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The general motion of an inclined impact damper with friction
Journal of Sound and Vibration (1995) 184(3), 417–427
THE GENERAL MOTION OF AN INCLINED IMPACT DAMPER WITH FRICTION C. N. B Mechanical Engineering Department, The City College of the City University of New York, Convent Avenue at 138 West, New York, New York, 10031, U.S.A. (Received 2 December 1993, and in final form 6 June 1994) An exact approach is presented to study N stable impacts per period of motion of an inclined impact damper with friction and collision on either one or both sides of the main mass with identical and non-identical coefficients of restitution. The modifications required to accommodate cases of pressurized damper mass and a damper mass oscillating on a spring-supported platform are presented. The comparison of theoretical predictions with results obtained using numerical simulation of motion on a digital computer and previous results indicate good agreement. 7 1995 Academic Press Limited
. 1. INTRODUCTION
An impact damper is a lightweight auxiliary device which consists of a container with a loose mass. This damper is effective for discrete and continuous systems in resonant horizontal operation and produces maximum displacement reduction during two equispaced impacts per cycle of motion under an external sinusoidal force [1–3]. However, in many practical applications, the damper may operate in an inclined direction [4, 5] with friction between the main mass and the damper mass. The speed of the damper mass increases as it moves in the direction of gravity, and decreases as it moves in the opposite direction. Hence, it is important to study the effect of gravity and friction on periodic motions and the displacement amplitude reduction during inclined operation. However, an exact analysis of general periodic motions with friction or gravity acting alone or acting together is not available and is presented in the following [2–7]. The 2N non-linear equations governing this motion have been developed. A single equation for a single unknown is developed for the practically important two impacts/cycle motion. The applications of the proposed theory to study cases of a damper mass oscillating on a spring-supported platform [8] and a pressurized damper mass [9] are also presented. Damping effectiveness of an inclined impact damper with friction during forced vibration is investigated. Comparison of theoretical predictions with previous results and with results obtained using a numerical simulation approach showed good agreement. 2. THEORY
An impact damper shown in Figure 1(a) consists of a primary mass M excited by an external sinusoidal force F sin (Vt), a linear spring with stiffness K and a viscous dashpot having a damping constant C. An impact damper consists of a loose mass m which can move uniaxially within a gap of length d. The system is assumed to vibrate along a line making an angle u with the horizontal plane. A free body diagram of m is shown in Figure 1(b). Let g be the coefficient of friction between m and M. The differential equations of motion of M and m between the ith and the (i + 1)th impacts can be written as [7] 417 0022–460X/95/280417 + 11 $12.00/0
7 1995 Academic Press Limited
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. .
Figure 1. A model of an impact damper (a) and a free body diagram of the secondary mass moving downwards (b).
X + 2jvn X + vn2 X = (F/M) sin Vt + li mgg cos u = (F/M) sin Vt + Fi , Z = g sin u − li gg cos u = gi ,
ti E t E ti + 1 ,
(1) (2)
respectively. The X and Z are the absolute displacements of M and m, respectively. The superscript dot represents the time derivative. The li has value +1 when the relative velocity of m with respect to M is downwards and −1 when it is upwards. The g and m are the gravitational acceleration and the mass ratio m/M, respectively. It is assumed that between impacts the relative velocity between m and M is non-zero and li does not change sign. With these assumptions, the Fi and gi remain constant between impacts and are defined by equations (1) and (2), respectively. The dynamics of this system with a joint motion of two masses cannot be studied using the theoretical approach presented here. 2.1. It is assumed that a steady state motion has been established with N impacts per period T0 (T0 =2pk/V, k=1, 2, . . . , ), and with a known sequence of impacts between m and M. 2N non-linear algebraic equations in 2N unknowns, which govern the assumed motion, are developed next by combining approaches developed previously by the author in references [3] and [7]. The 2N unknowns are the N dislacements Xi of M at impacts, N − 1 contact instants ti , i = 2, 3, . . . , N, and the phase angle t between the applid force and the displacement at the assumed first impact. It can be shown that the displacement and velocity of M just before the (i + 1)th impact can be expressed as Xi + 1 = C1i ai + C2i bi + A sin (ai + 1 + t) + Fi /K, X i + 1 = C3i ai + C4i bi + AV cos (ai + 1 + t),
i = 1, 2, . . . , N.
(3) (4)
V(ti + 1 − ti )/p
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Figure 2. The variation of time durations between impacts, V(ti + 1 − ti )/p, and Xmax /A with gap d in case of two, three and four impacts/cycle of motion of a frictionless horizontal operation (u = 0°, n = 0·0 and g = 1·0), - - - - ; frictionless inclined operation (u = 45°, n = 0·0 and g = 1·0), ——; and an inclined operation with friction (u = 45°, n = 0·5 and g = 1·0), – - – - – ; shown in (a)–(f), respectively, for r = 1·02.
The subscripts i, ia and ib, here and in the following represent the quantities at, just after and just before the ith impact, respectively. The undefined variables in this and subsequent expressions are given in Appendix I and, as far as possible, the variables used previously are reused for an easy comparison [3]. The displacement and the non-constant velocity of small mass m between the ith and the (i + 1)th impact can be obtained by integrating equation (2) as Z(t) = 0·5gi *t 2 + Z ia *t + Zia ,
Z (t) = gi *t + Z ia ,
0 Q *t E ti + 1 − ti ,
(5)
where *t = t − ti is the local time initialized after each impact. The Zia and Z ia are the displacement and the velocity of m just after the ith impact, respectively. The velocity of m just before the (i + 1)th impact, Z (i + 1)b , can be obtained from equation (5) by substituting *t = ti + 1 − ti as Z (i + 1)b = gi (ti + 1 − ti ) + Z ia .
(6)
. .
Xmax /A
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Figure 3. The effect on Xmax /A of: (a) different coefficients of restitution, r = 1·0 and g = 2·0; (b) angle of inclination u at r = 1·0, r = 1·02 and g = 9·81; (c) input frequency V, g = 9·81; (d) coefficient of friction, d = 50, r = 1·0 and g = 9·81. Same symbols are used as in Figure 2.
