The system shown in Fig. 1.10(a) consists of a mass m which is allowed to move only in the horizontal direction. The mass is connected to the ground by a spring and damper. The spring has a stiffness coefficient k, and the damper has a viscous damping coefficient c. The harmonic force Fx = F0

Question

The system shown in Fig. 1.10(a) consists of a mass m which is allowed to move only in the horizontal direction. The mass is connected to the ground by a spring and damper. The spring has a stiffness coefficient k, and the damper has a viscous damping coefficient c. The harmonic force [latex]F_{x} = F_{0}[/latex] sin [latex]\omega_{f}t[/latex] acts on the mass as shown in Fig. 1.10. Derive the differential equation of motion of this system, assuming that there is no friction force between the mass and the ground.

Accepted Answer

Since the motion is in the plane, we have the following equations

m[latex]\ddot{x}[/latex] = [latex]F_{x}[/latex], m[latex]\ddot{y}[/latex] = [latex]F_{y}[/latex]
From the free body diagram shown in the figure, we have
[latex]F_{x}[/latex] = [latex]F_{0}[/latex] sin [latex]\omega_{f}t[/latex] − c[latex]\dot{x}[/latex] − kx
[latex]F_{y}[/latex] = N − mg

where N is the reaction force and g is the gravitational constant. Since the motion is constrained to be only in the horizontal direction, we have [latex]\ddot{y}[/latex] = 0.
That is, the two equations of motion are given by

m[latex]\ddot{x}[/latex] = [latex]F_{0}[/latex] sin [latex]\omega_{f}t[/latex] − c[latex]\dot{x}[/latex] − kx
0 = N − mg
which can be written as
m[latex]\ddot{x}[/latex] + c[latex]\dot{x}[/latex] + kx = [latex]F_{0}[/latex] sin [latex]\omega _{f}t[/latex]
N = mg

There is only one second-order differential equation since the system has only one degree of freedom. The second equation which is an algebraic equation can be used to determine the reaction force N. The differential equation of motion given in the preceding equation is the standard equation for the linear damped forced vibration of single degree of freedom systems. This equation and its solution will be discussed in more detail in Chapter 4. The equation of forced vibration of undamped single degree of freedom systems can be extracted from this equation by letting the damping coefficient c equal zero.
In this special case, one has
m[latex]\ddot{x}[/latex]+ kx = [latex]F_{0}[/latex] sin [latex]\omega _{f}t[/latex]
If the force in this equation is equal to zero, one obtains the equation of motion for the linear free undamped vibration of single degree of freedom systems as
m[latex]\ddot{x}[/latex] + kx = 0
The solution of this equation is examined in Chapter 3 of this book.

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