According to Eq. (2), since the excitation signals are simple harmonic excitations with different frequencies and phases, assuming that the initial phase of the excitation signals is equal to zero, the vibration displacement equation of the box 5 can be written as21:
$$\ddotx_5 + \fracC_n \dotx_5 m_56 + \fracK_n x_5 m_56 = \fracF_ji m_56 $$
(4)
In the formula (4),
\(F_ji = F_cji + F_kji\),\(F_cji = c_15 (\dotx_1 – \dotx_2 ) + c_25 (\dotx_2 – \dotx_5 ) + c_45 (\dotx_4 – \dotx_5 )\),\(F_kji = k_15 (t)\xi_1 + k_25 (t)\xi_2 + k_45 (t)\xi_3\),\(F_ji\) is the excitation signal.\(F_cji\) is damping force.\(F_kji\) is elastic force.Assuming that the vibration displacement of the box 5 is the superposition of harmonic excitations with different frequencies and phases, Eq. (4) can be written as follows:
$$\ddotx_5 + \fracC_n \dotx_5 m_56 + \fracK_n x_5 m_56 = \frac\sum\limits_i = 1^n F_\max ^i \sin (\omega_i t) m_56 $$
(5)
In the formula (5), \(F_\max ^i\) is the \(i\) amplitude of the excitation signal. \(\omega_i\) is the \(i\) frequency of the excitation signal. \(n\) is the frequency number of the excitation signal.Then: \(B_s^i = \fracF_\max ^i K_n \) is the displacement caused by static force \(F_\max ^i\).\(\varsigma = \fracC_n 2\sqrt m_56 K_n \) is viscous damping ratio. \(\omega_n = \sqrt {\fracK_n {m_56 }}\) is natural angular frequency of undamped system.Accordingly, Formula (5) can be rewritten as:
$$\ddotx_5 + 2\varsigma \omega_n \dotx_5 + \omega_n^2 x_5 = B_s^i \omega_i^2 \sum\limits_i = 1^n \sin (\omega_i t)$$
(6)
Formula (6) is a second-order linear non-homogeneous differential equation with constant coefficients. According to the theory of ordinary differential equations, the full solution of the differential equation of formula (6) is the general solution of the corresponding homogeneous equation \(x^\prime_5\) and special solution of nonhomogeneous equation \(x^\prime\prime_5\) two parts are superimposed.The general solution of the homogeneous equation corresponding to formula (6) can be written as22.
$$x^\prime_5 = \sum\limits_i = 1^i = n e^ – \varsigma \omega_n t [C_1i \cos (\omega_i t) + C_2i \sin (\omega_i t)]$$
(7)
Obviously, in the bracket of formula (7) , \(C_1i \cos (\omega_i t) + C_2i \sin (\omega_i t) \le C_1i + C_2i\) with the increase of time, the exponential term of the general solution \(e^ – \varsigma \omega_n t \to 0\).Therefore, the general solution is a physical motion that decays with time, and it is a transient response of the system, indicating the free vibration response of the damped system. It is meaningful only for a short period of time after the vibration starts. With the increase of time, the amplitude of the general solution will decay to zero, and the greater the damping, the faster the amplitude of the general solution will decay.
According to \(\omega_n = \sqrt {\fracK_n m_5 }\),It can be seen that the natural frequencies of each order of composite box are higher than those of ordinary box in different degrees. Therefore, the decrease of the mass of composite box leads to the increase of viscous damping ratio to a certain extent, which leads to the exponential term of formula (7) \(\varsigma \omega_n\) enlarge, therefore, the decay process of transient response is accelerated and the decay time is shortened, and the dynamic characteristics are improved.
Since the excitation signal is a simple harmonic signal, the form of the special solution of the non-homogeneous equation in formula (6) can be assumed as follows:
$$x^\prime\prime_5 = \sum\limits_i = 1^n X_\max ^i \sin (\omega_i t – \psi_c^i )$$
(8)
In the formula (8), \(X_\max ^i\) is the amplitude of forced vibration. \(\psi_c^i\) is the phase difference between the displacement response and the excitation signal. Obviously, the excitation signal and the displacement response have the same frequency. The formula (8) is brought into the formula (6) to obtain:
$$\sum\limits_i = 1^n {X_\max ^i \left\ (\omega_n^2 – \omega_i^2 )\sin (\omega_i t – \psi_c^i ) + 2\varsigma \omega_n \left. \omega_i \cos (\omega_i t – \psi_c^i ) \right\ \right.} = \sum\limits_i = 1^n [B_s^i \omega_i^2 \sin (\omega_i t)]$$
(9)
Expand the excitation signal function on the right side of the equal sign of formula (9) into the form of trigonometric function, and get21:
$$\beginarray*20c {\sum\limits_i = 1^n B_s^i \omega_i^2 \sin (\omega_i t) = \sum\limits_i = 1^n {\left\ \left. B_s^i \omega_i^2 \sin (\omega_i t – \psi_c^i + \psi_c^i ) \right\ \right.} } \\ { = \sum\limits_i = 1^n {\left\ \left. B_s^i \omega_i^2 [\cos \psi_c^i \sin (\omega_i t – \psi_c^i ) + \sin \psi_c^i \cos (\omega_i t – \psi_c^i )] \right\ \right.} } \\ \endarray$$
(10)
Comparing Eqs. (9) and (10), we get:
$$\left\{ {\beginarray*20c {\sum\limits_i = 1^n {\left\ \left. (\omega_n^2 – \omega_i^2 )X_\max ^i \right\ \right.} = \sum\limits_i = 1^n {\left\ \left. B_s^i \omega_i^2 \cos \psi_c^i \right\ \right.} } \\ {\sum\limits_i = 1^n {\left\ \left. 2\varsigma \omega_n \omega_i X_\max ^i \right\ \right.} = \sum\limits_i = 1^n {\left\ \left. B_s^i \omega_i^2 \sin \psi_c^i \right\ \right.} } \\ \endarray } \right.$$
(11)
After the transformation of formula (11), it is brought into \(\sin^2 \psi_c^i + \cos^2 \psi_c^i = 1\),the formula for calculating the phase difference between the amplitude and displacement response of forced vibration lagging behind the excitation signal is obtained:
$$\left\{ {\beginarray*20c {X_\max = \sum\limits_i = 1^n {\left\{ {\left. {\fracB_s^i \omega_n^2 \sqrt (\omega_n^2 – \omega_i^2 )^2 + (2\varsigma \omega_n \omega_i )^2 } \right\}} \right.} } \\ {\psi_c = \sum\limits_i = 1^n {\left\{ {\left. {\arctan \left( \frac\sin \psi_c^i \cos \psi_c^i \right)} \right\} = \sum\limits_i = 1^n {\left\{ {\left. {\arctan \left( \frac2\varsigma \omega_n \omega_i \omega_n^2 – \omega_i^2 \right)} \right\}} \right.} } \right.} } \\ \endarray } \right.$$
(12)
In the formula (12),\(X_\max \) is the amplitude of forced vibration under the joint action of n excitation signals. \(\psi_c\) is n excitation signals, and the phase difference between the displacement response and the excitation signals.
Since the natural frequencies of the composite box are higher than those of the ordinary box in different degrees, according to Formula (12), the natural frequencies in the molecules of the formula for calculating the amplitude of the forced vibration show a square relationship, so that the amplitude of the forced vibration of the composite box is higher than that of the ordinary box under the action of the excitation signal. In addition, the increase of the natural frequencies of the composite box can, to a certain extent, lead to the decrease of the phase difference in Eq. (12) and the decrease of the system delay, thus improving the dynamic characteristics of the system.
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