A one-dimensional BK mass-spring model (Burridge and Knopoff, 1967) in the presence of velocity-weakening friction (in a linear form) is applied to dynamically simulate earthquakes. For theoretical analyses, a uniform distribution of the breaking strengths is considered. Results show that for an infinite mass-spring system, three types of the propagation of motion of mass elements can be deduced. The three types of propagation are related to three kinds of velocity-weakening friction, which depend upon the relation among three parameters: (1) the decreasing rate r of the dynamic frictional force with the sliding velocity, (2) the strength L of the leaf spring between the moving plate and a mass element, and (3) the mass m of a mass element. Hence, friction is called subsonic friction when r > 2 (Lm)1/2, sonic friction when r = 2 (Lm)1/2, and supersonic friction when r < 2 (Lm)1/2. In numerical simulations, a fractal distribution of the breaking strengths is considered. The magnitude of an event is defined on the basis of energy. Simulated results show a dependence of the frequency-magnitude (FM) relation on velocity-weakening friction. Velocity-weakening friction with large r or r = ∞ (i.e., classic static/dynamic friction) can produce large-sized events than that with small r. The frequency-magnitude relations for supersonic friction and sonic friction are almost the same, but they are somewhat different from that for subsonic friction. The scaling exponent of the frequency-magnitude relation to some extent decreases as r is increased. The roughness defined as the ratio of the difference between the maximum and minimum breaking strengths to the mean of the two quantities is a factor in affecting the frequency-magnitude relation, while the fractal dimension of the distribution of the breaking strengths is a minor one.

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