# 老干部象棋比赛: 应用于干摩擦振动系统中的Galerkin法Galerkin Method Applied to the Vibration System with Dry Friction

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In this paper, the approximate periodic solution of a vibration system subjected to dry friction is studied. The periodic solution of this system approximately represents the Fourier series, the dif-ferential equation is transformed into nonlinear algebraic equations by Galerkin method, and the nonlinear algebraic equations are solved by Broyden method. Finally, numerical integration results are used to verify the effectiveness of the method.

1. 引言

2. 模型的建立

Figure 1. Schematic diagram of vibration system with dry friction

$m x ¨ + c x ˙ + k x = F sin ω t + f$ (1)

$f = ? μ m g sign ( x ˙ )$ (2)

$m x ¨ + c x ˙ + k x + μ m g sign ( x ˙ ) = F sin ω t$ (3)

$ω 0 = k m$， $ξ = c ω 0 2 k$， $τ = ω t$， $Ω = ω ω 0$， $α = F k$， $β = μ m g k$

$Ω 2 x ¨ + 2 Ω ξ x ˙ + x + β sign ( x ˙ ) = α sin τ$ (4)

3. Galerkin法

$x = a 0 + ∑ n = 1 N ( a n cos ( n τ ) + b n sin ( n τ ) ) = V ( τ ) A$ (5)

$x ˙ = ( V ( τ ) A ) ′ = V ( τ ) C A$ (6)

$x ¨ = ( V ( τ ) C A ) ′ = V ( τ ) D A$ (7)

$r = Ω 2 V ( τ ) D A + 2 Ω ξ V ( τ ) C A + V ( τ ) A + β sign ( V ( τ ) C A ) ? α sin τ = 0$ (8)

$R = ∫ 0 2 π ( Ω 2 V ( τ ) D A + 2 Ω ξ V ( τ ) C A + V ( τ ) A + β sign ( V ( τ ) C A ) ? α sin τ ) V ( τ ) T d τ = 0$ (9)

$x ¯ = [ x ( τ 1 ) , x ( τ 2 ) , ? , x ( τ H ) ] T = Γ A i$ (10)

$r ¯ = [ r ( τ 1 ) , r ( τ 2 ) , ? , r ( τ H ) ] T = Γ r F$， $r F = [ r 0 i , 2 r 1 i , ? , 2 r N i , 2 q 1 i , ? , 2 q N i ] T$。

$Γ ? 1 = 1 H ( 1 ? 1 2 cos τ 1 ? 2 cos ( τ H ) ? ? ? 2 cos ( N τ 1 ) ? 2 cos ( N τ H ) 2 sin τ 1 ? 2 sin ( τ H ) ? ? ? 2 sin ( N τ 1 ) ? 2 sin ( N τ H ) )$，

$R i = 1 2 π ∫ 0 2 π r T T d t = [ r 0 i , r 1 i , ? , r N i , q 1 i , ? , q N i ] T$ (11)

$A i + 1 = A i + Δ A i$ (12)

$‖ Δ A i ‖ < δ$ 成立，则迭代结束，这里的 $‖ Δ A i ‖$ 是当前增量的模， $δ$ 为容许误差。

$B i + 1 = B i + ( R i + 1 ? R i ? B i Δ A i ) Δ A i T Δ A i T Δ A i$ (13)

4. 数值验证

$Ω = 1$， $ξ = 0.1$， $β = 1$， $α = 1.6$ 时，系统的稳态速度与时间的关系如图2所示，Galerkin表示本文的方法，*表示数值积分的方法?？梢钥吹?，系统在运动过程中不存在粘–滑状态转换，仅有滑动状态，两种方法的计算结果可以有比较好的吻合。

$Ω = 1$， $ξ = 0.1$， $β = 1$， $α = 1.4$ 时，系统的稳态速度与时间的关系如图3所示?？梢钥吹剑合低吃谡庾椴问?，其运动过程中存在粘–滑状态的转换，两种方法计算的结果也有很好的吻合。

Figure 2. The relationship between the system steady-state velocity and time

Figure 3. The relationship between the system steady-state velocity and time

5. 结论

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