Question 1.4: Find the nonlinear and linear differential equations of moti...
Find the nonlinear and linear differential equations of motion of the system shown in Fig. 1.17, assuming that z(t) is a specified known displacement.
Since the displacement z(t) of the slider block is specified, the number of degrees of freedom of the system reduces to one. From the free body diagram shown in the figure and by taking the moment about point O, one has
M_{a} = −mgl sin θ
M_{eff} = m\ddot{x}l cos θ + m\ddot{y}l sin θ
The mass m is assumed to be a point mass or a particle and therefore it has no moment of inertia about its center of mass. The coordinates x and y of this mass are given by
x = z + l sin θ, y = −l cos θ
Differentiating these coordinates with respect to time yields
\dot{x}= \dot{z} + \dot{θ}l cos θ, \dot{y} = \dot{θ}l sin θ
By differentiating these velocities with respect to time, one obtains
\ddot{x} = \ddot{z} + \ddot{θ}l cos θ − \dot{θ}^{2}l sin θ
\ddot{y} = \ddot{θ}l sin θ + \dot{θ²}l cos θ
Substituting these equations into the M_{eff} equation yields
M_{eff} = m(\ddot{z} +\ddot{θ}l cos θ − \dot{θ}² l sin θ)l cos θ + m(\ddot{θ}l sin θ + \dot{θ^{2}}l cos θ)l sin θ = m\ddot{z}l cos θ + ml²\ddot{θ}Because M_{eff} = M_{a}, the nonlinear differential equation of motion can be obtained as
ml² \ddot{θ} + mgl sin θ = −m\ddot{z}l cos θ
In order to obtain the linear differential equation, we assume small oscillations and use early linearization by linearizing the kinematic relationships. In this case,
x ≈ z + lθ, y ≈ −l
which yield
\dot{x} = \dot{z} + l \dot{θ}, \ddot{x} = \ddot{z} + l \ddot{θ}, \dot{y} = \ddot{y} = 0
The applied and effective moment equations can be written as
M_{a} = −mglθ, M_{eff} = m\ddot{x}l = m\ddot{z}l + ml² \ddot{θ}
Equating these two equations, one obtains the following linear differential equation of motion
ml² \ddot{θ} + mglθ = −m\ddot{z}l
Clearly, this equation can be obtained from the nonlinear equation previously obtained in this example by using the assumption of small oscillations.