The average velocity of m between the ith and the (i + 1)th impact Z iave , can be expressed as Z iave = (Z ia + Z (i + 1)b )/2 = (Xi + 1 + Yi + 1 − Xi − Yi )/(ti + 1 − ti ) = Z ia + gi (ti + 1 − ti )/2,
(7)
where Yi , i = 1, 2, . . . , N, represent the relative displacements at impacts and are assumed known, and take values d/2 or −d/2 depending on collision on side 1 or 2, respectively. The assumed pattern of impacts will determine the motion and also Fi and gi . Using equations (6) and (7), the Z ia and Z ib can be expressed in terms of Xi , Yi and ti as Z ia = Z iave − gi (ti + 1 − ti )/2 = (Xi + 1 + Yi + 1 − Xi − Yi )/(ti + 1 − ti ) − gi (ti + 1 − ti )/2, (8) Z ib = Z (i − 1)ave + gi − 1 (ti − ti − 1 )/2 = (Xi + Yi − Xi − 1 − Yi − 1 )/(ti − ti − 1 ) + gi − 1 (ti − ti − 1 )/2.
(9)
The velocities of M and m before and after the ith impact are related by MX ia + mZ ia = MX ib + mZ ib ,
Ri = −(X ia − Z ia )/(X ib − Z ib ),
(10, 11)
where Ri takes value R1 for collision on side 1 and R2 for collision on side 2. The values of X ia and X ib could be found in terms of Xi , ti and Ri using equations (8)–(11) as
421
Xmax /A
Figure 4. The effect of natural frequency v on Xmax /A for g = 9·81, d = 50, u = 15·0 and n = 0·5.
X ia = (1 + m)Ri /(1 + Ri )[(Xi + Yi − Xi − 1 − Yi − 1 )/(ti − ti − 1 ) + gi − 1 (ti − ti − 1 )/2 +(1 − mRi )/(1 + Ri )[(Xi + 1 + Yi + 1 − Xi − Yi )/(ti − ti − 1 ) − gi (ti + 1 − ti )/2],
(12)
X ib = (ei − m)/(1 + ei )([Xi + Yi − Xi − 1 − Yi − 1 )/(ti − ti − 1 ) + gi − 1 (ti − ti − 1 )/2] +(1 + m)/(1 + ei )[Xi + 1 + Yi + 1 − Xi − Yi )/(ti + 1 − ti ) − gi (ti + 1 − ti )/2].
(13)
The ai , given in Appendix I, can be obtained in terms of N Xi , N−1 ti and t by substituting the value of X ia from equation (12). Substituting the obtained expression of ai , and bi (also given in Appendi I) in the right side of equation (3) results after a lengthy algebraic manipulation in a set of governing equations W1i Xi + W2i Xi + 1 + W3i XN + i − 1 + W4i A cos t + W5i A sin t = W6i , i = 1, 2, . . . , N.
(14)
Equating the value of X (i + 1)b , obtained by changing every i + 1 for i in the right-hand side of equation (13) and using values of ai and bi in the right-hand side of equation (4), gives other N governing equations V1i Xi + V2i Xi + 1 V3i Xi + 2 + V4i Xi + N − 1 + V5i A cos t + V6i A sin t = V7i , i = 1, 2, . . . , N.
(15)
These equations are identical to those developed previously for a freely moving secondary mass [4] except for the coefficients W6i and V7i , which indicates that the effect of friction and gravity is reflected in these coefficients only. For the practically important case of two alternating unequispaced impacts per k cycles (N = 2, T0 = 2pk/V, k = 1, 2, . . . ), these equations simplify to cos t = M2 /AM3 ,
sin t = M1 /AM3 , 2 1
2 2
2 3
2
M +M −M A =0
or
A = [(M12 + M22 )/M32 ]1/2.
(16)
When the gap becomes so large that impacts occur only on side 1, the resulting model represents the vibratory conveyor, cast shake-out grits, vibroimpact test rigs, etc. Most of the previous studies dealt only with equispaced one impact per k cycle motions without friction [8]. The equations developed above can be used in this case by substituting Yi = 0, Ri = R and modifying period T0 . A device which uses compressed air to accelerate the
. .
Xmax /A
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Figure 5. The effect of angle of inclination u (a) optimized gap and (b) corresponding Xmax /A for g = 9·81 and r = 1.
damper mass between collisions, instead of cushioning impact as suggested by Cronin and Van [9], can be studied using the proposed theory by assuming that the air pressure remains constant between impacts and reverses immediately after each impact. The values of parameters l and g present in the second term in equations (1) and (2) need to be adjusted to get the required effective force due to pressure and friction. The stability of motion was checked by calculating the eigenvalues of a 4 × 4 stability-governing matrix P = Pi Pi + 1 · · · Pi + N . The elements of Pi are given in Appendix I. A periodic motion is asymptotically stable when all the moduli of eigenvalues of P are less than unity. Non-linear equations (14) and (15) were solved using a Harwell library subroutine NS01A [10]. Equation (16) was also solved numerically. Additionally, a digital simulation program was developed to confirm theoretical predictions. The complete time history of motion was obtained using zero initial conditions. A double precision arithmetic was used throughout. It was also confirmed that theoretically obtained periodic motion was viable, non-wall penetrating and without additional impacts between assumed impacts. 3. RESULTS AND DISCUSSION
Theoretical predictions of horizontal impact damper action based on the theory presented in section (2) agreed with previous results [1–3]. The results of an inclined impact damper with friction, gravity and non-identical coefficients of restitution also agreed with simulation results. The basic parameters were K = M = F = 1, m = 0·042, C = 0·01, d = 37 and R1 = R2 = 0·75, unless stated otherwise. The theoretical results with two, three and four impacts per cycle of motion of an impact damper are presented in Figure 2 where
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Xmax /A
Figure 6. The effect of using compressed air to accelerate the damper mass on Xmax /A for g = 9·81 and n = 0·5.
