Zp & Z ().1 FR of vibration aborber frequency (rad/ec) The graph belo how the aplitude (abolute) of the vibration aborber:.1 Aplitude of otion a 1 Note the null aplitude of otion for yte excitated at the ORIGINAL yte natural frequency. In the graph, Zp,Z > ean in phae with the external force, Zp, Z < ean 18 deg out of phae with external periodic force. Zp & Z ().75 a frequency (rad/ec) ec Note the aplitude of otion i zero for the yte when excited at it original natural frequency. The econdary yte doe have a large aplitude of otion and i out of phae 18 degree with the excitation force. Note that the addition of the econdary (K-M) yte render a -DOF yte with two natural frequencie, one above and one below the original natural frequency. In general, the aller the agnitude of the econdary tiffne and a, the larger the aplitude o otion for the econdary yte ince it i extreely flexible. The yte natural frequencie (1 and alo tend to approach that of the original natural frequency

6 The graph below how the FRF (aplitude and phae) of the vibration aborber: Zp & Z ().1 FR of vibration aborber frequency (rad/ec) The graph belo how the aplitude (abolute) of the vibration aborber:.1 Aplitude of otion a 5 Note the null aplitude of otion for yte excitated at the ORIGINAL yte natural frequency. In the graph, Zp,Z > ean in phae with the external force, Zp, Z < ean 18 deg out of phae with external periodic force. Zp & Z ().75 a frequency (rad/ec) ec

7 The graph below how the FRF (aplitude and phae) of the vibration aborber: Zp & Z ().1 FR of vibration aborber frequency (rad/ec) The graph belo how the aplitude (abolute) of the vibration aborber:.1 Aplitude of otion a Note the null aplitude of otion for yte excitated at the ORIGINAL yte natural frequency. In the graph, Zp,Z > ean in phae with the external force, Zp, Z < ean 18 deg out of phae with external periodic force. Zp & Z ().75 a frequency (rad/ec) ec

8 Deign of vibration aborber The equation of otion for the -DOF yte are: X (t) M S M d dt X + where yte ha:: + X F in Ωt [1] (t) F(t) : 5kg : N F 1 N : Range of excitation frequency: ω in : rad 16 ω ax : rad 4 ω n rad ω n : The yte repone i of the for: X Subtitution of Eq. [] into [1] lead to: Z p Z in( Ωt) [] + Ω Ω M Z p Z F [3] 1) A tuned aborber i deigned o that Z p i.e. no otion of the a. for operation at Ωωn Thu, fro the firt of eqn (3) Ω M F Z [4] Deign aborber (elect K & M) that atify operation within range of excitation frequencie: ω p : The deterinat of the yte of equation [3] i ω rad p Δ( Ω) ( K p + Ω ) Ω ( M ) [5]

9 Let: λ Ω a M tiffne ratio a ratio for tuned aborber Expand Eq. (5), i.e. the characteritic equation: 1 a Δ λ a 1 a Δ λ ( + λ 1 ) ( 1 λ) a a a 1 + a λ 1 ( + λ λ λa + λ a) a 1 λ ( 1 λ) a ( λa + λ ) a_ ( λ) a ax : a_ λ in : a in : a_ λ ax λ λ + a Δ λ + 1 The root of the characteritic equation are: Let lowet: λ 1 () a 1 + λ λ a ax. a in.134 [6] highet ( + a) ( 4a + a ) : λ () a λ in : ω in ω n λ ax : ω ax Given the ax and in value then, fro eqn. (6) ω n : ( + a) 4 ( + a) 4a + a 4a + a + for a : a in Z : Z p : F ak P Z.37 λ 1 () a ( + a) ( 4a + a ) : λ () a : ω 1 λ 1 () a K P : ω λ () a : ( + a) 4a + a + ω in 16 rad ω 1 rad ω rad 4 ω ax rad 4

10 for K in : ak P a : a ax M in : am P K in N M in 6.7 kg Z : Z p : Z.47 K ax : F ak P ak P M ax : ω 1 ω 1 : am P λ 1 () a : 16 rad ( + a) 4a + a λ 1 () a ω 5 rad ω : λ () a : λ () a ( + a) 4a + a + ω in ω ax 16 rad 4 rad DESIGNER elect econdary yte with ince natural frequencie are outide operating range a ax. K ax M ax 1.15 kg N Build Aborber FRF

11 Now let' graph the aplitude and phae lag of frequency repone function for both aborber ( and econdary a otion): Value > ean phae lag of degree, value < ean phae lag of -18 degree with repect to forcing function. Paing through the natural frequencie give a phae lag of -9 degree. Aplitude becoe unbounded while croing the yte natural frequencie. a : a ax a Aborber FRF ω np rad Z ωp : F ak Z ωp.47 [eter] frequency (rad/) Zp Z ω 1 () a rad 16 ω () a rad 5 ω in rad 16 ω ax rad 4