variation of (ti + 1 − ti )V/p and Xmax /A with gap size for three cases—frictionless horizontal (u = 0, n = 0, g = 2), inclined operation (u = 45°, n = 0 and g = 2) and inclined operation with friction (u = 45°, n = 0·5 and g = 2) at V = 1·02—is shown. This figure indicates that the effect of gravity and friction is detrimental as the corresponding Xma /A is larger than that corresponding to the frictionless horizontal opration. The effect of gravity and friction also results in reduction of stability ranges in most of the cases. However, the maximum reduction range occurs in two impacts per cycle motion. The effect of slightly differing coefficients of restitution on Xmax /A is shown in Figure 3(a), which indicates that the effect is detrimental but insignificant from the practical point of view. The effect of the angle of inclination on Xmax /A during two impacts per cycle of motion for r = 1·0 and r = 1·02 is shown in Figure 3(b) and it indicates that motion is more affected at r = 1 and is detrimental. The effect of inclination u on Xmax /A is presented in Figure 3(c) for given d and as the input frequency V changes for u = 0, 5, 10 and 15 degrees. It indicates that for small u the effect is small but detrimental. A similar effect was found at other input parameters for r q 1. The effect of the frictional force, which is maximum during horizontal operation, on amplitude reduction is shown in Figure 3(d) and it indicates that a large frictional force has a significant effect. Hence, sliding friction between two masses must be reduced as far as possible. The effect of the system’s natural frequency v on Xmax /A is shown in Figure 4. Non-dimensional parameters AK/F0 , j and r were kept constant at 1·0, 0·05 and 1·0, respectively, to have the same basis of comparison. It is indicated in Figure 4 that, as the natural frequency increases, the effect of friction and gravity on response become less significant. This results from the fact that the relative change in speed of the damper between impacts due to gravity and friction becomes insignificant and the response is closer to that of a horizontal frictionless damper. The effect of gravity on variation of optimal gap which produces maximum amplitude reduction, and on the corresponding Xmax /A, is shown in Figure 5. It can be seen that for small values of u, the optimal gap does not differ significantly from the corresponding value at u = 0 and, hence, previous results can be used when the angle u is small. The one impact per cycle of motion of a vertical system with the damper hitting only side 1 was also investigated. The stability range of a vertical impact damper in the high frequency range with small damper mass was studied. As an example, for K = M = 1·0,
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. .
F = 0·8, C = 0·1, m = 0·01, V = 10·0, R1 = 0·75, u = 90°, n = 0·0 and g = 1 unit/s2, R1 = 0·75, m = 0·01, the stability range was found to be 0·45 Q AV 2/g Q 1·12, which agrees with the previous results [11]. The effect of using compressed air to accelerate the damper mass between collisions (i.e., Fi = 2mgi , n = 0·5) was also studied using the theory presented in section 2 and the simulation approach. The results for the horizontal system are presented in Figure 6 and they indicate that the effect of small Fi is insignificant; however, at large Fi , the system becomes unstable. Hence from the practical point of view this idea is not promising. However, previous research indicates that using an active control to adjust the gap is a better alternative [12].
4. CONCLUSIONS
An approach to study the general motion of an inclined impact damper with friction with identical and non-identical coefficients of restitution is presented. The comparison of theoretical predictions with results obtained using a simulation approach and previous results indicated good agreement. Results of a mass oscillating on an elastically supported platform can be obtained from the general theory.
ACKNOWLEDGMENTS
The author gratefully acknowledges the support of PSC-CUNY.
REFERENCES 1. S. F. M and T. K. C 1966 Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics, 33, 586–592. On the stability of the impact damper. 2. S. F. M 1969 Journal of the Acoustical Society of America 47, 229–237. General motion of impact dampers. 3. N. P, C. N. B and K. ML 1983 Journal of Sound and Vibration 87, 41–59. Stable periodic vibroimpacts of an oscillator. 4. H. G. K 1961 Applied Science Research, Series A 10, 369–383. The behavior of a mass–spring system provided with a discontinuous dynamic vibration absorber. 5. M. M. S and B. M 1970 Journal of Mechanical Engineering Science, Institute of Mechanical Engineers 12, 268–287. Effect of gravity on the performance of an impact damper, part 1: steady state motion; part 2: stability of vibration modes. 6. W. M. M and D. R. T. F 1974 Journal of Sound and Vibration 33, 247–265. Impact damper with Coulomb friction. 7. C. N. B and S. S 1985 Journal of Sound and Vibration 103, 457–469. Multiunit impact damper: re-examined. 8. Y. A, I. Y and J. I 1989 Bulletin of the Japan Society of Mechanical Engineers 28, 1466–1472. Impact damper with granular material. 9. D. L. C and N. K. V 1975 ASME Paper No. 75-DET-17. Substitute for the impact damper. 10. H S L 1979 AERE-R 9185. Computer Science and Systems Division, AERE Harwell, Oxfordshire. 11. C. N. B, S. S and N. P 1986 Journal of Sound and Vibration 108, 99–115. Repeated impacts on a sinusoidally vibrating table reappraised. 12. S. F. M, R. K. M, T. J. D and T. K. C 1989 Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics 56, 658–666. Active parameter control of nonlinear vibrating structures.
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APPENDIX I: DEFINITIONS, COEFFICIENTS AND MATRIX ELEMENTS
The undefined variables in the main text are explicitly defined here: j = C/2(KM)1/2, ai = Vti ,
r = V/v,
v = (K/M)1/2,
a1 = 0,
h = (1 − j 2)1/2,
i = 2, 3 · · ·N,
c = tan−1 [2jr/(1 − r 2)],
A = (F/K)/[(1 − r 2)2 + (2jr)2]1/2,
bi = Xi − A sin (ai − c) − Fi /K, ai = (1/h){(1/v)X ia − Ar cos (ai − c) + jbi },
Et = exp[−(j/r)(Vt − ai )],
t = Vti − c, ft = (h/r)(Vt − ai ),
C1t = Et sin ft ,
C3t = v(hC2t − jC1t ),
C2t = Et cos ft ,
C4t = −v(jC2t + hC1t ).
The values of Ei and fi are obtained by substituting i for t and ai + 1 for Vt in the left and right-sides of the equation concerned, respectively. The values of C1i –C4i are obtained by changing t to i on both sides of C1t –C4i . The coefficients required in equations (14) and (15) are exlicitly given below: G1i = (1 + m)Ri V/[(1 + Ri )(ai + N − ai + N − 1 )], G3i = (Ri + 1 − m)V/[(1 + Ri + 1 )(ai + 1 − ai )], W1i = (C1i /hv)(G2i − G1i − jv) − C2i ,
G2i = (1 − mRi )V/[(1 + Ri )(ai + 1 − ai )], G4i = (1 + m)V/[(1 + Ri + 1 )(ai + 2 − ai + 1 )],
W2i = 1 − C1i G2i /hv,
W3i = C1i G1i /hv,
W4i = C1i {(r/h) cos ai + (j/h) sin ai } + C2i sin ai − sin ai + 1 , W5i = C1i {−(r/h) sin ai + (j/h) cos ai } + C2i cos ai − cos ai + 1 , W6i = (C1i /hv)(G1i {Yi + 1 − Yi + N − 1 + gi − 1 (ti − ti − 1 )2/2} + G2i {Yi + 1 − Yi −gi (ti + 1 − ti )2/2}) − (Fi /K)(C1i j/h + C2i − 1), V1i = −G3i − C4i − (C3i /hv)(G1i − G2i + jv),
V2i = G3i − G4i − C3i G2i /hv,
V3i = G4i , V4i = C3i G1i /hv,
V5i = C3i {(r/h) cos ai + (j/h) sin ai } + C4i sin ai − V cos ai + 1 ,
V6i = C3i {−(r/h) sin ai + (j/h) cos ai } + C4i cos ai + V cos ai + 1 , V7i = −G3i (Yi + 1 − Yi + gi (ti + 1 − ti )2/2) − G4i (Yi + 2 − Yi + 1 − gi + 1 (ti + 2 − ti + 1 )2/2) +(C3i /hv)(G1i {Yi + N − Yi + N − 1 + gi − 1 (ti − ti − 1 )2/2) +G2i {Yi + 1 − Yi − gi (ti + 1 − ti )2/2}) − (Fi /K)(C4i + jC3i /h). The undefined variables in equations (16) are given below: S1 = (W22 + W32 )W51 − W52 W11 ,
S2 = (W22 + W32 )W41 − W42 W11 ,
S3 = (W22 + W32 )W61 − W62 W11 , S4 = (V11 + V31 )(V12 + V32 ) − (V22 + V42 )(V21 + V41 ), S5 = (V11 + V31 )V62 − (V22 + V42 )V61 , S7 = (V11 + V31 )V72 − (V22 + V42 )V71 ,
S6 = (V11 + V31 )V52 − (V22 + V42 )V51 , S8 = (W22 + W32 )(W21 + W31 ) − W11 W12 ,
. .
426
S9 = (V11 + V31 − V21 − V41 )/(W11 − W21 − W31 ), H1 = −(W21 + W31 )W52 + W12 W51 ,
H2 = −(W21 + W31 )W42 + W41 W12 ,
H3 = −(W21 + W31 )W62 + W61 W12 , H5 = (V21 + V41 )V62 − (V12 + V32 )V61 ,
H4 = −S4 ,
H6 = (V21 + V41 )V52 − (V12 + V32 )V51 ,
H7 = (V21 + V41 )V72 − (V12 + V32 )V71 , L1 = S1 S4 − S5 S8 ,
L2 = S2 S4 − S6 S8 ,
H8 = −S8 , L3 = S7 S8 − S3 S4 ,
K1 = H1 H4 − H5 H8 ,
K2 = H2 H4 − H6 H8 ,
K 3 = H 7 H8 − H 3 H 4 ,
M1 = K2 L3 − K3 L2 ,
M2 = K3 L1 − K1 L3 ,
M3 = K1 L2 − K2 L1 .
The elements of a stability-governing matrix Pi are given below with a few additional variables: Q1i = (ai + 1 − ai )/V, k1i = (1 − mRi + 1 )/(1 + m),
Q2i = (gi (ai + 1 − ai )/V + Z ia )/V, k2i = m(1 + Ri + 1 )/(1 + m),
k4i = (m − Ri + 1 )/(1 + m),
Q3i = gi /V,
k3i = (1 + Ri + 1 )/(1 + m),
S1i = (jC1i + hC2i )/h,
S2i = C1i /(hv),
S3i = −vC1i /h, S4i = C3i /hv,
S5i = Q2i − A cos (ai + 1 + t) − S6i ,
S7i = S1i /S5i ,
S8i = S2i /S5i ,
S9i = −1/S5i ,
S6i = (ai C3i + bi C4i )/V, S10i = −(ai + 1 − ai )/(S5i V),
S11i = {−S6i − AS1i cos (ai + t) + AVS2i sin (ai + t) + Q2i }/S5i , S12i = A cos (ai + 1 + t) + S6i , S13i = {−vbi C3i + (hC4i − jC3i )[X ia − AV cos (ai + t)]}/hV, S14i = S13i − AV sin (ai + 1 + t), Pi (1, 1) = S1i + S12i (S7i + S8i ),
Pi (1, 2) = S2i + S12i S8i ,
Pia (1, 3) = S10i S12i ,
Pi (1, 4) = −AS1i cos (ai + t) + AVS2i sin (ai + t) − S6i + S11i S12i , Pi (2, 1) = k1i (S3i + S7i S14i + S9i S14i ) + k2i Q3i (S7i + S9i ), Pi (2, 2) = k1i (S4i + S8i S14i ) + k2i Q3i S8i ,
Pi (2, 3) = k1i S10i S14i + k2i (1 + Q3i S10i ),
Pi (2, 4) = k1i (−AS3i cos (ai + t) + AVS4i sin (ai + t) − S13i ) + K1i S11i S14i +k2i Q3i (S11i − 1), Pi (3, 1) = k3i (S3i + S7i S14i + S9i S14i ) + k4i Q3i (S7i + S9i ), Pi (3, 2) = k3i (S4i + S8i S14i ) + k4i Q3i S8i , Pi (3, 3) = k3i S10i S14i + k4i (1 + Q3i S10i ), Pi (3, 4) = k3i (−AS3i cos (ai + t) + AVS4i sin (ai + t) − S13i + S11i S14i ) + k4i Q2i (S11i − 1), Pi (4, 1) = S7i + S9i ,
Pi (4, 2) = S8i ,
Pi (4, 3) = S10i ,
Pi (4, 4) = S11i .
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APPENDIX II: LIST OF SYMBOLS A C d F g K M m N P Pi Ri R 1 , R2 r T0 ti *t X(t) Xi X ia X ib Xmax Yi Z(t) Z (t) Zi Z ia Z iave Z ib V v m g u t j ·
displacement amplitude of M without impacts viscous damping constant gap between M and m amplitude of the external sinusoidal force gravitational acceleration spring stiffness mass of the main system mass of the impact damper number of impacts in a period T0 =Pi Pi + 1 · · · Pi + N , stability-governing matrix a 4 × 4 matrix, relates perturbations at the (i + 1)th impact to those at the ith impact coefficient of restitution at the ith impact coefficients of restitution for sides 1 and 2, respectively =V/v frequency ratio period of motion time at which the ith impact occurs t−ti , time initialized after each impact absolute displacement of M displacement of M at the ith impact velocity of M just after the ith impact velocity of M just before the ith impact maximum displacement of M with impacts relative displacement at the ith impact absolute displacement of m non-constant velocity of m between impacts displacement of m at the ith impact velocity of m just after the ith impact average velocity of m between impacts i and i + 1 velocity of m just before the ith impact frequency of the external sinusoidal force =z(K/M), natural frequency =m/M, mass ratio coefficient of Coulomb friction direction of vibration phase angle between the applied force and the first impact =C/2Mv, damping ratio superscript, represents the time derivative
THE GENERAL MOTION OF AN INCLINED IMPACT DAMPER WITH FRICTION C. N. B Mechanical Engineering Department, The City College of the City University of New York, Convent Avenue at 138 West, New York, New York, 10031, U.S.A. (Received 2 December 1993, and in final form 6 June 1994) An exact approach is presented to study N stable impacts per period of motion of an inclined impact damper with friction and collision on either one or both sides of the main mass with identical and non-identical coefficients of restitution. The modifications required to accommodate cases of pressurized damper mass and a damper mass oscillating on a spring-supported platform are presented. The comparison of theoretical predictions with results obtained using numerical simulation of motion on a digital computer and previous results indicate good agreement. 7 1995 Academic Press Limited
. 1. INTRODUCTION
An impact damper is a lightweight auxiliary device which consists of a container with a loose mass. This damper is effective for discrete and continuous systems in resonant horizontal operation and produces maximum displacement reduction during two equispaced impacts per cycle of motion under an external sinusoidal force [1–3]. However, in many practical applications, the damper may operate in an inclined direction [4, 5] with friction between the main mass and the damper mass. The speed of the damper mass increases as it moves in the direction of gravity, and decreases as it moves in the opposite direction. Hence, it is important to study the effect of gravity and friction on periodic motions and the displacement amplitude reduction during inclined operation. However, an exact analysis of general periodic motions with friction or gravity acting alone or acting together is not available and is presented in the following [2–7]. The 2N non-linear equations governing this motion have been developed. A single equation for a single unknown is developed for the practically important two impacts/cycle motion. The applications of the proposed theory to study cases of a damper mass oscillating on a spring-supported platform [8] and a pressurized damper mass [9] are also presented. Damping effectiveness of an inclined impact damper with friction during forced vibration is investigated. Comparison of theoretical predictions with previous results and with results obtained using a numerical simulation approach showed good agreement. 2. THEORY
An impact damper shown in Figure 1(a) consists of a primary mass M excited by an external sinusoidal force F sin (Vt), a linear spring with stiffness K and a viscous dashpot having a damping constant C. An impact damper consists of a loose mass m which can move uniaxially within a gap of length d. The system is assumed to vibrate along a line making an angle u with the horizontal plane. A free body diagram of m is shown in Figure 1(b). Let g be the coefficient of friction between m and M. The differential equations of motion of M and m between the ith and the (i + 1)th impacts can be written as [7] 417 0022–460X/95/280417 + 11 $12.00/0
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Figure 1. A model of an impact damper (a) and a free body diagram of the secondary mass moving downwards (b).
X + 2jvn X + vn2 X = (F/M) sin Vt + li mgg cos u = (F/M) sin Vt + Fi , Z = g sin u − li gg cos u = gi ,
ti E t E ti + 1 ,
(1) (2)
respectively. The X and Z are the absolute displacements of M and m, respectively. The superscript dot represents the time derivative. The li has value +1 when the relative velocity of m with respect to M is downwards and −1 when it is upwards. The g and m are the gravitational acceleration and the mass ratio m/M, respectively. It is assumed that between impacts the relative velocity between m and M is non-zero and li does not change sign. With these assumptions, the Fi and gi remain constant between impacts and are defined by equations (1) and (2), respectively. The dynamics of this system with a joint motion of two masses cannot be studied using the theoretical approach presented here. 2.1. It is assumed that a steady state motion has been established with N impacts per period T0 (T0 =2pk/V, k=1, 2, . . . , ), and with a known sequence of impacts between m and M. 2N non-linear algebraic equations in 2N unknowns, which govern the assumed motion, are developed next by combining approaches developed previously by the author in references [3] and [7]. The 2N unknowns are the N dislacements Xi of M at impacts, N − 1 contact instants ti , i = 2, 3, . . . , N, and the phase angle t between the applid force and the displacement at the assumed first impact. It can be shown that the displacement and velocity of M just before the (i + 1)th impact can be expressed as Xi + 1 = C1i ai + C2i bi + A sin (ai + 1 + t) + Fi /K, X i + 1 = C3i ai + C4i bi + AV cos (ai + 1 + t),
i = 1, 2, . . . , N.
(3) (4)
V(ti + 1 − ti )/p
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Figure 2. The variation of time durations between impacts, V(ti + 1 − ti )/p, and Xmax /A with gap d in case of two, three and four impacts/cycle of motion of a frictionless horizontal operation (u = 0°, n = 0·0 and g = 1·0), - - - - ; frictionless inclined operation (u = 45°, n = 0·0 and g = 1·0), ——; and an inclined operation with friction (u = 45°, n = 0·5 and g = 1·0), – - – - – ; shown in (a)–(f), respectively, for r = 1·02.
The subscripts i, ia and ib, here and in the following represent the quantities at, just after and just before the ith impact, respectively. The undefined variables in this and subsequent expressions are given in Appendix I and, as far as possible, the variables used previously are reused for an easy comparison [3]. The displacement and the non-constant velocity of small mass m between the ith and the (i + 1)th impact can be obtained by integrating equation (2) as Z(t) = 0·5gi *t 2 + Z ia *t + Zia ,
Z (t) = gi *t + Z ia ,
0 Q *t E ti + 1 − ti ,
(5)
where *t = t − ti is the local time initialized after each impact. The Zia and Z ia are the displacement and the velocity of m just after the ith impact, respectively. The velocity of m just before the (i + 1)th impact, Z (i + 1)b , can be obtained from equation (5) by substituting *t = ti + 1 − ti as Z (i + 1)b = gi (ti + 1 − ti ) + Z ia .
(6)
. .
Xmax /A
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Figure 3. The effect on Xmax /A of: (a) different coefficients of restitution, r = 1·0 and g = 2·0; (b) angle of inclination u at r = 1·0, r = 1·02 and g = 9·81; (c) input frequency V, g = 9·81; (d) coefficient of friction, d = 50, r = 1·0 and g = 9·81. Same symbols are used as in Figure 2.
The average velocity of m between the ith and the (i + 1)th impact Z iave , can be expressed as Z iave = (Z ia + Z (i + 1)b )/2 = (Xi + 1 + Yi + 1 − Xi − Yi )/(ti + 1 − ti ) = Z ia + gi (ti + 1 − ti )/2,
(7)
where Yi , i = 1, 2, . . . , N, represent the relative displacements at impacts and are assumed known, and take values d/2 or −d/2 depending on collision on side 1 or 2, respectively. The assumed pattern of impacts will determine the motion and also Fi and gi . Using equations (6) and (7), the Z ia and Z ib can be expressed in terms of Xi , Yi and ti as Z ia = Z iave − gi (ti + 1 − ti )/2 = (Xi + 1 + Yi + 1 − Xi − Yi )/(ti + 1 − ti ) − gi (ti + 1 − ti )/2, (8) Z ib = Z (i − 1)ave + gi − 1 (ti − ti − 1 )/2 = (Xi + Yi − Xi − 1 − Yi − 1 )/(ti − ti − 1 ) + gi − 1 (ti − ti − 1 )/2.
(9)
The velocities of M and m before and after the ith impact are related by MX ia + mZ ia = MX ib + mZ ib ,
Ri = −(X ia − Z ia )/(X ib − Z ib ),
(10, 11)
where Ri takes value R1 for collision on side 1 and R2 for collision on side 2. The values of X ia and X ib could be found in terms of Xi , ti and Ri using equations (8)–(11) as
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Xmax /A
Figure 4. The effect of natural frequency v on Xmax /A for g = 9·81, d = 50, u = 15·0 and n = 0·5.
X ia = (1 + m)Ri /(1 + Ri )[(Xi + Yi − Xi − 1 − Yi − 1 )/(ti − ti − 1 ) + gi − 1 (ti − ti − 1 )/2 +(1 − mRi )/(1 + Ri )[(Xi + 1 + Yi + 1 − Xi − Yi )/(ti − ti − 1 ) − gi (ti + 1 − ti )/2],
(12)
X ib = (ei − m)/(1 + ei )([Xi + Yi − Xi − 1 − Yi − 1 )/(ti − ti − 1 ) + gi − 1 (ti − ti − 1 )/2] +(1 + m)/(1 + ei )[Xi + 1 + Yi + 1 − Xi − Yi )/(ti + 1 − ti ) − gi (ti + 1 − ti )/2].
(13)
The ai , given in Appendix I, can be obtained in terms of N Xi , N−1 ti and t by substituting the value of X ia from equation (12). Substituting the obtained expression of ai , and bi (also given in Appendi I) in the right side of equation (3) results after a lengthy algebraic manipulation in a set of governing equations W1i Xi + W2i Xi + 1 + W3i XN + i − 1 + W4i A cos t + W5i A sin t = W6i , i = 1, 2, . . . , N.
(14)
Equating the value of X (i + 1)b , obtained by changing every i + 1 for i in the right-hand side of equation (13) and using values of ai and bi in the right-hand side of equation (4), gives other N governing equations V1i Xi + V2i Xi + 1 V3i Xi + 2 + V4i Xi + N − 1 + V5i A cos t + V6i A sin t = V7i , i = 1, 2, . . . , N.
(15)
These equations are identical to those developed previously for a freely moving secondary mass [4] except for the coefficients W6i and V7i , which indicates that the effect of friction and gravity is reflected in these coefficients only. For the practically important case of two alternating unequispaced impacts per k cycles (N = 2, T0 = 2pk/V, k = 1, 2, . . . ), these equations simplify to cos t = M2 /AM3 ,
sin t = M1 /AM3 , 2 1
2 2
2 3
2
M +M −M A =0
or
A = [(M12 + M22 )/M32 ]1/2.
(16)
When the gap becomes so large that impacts occur only on side 1, the resulting model represents the vibratory conveyor, cast shake-out grits, vibroimpact test rigs, etc. Most of the previous studies dealt only with equispaced one impact per k cycle motions without friction [8]. The equations developed above can be used in this case by substituting Yi = 0, Ri = R and modifying period T0 . A device which uses compressed air to accelerate the
. .
Xmax /A
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Figure 5. The effect of angle of inclination u (a) optimized gap and (b) corresponding Xmax /A for g = 9·81 and r = 1.
damper mass between collisions, instead of cushioning impact as suggested by Cronin and Van [9], can be studied using the proposed theory by assuming that the air pressure remains constant between impacts and reverses immediately after each impact. The values of parameters l and g present in the second term in equations (1) and (2) need to be adjusted to get the required effective force due to pressure and friction. The stability of motion was checked by calculating the eigenvalues of a 4 × 4 stability-governing matrix P = Pi Pi + 1 · · · Pi + N . The elements of Pi are given in Appendix I. A periodic motion is asymptotically stable when all the moduli of eigenvalues of P are less than unity. Non-linear equations (14) and (15) were solved using a Harwell library subroutine NS01A [10]. Equation (16) was also solved numerically. Additionally, a digital simulation program was developed to confirm theoretical predictions. The complete time history of motion was obtained using zero initial conditions. A double precision arithmetic was used throughout. It was also confirmed that theoretically obtained periodic motion was viable, non-wall penetrating and without additional impacts between assumed impacts. 3. RESULTS AND DISCUSSION
Theoretical predictions of horizontal impact damper action based on the theory presented in section (2) agreed with previous results [1–3]. The results of an inclined impact damper with friction, gravity and non-identical coefficients of restitution also agreed with simulation results. The basic parameters were K = M = F = 1, m = 0·042, C = 0·01, d = 37 and R1 = R2 = 0·75, unless stated otherwise. The theoretical results with two, three and four impacts per cycle of motion of an impact damper are presented in Figure 2 where
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Xmax /A
Figure 6. The effect of using compressed air to accelerate the damper mass on Xmax /A for g = 9·81 and n = 0·5.
variation of (ti + 1 − ti )V/p and Xmax /A with gap size for three cases—frictionless horizontal (u = 0, n = 0, g = 2), inclined operation (u = 45°, n = 0 and g = 2) and inclined operation with friction (u = 45°, n = 0·5 and g = 2) at V = 1·02—is shown. This figure indicates that the effect of gravity and friction is detrimental as the corresponding Xma /A is larger than that corresponding to the frictionless horizontal opration. The effect of gravity and friction also results in reduction of stability ranges in most of the cases. However, the maximum reduction range occurs in two impacts per cycle motion. The effect of slightly differing coefficients of restitution on Xmax /A is shown in Figure 3(a), which indicates that the effect is detrimental but insignificant from the practical point of view. The effect of the angle of inclination on Xmax /A during two impacts per cycle of motion for r = 1·0 and r = 1·02 is shown in Figure 3(b) and it indicates that motion is more affected at r = 1 and is detrimental. The effect of inclination u on Xmax /A is presented in Figure 3(c) for given d and as the input frequency V changes for u = 0, 5, 10 and 15 degrees. It indicates that for small u the effect is small but detrimental. A similar effect was found at other input parameters for r q 1. The effect of the frictional force, which is maximum during horizontal operation, on amplitude reduction is shown in Figure 3(d) and it indicates that a large frictional force has a significant effect. Hence, sliding friction between two masses must be reduced as far as possible. The effect of the system’s natural frequency v on Xmax /A is shown in Figure 4. Non-dimensional parameters AK/F0 , j and r were kept constant at 1·0, 0·05 and 1·0, respectively, to have the same basis of comparison. It is indicated in Figure 4 that, as the natural frequency increases, the effect of friction and gravity on response become less significant. This results from the fact that the relative change in speed of the damper between impacts due to gravity and friction becomes insignificant and the response is closer to that of a horizontal frictionless damper. The effect of gravity on variation of optimal gap which produces maximum amplitude reduction, and on the corresponding Xmax /A, is shown in Figure 5. It can be seen that for small values of u, the optimal gap does not differ significantly from the corresponding value at u = 0 and, hence, previous results can be used when the angle u is small. The one impact per cycle of motion of a vertical system with the damper hitting only side 1 was also investigated. The stability range of a vertical impact damper in the high frequency range with small damper mass was studied. As an example, for K = M = 1·0,
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F = 0·8, C = 0·1, m = 0·01, V = 10·0, R1 = 0·75, u = 90°, n = 0·0 and g = 1 unit/s2, R1 = 0·75, m = 0·01, the stability range was found to be 0·45 Q AV 2/g Q 1·12, which agrees with the previous results [11]. The effect of using compressed air to accelerate the damper mass between collisions (i.e., Fi = 2mgi , n = 0·5) was also studied using the theory presented in section 2 and the simulation approach. The results for the horizontal system are presented in Figure 6 and they indicate that the effect of small Fi is insignificant; however, at large Fi , the system becomes unstable. Hence from the practical point of view this idea is not promising. However, previous research indicates that using an active control to adjust the gap is a better alternative [12].
4. CONCLUSIONS
An approach to study the general motion of an inclined impact damper with friction with identical and non-identical coefficients of restitution is presented. The comparison of theoretical predictions with results obtained using a simulation approach and previous results indicated good agreement. Results of a mass oscillating on an elastically supported platform can be obtained from the general theory.
ACKNOWLEDGMENTS
The author gratefully acknowledges the support of PSC-CUNY.
REFERENCES 1. S. F. M and T. K. C 1966 Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics, 33, 586–592. On the stability of the impact damper. 2. S. F. M 1969 Journal of the Acoustical Society of America 47, 229–237. General motion of impact dampers. 3. N. P, C. N. B and K. ML 1983 Journal of Sound and Vibration 87, 41–59. Stable periodic vibroimpacts of an oscillator. 4. H. G. K 1961 Applied Science Research, Series A 10, 369–383. The behavior of a mass–spring system provided with a discontinuous dynamic vibration absorber. 5. M. M. S and B. M 1970 Journal of Mechanical Engineering Science, Institute of Mechanical Engineers 12, 268–287. Effect of gravity on the performance of an impact damper, part 1: steady state motion; part 2: stability of vibration modes. 6. W. M. M and D. R. T. F 1974 Journal of Sound and Vibration 33, 247–265. Impact damper with Coulomb friction. 7. C. N. B and S. S 1985 Journal of Sound and Vibration 103, 457–469. Multiunit impact damper: re-examined. 8. Y. A, I. Y and J. I 1989 Bulletin of the Japan Society of Mechanical Engineers 28, 1466–1472. Impact damper with granular material. 9. D. L. C and N. K. V 1975 ASME Paper No. 75-DET-17. Substitute for the impact damper. 10. H S L 1979 AERE-R 9185. Computer Science and Systems Division, AERE Harwell, Oxfordshire. 11. C. N. B, S. S and N. P 1986 Journal of Sound and Vibration 108, 99–115. Repeated impacts on a sinusoidally vibrating table reappraised. 12. S. F. M, R. K. M, T. J. D and T. K. C 1989 Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics 56, 658–666. Active parameter control of nonlinear vibrating structures.
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APPENDIX I: DEFINITIONS, COEFFICIENTS AND MATRIX ELEMENTS
The undefined variables in the main text are explicitly defined here: j = C/2(KM)1/2, ai = Vti ,
r = V/v,
v = (K/M)1/2,
a1 = 0,
h = (1 − j 2)1/2,
i = 2, 3 · · ·N,
c = tan−1 [2jr/(1 − r 2)],
A = (F/K)/[(1 − r 2)2 + (2jr)2]1/2,
bi = Xi − A sin (ai − c) − Fi /K, ai = (1/h){(1/v)X ia − Ar cos (ai − c) + jbi },
Et = exp[−(j/r)(Vt − ai )],
t = Vti − c, ft = (h/r)(Vt − ai ),
C1t = Et sin ft ,
C3t = v(hC2t − jC1t ),
C2t = Et cos ft ,
C4t = −v(jC2t + hC1t ).
The values of Ei and fi are obtained by substituting i for t and ai + 1 for Vt in the left and right-sides of the equation concerned, respectively. The values of C1i –C4i are obtained by changing t to i on both sides of C1t –C4i . The coefficients required in equations (14) and (15) are exlicitly given below: G1i = (1 + m)Ri V/[(1 + Ri )(ai + N − ai + N − 1 )], G3i = (Ri + 1 − m)V/[(1 + Ri + 1 )(ai + 1 − ai )], W1i = (C1i /hv)(G2i − G1i − jv) − C2i ,
G2i = (1 − mRi )V/[(1 + Ri )(ai + 1 − ai )], G4i = (1 + m)V/[(1 + Ri + 1 )(ai + 2 − ai + 1 )],
W2i = 1 − C1i G2i /hv,
W3i = C1i G1i /hv,
W4i = C1i {(r/h) cos ai + (j/h) sin ai } + C2i sin ai − sin ai + 1 , W5i = C1i {−(r/h) sin ai + (j/h) cos ai } + C2i cos ai − cos ai + 1 , W6i = (C1i /hv)(G1i {Yi + 1 − Yi + N − 1 + gi − 1 (ti − ti − 1 )2/2} + G2i {Yi + 1 − Yi −gi (ti + 1 − ti )2/2}) − (Fi /K)(C1i j/h + C2i − 1), V1i = −G3i − C4i − (C3i /hv)(G1i − G2i + jv),
V2i = G3i − G4i − C3i G2i /hv,
V3i = G4i , V4i = C3i G1i /hv,
V5i = C3i {(r/h) cos ai + (j/h) sin ai } + C4i sin ai − V cos ai + 1 ,
V6i = C3i {−(r/h) sin ai + (j/h) cos ai } + C4i cos ai + V cos ai + 1 , V7i = −G3i (Yi + 1 − Yi + gi (ti + 1 − ti )2/2) − G4i (Yi + 2 − Yi + 1 − gi + 1 (ti + 2 − ti + 1 )2/2) +(C3i /hv)(G1i {Yi + N − Yi + N − 1 + gi − 1 (ti − ti − 1 )2/2) +G2i {Yi + 1 − Yi − gi (ti + 1 − ti )2/2}) − (Fi /K)(C4i + jC3i /h). The undefined variables in equations (16) are given below: S1 = (W22 + W32 )W51 − W52 W11 ,
S2 = (W22 + W32 )W41 − W42 W11 ,
S3 = (W22 + W32 )W61 − W62 W11 , S4 = (V11 + V31 )(V12 + V32 ) − (V22 + V42 )(V21 + V41 ), S5 = (V11 + V31 )V62 − (V22 + V42 )V61 , S7 = (V11 + V31 )V72 − (V22 + V42 )V71 ,
S6 = (V11 + V31 )V52 − (V22 + V42 )V51 , S8 = (W22 + W32 )(W21 + W31 ) − W11 W12 ,
. .
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S9 = (V11 + V31 − V21 − V41 )/(W11 − W21 − W31 ), H1 = −(W21 + W31 )W52 + W12 W51 ,
H2 = −(W21 + W31 )W42 + W41 W12 ,
H3 = −(W21 + W31 )W62 + W61 W12 , H5 = (V21 + V41 )V62 − (V12 + V32 )V61 ,
H4 = −S4 ,
H6 = (V21 + V41 )V52 − (V12 + V32 )V51 ,
H7 = (V21 + V41 )V72 − (V12 + V32 )V71 , L1 = S1 S4 − S5 S8 ,
L2 = S2 S4 − S6 S8 ,
H8 = −S8 , L3 = S7 S8 − S3 S4 ,
K1 = H1 H4 − H5 H8 ,
K2 = H2 H4 − H6 H8 ,
K 3 = H 7 H8 − H 3 H 4 ,
M1 = K2 L3 − K3 L2 ,
M2 = K3 L1 − K1 L3 ,
M3 = K1 L2 − K2 L1 .
The elements of a stability-governing matrix Pi are given below with a few additional variables: Q1i = (ai + 1 − ai )/V, k1i = (1 − mRi + 1 )/(1 + m),
Q2i = (gi (ai + 1 − ai )/V + Z ia )/V, k2i = m(1 + Ri + 1 )/(1 + m),
k4i = (m − Ri + 1 )/(1 + m),
Q3i = gi /V,
k3i = (1 + Ri + 1 )/(1 + m),
S1i = (jC1i + hC2i )/h,
S2i = C1i /(hv),
S3i = −vC1i /h, S4i = C3i /hv,
S5i = Q2i − A cos (ai + 1 + t) − S6i ,
S7i = S1i /S5i ,
S8i = S2i /S5i ,
S9i = −1/S5i ,
S6i = (ai C3i + bi C4i )/V, S10i = −(ai + 1 − ai )/(S5i V),
S11i = {−S6i − AS1i cos (ai + t) + AVS2i sin (ai + t) + Q2i }/S5i , S12i = A cos (ai + 1 + t) + S6i , S13i = {−vbi C3i + (hC4i − jC3i )[X ia − AV cos (ai + t)]}/hV, S14i = S13i − AV sin (ai + 1 + t), Pi (1, 1) = S1i + S12i (S7i + S8i ),
Pi (1, 2) = S2i + S12i S8i ,
Pia (1, 3) = S10i S12i ,
Pi (1, 4) = −AS1i cos (ai + t) + AVS2i sin (ai + t) − S6i + S11i S12i , Pi (2, 1) = k1i (S3i + S7i S14i + S9i S14i ) + k2i Q3i (S7i + S9i ), Pi (2, 2) = k1i (S4i + S8i S14i ) + k2i Q3i S8i ,
Pi (2, 3) = k1i S10i S14i + k2i (1 + Q3i S10i ),
Pi (2, 4) = k1i (−AS3i cos (ai + t) + AVS4i sin (ai + t) − S13i ) + K1i S11i S14i +k2i Q3i (S11i − 1), Pi (3, 1) = k3i (S3i + S7i S14i + S9i S14i ) + k4i Q3i (S7i + S9i ), Pi (3, 2) = k3i (S4i + S8i S14i ) + k4i Q3i S8i , Pi (3, 3) = k3i S10i S14i + k4i (1 + Q3i S10i ), Pi (3, 4) = k3i (−AS3i cos (ai + t) + AVS4i sin (ai + t) − S13i + S11i S14i ) + k4i Q2i (S11i − 1), Pi (4, 1) = S7i + S9i ,
Pi (4, 2) = S8i ,
Pi (4, 3) = S10i ,
Pi (4, 4) = S11i .
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APPENDIX II: LIST OF SYMBOLS A C d F g K M m N P Pi Ri R 1 , R2 r T0 ti *t X(t) Xi X ia X ib Xmax Yi Z(t) Z (t) Zi Z ia Z iave Z ib V v m g u t j ·
displacement amplitude of M without impacts viscous damping constant gap between M and m amplitude of the external sinusoidal force gravitational acceleration spring stiffness mass of the main system mass of the impact damper number of impacts in a period T0 =Pi Pi + 1 · · · Pi + N , stability-governing matrix a 4 × 4 matrix, relates perturbations at the (i + 1)th impact to those at the ith impact coefficient of restitution at the ith impact coefficients of restitution for sides 1 and 2, respectively =V/v frequency ratio period of motion time at which the ith impact occurs t−ti , time initialized after each impact absolute displacement of M displacement of M at the ith impact velocity of M just after the ith impact velocity of M just before the ith impact maximum displacement of M with impacts relative displacement at the ith impact absolute displacement of m non-constant velocity of m between impacts displacement of m at the ith impact velocity of m just after the ith impact average velocity of m between impacts i and i + 1 velocity of m just before the ith impact frequency of the external sinusoidal force =z(K/M), natural frequency =m/M, mass ratio coefficient of Coulomb friction direction of vibration phase angle between the applied force and the first impact =C/2Mv, damping ratio superscript, represents the time derivative